TTopological Full Groups
Leonhard Katzlinger a r X i v : . [ m a t h . G R ] J u l ontents Abstract ivIntroduction iv
Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivThe interest of group theory . . . . . . . . . . . . . . . . . . . . . . . . . . vOn the structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viOn this text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii ∧ -monoids and abstract pseudogroups . . . . . . . . . 452.5.3 Non-commutative Stone duality . . . . . . . . . . . . . . . . . 472.6 Resuming Cantor dynamics . . . . . . . . . . . . . . . . . . . . . . . 492.6.1 Homology of ´etale Cantor groupoids . . . . . . . . . . . . . . 492.6.2 Compact generation and expansive groupoids . . . . . . . . . 502.6.3 AF-groupoids and almost finite groupoids . . . . . . . . . . . 522.6.4 Purely infinite groupoids and groupoids of shifts of finite type 54 ONTENTS
Appendices 117A Some terms of geometric group theory 118
A.1 Growth of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118A.2 Grigorchuk groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119A.3 Graphs of actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120A.4 Amenability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122A.5 LEF groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124A.6 The Liouville property . . . . . . . . . . . . . . . . . . . . . . . . . . 125A.7 Higman-Thompson groups . . . . . . . . . . . . . . . . . . . . . . . . 126
B C*-algebras 129
B.1 Basic terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129B.2 Group C*-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133B.3 Hilbert bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135B.4 A hamfisted portrait of K-groups . . . . . . . . . . . . . . . . . . . . 135B.5 Von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Bibliography 139 iii bstract
Topological full groups originated from the theory of topological dynamical systemsand have been having considerable impact on group theory in recent years. This textrepresents an introduction/survey on topological full groups. After development ofthe theoretical and historical background, it gives an account of their significance intopological dynamics and discusses their group theoretical aspects.
Introduction
Historical remarks
Initially (measured) full groups grew out of von Neumann algebra theory and mea-sured dynamics. Henry Dye defined and studied full groups in the papers [Dye59]and [Dye63]. Full groups offer a way to classify automorphism groups of mea-sure algebras. In particular, every measure preserving action of a countable group G on a Lebesgue measure space ( X, λ ) gives rise to a group [ G ] consisting of allnon-singular, measurable transformations γ : X → X such that γ ( x ) ∈ Gx for λ -almost-every x ∈ X . Dye introduced the notion of approximate finitness of suchan automorphism group G i.e. every finite subset of elements in G can be approx-imated by elements in the full group [ G ], and showed that for ergodic actions ofapproximately finite groups by measure preserving transformations on ( X, λ ) theassociated full groups are complete isomorphism invariants for the arising von Neu-mann crossed products ([Dye59], Theorem 5). Furthermore, he demonstrated thatevery isomorphism between the full groups of ergodic actions of countable groupsby measure preserving transformations on (
X, λ ) is induced by an orbit equivalencebetween the systems ([Dye63], Theorem 2).Similar to the introduction of von Neumann algebras predating the consideration ofgeneral C*-algebras, the study of the interplay between measured dynamics and thetheory of von Neumann algebras, which goes back to the work of John von Neumannand Francis J. Murray, predates the study of the interplay between topologicaldynamics and C*-algebra theory. Dye’s ideas motivated efforts to find analogues inthe setting of topological dynamics. One of these efforts is represented by the amplegroups of Wolfgang Krieger defined in [Kri80], the definition of which recalls Dye’sfull groups. Some years later Thierry Giordano, Ian F. Putnam and Christian F.Skau obtained a complete classification of minimal topological dynamical Z -systemsover a Cantor space up to orbit equivalence (resp. strong orbit equivalence) in terms Preimages of null-sets are null-sets.
ONTENTS of their C*-algebra crossed products and the associated dimension groups [GPS95].Eli Glasner and Benjamin Weiss observed that some of the classification results of[GPS95] and weaker versions thereof can be shown without relying on C*-algebratheory by using a transferred version of Dye’s full group and a smaller, countablecousin of it – the topological full group T ( ϕ ).This group had already appeared implicitely 6 years earlier in [Put89] in terms ofunitary normalizers of crossed product C*-algebras. The results of [GW95] fore-shadowed that of [GPS99]: Full groups (resp. topological full groups) of minimaltopological dynamical Z -systems over a Cantor space are complete isomorphism in-variants for orbit equivalence (resp. flip conjugacy). Later on the concept of fullgroups proved to be fruitful in other settings e.g. minimal topological Z -systemsover locally compact Cantor spaces in [Mat02] or Borel actions of polish groups onpolish spaces in [MR07].In [Mat06] Hiroki Matui defined topological full groups of countable, ´etale equiv-alence relations on Cantor spaces. Those have the structure of a principal, ´etaleCantor groupoid as such encompassing free actions of discrete groups on Cantorspaces. The scope was even more generalized in [Mat12] where a notion of topo-logical full groups of vastly more general ´etale groupoids was introduced. In thismuch broader context topological full groups locate within the interplay between thetheories of C*-algebras, ´etale groupoids and inverse semigroups – the study of whichhad been initiated by Jean Renault in [Ren80] and developed further i.a. by AlanL.T. Patterson in [Pat99], Ruy Exel in [Exe08] and Mark V. Lawson in [Law10]. The interest of group theory
The first structural results on topological full groups were given in [GPS99] bytopological dynamical methods: The approximation of minimal Cantor systems byperiodic systems in terms of Kakutani-Rokhlin partitions, a fundamental tool in theclassification of such systems, induces a factorization of the associated topologicalfull groups as a product of a pair of direct limits of sequences of finite symmetricgroups each of which acts on atoms of Kakutani-Rokhlin partitions by permutation.These factors correspond precisely to the ample groups that have been introducedby Krieger. Furthermore, the topological full group of a minimal Cantor systemadmits a non-trivial homomorphism to Z called the index map, which describes theglobal transfer on orbits. Matui proved that the derived subgroup of the topologicalfull group of a minimal Cantor system is simple and in the case of minimal subshiftseven finitely generated in [Mat06]. This paper marks the first of a series of works([Mat12], [Mat13b], [Mat15], [Mat16a]) by Matui dealing with the subject, as wellas the awakening of geometric group theorist’s interest.In the first version of [GM14] Rostislav Grigorchuk and Kostya Medynets conjec-tured topological full groups of minimal Z -subshifts to be amenable, which was By no means “topological” refers to a topology on this group. v ONTENTS subsequently verified by Kate Juschenko and Nicolas Monod in [JM13]. This pro-duces an uncountably infinite number of non-isomorphic examples of infinite, finitelygenerated, simple, amenable groups, examples of which had hitherto been unknown.This triggered a number of publications dealing with the subject: In [Mat14a] and[Mat14b] Nicol´as Matte Bon gave examples of topological full groups of minimalsubshifts satisfying the Liouville property, thus providing first examples of infi-nite, finitely generated, simple groups with that property, and demonstrated thatGrigorchuk’s groups of intermediate growth admit embeddings into topological fullgroups of Cantor minimal systems. Volodymyr Nekrashevych dealt with topolog-ical full groups in [Nek17] and [Nek18], where he generalized Matui’s definition oftopological full groups, introduced a pair of subgroups consisting of somewhat “ele-mentary” elements and used these to give first examples of finitely generated, simplegroups with intermediate growth. It is moreover worth mentioning that topologicalfull groups associated to groupoids arising from one-sided Markov shifts cast newlight on Higman-Thompson groups. We recognize that the study of topological fullgroups has been a rewarding endeavour and is still ongoing.
On the structure
We finish the front matter with a short synopsis of what is to come. In the first twochapters we acquaint ourselves with the territory we are moving in. This is largelyprovided by the setting of ´etale groupoids. We start however in Chapter 1 with anintroduction to fundamental concepts of topological dynamical systems over Cantorspaces, which is the surrounding in which topological full groups originate:Section 1.1 lists basic definitions from topological and measured dynamics,gives basic results and examples of topological dynamical systems over Cantorspaces and recalls the fundamental features of crossed product C*-algebrasassociated with such systems.Section 1.2 introduces concepts of the theory of Cantor Z -systems. Keywordsare Bratteli-Vershik systems and Kakutani-Rokhlin partitions.In Chapter 2 we dwell on ´etale groupoids, inverse semigroups and the interrelationsbetween those types of objects:Section 2.1 contains basic definitions and examples of (topological) groupoids.Section 2.2 does the same for inverse semigroups.Section 2.3 introduces locally compact groupoids as setting for (functional)analysis on topological groupoids and recalls the definition of groupoid C*-algebras.Section 2.4 comprises the definition and characterization of ´etale groupoidsand the subclass of ample groupoids, moreover it gives a suitable version ofhomology of ´etale groupoids.vi ONTENTS
Section 2.5 recounts a general version of non-commutative Stone duality, whichis a duality between the categories of ´etale groupoids and inverse semigroups,and the historical developments leading up to it.Section 2.6 introduces ´etale Cantor groupoids as the setting of geheralizedCantor dynamics and the important subclasses of expansive groupoids, AFgroupoids, almost finite groupoids, purely infinite groupoids and groupoidsassociated to one-sided Markov shifts.Chapter 3 represents a survey on topological full groups and finishes with a concep-tual result on the irreducibility of certain Koopman representations:Section 3.1 starts with the definition of topological full groups (of Cantor sys-tems and more general of ´etale Cantor groupoids), it recalls the descriptionof topological full groups of minimal Cantor systems in terms of Kakutani-Rokhlin partitions, introduces important subgroups and outlines the signifi-cance of these groups in Cantor dynamics.Section 3.2 gives account of Matui’s work on topological full group in that itsketches amongst other things his proof of topological full groups being com-plete isomorphism invariants for ´etale Cantor groupoids, his proof of simplicityof the commutator subgroup of the topological full groups of a minimal Cantorsystem and his proof of exponential growth.Section 3.3 deals with the alternating full group introduced by Nekrashevych,in particular, with his proof of simplicity and finite generation.Section 3.4 accounts for the fact of topological full groups of ´etale groupoidsbeing groups of units of specific inverse ∧ -monoids and as such fall within thescope of non-commutative Stone duality. To this end it represents a dictio-nary of how principles from the groupoid world, and in very rough terms theisomorphism theorem of Matui, translate to the inverse semigroup setting.Section 3.5 represents a compilation of facts on topological full groups withregard to different notions from geometric group theory: Local embeddabilityinto finite groups, amenability, growth of groups, etc.Section 3.6 shows how Artem Dudko’s notion of measure contracting actionsproduces a short proof of irreducibility of the Koopman representation associ-ated with the natural action of the topological full group of a minimal, purelyinfinite, ´etale Cantor groupoid on the space of units.The Appendix consists of two parts:Appendix A lists basic terms of geometric group theory e.g. growth of groups,graphs associated with groups, amenability, Higman-Thompson groups etc.Appendix B is a collection of base vocabulary on C*-algebras, in particularit contains short accounts of Hilbert bundles, von-Neumann algebras and anon-formal adumbration of operator theoretic K-groups. vii ONTENTS
On this text
As it is the conversion of the author’s Master thesis at the University of Vienna,this text represents an undergraduate’s attempt to give an introduction on the topicof topological full groups with a focus on their impact in group theory. It waswritten with the goal in mind of being approachable, hence it includes lots of basicdefinitions, a big appendix and a dense system of references.For any crimes against style, grammar or spelling or more severely any misquotationsor misrepresentation I offer my earnest apologies, I am thankful for any suggestionsof improvement. Feel free to contact me by mail ([email protected]).I owe a debt of gratitude especially to Prof. Goulnara Arzhantseva and MartinFinn-Sell, but my thankfulness extends to everyone who supported me in any wayduring the last years.viii hapter 1Basic terms of dynamics andCantor systems
Dynamical systems theory branches out into the areas – amongst others – of topo-logical dynamics and ergodic theory. In Subsection 1.1.1 we recall basic definitionsfrom both contexts with an emphasis on the topological setting. In Subsection 1.1.2we turn to the specific case of Cantor systems. Subsection 1.1.3 comprises basicnotions of crossed product C*-algebras.
For a thourough introduction to topological dynamical systems the reader is referedto [Gla03] or [dV93], E.1.
Definition 1.1.1. (i) Let G be a group and let X be a set. An action of G on X is a map α : G × X → X such that:(a) α (1 , x ) = x for all x ∈ X .(b) α ( g, α ( h, x )) = α ( gh, x ) for all g, h ∈ G and x ∈ X .If the map α : G × X → X is continuous, the action is called continuous . Asfrom now we omit α from the notation assuming it to be implicit and write g · x instead of α ( g, x ) . Let a group G act on a set X .(ii) Let x ∈ X . The set Gx := { g · x | g ∈ G } is called the orbit of x .(iii) Let S be a subset of X . The group G S := { g ∈ G | gS ⊆ S } is called the stabilizer subgroup of S in G .(iv) The action is called faithful or effective if for every g, h ∈ G with g = h thereexists an x ∈ X such that gx = hx . .1. BASIC DEFINITIONS (v) The action is called free if gx = x for some x ∈ X and g ∈ G implies g = 1.(vi) A continuous action of a group G on a topological space X is called essentiallyfree , if for every g ∈ G \ { } the set of fixed points { x ∈ X | gx = x } has emptyinterior.(vii) A topological dynamical system or topological transformation group is a tuple( X, G ) where G is a discrete group, X is a compact Hausdorff space and G actscontinuously on X . We will use topological G -system over X as a synonymicterm to put emphasis on the involved topological space and group.Let ( X, G ) be a topological dynamical system.(viii) A pair (
Y, G ) is called a subsystem of ( X, G ), if Y is a closed, non-empty, G -invariant subset of X . Restricting the G -action on X to Y makes ( Y, G ) atopological G -system.(ix) The system ( X, G ) is said to be minimal , if it contains no proper subsys-tem. If (
X, G ) contains a unique, proper, minimal subsystem, it is said to be essentially minimal . Example 1.1.2.
Let X be a compact, metric space and let G be a discrete, count-able topological group. The natural continuous action of G on X G = { f : G → X } given by g · f ( h ) = f ( g · h ) for all g, h ∈ G is called a G -shift or topological X -Bernoulli system . A closed subsystem of a G -shift is called G -subshift .Minimal systems are fundamental “building blocks” of topological dynamical sys-tems. Closures of orbits induce subsystems, thus the following is immediate: Proposition 1.1.3.
Let ( X, G ) be a topological dynamical system. The followingare equivalent:(i) The system ( X, G ) is minimal.(ii) The orbit Gx is dense in X for every x ∈ X . Compactness and an application of Zorn’s Lemma imply a basic result by GarrettBirkhoff:
Proposition 1.1.4.
Every topological dynamical system contains a minimal subsys-tem.
For topological Z -systems we make adjustments to the notations chosen in Defini-tion 1.1.1: A topological Z -system ( X, Z ) is signified by the pair ( X, ϕ ), where ϕ isthe homeomorphism of X corresponding to the generator 1 ∈ Z . We deviate in thenotation of orbits by writing Orb ϕ ( x ) instead of Gx . Discreetness implies that the continuity of the action is equivalent to G acting by homeomor-phisms. Note, that in the literature the definition varies in constraints on G and X . .1.1 BASIC TERMS OF DYNAMICS Definition 1.1.5.
Let (
X, ϕ ) be a topological Z -system.(i) For every x ∈ X define the forward orbit of x as the set Orb + ϕ ( x ) := { ϕ n | n > } and the backward orbit of x as the set Orb − ϕ ( x ) := { ϕ n | n ≤ } .(ii) The system ( X, ϕ ) is called periodic if every orbit is finite and it is called aperiodic if every orbit is infinite.
Remark 1.1.6.
For a minimal Z -systems ( X, ϕ ) forward- and backward orbits aredense in X , because their respective sets of accumulation points are closed invariantsubsets. By definition minimal Z -systems are always examples of free actions. Definition 1.1.7.
Let ( X , G ) and ( X , G ) be topological G -systems.(i) A homomorphism of topological G -systems F : ( X , G ) → ( X , G ) consists ofa continuous map F : X → X that intertwines the respective G -actions i.e. F ( g · x ) = g · F ( x ) for all x ∈ X and g ∈ G .Let F : ( X , G ) → ( X , G ) be a homomorphism of topological G -systems.(ii) If the associated map F : X → X is surjective, it is called factor map . Inthis case the system ( X , G ) is said to be a factor of ( X , G ) or ( X , G ) is saidto be an extension of ( X , G ).(iii) It is called an isomorphism of topological G -systems if the associated map F : X → X is a homeomorphism. In this case the systems ( X , G ) and( X , G ) are said to be conjugate .The classification of systems up to conjugacy is often intractable and one turns tothe following weaker notions: Definition 1.1.8 ([GPS95], Definition 1.1 & 1.2 & 1.3) . Let ( X , ϕ ) and ( X , ϕ )be topological Z -systems.(i) The systems ( X , ϕ ) and ( X , ϕ ) are said to be orbit equivalent , if thereexists an orbit map F : X → X , i.e. a homeomorphism F : X → X suchthat F (Orb ϕ ( x )) = Orb ϕ ( F ( x )) for all x ∈ X .(ii) The systems ( X , ϕ ) and ( X , ϕ ) are said to be flip conjugate , if ( X , ϕ ) isconjugate to either ( X , ϕ ) or ( X , ϕ − ).(iii) Let ( X , ϕ ) and ( X , ϕ ) be minimal and let F : X → X be an orbit map.Then there exist unique n ( x ) , m ( x ) ∈ Z such that F ◦ ϕ ( x ) = ϕ n ( x )2 ◦ F ( x )and F ◦ ϕ m ( x )1 ( x ) = ϕ ◦ F ( x ) for all x ∈ X . The functions n, m : X → Z arecalled the orbit cocycles associated with F . This is why minimality is required. .1. BASIC DEFINITIONS (iv) Let ( X , ϕ ) and ( X , ϕ ) be minimal. They are said to be strongly orbitequivalent , if there exists an orbit map F : X → X such that the orbitcocyles associated with F each have at most one point of discontinuity.All above relations between topological Z -systems are equivalence relations satisfy-ing:conjugacy ⇒ flip conjugacy ⇒ orbit equivalence ⇐ strong orbit equivalenceWe finish this subsection with some basic terms from measured dynamics: Definition 1.1.9 ([BM00], Definition 1.1) . Let (
X, µ ) be a measure space such that µ is a σ -finite measure and let a locally compact, second countable group G act on X such that for every g ∈ G the map x g · x is measurable.(i) For every g ∈ G the pushforward measure g ∗ µ is defined by g ∗ µ ( A ) := µ ( g − A )for every measurable subset A ⊆ X .(ii) The measure µ is called quasi-invariant under the action of G if for all g ∈ G the measures µ and g ∗ µ are mutually absolutely continuous i.e. µ ( A ) = 0 ifand only if g ∗ µ ( A ) = 0 for every measurable subset A ⊆ X . Alternatively onespeaks of a measure class preserving action of G on ( X, µ ).(iii) The measure µ is called invariant under the action of G if for all g ∈ G themeasures µ and g ∗ µ coincide. Alternatively one speaks of a measure preservingaction of G on ( X, µ ).(iv) The action is called ergodic with respect to µ if every G -invariant measurablesubset A ⊆ X is either null or conull.Every measure preserving action of a group G on a probability space ( X, µ ) inducesa unitary representation of G on L ( X, µ ) by (˜ κ ( g ) f )( x ) := f ( g − x ) called the Koopman representation . For the quasi-invariant setting it is defined as:
Definition 1.1.10 ([Dud18]) . Let (
X, µ ) be a measure space such that µ is a σ -finitemeasure. Let G be a locally compact, second countable group acting measure-class-preserving on ( X, µ ). The
Koopman representation κ of G on L ( X, µ ) is givenby: ( κ ( g ) f )( x ) := s d g ∗ µ d µ ( x ) f ( g − x ) Example 1.1.11.
Let G be locally compact, second countable group. Then theregular representation of G is just the Koopman representation in the case of G acting on itself by left multiplication. See Definition B.2.1. See Appendix B.2. .1.2 MINIMAL CANTOR SYSTEMS The spectral properties of the Koopman representation allow to characterize ergod-icity or mixing properties of the underlying measured dynamical system (see e.g.[Gla03], Chapter 3.2).
The Cantor space is a paragon of an exciting object from mathematics – be it forits distinctive topological properties, its historical significance for the developmentof point-set topology and descriptive set theory or its habit to appear in a varietyof contexts and forms in dynamics.
Definition 1.1.12 ([Kec95], § . The topological space C := { , } N obtained as N -fold topological product of the discrete space { , } is called the Cantor space . Thesubsets C [( y , . . . , y n )] := n { x i } i ∈ N ∈ C : x i = y i , ∀ i ∈ , . . . , n o where ( y , . . . , y n ) ∈{ , } n for some n ∈ N are called cylinder sets .The cylinder sets constitute a countable basis of clopens of the topology on C i.e.the space C is zero-dimensional and second countable. Theorem 1.1.13 ([Kec95], Theorem 7.4) . The Cantor space C is uniquely charac-terized up to homeomorphisms by being a non-empty, compact, perfect, metrizableand zero-dimensional topological space. We call a topological space “a” Cantor space if it is homeomorphic to “the” Cantorspace.
Example 1.1.14. (i) Open compact subsets of Cantor spaces and countably infi-nite products of Cantor spaces are Cantor spaces. A countably infinite productof any finite discrete space is a Cantor space.(ii) Let d ∈ N ≥ . Let T be an infinite labelled directed d -ary rooted tree with root r i.e. an infinite tree together with a distinguished vertex point r called the root , such that every edge is oriented away from r and every vertex has exactly d children. The space ∂T of directed infinite paths in T starting at r endowedwith the topology arising by taking classes of paths which are identical onfinitely many first vertices as a basis is called the boundary of T . It is aCantor space.(iii) Let p be a prime number. The p -adic integers Z p with their conventionaltopology are a Cantor space. Remark 1.1.15. (i) Cantor spaces are by definition Stone spaces i.e. compact,Hausdorff, totally disconnected topological spaces, and Cantor spaces are Pol-ish spaces i.e. separable and completely metrizable. Every cylinder set is the complement of a finite union of cylinder sets. .1. BASIC DEFINITIONS (ii) Under Stone duality the characterization of the Cantor space up to homeo-morphism from Theorem 1.1.13 amounts to the fact that its Boolean algebraof clopen subsets is the up to Boolean isomorphism unique countable, atomlessBoolean algebra.For the sake of abbrevation we define: Definition 1.1.16. (i) A
Cantor system is a topological Z -system over a Cantorspace.(ii) Let ( X, ϕ ) be a Cantor system. Denote by M ϕ the set of all ϕ -invariantprobability measures on X .In the following we recall some fundamental examples of Cantor systems: Example 1.1.17. (i) Let A be a finite alphabet. The Z -shift on A Z is a Cantorsystem called the full Z -shift over A . This system is in general not minimaland the closed subsystems are called Z -subshifts over A .(ii) Let a = ( a n ) n ∈ N be a strictly increasing sequence of integers with a n ≥ a n divides a n +1 for all n ∈ N . This data gives rise to an inverse systemof surjective homomorphisms of finite cyclic groups ρ n : Z /a n +1 Z → Z /a n Z by ρ n ( z ) = z mod a n . By endowing every group Z /a n Z with the discretetopology, the corresponding inverse limit G a := lim ←− n Z /a n Z == (cid:26) z ∈ Y n Z /a n Z (cid:12)(cid:12)(cid:12) z n +1 ≡ z n mod a n for all n ∈ N (cid:27) is a compact, metric, zero-dimensional, abelian group called odometer of type a or adding machine which is a Cantor space. Every odometer G a contains theelement = (1 , , . . . ) inducing a minimal Cantor system ( G a , ϕ ) by ϕ : z z + . A survey on such systems can be found in [Dow05].We call a Cantor system an odometer if it is conjugate to an odometer of type a for some suitable a , and call it a subshift if it is conjugate to a subsystem of some Z -shift. Remark 1.1.18.
Any topological dynamical system (
X, G ) with a finite clopenpartition { P i } i ∈ I such that the G -translates of atoms separate points is conjugateto a G -shift in A G for some finite alphabet A – modelled from the G -action on the G -translates of atoms. This standard construction gives rise to the field of symbolicdynamics . Definition 1.1.19 ([Mat14a], p.1639) . Let A be a finite alphabet and let ( X, ϕ ) bea Z -subshift over A . The complexity ρ of ( X, ϕ ) is the function ρ : N → N where ρ ( n ) is defined as the number of words of length n over A that appear as subwordsin elements of X .6 .1.2 MINIMAL CANTOR SYSTEMS Remark 1.1.20. ([Gla03], Proposition 14.4.3 & Corollary 14.7)(i) For a subshift (
X, ϕ ) its topological entropy h top ( X, ϕ ) computes via its com-plexity ρ in that h top ( X, ϕ ) = lim n →∞ n log ρ ( n ) . (ii) Topological entropy is invariant under flip-conjugacy i.e. every topologicaldynamical Z -system satisfies h top ( X, ϕ ) = h top ( X, ϕ − ). Example 1.1.21.
In the following examples ϕ always denotes the shift operator.(i) ([Lot02], Chapter 2) Let α ∈ R > be irrational. Let the R α : [0 , → [0 ,
1) therotation given by R α : x x + α mod 1 and let A := { a, b } . Define a mapΦ : [0 , → { a, b } by Φ( x ) := a, x ∈ [0 , α ) b, elseFurthermore define:Σ α := {{ Φ( R iα ( x )) } i ∈ Z | x ∈ [0 , } ⊂ A Z Then (Σ α , ϕ ) is a minimal Cantor system with complexity ρ ( n ) = n + 1. Suchsystems are called Sturmian shifts . A look at the spectrum of the associatedKoopman operator shows that for irrational α, β ∈ (0 , ) with α = β theSturmian shifts (Σ α , ϕ ) and (Σ β , ϕ ) are not flip conjugate.(ii) ([Dow05], §
7) Let A := { a , . . . , a k } be a finite alphabet. A sequence s = { s i } i ∈ Z ∈ A Z is called a Toeplitz sequence if for every i ∈ Z there exists an p ∈ N such that s i = s i + pk for all k ∈ Z . For such an s define T s := Orb ϕ ( s ) ⊂ A Z . If s is not periodic, the arising subshift ( T s , ϕ ) is a minimal Cantor system calleda Toeplitz shift . For every α ∈ R > there exists a Toeplitz shift ( T s , ϕ ) suchthat the entropy satisfies h ( T s , ϕ ) = α .Odometers are in some sense opposites to minimal shifts in that every homomor-phism from an odometer to a subshift has finite image. A minimal Cantor system isan odometer if and only if it is the inverse limit of a sequence of periodic systems.In particular one has the following: Lemma 1.1.22 ([Mat13b], Lemma 2.2) . Let ( X, ϕ ) be a minimal Cantor system.If it is not an odometer, there exists a homomorphism from ( X, ϕ ) to the full -shift ( { , } Z , σ ) with infinite image.Proof. Let { C n } n ∈ N be the set of all clopen sets in X – it is countable since X is aCantor space – and define a family of maps { π n : X → { , } Z } n ∈ N by ( π n ( x ) k ) k ∈ Z := ϕ − k ( C n ) ( x ). Every such map is continuous and π n ◦ ϕ = σ ◦ π n for all n ∈ N . For a formal definition of topological entropy see [Gla03], § .1. BASIC DEFINITIONS The conclusion follows, if finiteness of | π n ( X ) | for all n ∈ N implies that ( X, ϕ )is conjugate to an odometer G a for some sufficient sequence a = ( a n ) n ∈ N . Hencewe need to find a sufficient sequence a = ( a n ) n ∈ N with associated inverse system ρ n : Z /a n +1 Z → Z /a n Z defined by ρ n ( z ) = z mod a n and a family of continuousmaps { ˜ π n : X → Z /a n Z } n ∈ N such that ˜ π n ◦ ϕ ( x ) = ˜ π n +1 for all x ∈ X , ˜ π n +1 = ρ n ◦ ˜ π n and π n factors through ˜ π n for all n ∈ N . To this end assume | π n ( X ) | is finite for all n ∈ N . Let a = | π | . Then there exists a bijection b : π ( X ) → Z /a Z such that b ◦ σ ( z ) = 1+ b ( z ) for z ∈ π ( X ). It is easy to show ˜ π := b ◦ π is sufficient. Assume˜ π n has already been defined, then π n +1 × ˜ π n is a continuous map. Define a n +1 := | ( π n +1 × ˜ π n )( X ) | . Then there exists a bijection b n +1 : ( π n +1 × ˜ π n )( X ) → Z /a n +1 Z such that ˜ π n +1 := b n +1 ◦ ( π n +1 × ˜ π n ) is sufficient. The map f : X → G a defined by f ( x ) := (˜ π n ) n ∈ N for x ∈ X is continuous and injective – since the functions { π n } n ∈ N separate points in X – and thus ( X, ϕ ) is conjugate to the odometer of type a .Minimal Cantor systems play an exceptional role for minimal Z -systems over com-pact, metrizable spaces steming from the exceptional role of the Cantor space C inthe category of compact metric spaces: Theorem 1.1.23 ([Kec95], 4.18) . Every non-empty, compact, metrizable topologicalspace is a continuous image of C . This implies for minimal Cantor systems:
Theorem 1.1.24 ([GPS95], p. 55) . Let ( X, ψ ) be a minimal topological Z -system,such that X is metrizable (and compact). Then ( X, ψ ) is the factor of a minimalCantor system.Proof. By Theorem 1.1.23 for every non-empy, compact, metrizeable space X thereexists a continuous surjection from C onto X . Let π be such a surjection. Theproduct space C Z is a Cantor space by Example 1.1.14 (ii). The subset C := { ( c i ) i ∈ Z : c i ∈ C , π ( c i +1 ) = ψ ( π ( c i )) } is closed, as it is an intersection of cylinder sets, hencecompact. It is invariant under the natural Z -shift σ on C Z . By Proposition 1.1.4, thesystem ( C, σ | C ) must contain a minimal subsystem ( M, σ | M ). Let p | M denote therestriction of the projection p on the 0’th component in C Z to M . Then ( M, σ ) is aminimal Cantor system and π ◦ p | M is continuous, surjective and ( π ◦ p | M ) ◦ σ M = ψ ◦ ( π ◦ p | M ) holds. In dynamics often the study of group actions on spaces is replaced by the studyof automorphism groups of algebras. Crossed product C*-algebras are operatoralgebras that arise from C*-dynamical systems and, in particular, from topologicaldynamical systems, thus providing a rich source of C*-algebras. The phrase “exceptional role” is chosen, since we can not speak of a universal property in theprecise meaning of category theory. See Appendix B for basic terms of C*-algebras. .1.3 SOME PRELIMINARIES OF CROSSED PRODUCTS Definition 1.1.25 ([Dav96], p. 216) . A C*-dynamical system is a triple ( A , G, α )comprising a C*-algebra A , a locally compact group G and a continuous group-homomorphism α : G → Aut( A ) i.e. a continuous action of G on A by automor-phisms. Example 1.1.26.
Let (
X, G ) be a topological dynamical system given by an action α : G → Homeo( X ). Then α gives rise to a C*-dynamical system ( C ( X ) , G, ˜ α ) bythe induced map ˜ α : G → Aut( C ( X )) given by (cid:16) ˜ α ( g )( f ) (cid:17) ( x ) := f ( α ( g − ) x ). Definition 1.1.27 ([Dav96], p. 216) . Let ( A , G, α ) be a C*-dynamical system. A covariant representation of ( A , G, α ) is a pair ( π, u ) for which there exists a Hilbertspace H such that π is a ∗ -algebra-representation of A on H and u is a unitaryrepresentation of G on H satisfying π ( α ( g ) A ) = u ( g ) π ( A ) u ( g ) ∗ for all g ∈ G, A ∈ A .Let ( A , G, α ) be a C*-dynamical system. The space of functions C c ( G, A ) becomesa ∗ -algebra by defining : f ∗ ( g ) = ∆( g − ) α ( g )( f ( g − ) ∗ )( f ∗ f )( g ) := Z h ∈ G f ( h ) α ( h ) (cid:16) f ( h − g ) (cid:17) d µ G ( h )Define L p ( G, A ) to be the completion of C c ( G, A ) with respect to the p -norm givenby k f k p := (cid:16) Z h ∈ G k f ( g ) k p d µ G ( h ) (cid:17) /p Any covariant representation ( π, u ) of the system ( A , G, α ) induces a representationof C c ( G, A ) by integration: π (cid:111) u ( f ) := Z g ∈ G π (cid:16) f ( g ) (cid:17) u ( g )d µ G ( g ) Definition 1.1.28 ([Dav96], p. 217) . Let ( A , G, α ) be a C*-dynamical system.Thecompletion of C c ( G, A ) with respect to the norm k f k C ∗ := sup {k π (cid:111) u ( f ) k : ( π, u ) is a covariant representation of ( A , G, α ) } is a C*-algebra, called the (maximal) crossed product C*-algebra of ( A , G, α ) denotedby A (cid:111) α G . In the case of topological dynamical system ( X, G ), we write C ( X ) (cid:111) G and C ( X ) (cid:111) ϕ Z in case of a Z -system ( X, ϕ ) for the associated crossed productC*-algebras. ∆ denotes the modular function of G , see Subsection B.2. The definition of reduced crossed product C*-algebras has been left out. For minimal Cantorsystems the reduced and the maximal crossed product coincide. .1. BASIC DEFINITIONS Remark 1.1.29.
Every locally compact group G gives rise to a degenerate C*-dynamical system ( C , G, triv) and every C*-algebra A a degenerate C*-dynamicalsystem ( A , { e } , triv) where triv denotes the trivial action. Group C*-algebras arecrossed product C*-algebras for a degenerate C*-dynamical system in that C ∗ ( G ) = C (cid:111) triv G . Remark 1.1.30 ([Dav96], VIII.2) . In the case of a topological Z -system ( X, ϕ ) thecrossed product C*-algebra C ( X ) (cid:111) ϕ Z is the universal C*-algebra generated by C ( X ) and a unitary u such that uf u ∗ = f ◦ ϕ − . The set of “Laurent polynomials” { n X k = − n f k u k | n ∈ N , f k ∈ C ( X ) } is a dense subalgebra of C ( X ) (cid:111) ϕ Z . There is an action of the dual group ˆ Z = T by automorphisms on C ( X ) (cid:111) ϕ Z : Every λ ∈ T induces an automorphism ρ λ of C ( X ) (cid:111) ϕ Z satisfying ρ λ | C ( X ) = id and ρ λ : u k λ k u k for k ∈ Z . This action of T , also denoted by ˆ ϕ , induces a conditional expectation E : C ( X ) (cid:111) ϕ Z → C ( X ) by E ( a ) := Z λ ∈ T ρ λ ( a ) d µ T If the system is essentially free the above constitutes ( C ( X ) , C ( X ) (cid:111) ϕ Z ) as a Cartanpair. The conditional expectation E correponds to the computation of the zero’thFourier coefficient of a function on T in that on the dense subalgebra of “Laurentpolynomials” we have E ( n X k = − n f k u k ) = Z λ ∈ T ρ λ ( n X k = − n f k u k ) d µ T = n X k = − n Z ρ exp(2 πit ) ( f k u k ) d t = n X k = − n f k Z exp(2 πikt )( u k ) d t n X k = − n f k Z exp(2 πikt ) u k d t = f Analogously, for every k ∈ Z the maps E k : C ( X ) (cid:111) ϕ Z → C ( X ) given by a E ( au − k ) compute the “k’th Fourier coefficient”.For a C*-dynamical Z -system ( A , Z , α ) the cyclical 6-term exact sequence of K - See Appendix B.2. The map λu : z → λ z u z defines a unitary representation of Z . The pair (id , λu ) is a covariantrepresentation thus inducing a canonical ∗ -algebra representation of C ( X ) (cid:111) ϕ Z on itself. See Definition 2.5.1. gives rise to the Pimsner-Voiculescu sequence: K ( A ) K ( A ) K ( A (cid:111) α Z ) K ( A (cid:111) α Z ) K ( A ) K ( A ) id − α − ∗ ι ∗ δ δ ι ∗ id − α − ∗ Here ι denotes the canonical embedding A , → A (cid:111) α Z . For the crossed prod-uct C*-algebra C ( X ) (cid:111) ϕ Z associated with a Cantor system ( X, ϕ ), it holds that K ( C ( X )) = C ( X, Z ) and K ( C ( X )) = 0 (Example 1.2.19) and above sequencebecomes0 −→ K ( C ( X ) (cid:111) ϕ Z ) δ −→ C ( X, Z ) id − ϕ ∗ −→ C ( X, Z ) ι ∗ −→ K ( C ( X ) (cid:111) ϕ Z ) −→ K ( C ( X ) (cid:111) ϕ Z ) ∼ = C ( X, Z ) / Im(id − ϕ ∗ ) and K ( C ( X ) (cid:111) ϕ Z ) ∼ = Z . Subsection 1.2.1 brushes the theory of AF C*-algebras and Bratteli diagrams. Sub-section 1.2.2 provides a description of Kakutani-Rokhlin partitions. Nested se-quences of such partitions allow to approximate minimal Cantor systems, show-ing these are conjugate to Bratteli-Vershik systems (see Subsection 1.2.3). Subsec-tion 1.2.4 gives the definition of dimension groups. In Subsection 1.2.5 cites Krieger’swork on ample groups. These notions are important to the classification of minimalCantor systems developed in [Put89], [HPS92], [GPS95] and [GPS99] (see Subsec-tion 1.2.6) and the works of Matui on topological full groups (see Subsection 3.2.2).
AF C*-algebras are limits of sequences of finite dimensional C*-algebras. The spe-cial case of uniformly hyperfinite algebras i.e. limits of sequences of simple finitedimensional C*-algebras, had already been classified by James G. Glimm in [Gli60]and Jaques Dixmier in [Dix67] in terms of their dimension range i.e. in terms oftheir K -groups. Ola Bratteli related AF C*-algebras with certain diagrams: Definition 1.2.1 ([HPS92], Definition 2.1 & 2.2) . See Appendix B.4. The cylic 6-term exact sequence is applied to a Toeplitz extension of A ⊗ K by A (cid:111) α Z . Theresulting sequence can be shown to correspond to the Pimsner-Voiculescu sequence via diagramchasing. .2. A TOOL BOX OF MINIMAL CANTOR SYSTEMS (i) A Bratteli diagram
Γ = (
V, E ) consists of the following data:(a) a vertex set V , which is a disjoint union F n ∈ N V n of finite, non-empty sets V n with V a singleton(b) a set of edges E , which is a disjoint union F n ∈ N E n of finite, non-emptysets E n (c) a pair of maps r, s : E → V called range resp. source map , such that r ( E n ) ⊆ V n +1 , s ( E n ) ⊆ V n , s − ( v ) = ∅ for all v ∈ V and r − ( v ) = ∅ forall v ∈ V \ V . (ii) An isomorphism of Bratteli diagrams consists of a bijection between the vertexsets and a bijection between the sets of edges intertwining the range and sourcemaps such that the grading on these sets is preserved.Let Γ = ( V, E ) be a Bratteli diagram.(iii) Let v ∈ V i and v ∈ V j with i < j . A path from v to v is an edge tuple { ( e i , e i +1 , . . . , e j − ) } such that e k ∈ E k for i ≤ k ≤ j − s ( e i ) = v , r ( e j − ) = v and r ( e k ) = s ( e k +1 ) for i ≤ k ≤ j −
2. The set of all paths from v to v isdenoted by P v,v An infinite path P in Γ is a sequence { e n } n ∈ N of edges e n ∈ E n such that r ( e n ) = s ( e n +1 ) for all n ∈ N . The set of all infinite paths in Γ isdenoted P Γ .(iv) Let { m n } n ∈ N be a sequence with m n ∈ N and m n < m n +1 for all n ∈ N and m = 0. Define a Bratteli diagram ˜Γ = ( ˜ V , ˜ E ) by setting ˜ V n := V m n and ˜ E n to be the set of paths from vertices in V m n to vertices in V m n +1 and by definingrange and source of such an edge by r (( e m n , e m n +1 , . . . , e m n +1 − )) := r ( e m n +1 − ) s (( e m n , e m n +1 , . . . , e m n +1 − )) := s ( e m n ) . The Bratteli diagram ˜Γ is called a contraction or a telescoping of
Γ. TwoBratteli diagrams are equivalent if one can be obtained from the other via afinite chain of telescopings and isomorphisms.
Remark 1.2.2.
Microscoping , a notion dual to telescoping, iteratively inserts newlayers of vertices between consecutive vertex sets V n and V n +1 by splitting each edgebetween V n and V n +1 by a new vertex.Ola Bratteli introduced Bratteli diagrams to study AF C*-algebras in [Bra72]: Definition 1.2.3 ([Bra72], Definition 1.1 & § . A C*-algebra is said to be approx-imately finite-dimensional (abbreviated as AF), if it is the direct limit of a sequenceof finite dimensional C*-algebras. The representation as an arrow diagram follows by interpreting an edge e as an arrow from s ( e ) to r ( e ). Finite dimensional C*-algebras are finite direct sums of full matrix algebras over C . .2.1 AF C*-ALGEBRAS AND BRATTELI DIAGRAMS Remark 1.2.4.
By continuity of the K -functor and K ( C ) = 0 the K -group of AFC*-algebras always vanishes. In particular the crossed product C*-algebra C ( X ) (cid:111) ϕ Z of a Cantor system ( X, ϕ ) is never
AF, however, every minimal Cantor system C ( X ) (cid:111) ϕ Z admits unital embeddings into AF C*algebras (see Theorem 1.2.28).Every AF C*-algebra admits a class of Bratteli diagrams depending on the directedsystems of embeddings of finite dimensional C*-subalgebras. Conversely, every Brat-teli diagram Γ = ( V, E ) induces a unique AF C*-algebra C ∗ ( V, E ): Denote for every v ∈ V by M v the full matrix algebra M |P v ,v | ( C ). The AF C*-algebra C ∗ ( V, E ) isdefined as the direct limit of the sequence C i , → M v ∈ V M v i , → M v ∈ V M v i , → . . . where i k : L v ∈ V k M v i k , → L v ∈ V k +1 M v is given on a summand M v as a sum of diagonalmaps M v → M v over V k +1 with multiplicity corresponding to the number of con-necting edges between v and v . Information on properties of C ∗ ( V, E ) like the idealstructure can be read in the underlying Bratteli diagram. Theorem 2.7 of [Bra72]essentially implies the following:
Theorem 1.2.5.
The just descibed procedures induce a 1-1 correspondence betweenisomorphism classes of AF C*-algebras and equivalence classes of Bratteli diagrams.
This is a manifestation of a much more general idea where C*-algebras are modelledon combinatorial structure. S¸erban Str˘atil˘a and Dan-Virgil Voiculescu showedthat AF C*-algebras can be diagonalized, which offers means to study their idealstructure. This was one of the key motivations for [Ren80] by Jean Renault (– seeChapter 2):
Theorem 1.2.6 ([SV75], p. 17) . For every AF C*-algebra A there exist a maximalabelian subalgebra C ⊆ A , a conditional expectation P : A → C and a locally finiteunitary subgroup U ⊆ U ( A ) such that:(i) u ∗ Cu = C for all u ∈ U (ii) P ( u ∗ Au ) = u ∗ P ( A ) u for all A ∈ A and u ∈ U (iii) The C*-algebra A is the closed linear span of the set { uC | u ∈ U, C ∈ C } .(iv) Let X be the spectrum of C i.e. C = C ( X ) , and let Γ U be the group of homeo-morphisms of X identified with the group of ∗ -automorphisms of C given bythe set { γ u : c u ∗ cu | u ∈ U } . The C*-algebra A is ∗ -isomorphic to a quotientof C ( X ) (cid:111) Γ U . The group Γ U is actually a first glimpse into the realms of topological full groupsan thus of Chapter 3. A bit more on this in Subsection 2.5.1. By Gelfand duality – see Appendix B.1. .2. A TOOL BOX OF MINIMAL CANTOR SYSTEMS Kakutani-Rokhlin partitions are a notion transferred from the measured dynamicssetting: The
Rokhlin-Kakutani lemma is a result in ergodic theory proven inde-pendently by Vladimir A. Rokhlin and Shizuo Kakutani and was used by Dye toapproximate measured dynamical systems. We conform in our choice of notationwith [BK00] and [GM14].
Definition 1.2.7 ([GM14], Definition 3.1) . Let (
X, ϕ ) be a minimal Cantor system.(i) Let A be a non-empty clopen subset of X . The first return function of A is thecontinuous function t ϕ,A : A → N defined by t ϕ,A ( x ) := min { n ∈ N | ϕ n ( x ) ∈ A } . (ii) Let A be a clopen subset of X and n ∈ N such that the sets A, ϕ ( A ) , . . . ϕ n − ( A )are mutually disjoint. The collection { A, ϕ ( A ) , . . . ϕ n − ( A ) } is called a towerof height n . The set A is called base of the tower , the set ϕ n − ( A ) its roof .(iii) A disjoint clopen partition A of X of the form A = { ϕ k ( A i ) | ≤ k ≤ h i − , i ∈ [1 , n ] } for some n ∈ N is called Kakutani-Rohklin partition . Every atom of A corresponds to a pair ( k, i ) with 0 ≤ k ≤ h i − , i ∈ { , . . . , n } . We set D k,i = ϕ k ( A i ) and we define U ( A ) to be the set of all such pairs ( k, i ). Denoteby D ( i ) := F ≤ k ≤ h i − D k,i the i -th tower and denote the minimal height of atower in A by h A := min { h i | i ∈ { , . . . , n }} .(iv) Let A be a Kakutani-Rohklin partition of X . Then B ( A ) := F i A i = F i D ,i is called the base of A and R ( A ) := F i ϕ h i − ( A i ) = F i D h i − ,i is called the roofof A . Remark 1.2.8.
Such partitions exist: Let (
X, ϕ ) be a minimal Cantor systemand let A be a proper non-empty clopen subset of X . By continuity of the firstreturn function, there exists a finite family { n i } of positive integers and a finiteclopen partition A = F i A i such that t ϕ,A ( A i ) = n i . Since ϕ k ( A i ) ∩ ϕ l ( A j ) = ∅ for0 ≤ k ≤ n i −
1, 0 ≤ l ≤ n j − i = j , The partition A = { ϕ k ( A i ) | ≤ k ≤ n i − , i ∈ { , . . . , n }} is a Kakutani-Rohklin partition of X .Let A = { D k,i | ≤ k ≤ h i − , i ∈ { , . . . , n }} be a Kakutani-Rohklin partition ofa minimal Cantor system ( X, ϕ ). It partitions X into n disjoint towers, where thenumber h i signifies the height of the i -th tower. Definition 1.2.9 ([GPS95], Definition 1.5) . Let (
X, ϕ ) be a minimal Cantor systemand let A be a non-empty clopen subset of X . The function t ϕ,A is well-defined as forward-orbits are dense in minimal systems by Re-mark 1.1.6. Furthermore, it is continuous, since t − ϕ,A ( n ) = ϕ − n ( A ∩ ϕ n ( A )) \ S n − i =1 t − ϕ,A ( i ). .2.2 KAKUTANI-ROKHLIN PARTITIONS (i) The homeomorphism ϕ A ∈ Homeo( X ) defined by ϕ A ( x ) := ϕ t ϕ,A ( x ) ( x ) , x ∈ Ax, elsewhere t A is the first return function on A , is called the induced transformationof ϕ on A .(ii) The restriction of the induced transformation ϕ A to A is a minimal homeomor-phism of A by the minimality of ϕ and the minimal Cantor system ( A, ϕ A | A )is called the induced or derivative Cantor system of ( X, ϕ ) over A .As their measure theoretic counterparts Kakutani-Rokhlin partitions are used toapproximate dynamical systems. The definition is essentially implicit in Theorem4.2 of [HPS92]. Definition 1.2.10.
Let (
X, ϕ ) be a minimal Cantor system.(i) Let A = { D k,i | ≤ k ≤ h i − , i ∈ { , . . . , n }} be a Kakutani-Rokhlin partition.Let P be a finite partition of X . Let P ∈ P and let U P, A ⊆ U ( A ) be the setoff all pairs ( k, i ) such that P ∩ D k,i = ∅ and P D k,i = ∅ . Then { ϕ j ( D ,i ∩ ϕ − k ( P )) | ≤ j ≤ k, ( k, i ) ∈ S P, A , P ∈ P} is a Kakutani-Rokhlin partition of X called the refinement of A by P .(ii) Let Y ⊆ X such that Y is either clopen or a singleton. A sequence {A n } n ∈ N = { D nk,i | ≤ k ≤ h ni − , i ∈ { , . . . , i n }} of Kakutani-Rokhlin partitions is calleda nested sequence of Kakutani-Rokhlin partitions around Y if it satisfies thefollowing properties:(a) S n {A n } generates the topology of X .(b) A n +1 refines A n .(c) B ( A n +1 ) ⊂ B ( A n ).(d) T n B ( A n ) = Y .(iii) Let x ∈ X . A nested sequence {A n } n ∈ N of Kakutani-Rokhlin partitions around x is said to have property (H) if the following holds:(H) h A n → ∞ for n → ∞ . Remark 1.2.11.
Let Y be a subset of X that is either clopen or a singleton and let { B n } n ∈ N be a descending filtration of clopen sets B n with T n B n = Y and B = X .Applying the construction from Remark 1.2.8 to each set B n and taking sufficientrefinements, generates a nested sequence of Kakutani-Rohklin partitions {A n } n ∈ N = { D nk,i | ≤ k ≤ h ni − , i ∈ { , . . . , i n }} around Y . This construction also works forthe more general systems considered in [HPS92] and by restriction to subsequences,hereafter, property (H) can always be assumed. 15 .2. A TOOL BOX OF MINIMAL CANTOR SYSTEMS Example 1.2.12.
Let G a be the odometer of type a (see Example 1.1.17(ii)) for asufficient sequence a = ( a n ) n ∈ N . Then {A n } n ∈ N := { D nk } n ∈ N , ≤ k ≤ a n − :== { ( z i ) i ∈ N + k · | ( z i ) i ∈ N ∈ G a , z = · · · = z n = 0 } n ∈ N , ≤ k ≤ a n − = { ( z i ) i ∈ N ∈ G a | z n = k } n ∈ N , ≤ k ≤ a n − is a nested sequence of Kakutani-Rokhlin partitions around (0 , , . . . ) with prop-erty (H) in which every partition A n consists of one tower of height a n . Motivated by work of Anatoly M. Vershik in the measured dynamics context, Cantorsystems are described in [HPS92] in terms of Bratteli diagrams. When endowed withan order structure Bratteli diagrams are inherently dynamical objects:
Definition 1.2.13 ([HPS92], Definition 2.3. & 2.6.) . Let Γ = (
V, E ) be a Brattelidiagram.(i) If Γ is endowed with a partial order ≤ on E such that e, ˜ e ∈ E are comparable if and only if r ( e ) = r (˜ e ), it is called an ordered Bratelli diagram . This partialorder induces a total order on r − ( r ( e )) for all e ∈ E .(ii) If Γ is ordered, an infinite path is called minimal (resp. maximal) , if e n isminimal (resp. maximal) in r − ( r ( e n )) for all n ∈ N and such paths necessarilyexist.(iii) Let Γ be ordered. Γ is said to be essentially simple , if it has unique minimal andmaximal paths P min , P max and it is simple if in addition Γ has a telescoping˜Γ such that for all n ∈ N and v ∈ ˜ V n , v ∈ ˜ V n +1 the set { e n ∈ ˜ E n | s ( e n ) = v, r ( e n ) = v } is non-empty.Let Γ be an essentially simple ordered Bratteli diagram. The space of infinite paths P Γ carries a topology given by cylinder sets, resulting in a compact, totally discon-nected, metrizable space. If Γ is simple, P Γ is either finite or a Cantor space. Weneed to define a homeomorphism ϕ Γ : P Γ → P Γ . To this end we set ϕ Γ ( P max ) = P min .Let P = { e n } n ∈ N ∈ P Γ with P = P max . Then P contains edges e i not maximal in r − ( r ( e i )). Let e j be the first such edge appearing in P . Then there exists an edge f j that is a successor of e j in r − ( r ( e j )) with respect to the induced total ordering.There exists a unique path ( f , . . . , f j − ) from v to s ( f j ) such that every containededge f k is minimal in r − ( r ( f k )). Define ϕ Γ ( P ) := ( f , . . . , f j , e j +1 , . . . ). The ob-tained map ϕ Γ is indeed a homeomorphism of P Γ called the Vershik map , namedafter Vershik, whose slightly different version – the adic transformations in [Ver81] – By definition the atoms D nk form a basis of clopens for the topology on X = G a . .2.4 DIMENSION GROUPS inspired this construction. The associated topological dynamical system ( P Γ , ϕ Γ ) iscalled Bratteli-Vershik dynamical system . Fixing P max , makes it a canonical pointedsystem. Theorem 1.2.14 ([HPS92], Theorem 4.7.) . Conjugacy classes of pointed essen-tially minimal Z -systems over compact, totally disconnected, metrizable spaces arein 1-1 correspondence with equivalence classes of essentially simple, ordered Brattelidiagrams. The tool to establish the other direction are nested sequences of Kakutani-Rokhlinpartitions (see Section 4 of [HPS92]): Let (
X, ϕ ) be a minimal Cantor system withdistinguished point x . There exists a nested sequence {A n } n ∈ N of Kakutani-Rokhlinpartitions around x arising from a sufficient descending filtration { B n } n ∈ N of clopensets. The set of vertices in V n is the set of towers of the Kakutani-Rokhlin partition A n . Since B = X , the set V consists of a single vertex. By definition A n +1 is arefinement of A n , following the orbits of points from the base of a tower D n +1 ( i )in V n +1 in positive direction up to the top of D n +1 ( i ) corresponds to following thepositive orbits of these points through a sequence of towers in V n . For every pass-ing of a tower D n ( j ) in V n , we draw a directed edge from D n ( j ) to D n +1 ( i ). Theorder in which towers are passed induces a linear order on these edges. Property(d) in the definition of nested sequences implies that minimal and maximal infinitepaths of the resulting ordered Bratteli diagram are unique. Furthermore, the dia-gram is essentially simple. Choosing another nested sequence of Kakutani-Rokhlinpartition yields an equivalent Bratteli diagram, since restricting to subsequencescorresponds to telescoping. The Bratteli-Vershik system of the associated orderedBratteli diagram is conjugate to the initial dynamical system, eventually implyingTheorem 1.2.14. Remark 1.2.15.
Let (
X, ϕ ) be a minimal Cantor system with a nested sequence ofKakutani-Rokhlin partitions {A n } n ∈ N = { D nk,i | ≤ k ≤ h ni − , i ∈ { , . . . , i n }} withproperty (H). Then for every tower D in some A n there exists an l > n such thatfor every k ≥ n every tower in A k runs through D . This implies that the associatedBratteli diagram is simple. Dimension functions are real valued, non-negative functions on the positive elementsin a C*-algebra (thus in particular on the projections), that preserve the order struc-ture, that are additive on orthogonal elements and invariant under the equivalenceof projections. These functions are in close correspondence with the C*-algebrastraces and can in some sense be seen as measuring the supports of positive elements– see [Bla06], II.6.8. Dimension functions and their ranges had been used in theworks Glimm and Dixmier, inspired by which George A. Elliott gave a classificationof AF C*-algebras in terms of dimension groups – these are a particular kind ofabelian ordered group in which the dimension ranges embed and which have subse-quently become identified as operator K-groups. Elliott’s classification in turn was17 .2. A TOOL BOX OF MINIMAL CANTOR SYSTEMS a building block in the classification of essentially minimal pointed Cantor systems(see Subsection 1.2.3). Bratteli diagrams as well as minimal Cantor systems giverise to dimension groups. These turn out to be K-groups in disguise and have beenfundamental in the classification of minimal Cantor systems: Definition 1.2.16 ([Bla06], § V.2.4) . (i) An ordered group ( G, G + ) is a pair con-sisting of an abelian group G and a positive cone G + i.e. a subset of G suchthat:(a) G + + G + ⊆ G + (b) G + − G + = G (c) G + ∩ ( − G + ) = { } .The positive cone G + induces an G -invariant partial ordering ≤ on G by setting y ≤ x for x, y ∈ G if x − y ∈ G + .(ii) Let ( G , G +1 ) and ( G , G +2 ) be a pair of ordered groups. A positive homomor-phism ϕ : ( G , G +1 ) → ( G , G +2 ) is a group-homomorphism ϕ : G → G with ϕ ( G +1 ) ⊆ G +2 . Ordered groups with positive homomorphisms as morphismsform a category.Let ( G, G + ) be an ordered group.(iii) An element u ∈ G + is called an order unit if for every x ∈ G there exists an n ∈ Z , n > x ≤ n · u i.e. the order ideal generated by u is all of G .(iv) The ordered group ( G, G + ) is said to be simple if it has no proper order idealsi.e. every non-zero positive element is an order unit.(v) It is called unperforated , if for every g ∈ G and positive integer n , n · g ∈ G + implies g ∈ G + .(vi) It is said to have the Riesz interpolation property , if for every g , g , h , h ∈ G with g i ≤ h j for j, i ∈ { , } , there exists an l ∈ G with g i ≤ l ≤ h j for j, i ∈ { , } .(vii) A dimension group is an countable, unperforated, ordered group that has theRiesz interpolation property.(viii) Let u be a fixed order unit of ( G, G + ). The triple ( G, G + , u ) is called a scaledordered group . An isomorphism of scaled ordered groups is a positive isomor-phism that preserves the order unit.(ix) Let ( G, G + , u ) be a scaled ordered group. A state on ( G, G + , u ) is a positivehomorphism f : ( G, G + ) → ( R , R ≥ ) with f ( u ) = 1. Denote by S ( G, G + , u )the set of all states on ( G, G + , u ). See appendix B.4. .2.4 DIMENSION GROUPS (x) Let ( G, G + , u ) be a simple scaled ordered dimension group. An element g ∈ G is called infinitesimal if s ( g ) = 0 for all s ∈ S ( G, G + , u ). The set ofinfinitesimals is denoted by Inf( G ). Remark 1.2.17. (i) For all n ∈ N the group Z n the standard order is defined by( Z n ) + := { ( z , . . . , z n ) ∈ Z n | z i ≥ ≤ i ≤ n } for which the element(1 , . . . ,
1) is an order unit. George A. Elliott introduced dimension groups in[Ell76] as direct limits of sequences of free abelian groups of finite rank withstandard order structure and positive group homomorphisms as maps. The
Effros-Handelman-Shen theorem ([EHS80], Theorem 2.2.) characterizes suchordered groups as the class of groups from Definition 1.2.16(vi).(ii) The order structure of a simple dimension group (
G, G + ) with order unit u isdetermined by the set of states, since G + = { } ∪ { g ∈ G | f ( g ) > f ∈S ( G, G + , u ) } .(iii) We omit the notation of (scaled) order groups as tuples (resp. triples) whenthe order structure is clear.Let Γ = ( V, E ) be a Bratteli diagram. The limit of Z ι , → Z | V | ι , → Z | V | ι , → . . . where ι k : Z | V k | → Z | V k +1 | is given by ι k ( v ) := P e ∈ s − ( v ) r ( e ) for v ∈ V k definesa dimension group denoted by K ( V, E ). Since Z | V k | ∼ = K ( L v ∈ V k M v ), we have K ( V, E ) ∼ = K ( C ∗ ( V, E )) by continuity of the functor K .Later Elliott would go on to conjecture a classification of a wide class of C*-algebrasup to isomorphism by considering both K-groups with some additional informa-tion. This program, which was highly influential in C*-algebra theory, was in partmotivated by his result on AF C*-algebras established in [Ell76]: Theorem 1.2.18.
The scaled ordered K -groups of AF C*-algebras are completeisomorphism invariants. Example 1.2.19.
Let Γ = (
V, E ) be the Bratteli diagram given by an infinite com-plete lexicographically labelled directed binary rooted tree. Then L v ∈ V k M v = L k C for all k ∈ N , which is isomorphic to the C*-algebra of complex-valued functions onthe boundary constant on the cylinder sets C [ { , } k ]. The limit is the C*-algebra C ( X ) of complex-valued continuous functions on the Cantor space given by the in-finite paths. We have K ( C ( X )) = C ( X, Z ) and K ( C ( X )) = 0 by continuity andthe positive cone of C ( X, Z ) is just the subset of non-negative functions. Remark 1.2.20. ([Dav96], Example III.2.5) Commutative AF C*-algebras are pre-cisely the commutative C*-algebras with totally disconnected spectrum.Cantor systems admit dimension groups as follows: 19 .2. A TOOL BOX OF MINIMAL CANTOR SYSTEMS
Theorem 1.2.21 ([GPS95], p. 59) . Let ( X, ϕ ) be a Cantor system. Define K ( X, ϕ ) := C ( X, Z ) / { f − f ◦ ϕ − | f ∈ C ( X, Z ) } Then ( K ( X, ϕ ) , K ( X, ϕ ) + , X ) is a scaled ordered dimension group with positivecone K ( X, ϕ ) + := { [ f ] | f ≥ , f ∈ C ( X, Z ) } . This makes K ( X, ϕ ) / Inf( K ( X, ϕ )) a simple scaled ordered dimension group. This group of coinvariants K ( X, ϕ ) is a K-group in disguise, as noted in Subsec-tion 1.1.3 we have K ( X, ϕ ) ∼ = K ( C ( X ) (cid:111) ϕ Z ). The states of K ( X, ϕ ) have thefollowing description:
Theorem 1.2.22 ([Put89], Corollary 5.7) . Let ( X, ϕ ) be a Cantor system. Every µ ∈ M ϕ induces a state s ( µ ) of the scaled ordered group ( K ( X, ϕ ) , K ( X, ϕ ) + , X ) defined by s ( µ ) : f Z X f d µ for f ∈ C ( X, Z ) and a tracial state of C ( X ) (cid:111) ϕ Z by s ( µ ) ◦ E , where E is the condi-tional expectation defined in Remark 1.1.30(iii), producing a 1-1-1 correspondencebetween:(i) ϕ -invariant probability measures on X (ii) the set of states S ( K ( X, ϕ ) , K ( X, ϕ ) + , X ) (iii) tracial states of C ( X ) (cid:111) ϕ Z This implies:
Corollary 1.2.23.
There is an isomorphism of simple scaled ordered dimensiongroup between K ( X, ϕ ) / Inf( K ( X, ϕ )) and C ( X, Z ) / { f ∈ C ( X, Z ) | R X f d µ =0 for all µ ∈ M ϕ } . Wolfgang Krieger introduced in [Kri80] ample groups – of which the definition re-minds of Dye’s full group, and gave – in the wake of Elliot’s work – a classificationin terms of dimension.
Definition 1.2.24 ([Kri80], p. 88) . Let X be a Cantor space. Let A be a subalgebraof the Boolean algebra B X of clopen sets of X and let G be a countable group ofhomeomorphisms of X that leaves A invariant.(i) Denote by [ G, A ] the group of homeomorphisms h ∈ Homeo( X ) such thatthere exists a finite clopen partition { X i } i ∈ I of X with X i ∈ A for all i ∈ I and a finite collection { g i } i ∈ I of elements in G with X = G i ∈ i X i = G i ∈ i g i ( X i ) , such that h | X i = g i | X i for all i ∈ I .20 .2.6 CLASSIFICATION OF CANTOR SYSTEMS (ii) The pair ( G, A ) is called a unit system if G is a locally finite group such thatthe set of fixed points is in A for all g ∈ G , the map g → g | A is an isomorphismof groups and G = [ G, A ]. A unit system is called finite if A is finite.(iii) Let ( A , G ) and ( B , H ) be unit systems. The unit system ( A , G ) is said to be finer than ( B , H ) if B ⊆ A and H ≤ G .(iv) A countable, locally finite group G of homeomorphisms of X is called ample if ( B X , G ) is a unit system i.e. if the set of fixed points is clopen for all g ∈ G and for any finite clopen partition { X i } i ∈ I of X and finite collection { g i } i ∈ I ofelements in G with X = G i ∈ i X i = G i ∈ i g i ( X i ) , the element g ∈ Homeo( X ) defined by g | X i = g i | X i is in G .(v) The topological dynamical system ( X, G ) given by the action of an amplegroup G of homeomorphisms on X is called an AF-system .Every unit system ( A , G ) can be represented as limit ( S n ∈ N A n , S n ∈ N G n ) of a refiningsequence { ( A n , G n ) } n ∈ N of finite unit systems ([Kri80], Lemma 2.1). For every amplegroup G the associated refining sequence of unit systems { ( A n , G n ) } n ∈ N induces anaction of G on a Bratteli diagram ( V, E ) where the level sets V n are given by the G n -orbits of atoms in A n and the edge structure arises from the relations betweenthese orbits. Under Theorem 1.2.5 this Bratteli diagram corresponds to the crossedproduct C*-algebra C ( X ) (cid:111) G associated to the AF-system ( X, G ) with associateddimension group K ( X, G ) := K ( C ( X ) (cid:111) G ). Drawing from Elliot’s classificationof AF C*-algebras, Krieger obtained the following: Theorem 1.2.25 ([Kri80], Corollary 3.6) . Let G , G be ample groups of homeo-morphisms of a Cantor space X . Then G and G are spatially isomorphic if andonly if the dimension groups K ( X, G ) and K ( X, G ) are isomorphic as scaledordered groups. In [Put89] Putnam used a specific subalgebra of C ( X ) (cid:111) ϕ Z associated to a minimalCantor system to obtain information on the K-theory of C ( X ) (cid:111) ϕ Z . Lemma 1.2.26 ([Put89], Lemma 3.1) . Let ( X, ϕ ) be a minimal Cantor system, let Y be a clopen subset of X and let P = { P i } i ∈ I be a finite clopen partition of X .Then the C*-subalgebra A Y, P of C ( X ) (cid:111) ϕ Z generated by C ( P ) := h{ χ P i } i ∈ I i and uχ X \ Y is finite dimensional. A group is called locally finite if every finitely generated subgroup is finite. Equivalently it isa group that is a direct limit of a directed system of finite groups. See Definition 3.1.50. .2. A TOOL BOX OF MINIMAL CANTOR SYSTEMS Sketch of Proof.
Let A be a Kakutani-Rohklin partition of ( X, ϕ ) which has Y asits base and let A = { D k,i | ≤ k ≤ h i − , i ∈ { , . . . , n }} be the refinement of A by P . Let P be the clopen finite partition of X induced by A . One can show that A Y, P is contained in a subalgebra A ( Y, P ) of C ( X ) (cid:111) ϕ Z isomorphic to L ni =1 M h i ( C )given by matrix units E ( k ) i,j = u i − j χ D k,j = χ D k,i u i − j . Note that in particular diagonalentries generate C ( P ).This then implies: Theorem 1.2.27 ([Put89], Theorem 3.3) . Let ( X, ϕ ) be a minimal Cantor systemand let Y be a closed subset of X . Then the C*-subalgebra A Y of C ( X ) (cid:111) ϕ Z generatedby C ( X ) and uC ( X \ Y ) is an AF C*-algebra.Sketch of Proof. Let {A n } n ∈ N be a nested sequence of Kakutani-Rokhlin partitionsaround Y induced by a descending filtration { B n } n ∈ N around Y . Let P n be the finiteclopen partition of X induced by A n . Then one can show that A Y is the closed unionof the finite dimensional C*-subalgebras A ( B n , P n ) from Lemma 1.2.26.The above observations are fundamental for the classification obtained in [GPS95]in that they imply the following theorem: Theorem 1.2.28 ([Put89], Theorem 6.7) . Let ( X, ϕ ) be a minimal Cantor systemand let x ∈ X . Then there is a unital embedding ι : C ( X ) (cid:111) ϕ Z , → A { x } of whichthe induced map in K-theory K ( ι ) : K ( C ∗ ( X, ϕ )) → K ( A { x } ) is an isomorphismof ordered groups. We finish this chapter with a quick recollection of the classification of minimal Cantorsystems from [GPS95]:
Theorem 1.2.29 ([GPS95], Theorem 2.1) . Let ( X , ϕ ) and ( X , ϕ ) be minimalCantor systems. Then the following are equivalent:(i) The systems ( X , ϕ ) and ( X , ϕ ) are strongly orbit equivalent.(ii) The K -groups K ( X , ϕ ) and K ( X , ϕ ) are isomorphic as scaled orderedgroups.(iii) C ( X ) (cid:111) ϕ Z and C ( X ) (cid:111) ϕ Z are isomorphic. For ( ii ) ⇒ ( i ) the orbit map is derived from the representations as Bratteli-Vershiksystems. The inverse direction ( i ) ⇒ ( ii ) is a consequence of the work of Putnamon the AF C*-subalgebras A Y . ( ii ) ⇔ ( iii ) follows from work of Elliott. He had obtained a characterization of crossed products C*-algebras associated with minimalCantor systems as the simple circle algebras of real rank zero with K -group equal to Z and hadshown that K -groups as scaled ordered groups are a complete isomorphism invariant of simplecircle algebras of real rank zero. .2.6 CLASSIFICATION OF CANTOR SYSTEMS Another theorem obtained in [GPS95] is directly influenced by results in measureddynamics. Every non-singular transformation T on a Lebesgue space ( X, λ ) givesrise to an R -action on the weights of the associated von Neumann factor and furtherto a von Neumann crossed product called the flow of weights . In [Kri76] WolfgangKrieger showed that orbit equivalence of non-singular transformations T and T is equivalent to conjugacy of their associated flows. In the topological dynamicssetting, the role of the flow of weights is taken by the simple scaled ordered dimensiongroup K ( X, ϕ ) / Inf( K ( X, ϕ )) – see Corollary 1.2.23.
Theorem 1.2.30 ([GPS95], Theorem 2.2) . Let ( X , ϕ ) and ( X , ϕ ) be minimalCantor systems. Then the following are equivalent:(i) The systems ( X , ϕ ) and ( X , ϕ ) are orbit equivalent.(ii) The groups K ( X , ϕ ) / Inf( K ( X , ϕ )) and K ( X , ϕ ) / Inf( K ( X , ϕ )) areisomorphic as scaled ordered groups.(iii) There exists a homeomorphisms F : X → X such that M ϕ is mapped to M ϕ . ( i ) ⇒ ( iii ) is straightforward and ( iii ) ⇒ ( ii ) follows from Theorem 1.2.22. Im-plication ( ii ) ⇒ ( i ), however, is by far the most technical part in all of [GPS95]and requires besides Bratteli-Vershik systems and Putnam’s results in [Put89] anexistence theorem by Cartan-Eilenberg from homological algebra. Remark 1.2.31.
Theorem 2.3 in [GPS95] extends Theorem 1.2.30 to minimal AF-systems.Closest to our concern is the following classification theorem, which clarifies underwhich assumptions minimal Cantor systems are flip conjugate. Topological fullgroups (see Chapter 3) – they are already implicit in the works [Put89] and [GPS95]– were later on recognized as complete invariants for flip conjugacy of minimalCantor systems:
Theorem 1.2.32 ([GPS95], Theorem 2.4) . Let ( X , ϕ ) and ( X , ϕ ) be minimalCantor systems. Then the following are equivalent:(i) The systems ( X , ϕ ) and ( X , ϕ ) are flip conjugate.(ii) There exists an orbit map F : X → X such that the orbit cocycles associatedwith F are continuous.(iii) There exists an isomorphism α : C ( X ) (cid:111) ϕ Z → C ( X ) (cid:111) ϕ Z such that α ( C ( X )) = C ( X ) . hapter 2´Etale groupoids and inversesemigroups In this chapter we roam around the playgrounds of groupoids and inverse semigroups .While it is the study of Cantor systems from which topological full groups emanated,this setting is embedded in the broader context of ´etale groupoids.
As references for the basic definitions in this section see [Ren80] and [Pat99] exceptwhere noted. Subsection 2.1.1 lists the basic definitions and examples of groupoidsand Subsection 2.1.2 does the same for topological groupoids.
In the twentieth century especially groupoids have become objects of utmost impor-tance – naturally in step with the rise of category theory.
Definition 2.1.1. (i) A groupoid is a set G with a product map ( g , g ) g g defined on a subset G (2) ⊆ G × G and an inverse map g g − defined on allof G satisfying the following conditions for all g, g , g , g ∈ G :(a) If ( g , g ) , ( g , g ) ∈ G (2) , then ( g g , g ) , ( g , g g ) ∈ G (2) and ( g g ) g = g ( g g )(b) ( g, g − ) , ( g − , g ) ∈ G (2) (c) If ( g , g ) ∈ G (2) , then ( g , g ) g − = g and g − ( g , g ) = g Let G be a groupoid.(ii) A subset H ⊆ G is called a subgroupoid of G if it is closed with respect to theproduct and inverse map. .1.1 GROUPOIDS (iii) The structure maps s : G → G (0) , g g − g resp. r : G → G (0) , g gg − arecalled source map resp. range map .(iv) The elements in G (2) are called composable pairs . Denote more generally for all n ∈ N by G ( n ) the set of sequences ( g , g , . . . , g n ) ∈ G n such that the product g g . . . g n is defined, i.e. s ( g i ) = r ( g i +1 ) for all i ∈ , , . . . , n − product of two subsets G , G ⊆ G is defined as the set G G := { ( g , g ) ∈G (2) | g ∈ G , g ∈ G } .(vi) The elements of { gg − | g ∈ G} are called units , the set of all units is denotedby G (0) .(vii) Let U be a subset of G (0) . We set G U := r − ( U ) and G U := s − ( U ), in particularthe range fiber of a unit u ∈ G (0) is denoted by G u and its source fiber by G u .The set G U ∩ G U forms a subgroupoid of G when endowed with the restrictedproduct- and inverse map called reduction of G to U denoted by G| U . Inparticular for any u ∈ G (0) , the reduction groupoid G| u is a group of pairwisecomposeable elements of G called the isotropy group of u . Denote by G triv theset of units with trivial isotropy groups.(viii) A subset B ⊆ G is called slice or G -bisection , if s | B and r | B are injective maps.Denote the set of all slices in G by B G .(ix) The groupoid G is called principal , if G triv = G (0) holds.(x) The set Iso( G ) := { g ∈ G : s ( g ) = r ( g ) } = S u ∈G (0) G| u forms a subgroupoid of G , called the isotropy bundle of G .(xi) A pair of units u , u ∈ G (0) is contained in the same G -orbit , if there existsa g ∈ G such that s ( g ) = u and r ( g ) = u . Let u ∈ G (0) , denote the orbitcontaining u by G ( u ).(xii) ([Nek15], p. 31) If a subset S ⊆ G intersects every G -orbit it is called a G -transversal .(xiii) Let G , G be groupoids. A groupoid homomorphism is a map ϕ : G → G suchthat ϕ ( G (0)1 ) ⊆ G (0)2 and for all ( g, h ) ∈ G (2)1 it holds that (cid:16) ϕ ( g ) , ϕ ( h ) (cid:17) ∈ G (2)2 and ϕ ( gh ) = ϕ ( g ) ϕ ( h ). If ϕ has an inverse, it is a groupoid isomorphism . Inparticular, groupoid homomorphisms intertwine the structure maps s and r .(xiv) A subset S is said to be a generating set of G if G = S n ∈ N ( S ∪ S − ) n . Remark 2.1.2. (i) Groupoids are essentially the same as a small category inwhich every morphism is invertible, generalizing the representation of groupsas categories of the same kind with only one object by Cayley’s theorem.Groupoid elements translate to morphisms in the category theoretic descrip-tion, the units to identity morphisms, range- and source map to the respective25 .1. BASICS OF GROUPOIDS category theoretic term, groupoid homorphisms to functors between categoriesof the described kind etc. Let G be a groupoid.(ii) A set B ⊆ G is a slice if and only if BB − ⊆ G (0) or equivalently B − B ⊆ G (0) holds.(iii) Note that if two units u, v ∈ G (0) are contained in the same orbit, then theirisotropy groups G| u and G| v are isomorphic. Example 2.1.3 ([Ren80], Examples 1.2.c) . Let ∼ be an equivalence relation on aset X . Then the set G ∼ := { ( x, y ) ∈ X × X | x ∼ y } becomes a groupoid by defininga pair (( x , y ) , ( x , y )) to be composeable if y = x . Their product is defined by( x , y ) · ( x y ) = ( x , y ) and the inverse by ( x, y ) − = ( y, x ). The arising groupoid G ∼ is principal and G (0) ∼ ∼ = X .Equivalence relations on spaces with structure can be studied by looking at thequotient by collapsing equivalence classes, however, if the quotient map is not suf-ficiently “nice”, the structure of the quotient space might turn out to be “bad”,e.g. for the Kronecker foliation , a foliation of the torus T = S × S induced by a1-dimensional subbundle of the tangent bundle with constant irrational slope, theleaf space carries no manifold structure. Groupoids offer a way to encode the datawithout the – possibly brutal – application of a quotient map e.g. in the contextof foliations this role is played by the holonomy groupoid . Some of the terminologyfrom Definition 2.1.1 is accounted for by the capability of groupoids to picture groupactions: Definition 2.1.4 ([Ren80], Examples 1.2.a) . Let α : G (cid:121) X be the action of a group G on a set X . The set G × X carries the structure of a groupoid: A tuple of elements( g , x ) , ( g , x ) ∈ G × X is composeable, if g ( x ) = x . Its product is defined by( g , x )( g , x ) = ( g g , x ) and the inverse by ( g, x ) − = (cid:16) g − , g ( x ) (cid:17) . The resultinggroupoid is called the groupoid of the action or transformation groupoid and isdenoted by G ( X,G ) and by G ϕ in the case of Z -action generated by a homeomorphism ϕ . Its unit space G (0)( X,G ) coincides with X . A transformation groupoid is principal ifand only if the action is free. The groupoids we consider are more than algebraic objects – pre-eminently they aretopological objects and under sufficient conditions open to the methods of analysis. Note however, when interpreted as ”arrows”, products must be read from right to left to makesense! .1.2 TOPOLOGICAL GROUPOIDS Definition 2.1.5. (i) Groupoids endowed with a topology such that product mapand inverse map are continuous are called topological groupoids . Let G be a topological groupoid.(ii) If G triv is dense in G (0) , the groupoid G is called essentially-principal .(iii) If all G -orbits are dense in G (0) , the groupoid G is called minimal .(iv) Denote the set of all open slices by B o G and the set of all open, compact slicesby B o,k G . Example 2.1.6. (i) Topological groups are topological groupoids.(ii) For an equivalence relation ∼ on a topological space X , the groupoid G ∼ becomes a topological groupoid when endowed with the subspace topologycoming from X × X .(iii) Let α : G → Homeo( X ) define a continuous action of a group G on a topo-logical space X and let { U i } i be a basis of the topology on X . Then thetransformation groupoid can be endowed with a topology: The sets { ( g, U i ) } form the basis of a topology on G ( X,G ) for which the product and inverse mapsare trivially continuous.(iv) If in the above example G is a discrete group, an equivalence relation can bedefined on G ( X,G ) by ( g , x ) ∼ ( g , x ) if x = x and there exists a neighbour-hood U of x such that g | U = g | U . The quotient Germ( X, G ) := G ( X,G ) / ∼ inherits the structure of a topological groupoid. The resulting groupoid iscalled the groupoid of germs of the action . Remark 2.1.7.
Let G be a topological groupoid.(i) If G (0) is Hausdorff, the isotropy bundle Iso( G ) is closed, since it is the preimageof the diagonal ∆ ⊆ G (0) × G (0) under the continuous map g ( s ( g ) , r ( g )).(ii) If a topological groupoid G is Hausdorff, then G (0) is closed, since it is thepreimage of ∆ ⊆ G × G under the continuous map g ( g, g − ).The following lemma is crucial in the characterization of ´etale groupoids: Lemma 2.1.8 ([Res07], Lemma 5.16) . Let G be a topological groupoid and let Ω( G ) be its collection of open sets. Then every U ∈ Ω( G ) satisfies:(i) ( U ∩ G (0) ) G ⊆ S { X ∩ Y | X, Y ∈ Ω( G ) , XY − ⊆ U } ⊆ S { V ∈ Ω( G ) | V V − ⊆ U } (ii) G ( U ∩ G (0) ) ⊆ S { X ∩ Y | X, Y ∈ Ω( G ) , X − Y ⊆ U } ⊆ S { V ∈ Ω( G ) | V − V ⊆ U } This implies continuity of the range and source maps. .2. BASICS OF INVERSE SEMIGROUPS With Felix Klein’s proposal from the Erlangen program, to study geometries bythe means of transformations under which geometric structure is invariant, groupsbecame the predominant algebraic structure to describe symmetries. While theyare a proper algebraic tool when dealing with “global” symmetries, they fail tocharacterize symmetries of “local” nature. In differential geometry local structures(e.g. foliations) are defined in terms of atlantes of coordinate charts which areinvariant under a family of chart transition maps. The transition maps are partialfunctions and cannot be abstractly described by groups. Space groups are fit todescribe symmetries of crystals and periodic tilings, but fail to do so for quasicrystalsand aperiodic tilings. Scaling symmetries of regions in self-similar structures (e.g.Julia sets in holomorphic dynamics) warrant a structure more general then groups.One way to deal with this insufficiency is provided by pseudogroups , another bygroupoids. More on the motivation and historical background can be found in § Definition 2.2.1 ([Joh82]) . (i) A poset is a set P with a binary relation ≤ thatis reflexive, anti-symmetric and transitive.Let P be a poset and let S be a subset of P .(ii) A join (resp. meet) of S is an element a such that(a) s ≤ a (resp. a ≤ s ) for all s ∈ S (b) s ≤ b (resp. b ≤ s ) for all s ∈ S implies a ≤ b (resp. b ≤ a ) for all b ∈ B .If a join (resp. meet) of S exists it is unique and denoted by W S (resp. V S ).(iii) Denote by S ↑ the set { p ∈ P | p ≤ s for all s ∈ S } and by S ↓ the set { p ∈ P | s ≤ p for all s ∈ S } . The subset S is called (downward) directed if for everypair a, b ∈ S there exists a c ∈ S with c < a and c < b .(iv) If S = S ↓ holds, S is called an order ideal . If S is a directed set with S = S ↑ ,it is called a filter .(v) A join (resp. meet) semi-lattice is a poset in which every finite subset hasa join (resp. meet) and thus ∨ (resp. ∧ ) defines a binary operation. Everyjoin (resp. meet) semi-lattice contains a unique least element 0 (resp. uniquegreatest element 1) with l ∨ l (resp. l ∧ l ) for all l ∈ L .(vi) A lattice is a poset in which every finite subset has a join and a meet. Let
X, Y be sets. A partial function f : X → Y is a function f : A → B where A ⊆ X and B ⊆ Y . .2.2 INVERSE SEMIGROUPS Let L be a lattice.(vii) The lattice L is said to be complete if every subset has a join and a meet.(viii) The lattice L is called distributive , if l ∧ ( m ∨ n ) = ( l ∧ m ) ∨ ( l ∧ n ) holds for all l, m, n ∈ L . It is said to be infinitely distributive if for every x ∈ L and every S ⊆ L the following holds: x ∧ _ s ∈ S s = _ s ∈ S ( x ∧ s )(ix) Let a ∈ L . Any element b ∈ L with a ∧ b = 0 and a ∨ b = 1 is called a complement of a .(x) A Boolean algebra is a distributive lattice B with an unary operation ¬ : B → B such that ¬ b is a complement of b for all b ∈ B .(xi) Let A, B be Boolean algebras. A function f : A → B is called a homomorphismof boolean algebras if it satisfies f (0) = 0, f (1) = 1, f ( a ∧ a ) = f ( a ) ∧ f ( a )and f ( a ∨ a ) = f ( a ) ∨ f ( a ) for all a , a ∈ A .(xii) A frame is a complete infinitely distributive lattice. Let F, R be frames. Afunction f : F → R is called a morphism of frames if it preserves finite meetsand arbitrary joins.(xiii) The category of frames Frm is the category whose class of objects consistsof frames and whose morphisms consist of morphisms of frames. Its oppositecategory is called the category of locales denoted by
Loc , whose objects arecalled locales and whose morphisms are called continuous maps . Example 2.2.2.
Let X be a topological space. Taking union and intersection asjoin and meet the set Ω( X ) of open subsets of X becomes a frame. A continuousmap f : X → Y induces a morphism of frames Ω( f ) : Ω( Y ) → Ω( X ). The abstract setting for the aforementioned pseudogroups is provided by inversesemigroups:
Definition 2.2.3 ([Law98]) . (i) A semigroup is a set S together with an associa-tive binary operation.(ii) Let S be a semigroup. A subsemigroup of S is a subset which is closed withrespect to the semigroup operation. On the level of objects frames and locales are completely synonymous, it is at the level ofmorphisms at which these notions are different. .2. BASICS OF INVERSE SEMIGROUPS (iii) A monoid is a semigroup S , that contains a unique element 1 ∈ S called the identity such that 1 s = s = 1 s holds for all s ∈ S .(iv) Let S be a monoid. An element s ∈ S is called a unit if there exists an element t ∈ S such that 1 = st = ts . Denote by U ( S ) the set of all units in S .(v) A semigroup S is called a semigroup with zero , if it contains an element 0 suchthat 0 · s = 0 = s · s ∈ S .Let S be a semigroup.(vi) It is said to be an inverse semigroup if for every s ∈ S there exists a unique t ∈ S such that sts = s and tst = t . This element is called the inverse of s and denoted by s − . The map S → S induced by taking inverses is called involution .(vii) An element s ∈ S is called an idempotent if s = s . The set of all idempotentsis denoted by E ( S ).(viii) Let S and T be semigroups. A semigroup homomorphism is a map f : S → T such that f ( s s ) = f ( s ) f ( s ) for all s , s ∈ S .Let S be an inverse semigroup.(ix) For all s ∈ S define the source- (resp. range) maps d, r : S → E ( S ) by d : s s − s (resp. r : s ss − ). Remark 2.2.4. (i) The set of units U ( S ) in a monoid S is a group with respectto the semigroup operation. Topological full groups turn out to be groups ofsuch kind.(ii) All inverse semigroup are from now on assumed to be inverse semigroups withzero!
Inverse semigroups carry a natural poset structure: Let S be an inverse semigroup.Define a partial order on S by s ≤ t if and only if there exists an idempotent e suchthat s = et or equivalently s = ss − t . For all s ∈ S the elements ss − and s − s areidempotents. The set of idempotents E ( S ) is a commutative idempotent semigroup.Commutative idempotent semigroups are essentially the same as semilattices andthe product of two elements is given by their meet. The following are importantbasic examples of inverse semigroups: Example 2.2.5. (i) ([Hae02], p.276) Historically the motivation for the intro-duction of abstract inverse semigroups had been pseudogroups in differentialgeometry. The definition of pseudogroups varies in the literature, we fix a tra-ditional pseudogroup to be a set S of partial homeomorphisms between opensets of a topological space X such that the following hold:30 .2.2 INVERSE SEMIGROUPS (a) For every s : U → V and s : U → V in S the inverse s − and the com-postion s s given by the partial homeomorphism s | U ∩ V ◦ s | ( s ) − ( U ∩ V ) are contained in S .(b) The homeomorphism id X is in S .(c) Any partial homeomorphism between open subsets of X that is locallyin S , is in S .(ii) The inverse of an element s in an inverse semigroup is often denoted by s ∗ reminding of the adjoint in C*-algebras. This is not just a coincidence: Let A be a C*-algebra. The set Par( A ) of all partial isometries in A is an inversesemigroup with respect to multiplication, where the inverse is given by theadjoint. If A is unital, it is an inverse monoid. Any subset of Par( A ) which isclosed with respect to multiplication and involution is an inverse semigroup.In fact every inverse semigroup has a representation as an inverse semigroupof partial isometries over a Hilbert space ([Pat99], Proposition 2.1.4).(iii) ([Law98], p.5 & p.36) Let X be a set. Denote by I ( X ) the set of all bijectivepartial functions of X . In particular I ( X ) contains all partial empty functionsand all partial identities. Let f : X → Y and g : Y → Z be partial functions.Define the partial function g ◦ f : X → Z by setting dom( g ◦ f ) := f − (dom f ∩ Im g ) and ( g ◦ f )( x ) := g ( f ( x )) for all x ∈ dom( g ◦ f ). Endowing I ( X )with the multiplication given by composition makes it an inverse monoid,called the symmetric inverse monoid of X . Generalizing Cayley’s theorem ongroups the Wagner-Preston theorem shows that every inverse semigroup S hasa representation as a subsemigroup of I ( S ).(iv) Let G be a groupoid. Then the set of slices B G is an inverse monoid withrespect to the product and inversion of subsets of G , where the identity isgiven by G (0) . Definition 2.2.6 ([Exe08], Definition 4.3) . (i) Let X be a locally compact Haus-dorff space and S an inverse semigroup. A continuous action α of S on X is aninverse semigroup homomorphism α : S → I ( X ) such that for every s ∈ S thepartial function α ( s ) is continuous with open domain and { dom( α ( s )) | s ∈ S } covers X .(ii) Let α be a continuous action of an inverse semigroup S on a compact, Hausdorffspace X . It is called faithful , if α is injective. The action is called relativelyfree if for all s ∈ S the set of fixed points of α ( s ) in dom( α ( s )) is compact andopen.The groupoid of germs is a standard construction in the theory of foliations. Exam-ple 2.1.6(iv) is a particular case of this more general construction: Definition 2.2.7 ([Exe08], p. 208-213) . Let α be a continuous action of an inversesemigroup S on a locally compact, Hausdorff space X . Define the subset Ξ := { ( s, x ) ∈ S × X | x ∈ dom( α ( s )) } . Define an equivalence relation on Ξ by ( s , x ) ∼ .3. ANALYSIS ON TOPOLOGICAL GROUPOIDS ( s , x ) if x = x and there exists an idempotent e ∈ E ( S ) such that x ∈ dom( α ( e ))and s e = s e . An equivalence class [( s, x )] in Ξ / ∼ is called the germ of s at x .This set of germs can be endowed with a groupoid structure by defining a tuple ofequivalence classes ([( s , x )] , [( s , x )]) to be composable if x = α ( s ) x and definethe product as [( s s , x )]. Define the inverse by [( s, x )] − = [( s − , α ( s ) x )]. Theobtained groupoid is called the groupoid of germs of the action and is denoted byGerm(
X, S ). It is a topological groupoid with respect to the basis given by sets τ ( s, U ) = { [( s, x )] : x ∈ U } where s ∈ S and U is an open subset of dom( α ( s )) andits unit space is homeomorphic to X . In this section we give basic aspects and definitions that allow for analysis ongroupoids. Subsection 2.3.1 explains what is meant by “locally compact groupoids”.Subsection 2.3.2 looks at Haar systems. These are a fibered generalization ofHaar measures on locally compact groups that enable analysis on locally compactgroupoids. Subsection 2.3.3 gives the definition of groupoid C*-algebras and the –compared to groups more fiddly – regular representations of groupoids. In particu-lar unitary representations of groupoids are defined in terms of
Hilbert bundles andHilbert modules – see Appendix B.3.
As topological spaces topological groupoids are bit “nastier” than topological groups.The following example may seem affected, but non-Hausdorff groupoids do emergein the wild e.g. in foliation theory:
Example 2.3.1 ([KS02], Example 1.2) . Let X be a compact space together with anon-isolated point x ∈ X and let Γ be a non-trivial discrete group acting trivially on X i.e γ · x = x for all γ ∈ Γ , x ∈ X . Consider the equivalence relation ( γ , x ) ∼ ( γ , x )if x = x . The topological quotient G := (Γ × X ) / ∼ of the transformation groupoidcarries the structure of a topological groupoid which is non-Hausdorff, because if x → x , then ( g, x ) = ( e, x ) → ( g, x ) for every g ∈ Γ and equivalence classesof points in Γ × { x } cannot be seperated by open neighbourhoods. Continuousfunctions must be constant on Γ × { x } and thus if Γ is infinite every compactlysupported continuous function vanishes on Γ × { x } . This means C c ( G ) no longersufficiently encodes the topological data of G .Alain Connes has shown that in groupoids from foliation theory enough “Hausdorff-ness” is present to render analysis possible – such groupoids are “locally Hausdorff”i.e. every groupoid element has a compact, Hausdorff neighbourhood. With this inmind one makes the following replacement of C c ( G ): One has to verify this is well defined i.e. does not depend on the choice of representants. .3.2 HAAR SYSTEMS AND MEASURES ON GROUPOIDS Definition 2.3.2 ([KS02], p. 52) . Let G be a topological groupoid, such that every g ∈ G has a compact Hausdorff neighbourhood. Let C c ( U G ) denote the set of complexvalued functions f such that there exists an open Hausdorff subset U f ⊆ G for which f | U f is compactly supported and continuous with respect to the subspace topologyon U f and f is zero-valued on the outside of U f . Define C ( G ) := h C c ( U G ) i .If G is Hausdorff, then C ( G ) = C c ( G ), if G is non-Hausdorff, functions in C ( G )need not be continuous! The definition of locally compactness for groupoids hasto be chosen such that it guarantees the existence of sufficiently many compactHausdorff neighbourhoods in the non-Hausdorff case enabling the construction fromDefinition 2.3.2. Definition 2.3.3 ([KS02], Definition 1.1) . Let G be a topological groupoid. It issaid to be locally compact , if it satisfies the following properties:(i) The topological subspace G (0) is Hausdorff.(ii) Each g ∈ G has a compact, Hausdorff neighbourhood.(iii) The groupoid G is σ -compact i.e. it is a countable union of compact subspaces.(iv) The source and range maps s and r are open. Remark 2.3.4. (i) The definitions of locally compact groupoids vary in the lit-erature e.g. compare above definition with Definition 2.2.1 of [Pat99].Let G be a locally compact groupoid.(ii) By condition (ii) in the definition every singleton in G is closed. The groupoid G is Hausdorff if and only if G (0) is closed. Compact subsets of G need notbe closed. The range fibers G u and source fibers G u are Hausdorff in therelative topology for all u ∈ G (0) , since their respective diagonal { ( g, g ) ∈ G u ×G u | r ( g ) = r ( g ) , g − g = u } (resp. { ( g, g ) ∈ G u × G u | s ( g ) = s ( g ) , gg = u } )is closed as the preimage of a singleton.(iii) Functions in C ( G ) are not necessarily continuous, however all of them arebounded Borel functions.(iv) Let U jj ∈ J be an open Hausdorff cover of G . Then C ( G ) consists of all finitesums f = P i ∈ I f i where for every i ∈ I the function f i is the extension of afunction in C c ( U i ) by zero outside of U i . The tool that enables analysis on groupoids is a construction in the vein of the Haarmeasure for locally compact groups. Instead of a single measure, one needs to workwith a family of measures parametrized by the units of the groupoid with supportsin the respective source fiber: 33 .3. ANALYSIS ON TOPOLOGICAL GROUPOIDS
Definition 2.3.5 ([KS02], p. 50) . Let G be a locally compact groupoid.(i) A right Haar system on G is a family { ν u } u ∈G (0) , where every ν u is a positive,regular, locally finite Borel measures on G u , such that:(a) supp( ν u ) = G u for all u ∈ G (0) .(b) R G r ( g ) f ( hg ) d ν r ( g ) ( h ) = R G s ( g ) f ( h ) d ν s ( g ) ( h ) for all g ∈ G and f ∈ C ( G ).(c) The maps α f : G (0) → R + defined by u R G u f ( g ) d ν u ( g ) are containedin C c ( G (0) ) for all f ∈ C ( G ).(ii) Let ν = { ν u } u ∈G (0) be a right Haar system. Denote for u ∈ G (0) by ν u themeasure on G u defined by ν u ( A ) = ν u ( A − ) for every Borel subset A ⊆ G u .Then ν − := { ν u } u ∈G (0) is the left Haar system induced by ν .This makes C ( G ) a convolution algebra: Theorem 2.3.6 ([Pat99], Theorem 2.2.1) . Let G be a locally compact groupoid andlet ν = { ν u } u ∈G (0) be a fixed left Haar system. The space C ( G ) can be given thestructure of a normed ∗ -algebra by setting for all f, f , f ∈ C ( G ) and g, h ∈ G f ∗ ( g ) := f ( g − )( f ∗ f )( g ) := Z G s ( g ) f ( gh − ) f ( h ) d ν s ( g ) ( h ) k f k I := sup u ∈G (0) n max (cid:16) Z G u | f ( g ) | d ν u ( g ) , Z G u | f ( g − ) | d ν u ( g ) (cid:17)o While the Haar measure of a locally compact group necessarily exists and is uniqueup to multiplication by a constant factor, neither of which is necessarily true for theHaar systems of locally compact groupoids.
Definition 2.3.7 ([Pat99], p. 86) . Let G be a locally compact groupoid with a fixedHaar system { ν u } u ∈G (0) and let µ be a probability measure on the space G (0) .(i) The measure µ induces a positive regular Borel measure ν µ on G by ν µ := Z G (0) ν u d µ (ii) Denote by ν − µ the positive regular Borel measure on G defined by ν − µ ( B ) = ν µ ( B − ) for every Borel set B ⊆ G . Note that ν − µ = Z G (0) ν u d µ (iii) Define a regular Borel measure ν µ on G (2) by Z G (2) f ( g, h ) d ν µ = Z G (0) d µ Z Z f ( g, h ) d ν u ( g ) d ν u ( h )34 .3.2 HAAR SYSTEMS AND MEASURES ON GROUPOIDS for f ∈ C c ( G (2) ). Remark 2.3.8.
To set up ν µ , one needs to show for every open set U ⊆ G withcompact closure in a Hausdorff subset U ⊆ G the linear functional Φ U on C c ( U )defined by Φ U ( f ) := Z G (0) d µ Z G u f d ν u is continuous. By the Riesz representation theorem, there exists a regular Borelmeasure ν U such that Z f d ν U = Φ U ( f )It can be shown that the resultant measures ν U coincide on intersections and thusthere exists a regular Borel measure ν µ such that ν µ | U = ν U . Similar argumentsmust be applied to obtain ν µ . Example 2.3.9.
Let G be a locally compact groupoid with a fixed Haar system { ν u } u ∈G (0) and let v ∈ G (0) . Let δ v denote the Dirac measure at v . Its induced Borelmeasure is given by ν δ v = ν v .As with the special case of transformation groups, not all probability measures onthe unit space are meaningful for the groupoids representation theory, one restrictsto probability measures that satisfy a specific invariance property: Definition 2.3.10 ([Ren80], Definition 3.2 & 3.5 & 3.7) . Let G be a locally compactgroupoid with a fixed Haar system { ν u } u ∈G (0) and let µ be a measure on G (0) .(i) The measure µ is said to be quasi-invariant if the measure ν µ is equivalent tothe measure ν − µ and it is called invariant if ν µ = ν − µ .(ii) A µ -measurable subset U ⊆ G (0) is called almost invariant if s ( g ) ∈ U ⇔ r ( g ) ∈ U for ν µ -a.e. g ∈ G (iii) Let µ be quasi-invariant. It is called ergodic if every almost invariant measur-able set U ⊆ G (0) satisfies either µ ( U ) = 0 or µ ( G (0) \ U ) = 0.(iv) The measure [ µ ] := r ∗ ν µ i.e. for a µ -measurable set E ⊆ G (0) define r ∗ ν µ ( E ) = ν µ ( r − ( E )), is called the saturation of µ .Ergodic, quasi-invariant measures always exist: Proposition 2.3.11 ([Ren80], Proposition 3.6) . Let G be a locally compact groupoidwith a fixed Haar system { ν u } u ∈G (0) and let µ be a measure on G (0) . The saturation [ µ ] is a quasi-invariant measure. Proposition 2.3.12 ([Ren80], Proposition 3.8) . Let G be a locally compact groupoidwith a fixed Haar system { ν u } u ∈G (0) and let v ∈ G (0) . Then the saturation [ δ v ] = r ∗ ν v is an ergodic measure. .3. ANALYSIS ON TOPOLOGICAL GROUPOIDS The presence of multiple units in a groupoid G entails that its unitary representationstake range over fibered objects with G (0) as the base space i.e. Hilbert bundles insteadof Hilbert spaces. Definition 2.3.13 ([Pat99], Definition 3.1.1) . Let G be a locally compact groupoidwith a fixed Haar system { ν u } u ∈G (0) and let µ be probability measure on G (0) . A unitary representation L of G on a measured Hilbert bundle ( G (0) , { H u } u ∈G (0) , µ ) is afunction L : G → S u,v ∈G (0) U ( H u , H v ) with the following properties :(i) The probability measure µ is quasi-invariant.(ii) L ( g ) ∈ U ( H s ( g ) , H r ( g ) ) for every g ∈ G .(iii) L ( u ) = id H u for all u ∈ G (0) .(iv) L ( g ) L ( h ) = L ( gh ) holds ν µ -almost everywhere in G (2) .(v) L ( g ) − = L ( g − ) holds ν µ -almost everywhere in G .(vi) For every pair σ , σ ∈ R ⊕G (0) H u d µ the function on G defined by g
7→ h L ( g ) σ ( s ( g )) , σ ( r ( g )) i is ν µ -measurable. Example 2.3.14 ([Pat99], p. 93) . Let G be a locally compact groupoid with a fixedHaar system ν = { ν u } u ∈G (0) and let µ be probability measure on G (0) . Then the rightregular representation of G with respect to ν and µ is given on the measured Hilbertbundle ( G (0) , { L ( G u , ν u ) } u ∈G (0) , µ ) by: L red ( g )( f )( h ) = f ( hg )for all g ∈ G , f ∈ L ( G s ( g ) and h ∈ G r ( g ) . The left regular representation of G with re-spect to ν and µ is given on the measured Hilbert bundle ( G (0) , { L ( G u , ν u ) } u ∈G (0) , µ )by: L red ( g )( f )( h ) = f ( g − h )for all g ∈ G , f ∈ L ( G s ( g ) and h ∈ G r ( g ) . See [KS02] for a more detailed description.
Definition 2.3.15.
Let G be a locally compact groupoid and let ν = { ν u } u ∈G (0) bea fixed left Haar system. Let U ( H x , H y ) denote the set of unitary bounded operators linear operators from H x to H y . C ( G ) with respect to the norm k · k I is a Banach ∗ -algebraof which the enveloping C*-algebra is called the full groupoid C*-algebra of G with respect to ν denoted by C ∗ ( G , ν ).(ii) Let u ∈ G (0) . Define a bounded ∗ -representation λ u of C ( G ) on L ( G u , ν u ) by λ u ( f ) ξ ( g ) = Z G u f ( gh − ) ξ ( h ) d ν u ( h )for every ξ ∈ L ( G u , ν u ) , f ∈ C c ( G ). It holds that k λ u ( f ) k ≤ k f k I .(iii) The reduced C*-algebra of G with respect to ν denoted by C ∗ r ( G , ν ) is thecompletion of C ( G ) with respect to the norm k · k r defined by k f k r := sup u ∈G (0) {k λ u ( f ) k} for f ∈ C ( G ) . Example 2.3.16.
Let G α be the transformation groupoid associated with an action α : G → Homeo( X ) of a countable, discrete group G on a locally compact, Hausdorffspace X . Then C ∗ ( G α ) ∼ = C ( X ) (cid:111) G .If G is a locally compact group the definitions of the adjoint and the norm differfrom the conventional definitions for group C*-algebras and consequently set updifferent normed ∗ -algebra structures on C c ( G ) resp. C ( G ). Passing over to theC*-completions the differences vanish and the associated (reduced) C*-algebras areisomorphic. In [Ren80] Renault gives special attentation to r-discrete groupouids which generalizetransformation groupoids arising from actions of discrete groups.
Definition 2.4.1 ([Ren80], Definition 2.6) . Let G be a locally compact groupoid.It is called r -discrete , if G (0) is an open subset of G .The denotation is justified – for such a groupoid, the subspaces G u and G u arediscrete for every u ∈ G (0) . If a Haar system of an r -discrete groupoid exists – thishowever is not necessarily the case – it is essentially unique i.e. it is equivalent to thesystem given by the system of counting measures (see [Ren80], Lemma 2.7). ´Etalegroupoids are more convenient in that a Haar system always exists and it is uniqueup to the equivalence of measures: Proposition 2.4.2.
Let G be a locally compact, ´etale groupoid. Then the system ofcounting measures on source fibers is a unique Haar system (up to equivalence). Subsection 2.4.1 discusses the structure and characterization of ´etale groupoids.Subsection 2.4.2 recalls a suitable version of homology for ´etale groupoids. 37 .4. ´ETALE GROUPOIDS
Definition 2.4.3 ([Res07], p.162) . A topological groupoid G is said to be ´etale ,if the range map r (or equivalenty the source map s ) is ´etale i.e. r is a localhomeomorphism . Example 2.4.4. (i) Let ϕ : Γ → Homeo( X ) define a action of a countable,discrete group Γ on a locally compact, Hausdorff space X . Then the transfor-mation groupoid G ϕ is `etale.(ii) Let X be a compact, metrizable, zero-dimensional space and ∼ be a countableequivalence relation on X . Then the topological groupoid G ∼ is ´etale.(iii) Let α be a continuous action of an inverse semigroup S on a locally compact,Hausdorff space X . The groupoid of germs Germ( X, S ) is ´etale, since theelements τ ( s, U ) of its topological basis are slices. Lemma 2.4.5.
Let G be an ´etale groupoid. Then the set B o G of open slices is a basisof the topology on G .Proof. Let g ∈ G be contained in some open set V . Then there exist open neighbour-hoods U g,s resp. U g,r on which the source resp. range maps are a homeomorphismwith open image. The the set U g,s ∩ U g,r ∩ V is a non-empty open slice containing g .The following lemma follows by plugging in U = G (0) in Lemma 2.1.8: Lemma 2.4.6 ([Res07], Lemma 5.17) . Let G be a topological groupoid such that G (0) is open. Then the open slices B o G form an open cover of G . This lemma implies in particular the following characterization of ´etale groupoids,which shows that ´etale groupoids are precisely those, whose collection of open set isinherently algebraic:
Theorem 2.4.7 ([Res07], Theorem 5.18) . Let G be a topological groupoid. Then thefollowing are equivalent:(i) G is ´etale.(ii) G (0) is an open subspace and the product of any two open subsets is open i.e.the frame of open subsets of G forms a monoid.(iii) G (0) is an open subspace and the structure maps s and r are open. A continuous function f : X → Y between topological spaces is said to be a local homeomor-phism , if every x ∈ X has an open neighbourhood U x such that f ( U x ) is open and f | U x is ahomeomorphism. .4.1 DEFINITION AND PROPERTIES Already Renault had observed in [Ren80], that locally compact, ´etale groupoidsgive rise to inverse monoids by looking at the sets of open slices with multiplicationinherited from the groupoid.
Proposition 2.4.8.
Let G be an ´etale groupoid. Then the following hold:(i) The open slices B o G form an inverse submonoid of the inverse monoid B G .(ii) The map α : B o G → I ( G (0) ) given by α : B r | B ◦ ( s | B ) − defines a continuousaction of B o G on G (0) by partial homeomorphisms. This raises the question how the original groupoid G relates to the induced groupoidof germs Germ( G (0) , B o G ). Proposition 2.4.9 ([Ren08], Proposition 3.2) . Let G be an ´etale groupoid. Thenthe following is a short exact sequence of ´etale groupoids: G (0) −→ Iso( G ) ◦ −→ G α ∗ −→ Germ( G (0) , α ( B o G )) −→ G (0) Furthermore, the action α is faithful if and only if Iso( G ) ◦ = G (0) . Remark 2.4.10.
The map α ∗ : G −→
Germ( G (0) , α ( B o G )) is defined by sending anelement g ∈ G to the germ [( α ( B ) , s ( g )] where B is an open slice that contains g .This does not depend on the choice of B and sets up a continuous surjection. Definition 2.4.11 ([Nek17], Definition 2.4) . Let G be an ´etale groupoid. it is saidto be effective or a groupoid of germs if every non-empty open slice B ∈ B o G forwhich B ∩ ( G \ G (0) ) = ∅ holds satisfies B ∩ ( G \
Iso( G )) = ∅ . Remark 2.4.12.
If an ´etale groupoid G is effective, it necessarily satisfies Iso( G ) ◦ = G (0) . The converse holds if G is Hausdorff. Effectiveness is a kind of “freeness” condition in that it is closely related to beingessentially principal:
Proposition 2.4.13 ([Ren08], Proposition 3.6.(i)) . Let G be an essentially prinipal,´etale groupoid. If it is Hausdorff, it is effective. Proposition 2.4.14 ([Ren08], Proposition 3.6.(ii)) . Let G be an effective, ´etalegroupoid. If G is second-countable and G (0) is a Baire space, it is essentially princi-pal. As the definition suggests, ´etalness entails strong ties between the topology of thewhole groupoid G and the subspace topology on G (0) : There is no consensus on the terminology in the literature. In [Ren09] the property of beingessentially principal is called topologically principal , another term in use is essentially free . Some-times (as in [Mat12]) a groupoid G is defined to be essentially principal if it is effective in ourterms. This stems from the fact that all of this notions are closely related. In this case G (0) is clopen. .4. ´ETALE GROUPOIDS Proposition 2.4.15.
Let G be an ´etale groupoid. The topology on G has a basis ofcompact, open slices if and only if G (0) has a basis of compact, open sets. Ample semigroups were introduced in [Ren80] to characterize groupoids which admita basis of compact open slices, the terminology was inherited by Krieger’s definitionof ample groups (compare with Definition 1.2.24). Subsequently such groupoidshave been termed ample.
Definition 2.4.16 ([Ren80], Definition 2.12 & [Pat99], Definition 2.2.4) . (i) Let α be the continuous action of an inverse subsemigroup S ⊆ I ( X ) on acompact, Hausdorff space X . Then S is called ample if the following hold:(a) For every compact open subset U ⊆ X the partial homeomorphism id U is induced by an element in S .(b) For every finite collection { s i } i ∈ I of elements in S with Im( α ( s i )) ∩ Im( α ( s j )) = ∅ and dom( α ( s i )) ∩ dom( α ( s j )) = ∅ for all i, j ∈ I with i = j ,there exists an s ∈ S such that α ( s )( x ) = α ( s i )( x ) for all x ∈ dom( α ( s i )).(ii) A topological groupoid G is called ample , if B o,k G is a basis of the topology.Every ample groupoid is by definition ´etale. Conversely we have: Proposition 2.4.17 ([LL13], Lemma 3.13) . Let G be an ´etale groupoid. Then G isample if and only if the space of units G (0) has a basis of compact, open sets. Remark 2.4.18.
This implies that every ´etale groupoid G such that G (0) is a Stonespace is ample. Proposition 2.4.19.
Let G be an ample groupoid. Then the set of compact, openslices B o,k G is an inverse subsemigroup of the inverse monoid B o G . In Section 2.5 we give a quick rundown on the connection between ´etale groupoidsand certain inverse semigroups and how this connection had played a role in C*-algebra theory. We finish this subsections with some remarks on locally compact,´etale groupoids which will be helpful in Subsection 3.6.2:
Proposition 2.4.20 ([Pat99], Proposition 3.2.1) . Let G be a locally compact, ´etalegroupoid and let µ be a probability measure on G (0) . Let B ∈ B o G be an open sliceand let f be a bounded Borel function on B . Then f is ν µ integrable if and only if f ◦ ( s | B ) − is ν µ integrable and we have Z f d ν µ = Z f ◦ ( s | B ) − d µ and Z f d ν − µ = Z f ◦ ( r | B ) − d µ Proof.
It follows immediatly from Z G u f d ν u = f ◦ ( s | B ) − ( u ) , if u ∈ s ( B )0 , else40 .4.2 HOMOLOGY OF ´ETALE GROUPOIDS Proposition 2.4.21.
Let G be a locally compact, ´etale groupoid and let µ be aprobability measure on G (0) . Then µ is quasi-invariant if and only if for every openslice B ∈ B o G we have µ ◦ r | B ◦ ( s | B ) − ∼ µ on r ( B ) .Proof. By definition G is σ -compact and in consequence Lindel¨of . It follows thereexists a countable cover of G by open slices and thus ν µ ∼ ( ν µ ) − if and only if ν µ | B ∼ ( ν µ ) − | B for all B ∈ B o G . Let B ∈ B o G and let f be a positive µ -measurablefunction. Then by Proposition 2.4.20 we have Z f d ν µ = Z f ◦ ( s | B ) − d µ and furthermore Z G f d ν − µ = Z G (0) f ◦ ( r | B ) − d µ = Z G (0) f ◦ ( s | B ) − ◦ s | B ◦ ( r | B ) − d µ == Z G (0) f ◦ ( s | B ) − d( µ ◦ s | B ◦ ( r | B ) − )Thus ν µ | B ∼ ( ν µ ) − | B if and only if µ ◦ s | B ◦ ( r | B ) − ∼ µ on s ( B ) and equivalently µ ◦ r | B ◦ ( s | B ) − ∼ µ on r ( B ). Remark 2.4.22.
Correspondingly a probability measure µ is invariant if and onlyif for every open slice B ∈ B o G we have µ ( s ( B )) = µ ( r ( B )). For ´etale groupoids homology groups with coefficients in a sheaf over the unit spacewhere introduced in [CM00] – dual to an already existing cohomology theory of ´etalegroupoids. Matui studied the case of groupoid homology with constant coefficientsof second countable, locally compact, Hausdorff, ´etale Cantor groupoids in a seriesof papers [Mat12] and applied those results to get information on their associatedtopological full groups. The following is a slightly generalized version descibed in[Nek17] to include non-Hausdorff groupoids:
Definition 2.4.23 ([Nek17], § . Let G be a locally compact, ´etale groupoidand let A a topological, abelian group. Analogous to Definition 2.3.2, denote by C ( G ( n ) , A ) the subgroup of A G ( n ) generated by all functions which are constant onsome compact open Hausdorff subset S ⊆ G ( n ) and map to 0 if restricted to G ( n ) \ S . The spaces of composable elements {G ( n ) } n ∈ N form simplicial spaces with face maps d i : G ( n ) → G ( n − for i = 0 , , . . . , n defined by: Every open cover has a countable subcover. If G is Hausdorff, this is just the set of compactly supported continuous functions, where A isendowed with the discrete toplogy. .5. OF THE RELATION BETWEEN INVERSE SEMIGROUPS AND ´ETALEGROUPOIDS (i) If n = 1: d i ( g ) := s ( g ) , for i = 0 r ( g ) , for i = 1(ii) Otherwise: d i ( g , g , . . . , g n ) := ( g , . . . , g n ) , for i = 0( g , . . . , g i g i +1 , . . . , g n ) , for 1 ≤ i ≤ n − g , . . . , g n − ) , for i = n These give families of morphisms d i ∗ : C ( G ( n ) , A ) → C ( G ( n − , A ), defined by: d i ∗ ( f )( g ) := X g ∈ d − i ( g ) f ( g )which in turn give the boundary maps δ n : C ( G ( n ) , A ) → C ( G ( n − , A ), defined by: δ n := n X i =0 ( − i d i ∗ Then the following defines a chain complex0 δ ← C ( G (0) , A ) δ ← C ( G (1) , A ) δ ← C ( G (2) , A ) δ ← · · · The homology groups H n ( G , A ) := ker δ n / Im δ n +1 of this chain complex are calledthe homology groups of G with constant coefficients A . In the case of A = Z , weabbreviate H n ( G ) := H n ( G , Z ). Furthermore define H ( G ) + := { [ f ] ∈ H ( G ) | f ≥ , f ∈ C ( G (0) , A ) } . Example 2.4.24 ([Mat12], § . Let G α be the transformation groupoid associatedwith an action α : G → Homeo( X ) of a countable, discrete group G on a Cantorspace X . Then the homology groups can be expressed in terms of group homology as H n ( G α ) ∼ = H n ( G, C ( X, Z )) and Poincar´e-duality holds – correspondingly H n ( G α ) ∼ = H n ( G, C ( X, Z )) holds for the cohomology groups. These groups are respectivelytermed the dynamical (co-)homology groups of the system ( X, G ). In case of aminimal Cantor system (
X, ϕ ) we have H ∗ ( G ϕ ) ∼ = K ∗ ( C ( X ) (cid:111) ϕ Z ) (– see [FH99]). Subsection 2.5.1 recollects aspects that tie together some of the content of precedingsubsections and that motivated non-sommutative Stone duality. Subsection 2.5.2 This set is not necessarily a positive cone! .5.1 SOME HISTORICAL REMARKS introduces an abstract definition of pseudogroups. Subsection 2.5.3 recalls a verygeneral version of non-commutative duality i.e. between sober ´etale groupoids andspatial pseudogroups. Reconsider Example 1.1.30 of the crossed product C*-algebra C ( X ) (cid:111) ϕ Z associatedwith a topological Z -system ( X, ϕ ) or the case of AF C*-algebras in Theorem 1.2.6.Both settings comprise a pair ( A , B ) of operator algebras where B is an abeliansubalgebra of A the inclusion of which reflects the nature of the underlying system.The Feldman-Moore theorem gave a description for this in the context of vonNeumann algebras, in that Cartan pairs were shown to precisely correspond totwisted countable standard measured equivalence relations i.e. in combination witha cohomology class on the relation.In [Ren80] Renault took first steps towards a C*-algebra theoretic/topological ver-sion and adopted the viewpoint of using topological groupoids as model spaces forC*-algebras. He did not succeed in finding an analog to [FM75] – in part due tothe fact that some definitions coming from the measured context needed an adap-tion, nevertheless, much of the ingredients were already present. In [Kum86] AlexKumjian showed that C*-diagonals, the definition of which is motivated by Cartansubalgebras in the measured context, translate to twisted ´etale equivalence relations,which correspond to twisted principial, ´etale groupoids. This result, however, doesnot cover some fundamental examples e.g. Cuntz-Krieger algebras. In [Ren08] Re-nault was able to extend this to the setting of effective, ´etale groupoids in [Ren08].
Definition 2.5.1 ([Ren08], Definition 5.1) . An abelian C*-subalgebra B of a C*-algebra A is called a Cartan subalgebra if it satisfies the following properties:(i) B contains an approximate unit of A (ii) B is a maximal abelian subalgebra(iii) B is regular i.e. the normalizer N ( B , A ) generates A .(iv) There exists a faithful conditional expectation P of A onto B .The pair ( B , A ) is then called a Cartan pair . Example 2.5.2 ([Ren80], § . Let G be a locally compact, second countable, Haus-dorff, ´etale groupoid. Then C ( G (0) ) is an abelian subalgebra of C ∗ r ( G ) that containsan approximate unit. Moreover the map E : C ∗ r ( G ) → C ( G (0) ) induced by the See [FM75] and [FM77]. Analogous to the case of groups, groupoid extensions are parametrized by second cohomology([Ren80], Proposition 1.14). A twist is a groupoid extensions arising from a 2-cocycles with valuesin the circle group T – we won’t go into details. .5. OF THE RELATION BETWEEN INVERSE SEMIGROUPS AND ´ETALEGROUPOIDS restriction map C c ( G ) → C ( G (0) ) given by f f | G (0) is a faithful conditional ex-pectation. The pair ( C ( G (0) ) , C ∗ r ( G )) constitutes a Cartan pair i.e. if and only if G is effective.Let ( B , A ) be such a Cartan pair, let n ∈ N ( B , A ) and let X denote the spec-trum of B . The adjoint action of n on B induces a uniquely defined partialhomeomorphism α ( n ) between the open sets d( n ) := { x ∈ X | n ∗ n ( x ) > } andr( n ) := { x ∈ X | nn ∗ ( x ) > } such that n ∗ bn ( x ) = b ( α ( n )( x )) n ∗ n ( x ) for all b ∈ B and x ∈ d( n ) ([Kum86], Proposition 6). This defines a continuous action of an in-verse semigroup on X . The associated groupoid of germs Germ( X, α ( N ( B , A ))) to-gether with a twist constructed from the pair ( B , A ) gives rise to a twisted groupoidC*-algebra which is precisely the original Cartan pair. For an effective groupoid G as in Example 2.5.2 the groupoid Germ( X, α ( N ( C ( G (0) ) , C ∗ r ( G )))) is canonicallyisomorphic to G .Already in [Ren80] Renault had observed that inverse semigroups play a crucialrole in the theory. The idea to model a C*-algebra on an ´etale groupoid whichcomes from an inverse semigroup proved to be very fruitful, in particular it providesmeans to tackle C*-algebras that arise from combinatorial structure e.g. directedgraphs, higher-rank graphs or tilings. Examples of such combinatorial C*-algebrasare Cuntz-Krieger algebras (see Definition B.1.11) or AF C*-algebras (see Proposi-tion 2.6.16). Renault’s investigations where developed further by Alan L. T. Pater-son in [Pat99]. By Theorem 3.3.2 in [Pat99] every localization ( X, S ) i.e. a locallycompact Hausdorff space X acted upon by a countable inverse semigroup S suchthat the domains of elements generate the topology of S , induces an inverse semi-group homomorphism ψ : S → B o Germ(
X,S ) , which raises the question how an ´etalegroupoid can be built from a general inverse semigroup S e.g. by an action thatis intrinsic to S . Examples of such constructions are the universal groupoid of Pa-terson described in [Pat99] or – since in many cases this is too big – a reductionof it, the tight groupoid of Exel from [Exe08]. Another viewpoint was provided byKellendonk in [Kel97] in the context of tilings.This brought the attentation to the relations between groupoids and inverse semi-groups and to the question if there is some unified approach subsuming the aboveconstructions. David Lenz observed in [Len08] that the approaches are reconciledwhen considering the order structures on inverse semigroups, in that the constructedgroupoids are equivalence classes of downward directed sets. Mark V. Lawson re-alized that in the case of Boolean ´etale groupoids and Boolean inverse monoidsthe consideration of inverse semigroups of slices and groupoids of filters producesa duality that generalizes classical Stone duality i.e. the Boolean ´etale groupoidsare ”non-commutative Stone spaces” and Boolean inverse monoids the corresponding”non-commutative Boolean algebra”. Similar to classical Stone duality this dualitygeneralizes to much broader contexts – that of pseudogroups and ´etale groupoids. Motivated by Renault’s observations, localizations where introduced by Kumjian in [Kum84]– there the inverse semigroup S is assumed to be a traditional pseudogroup. A quick description of this will be given in Subsection 3.4.1. .5.2 INVERSE ∧ -MONOIDS AND ABSTRACT PSEUDOGROUPS Another question relevant in this context is, which inverse subsemigroups of B o G allow a reconstruction of the original groupoid G . This lets us already glimpse atChapter 3, where the topological full group of an ´etale Cantor groupoid G turnsout to be the unit group of the inverse semigroup of open, compact slices U ( B o,k G ).The question of reconstructability of an ´etale Cantor groupoid up to isomorphismfrom its topological full group is situated in midst of the interplay between ´etalegroupoids and inverse semigroups. ∧ -monoids and abstract pseudogroups The following definitions introduce the objects on the “inverse semigroup side” ofnon-commutative Stone duality, they are inverse semigroups with a frame-like order.Frames are order structures that in particular describe how open sets of topologicalspaces join and intersect. The formal dual to the category of frames
Frm is thecategory of locales
Loc , which in some sense are generalized topological spaces thatdo not necessarily have “enough points” – the reason why the theory has beentermed pointless topology. One can adopt the view of this formal dual relationas the most general and obviously most superficial form of Stone duality. Resultsof deeper meaning arise, when one turns to subcategories i.e. dualities betweenclasses of topological spaces and frame subcategories, the most general in this contextbeing the duality between sober spaces and spatial frames. The historically firstiteration of such a duality came in form of the Stone representation theorem forBoolean algebras. Marshall H. Stone discovered that the space of ultrafilters ina Boolean algebra carries a topology and moreover that the Boolean algebra ofclopen sets in this space is precisely the initial Boolean algebra. He was able tocharacterize the topological spaces which arise in such a way as compact, Hausdorff,totally disonnected spaces (these are now termed
Boolean spaces or Stone spaces )and showed that the correspondence between these classes of objects lifts to the levelof morphism. Stone’s remarkable achievements compelled mathematicians to namethe subject it inspired in his honour – Stone’s glory engraved in stone. See [Joh82]for a monograph on Stone duality.
Definition 2.5.3 ([Law16], § . Let S be an inverse semigroup. If every binarymeet in S exists, it is called inverse ∧ -semigroup. Inverse ∧ -monoids thus are inverse monoids with a (meet-)semilattice-like orderstructure and as such had already been identified as worthwhile to study by Leechin [Lee95]. Leech observed that a characteristic feature of inverse ∧ -monoids is thepresence of a fixed-point operator: Definition 2.5.4 ([Law16], § . Let S be an inverse monoid. A map f : S → E ( S )is called a fixed point operator if it satisfies the following properties:(i) s ≥ f ( s ) ∀ s ∈ S (ii) Every s ∈ S and e ∈ E ( S ) such that e ≤ s satisfy e ≤ f ( s ). 45 .5. OF THE RELATION BETWEEN INVERSE SEMIGROUPS AND ´ETALEGROUPOIDS Proposition 2.5.5 ([Law16], Proposition 2.2) . An inverse monoid is an inverse ∧ -monoid if and only if it has a fixed point operator. The fixed point operator of aninverse ∧ -monoid S is unique and given by f : s s ∧ . The abstract frame theoretic definition of pseudogroups goes back to Pedro Resende.
Definition 2.5.6 ([Res07], Definition 2.5 & 2.8 & 2.9) . Let S be an inverse semi-group.(i) A pair of elements s, t ∈ S is said to be compatible if s − t and st − are idem-potents. A subset A ⊆ S is called compatible if all its elements are pairwisecompatible. Note, that if s ∨ t exists, s and t are necessarily compatible.(ii) It is called complete if every compatible subset has a join.(iii) It is called distributive if s ∨ t exists for every compatible pair s, t ∈ S andevery binary join s ∨ t ∈ S and every u ∈ S the joins us ∨ ut and su ∨ tu existand us ∨ ut = u ( s ∨ t ) and su ∨ tu = ( s ∨ t ) u (iv) It is called infinitely distributive if for any subset T ⊆ S that has a join andevery s ∈ S the following hold:(a) W t ∈ T st and W t ∈ T ts exist.(b) s ( W t ∈ T t ) = W t ∈ T st .(c) ( W t ∈ T t ) s = W t ∈ T ts .(v) A pseudogroup is a complete, infinitely distributive inverse monoid.(vi) A semigroup homomorphism between pseudogroups that preserves all com-patible joins is called a pseudogroup homomorphism . Proposition 2.5.7 ([Res07], Proposition 2.10) . Let S be a pseudogroup. Then S isan inverse ∧ -semigroup and E ( S ) is a frame. The above definition of pseudogroup is the ’pointless topology’-version of whatwas historically the motivation to consider abstract inverse semigroups – the pseu-dogroups of partial homeomorphisms between open sets in a topological space. Anal-ogous to the bridge from algebra to topology in Stone’s representation theorem, thefundamental notion to get from pseudogroups to ´etale groupoids are filters:
Definition 2.5.8 ([LL13]) . Let S be a pseudogroup.(i) Denote by L( S ) the set of all filters in S .(ii) A filter F in L( S ) is proper , if 0 / ∈ F .(iii) A filter F in L( S ) is called completely prime if for every join ∨ i ∈ I a i ∈ F thereexists an i ∈ I such that a i ∈ F .46 .5.3 NON-COMMUTATIVE STONE DUALITY (iv) Denote by G( S ) the set of all completely prime filters in S .(v) Let s ∈ S . Denote by X s the set of all completely prime filters that contain s .(vi) A pseudogroup S is called spatial if X s = X t implies s = t for all s, t ∈ S .(vii) A maximal proper filter F in L( S ) is called an ultrafilter .(viii) Let s ∈ S . Denote by V s the set of all ultrafilters that contain s .(ix) Let S and T be pseudogroups. A map f : S → T is called callitic if:(a) It is a pseudogroup homomorphism that preserves all meets.(b) F ∩ Im( f ) = ∅ for all completely prime filters F ⊆ T .(x) Denote by Pseu the category which has pseudogroups as objects and calliticmaps as morphisms.
On the level of objects this general duality had already established in [MR10]. En-dowed with a sufficient notion of morphisms the class of ´etale groupoids becomes acategory:
Definition 2.5.9 ([LL13]) . (i) Let G and H be ´etale groupoids and let F : G → H be a groupoid homomorphism. It is called a covering functor if F | G u is bijectivefor all u ∈ G (0) .(ii) Denote by Etale the category which has ´etale groupoids as objects and con-tinuous covering functors as morphisms.We start with the very general formulation of [LL13]: The aim is to find a pairof functors B :
Etale → Pseu op and G : Pseu op → Etale between the categoryof ´etale groupoids
Etale and the category of pseudogroups
Pseu where G is rightadjoint to B. This adjunction restricts to a duality between the categories of spatialpseudogroups
Pseu sp and sober ´etale groupoids Etale sob . We start with the de-scription of the functors B :
Etale → Pseu op and G : Pseu op → Etale on the levelof objects:
Proposition 2.5.10 ([LL13], Proposition 2.1) . Let G be an ´etale groupoid. Thenits set of open slices B o G is a pseudogroup with respect to multiplication of subsets. The functor B : Etale → Pseu op is defined on objects as B : G 7→ B o G . The definitionof G on objects is given by means of filters: By setting s ( F ) = ( F ∗ F ) ↑ and r ( F ) =( F F ∗ ) ↑ for F ∈ L( S ), the set of composable pairs i.e. { ( F, G ) ∈ L( S ) | s ( F ) = r ( G ) } admits a product F · G := ( F G ) ↑ with respect to which L( S ) is a groupoid. All thedifferent groupoids constructed from an inverse semigroup ([Pat99], [Kel97], [Exe08],etc.) are specific subgroupoids of L( S ) ([LL13], p.6). We focus on the subgroupoidof completely prime G( S ). This groupoid becomes a topological groupoid by takingthe sets { X s | s ∈ S } as a basis. 47 .5. OF THE RELATION BETWEEN INVERSE SEMIGROUPS AND ´ETALEGROUPOIDS Proposition 2.5.11 ([LL13], Proposition 2.8) . Let S be a pseudogroup. Then G( S ) is an ´etale groupoid. One needs to inspect how B and G behave in conjunction with morphisms:
Lemma 2.5.12 ([LL13], Lemma 2.13 & 2.14) . Let f : S → T be a callitic map be-tween pseudogroups. Then the inverse image f − ( F ) of every completely prime filter F in T is a non-empty completely prime filter and the induced map f − : G( T ) → G( S ) is a continuous covering functor. The above lemma “shows” that the first notion of morphisms between pseudogroupsone might think of i.e. just combining the notions of frame morphisms and mor-phisms of inverse semigroups is not sufficient. The second condition in the definitionof callitic maps is needed, because it assures that inverse images of completely primefilters are non-empty. In inverse direction one has:
Lemma 2.5.13 ([LL13], Lemma 2.19) . Let f : G → H be a continuous coveringfunctor between ´etale groupoids. Then the induced map f − : B( H ) → B( G ) on openslices is a callitic map. The obtained functors B :
Etale → Pseu op and G : Pseu op → Etale then set upan adjunction:
Theorem 2.5.14 ([LL13], Theorem 2.22) . The functor
G :
Pseu op → Etale is rightadjoint to the functor
B :
Etale → Pseu op . This adjunction is a generalization of the adjunction between topological spaces andframes in that every ´etale groupoid that only consists of units is simply a topologicalspace and its dual a pseudogroup that contains only idempotents, hence a frame.From this generalization one derives a duality in the same manner as the dualitybetween sober topological spaces and spatial frames comes from the adjunctionbetween topological spaces and frames.
Definition 2.5.15. (i) An ´etale groupoid G is said to be sober if η : G →
G(B( G ))is a homeomorphism.(ii) Denote by Pseu sp the category which has spatial pseudogroups as objects andcallitic maps as morphisms and by Etale sob the category which has sober ´etalegroupoids as objects and continuous covering functors as morphisms.
Lemma 2.5.16 ([LL13], Proposition 2.12) . For every sober ´etale groupoid G thepseudogroup B( G ) is a spatial pseudogroup and for every spatial pseudogroup S the´etale groupoid G( S ) is sober. Above lemma then implies:
Theorem 2.5.17 ([LL13], Theorem 2.23) . The adjunction in Theorem 2.5.14 in-duces an equivalence of categories between
Pseu opsp and
Etale sob . In Subsection 3.4.1 we turn to a refinements of this correspondence.48 .6 Resuming Cantor dynamics
For the remainder of this text with the exception of Section 3.4 every´etale groupoid is assumed to be second countable. ´Etale groupoids with a Cantor space as space of units provide Cantor dynamics witha broader scope. For the sake of abbreviation we define:
Definition 2.6.1.
A topological groupoid G is called a Cantor groupoid if the topo-logical subspace G (0) is a Cantor space. Remark 2.6.2. ´Etale Cantor groupoids are by definition locally compact groupoids.In Subsection 2.6.1 we take a quick look at the homology of ´etale Cantor groupoids.Subsection 2.6.2 showcases generalized subshifts. Subsection 2.6.3
Matui studied ´etale Cantor groupoids and their topological full group in a series ofpapers ([Mat12],[Mat15],[Mat16a]). In [Mat16a] Matui made a conjecture, whichhe termed
HK-conjecture , that relates the groupoid homology with the K -group ofthe groupoid C*-algebra: Conjecture 2.6.3 ([Mat16a], Conjecture 2.6) . Let G be an effective, minimal, ´etaleCantor groupoid. Then K ( C ∗ r ( G )) ∼ = ∞ M i =0 H i ( G ) and K ( C ∗ r ( G )) ∼ = ∞ M i =0 H i +1 ( G ) . The following propositions will be of use in later sections:
Proposition 2.6.4 ([Mat12], Lemma 7.3) . Let G be an effective, ´etale Cantorgroupoid. Then the following holds:(i) Let B, B be compact, open slices of G with r ( B ) = s ( B ) and let O ⊆ G (2) be the set of composable pairs ( b, b ) with b ∈ B and b ∈ B . Then δ ( O ) = B − BB + B As immediate consequences of (i), the following hold:(ii) [ U ] = 0 in H ( G ) for every compact, open subset U ⊆ G (0) .(iii) [ B ] + [ B − ] = 0 in H ( G ) for every compact, open slice B of G . .6. RESUMING CANTOR DYNAMICS In Chapter 3, the group H ( G ) ⊗ ( Z / Z ) = H ( G , Z / Z ) will play a role in thedescription of the abelization of the topological full group of certain ´etale Cantorgroupoids. Let f ∈ C ( G (0) , Z ). The set U f := { u ∈ G (0) | f ( u ) / ∈ Z } is compact,open. The class [ f ] + 2 H ( G ) in H ( G ) ⊗ ( Z / Z ) is uniquely determined by the classof [ U f ] + 2 H ( G ). Thus one obtains the following description: Proposition 2.6.5 ([Nek15], Proposition 7.1) . Let G be an ´etale Cantor groupoid.Let ˜ H ( G ) be the abelian group given by the following representation:(i) The set of generators is given by all U , where U is a clopen subset of G (0) .(ii) The set of relations consist of:(a) U + U = 0 for all clopen U ⊆ G (0) .(b) For every finite disjoint union of clopen subsets U = F U i , it holds that U = P i U i .(c) For every clopen bisection B ∈ B o,k G , it holds that s ( B ) = r ( B ) .Then H ( G ) ⊗ ( Z / Z ) ∼ = ˜ H ( G ) . The following is a translation of the definition in [Hae02] – there given in the contextof pseudogroups:
Definition 2.6.6 ([Nek15], Definition 2.3.1) . Let G be an ´etale groupoid.(i) Let ( S, U ) be a pair of compact sets S ⊆ G and U ⊆ G (0) . The pair ( S, U )called a compact generating pair if U contains an open G -transversal and forall g ∈ G| U there exists an n ∈ N such that S nk =1 ( S ∪ S − ) k is an openneighbourhood of g in G| U .(ii) The groupoid G is said to be compactly generated if it contains a compactgenerating pair.The case of an ´etale groupoid with compact unit space, in particular the case of an´etale Cantor groupoid, greatly simplifies above definition Definition 2.6.7.
Let G be an ´etale Cantor groupoid. It is called compactly gener-ated if it contains a compact generating set i.e. there exists a compact subset S ⊆ G such that G = S ∞ n =1 ( S ∪ S − ) n .By the assumptions S admits a finite cover by open, compact slices, the union ofwhich may replace S in the definition. Expansiveness is given in terms of such acover and constitutes a class of compactly generated ´etale Cantor groupoids thatgeneralize the notion of dynamical shifts:50 .6.2 COMPACT GENERATION AND EXPANSIVE GROUPOIDS Definition 2.6.8 ([Nek17], Definition 5.2) . Let G be a compactly generated, ´etaleCantor groupoid.(i) Let S be a compact generating set of G and let B be a finite cover of S byopen, compact slices. The cover B is called expansive if S n ∈ N ( B ∪ B − ) n is atopological basis for G .(ii) The groupoid G is said to be expansive , if it has a compact generating setwhich has an expansive cover. Remark 2.6.9 ([Nek17], Proposition 5.2) . If an ´etale Cantor groupoid G is expan-sive, all compact generating sets of G admit expansive covers.Expansive covers can be characterized as follows: Proposition 2.6.10 ([Nek17], Proposition 5.3) . Let G be a compactly generated,´etale Cantor groupoid and let S be a compact generating set. Let B be a finite coverof S by open slices. Then the following are equivalent:(i) The cover B is expansive.(ii) The set { s ( F ) | F ∈ S n ∈ Z + ( B ∪ B − ) n } is a topological basis of G (0) .(iii) For every pair u , u ∈ G (0) with u = u there exist F , F ∈ S n ∈ Z + ( B ∪ B − ) n such that u ∈ s ( F ) and u ∈ s ( F ) and s ( F ) ∩ s ( F ) = ∅ .(iv) For every u ∈ G (0) , it holds that: \ n ∈ Z + \ { s ( F ) | k ≤ n, F ∈ ( B ∪ B − ) k , u ∈ s ( F ) } = { u } The naming derives from the following class of group actions:
Definition 2.6.11.
The action of a finitely generated group G on a uniform space X is called expansive , if there exists a neighbourhood of the diagonal ∆ ⊆ W ⊆ X × X such that x, y ∈ X and ( g ( x ) , g ( y )) ∈ W for all g ∈ G implies x = y .In the case of the action of a finitely generated group G on a Cantor space X above definition allows to construct a finite clopen partition P of X such that G -translates of atoms separate points, by which the system is conjugate to a G -shift(Remark 1.1.18). Expansive Cantor groupoids can be seen as “generalized subshifts”and in the case of transformation groupoids on has: Proposition 2.6.12 ([Nek17], Proposition 5.5) . Let a finitely generated group G act by α : G → Homeo( X ) on a Cantor space X . Then the following are equivalent:(i) The action of G on X is expansive.(ii) The groupoid G ( X,G ) is expansive. .6. RESUMING CANTOR DYNAMICS (iii) The groupoid Germ(
X, G ) is expansive.(iv) The dynamical system ( X, G ) is a subshift. Example 2.6.13.
Let A be a finite set and let G be a group with finite, symmetricgenerating set T . Let ( X, G ) be a subsystem of ( A G , G ) i.e. a G -subshift. Then S := { ( t, x ) | s ∈ T, x ∈ X } is a compact generating set of G ( X,G ) . The set S := { ( t, C ( a )) | t ∈ T, a ∈ A } where C ( a ) denotes the cylinder set { x ∈ X | x = a } is anexpansive cover of S by Proposition 2.6.10[(iv) ⇒ (i)].The following applies in particular to minimal groupoids: Lemma 2.6.14 ([Nek17], Lemma 5.1) . Let G be a compactly generated ´etale Cantorgroupoid, such that every G -orbit contains or more elements. Then there exists anopen compact generating set contained in G \
Iso( G ) .Proof. Let S be a open compact generating set containing G (0) . By the assumptions S has a finite cover S of by slices. Let S i be a component of S and let g ∈ S i .Given g is contained in Iso( G ), then there exists an element h ∈ G \ Iso( G ) with s ( g ) = s ( h ). Since G is ´etale, there exists a slice B with B ⊆ G \ Iso( G ) and h ∈ B such that s ( BS i ) ∩ r ( BS i ). Then the slice B g := B ∪ BS i is contained in G \
Iso( G ) and N g := B − BS i is a neighbourhood of g contained in S n ∈ N ( B g ∪ B − g ) n (i.e the subgroupoid of G generated by B g ). Given g ∈ G \ Iso( G ), let N g ⊆ S i bea neighbourhood of g contained in G \
Iso( G ) and define B g := N g . The collection { N g } g ∈ S i is a cover of S i , hence there exists a finite subcover { N g k } . Then the set S := S k B g k is compact, it generates S and S ⊆ G \ Iso( G ). The following definition is originally due to Renault:
Definition 2.6.15 ([Mat12], Definition 2.2) . Let G be an ´etale Cantor groupoid.(i) A subgroupoid H ≤ G is called an elementary subgroupoid , if H is compact,open and principal with H (0) = G (0) .(ii) The groupoid G is called an AF-groupoid if there is a family of nested elemen-tary subgroupoids {H i } such that G = S i H i .These are the first examples which arise as an application of Renault’s program in[Ren80] in that: Proposition 2.6.16 ([Ren80], Proposition III.1.15) . Let A be a C*-algebra. Thenthe following are equivalent:(i) The C*-algebra A is AF.(ii) The C*-algebra A is isomorphic to C ∗ ( G ) of an AF-groupoid G where G isuniquely determined up to isomorphism. .6.3 AF-GROUPOIDS AND ALMOST FINITE GROUPOIDS Furthermore AF-groupoids are in 1-1 correspondence to inductive limits of com-pact, ´etale equivalence relations (abbr.: CEER) which are termed
AF-equivalencerelations . In this disguise they are intrinsically studied in [GPS04]. The following isthe prime example of this correspondence:
Example 2.6.17.
Let Γ = (
V, E ) be a Bratteli diagram and let P Γ be its space ofinfinite paths. Define for all k ∈ N equivalence relations ∼ k by ( e n ) n ∈ N ∼ k ( f n ) n ∈ N if and only if e n = f n for all n ≥ k . The sequence of associated groupoids {G ∼ k } k ∈ is a nested sequence of ´etale groupoids. The inductive limit of this sequence is perdefinition an AF groupoid. By Theorem 3.9 of [GPS04] for every AF-equivalencerelation and hence every AF-groupoid there exists a Bratteli diagram upon which itcan be modelled.AF-groupoids generalize Krieger’s AF-systems (see Subsection 1.2.5), satisfy theHK-conjecture and fall within Krieger’s classification via dimension groups: Theorem 2.6.18 ([Mat12], Theorem 4.10 & 4.11) . Let G be an AF-groupoid. Thenthe following hold:(i) ( H ( G ) , H ( G ) + , [1 G (0) ]) is an ordered group with order unit and there exists anisomorphism of ordered groups with order unit ( H ( G ) , H ( G ) + , [1 G (0) ]) ∼ = ( K ( C ∗ r ( G )) , K ( C ∗ r ( G )) + , [1 C ∗ r ( G ) ]) (ii) The ordered groups with order unit ( H ( G ) , H ( G ) + , [1 G (0) ]) are a complete iso-morphism invariant for AF-groupoids.(iii) For any topological abelian group A , the groups H n ( G , A ) are trivial for n ≥ . In [Mat12] Matui introduced the class of almost finite groupoids:
Definition 2.6.19 ([Mat12], Definition 6.2) . Let G be an effective, ´etale Cantorgroupoid. It is said to be almost finite , if for every compact subset K ⊆ G and every ε > H ≤ G such that for all u ∈ G (0) : | K H u \ H u ||H u | < (cid:15) Almost finite groupoids encompass AF-groupoids and transformation groupoids as-sociated with free Z n -actions on a Cantor space by Lemma 6.3 in [Mat12]. Proposition 2.6.20 ([Mat12], Lemma 6.3) . Let n ∈ N and let ϕ be a free continuousaction of Z n on a Cantor space. Then the following hold:(i) The transformation groupoid G ϕ is almost finite.(ii) The transformation groupoid G ϕ satisfies the HK-conjecture. See Definition 1.2.1. .6. RESUMING CANTOR DYNAMICS Matui defined purely infinite groupoids in [Mat15] based on [RS12] where conditionon dynamical systems are given such that the arising crossed product C*-algebra ispurely infinite. For a treatment of purely infinite C*-algebras see [Bla06], V.2.2.
Definition 2.6.21 ([Mat15], Definition 4.9.) . Let G be an ´etale Cantor groupoid.(i) A clopen subset U ⊆ G (0) is called properly infinite , if there exist compact openslices S , S satisfying s ( S ) = s ( S ) = U , r ( S ) ∪ r ( S ) ⊆ U and r ( S ) ∩ r ( S ) = ∅ .(ii) The groupoid G is called purely infinite , if every clopen subset of G (0) is properlyinfinite. Remark 2.6.22.
If the unit space G (0) of a Hausdorff, ´etale Cantor groupoid G isproperly infinite, there exists no G -invariant probability measure on G (0) .The class of purely infinite groupoids contains in particular groupoids that arisefrom one-sided shifts of finite type. Such groupoids are of significance for opera-tor theory in that they are groupoids upon which Cuntz-Krieger algebras can bemodeled. It is important to mention the name of Kengo Matsumoto who has beenstudying orbit equivalence and full groups in this context – see [Mat10], [Mat13a]. Definition 2.6.23 ([Mat15], § . Let n ∈ N and let { A i,j } i,j ∈{ ,...,n } ∈ M n × n ( Z ≥ ).(i) The matrix A is said to satisfy (Inp) if it is irreducible i.e. for all i, j ∈{ , . . . , n } there exists an n ∈ N such that ( A n ) i,j >
0, and A is not a permu-tation matrix.Let A = { A i,j } i,j ∈{ ,...,n } ∈ M n × n ( Z ≥ ) be such that it satisfies (Inp).(ii) Let Γ A be a finite, directed graph with a finite vertex set V and a finite set ofedges E such that A is the adjacency matrix of Γ. The space of one-endedinfinite directed paths X A := { ( e k ) k ∈ N ∈ E N | r ( e k ) = s ( e k +1 ) for all k ∈ N } isa Cantor space. For every finite word ω = ( ω , . . . , ω l ) ∈ E ∗ the cylinder set C ω := { ( e k ) k ∈ N ∈ X A | e i = ω i for all i ∈ { , . . . , l }} is a clopen subset of X A . For details on such systems see [LM95], Capter 2. See Definition B.1.11. This means the vertex set is associated with { , . . . , n } and for every pair of vertices v, w ∈ V define A v,w := { e ∈ E | s ( e ) = v, r ( e ) = w } . Note that property (Inp) implies that the arisingdirected graph is strongly connected and is not a cycle. .6.4 PURELY INFINITE GROUPOIDS AND GROUPOIDS OF SHIFTS OFFINITE TYPE (iii) Define a continuous surjection σ A : X A → X A by σ A ( x ) i = e i +1 for every i ∈ N and x = ( e k ) k ∈ N ∈ X A . The pair ( X A , σ A ) is called an irreducible one-sidedtopological Markov shift or irreducible one-sided shift of finite type .(iv) If A = n ∈ M × ( Z ≥ ), the arising one-sided shift of finite type is called full .Since it only depends on n it is denoted by ( X n , σ n ).(v) Let ( X A , σ A ) be a one-sided shift of finite type. Denote by G A the topologicalgroupoid given on the set { ( x, k, y ) ∈ X A × Z × X A |∃ m, n ∈ Z ≥ : σ nA ( x ) = σ mA ( y ) , k = n − m } by defining pairs (( x , k , y ) , ( x , k , y )) ∈ G A as compatible when y = x , the product by ( x , k , y ) · ( x , k , y ) = ( x , k + k , y ), the inverse by( x, k, y ) − = ( y, − k, x ) for all ( x, k, y ) ∈ G A and by declaring sets of the form { ( x, k − l, y ) | x ∈ U, y ∈ V, σ kA ( x ) = σ lA ( y ) } where U, V are open subsets of X A to be a topological basis. We call a groupoid a SFT-groupoid if it is G A forsome one-sided shift of finite type.Let G A be SFT-groupoid. The space of units G (0) A given by the elements ( x, , x ) ∈ G A canonically corresponds to X A making the groupoid G A a Cantor groupoid. For every x ∈ X A the orbit G A ( x ) is given by the set { y ∈ X A |∃ l, m ∈ Z ≥ : σ nA ( x ) = σ mA ( y ) } .Irreducibility of A then assures that G A is minimal. For every pair of finite directedpaths ω = ( ω , . . . , ω l ) , υ = ( υ , . . . , υ m ) ∈ E ∗ which satisfy r ( ω l ) = r ( υ m ) the set B ω,υ := { ( x, l − m, y ) ∈ G A | σ lA ( x ) = σ mA ( y ) , x ∈ C ω , y ∈ C υ } is a compact open slice. Slices of this form generate the topology of G A by whichit is ´etale. Since A satisfies (Inp), the set of non eventually periodic paths in X A isdense and the groupoid G A is essentially principal. Lemma 2.6.24 ([Mat15], Lemma 6.1) . SFT-groupoids are purely infinite, Haus-dorff, minimal, effective, ´etale Cantor groupoids.
Definition 2.6.25.
Let n ∈ N , let { A i,j } i,j ∈{ ,...,n } ∈ M n × n ( Z ≥ ) be such that itsatisfies (Inp) and let Γ = ( V, E ) be the associated directed, finite graph. Define ˜ A to be the edge matrix of this graph i.e. the matrix ˜ A ∈ M | E |×| E | ( Z / Z ) given by˜ A e,f := , if r ( e ) = s ( f )0 , else Remark 2.6.26.
If a matrix A satisfies (Inp), then the matrix ˜ A is a matrix satis-fying the assumptions in Definition B.1.11. Theorem 2.6.27 ([Ren00], Proposition 4.8) . Let ( X A , σ A ) be a one-sided shift offinite type. Then O ˜ A ∼ = C ∗ r ( G A ) . .6. RESUMING CANTOR DYNAMICS Example 2.6.28.
Let A = n ∈ M × ( Z ≥ ). Then the associated graph Γ A is just a“directed bouquet graph”– it consists of one vertex with n attached directed loops.Its edge matrix ˜ A is a n × n -matrix with all entries 1. Correspondingly the Cuntzalgebra of order n is the groupoid C*-algebra of an SFT-groupoid associated to afull one-sided shift of finite type.All of this considerations embed into the broader context of graph groupoids, graphinverse semigroups and graph C*-algebras.56 hapter 3Topological full groups We start in Subsection 3.1.1 with the definition of full groups and topological fullgroups and some immediate observations. Subsection 3.1.2 contains description oftopological full groups of minimal Cantor systems in terms of Kakutani-Rokhlin par-titions. Subsection 3.1.3 traces back the first glimpses on topological full groups inwork of Krieger and Putnam and introduces important subgroups. Subsection 3.1.5cites a lemma by Glasner and Weiss. Subsection 3.1.7 concludes this section with alook on full groups as isomorphism invariants in topological dynamics.
Definition 3.1.1.
Let (
X, G ) be a topological dynamical system such that G iscountable.(i) The full group of ( X, G ) denoted by F ( G ) is the subgroup of homeomorphisms γ ∈ Homeo( X ) for which there exists a function f γ : X → G such that γ ( x ) = f γ ( x ) · x . Such a function f γ is called an orbit cocycle of γ .(ii) The topological full group of ( X, G ) denoted by T ( G ) is the subgroup of home-omorphisms γ ∈ Homeo( X ) for which there exists a continuous orbit cocycle. In the case of a Z -system ( X, ϕ ) we write F ( ϕ ) (resp. T ( ϕ )) instead. In most references e.g. in [GPS99] and all of Matui’s works, the full group (resp. topologicalfull group) of a topological dynamical system (
X, G ) is denoted by [ G ] (resp. [[ G ]]) or by [ ϕ ] (resp.[[ ϕ ]]) in the context of Z -systems ( X, ϕ ). We choose a notation close to the one for topological fullgroups of ´etale groupoids used in [Nek15] and [Nek17] to avoid confusion with commutator bracketsand to subsequently have a natural notation for important subgroups. The naming “topologicalfull group” is a little awkward, in that the word “topological” serves just as a reference to the fieldof topological dynamics and signifies the difference between F ( G ) and T ( G ), but it does not referto a topology on T ( G ). .1. BASIC STRUCTURE AND SIGNIFICANCE IN CANTOR DYNAMICS We begin with some immediate observations:
Remark 3.1.2. (i) The groups G , T ( G ) and F ( G ) have the same orbits.(ii) In general orbit cocycles and continuous orbit cocycles are not unique. Notethat if the action is free, the orbit cocycle f γ of every element γ ∈ F ( G ) isuniquely determined.(iii) The groups F ( G ) and T ( G ) are not necessarily countable and interesting toconsider. Let γ ∈ T ( G ). Since G was assumed countable, the fibers of f γ induce a countable partition X = F g ∈ G X g by closed sets, thus if G acts freelyon a connected space X the topological full group is just isomorphic to G .(iv) In the following we restrict ourselves to the case where the space X is a Cantorspace and thus totally disconnected. Note that orbit cocycles in general are notuniquely determined, thus for T ( G ) to be graspable, we require the “freenesscondition” of minimality on the underlying system.The full group F ( ϕ ) of a minimal Cantor system ( X, ϕ ) is in general uncountable,whereas we have for topological full groups:
Proposition 3.1.3.
Let ( X, ϕ ) be a minimal Cantor system.(i) For every γ ∈ F ( ϕ ) the orbit cocycle f γ is uniquely determined.(ii) A homeomorphism γ ∈ Homeo(X) satisfies γ ∈ T ( ϕ ) if and only if there existsa finite clopen partition X = F i ∈ I ⊂ Z X γi such that γ | X γi = ϕ i | X γi .In particular T ( ϕ ) is countable and supp( γ ) = { x ∈ X | γ ( x ) = x } is clopen for all γ ∈ T ( ϕ ) .Proof. (i) Assume that both f γ , g γ are orbit cocycles of γ such that f γ ( x ) = g γ ( x )for some x ∈ X . This is only possible if x is a periodic point.(ii) Let γ in T ( ϕ ). Since f γ is continuous, its image must be compact, hencefinite. Since Z is discrete, fibers of the orbit cocycle f γ are clopen and thus X γi = f − γ ( { i } ) for i ∈ Im( f γ ) ⊂ Z is the required partition. Conversely, thefunction f γ : X → Z defined by f γ ( x ) = i for x ∈ X γi is continuous, sinceevery X γi is clopen. Since X is zero-dimensional and has a countable basis,there are only countably many clopen sets in X and consequently T ( G ) mustbe countable. Remark 3.1.4.
For every γ ∈ T ( ϕ ) the orbit cocycle f γ can be represented as f γ = P k ∈ Z k X γk and for γ , γ ∈ T ( ϕ ) the associated orbit cocycles satisfy true totheir names f γ γ = f γ ◦ γ + f γ .58 .1.1 DEFINITION Already in [Mat06] Matui defined topological full groups of ´etale equivalence rela-tions on a Cantor space – the associated equivalence groupoid is a principal, ´etaleCantor groupoid. In [Mat12] he generalized the definition to ´etale groupoids withcompact unit space as T ( G ) := { γ ∈ Homeo( G (0) ) |∃ B ∈ B o,k G : γ = r ◦ ( s | B ) − } Analogous to the above we mostly restrict to the case of Cantor groupoids:
Definition 3.1.5 ([Nek17], Definition 2.3) . Let G be an ´etale Cantor groupoid. Theset { B ∈ B o,k G : s ( B ) = r ( B ) = G (0) } forms a group with respect to multiplicationand inversion of subsets, called the topological full group of G , denoted by T ( G ). Remark 3.1.6. (i) Put differently, the topological full group T ( G ) is by definitionthe unit group U( B o,k G ) of the inverse monoid of compact open slices.(ii) If G is a second countable, ´etale Cantor groupoid, then T ( G ) is countable perdefinition.(iii) If G is effective, T ( G ) is isomorphic to the group of Matui’s definition i.e. agroup of homeomorphisms of G (0) via the injection B r ◦ ( s | B ) − . We indulge ourselves with leaving it to the context in the effec-tive case if we refer to T ( G ) as a group of slices or of home-omorphisms! As an al levation we wil l stand by the conventionof writing upper case letters for slices and lower case letters forhomeomorphisms. (iv) In line with Proposition 2.4.8 we denote the homeomorphism of G (0) inducedby a slice B ∈ T ( G ) by α B .(v) In the case of a continuous action of a group G on a Cantor space X via α : G → Homeo( X ), the groups T ( G ) and T ( G α ) coincide.As in the reduced context of Cantor systems minimality is key in a lot of results.It allows for the construction of certain elementary elements in the topological fullgroup: Lemma 3.1.7.
Let G be a minimal, ´etale Cantor groupoid and let U, V ⊆ G (0) benon-empty and clopen. Then there exists a finite family of open, compact slices { B i } with r ( B i ) ⊂ V , U = S i s ( B i ) and s ( B i ) ∩ s ( B j ) = ∅ for i = j .Proof. The G -orbit of every element in U is dense in G (0) by minimality, thus forevery u ∈ U there exist an g u ∈ G with s ( g u ) = u und r ( g u ) ∈ V . Since G is ´etale,there exists an open compact slice B u containing g u such that r ( B u ) ⊆ V . The family { s ( B u ) } u ∈ U covers U . By compactness, there exists a finite family { B i } with U = S i s ( B i ) and r ( B i ) ⊂ V , which we can assume to be disjoint, thus s ( B i ) ∩ s ( B j ) = ∅ for i = j . 59 .1. BASIC STRUCTURE AND SIGNIFICANCE IN CANTOR DYNAMICS Definition 3.1.8.
Let G be an ´etale Cantor groupoid and let B ∈ B o,k G be non-empty such that s ( B ) ∩ r ( B ) = ∅ . Denote by T B (resp. τ B ) the induced involutionin T ( G ) given by B ∪ B − ∪ ( G (0) \ ( s ( B ) ∪ r ( B ))). Corollary 3.1.9.
Let G be a minimal, ´etale Cantor groupoid and let U ⊆ G (0) benon-empty and clopen. Then there exists an element T ∈ T ( G ) with T = 1 suchthat G (0) \ ( T ∩ G (0) ) ⊆ U .Proof. Let x, y ∈ U with x = y . Since G (0) is a Cantor space, there exist clopenneighbourhoods V, W separating them. Then for every compact open bisection B i as chosen in Lemma 3.1.7 for the non-empty clopen sets V ∩ U and W ∩ U theelement T B i is sufficient. Minimal Cantor systems can be approximated by Kakutani-Rokhlin partitions andthose allow for a convenient representation of T ( ϕ ) as originally demonstrated in[BK00]:Let ( X, ϕ ) be a minimal Cantor system with a fixed Kakutani-Rokhlin partition A = { D k,i | ≤ k ≤ h i − , i ∈ { , . . . , n }} . Then A induces partitions α ( A ) , α ( A )of { , . . . , n } in the following way: A subset J ⊆ { , . . . , n } is an atom of α ( A ) if andonly if there exists a subset J ⊆ { , . . . , n } such that ϕ ( F j ∈ J D h j − ,j ) = F j ∈ J D ,j and for any proper subset J ⊂ J the set ϕ ( F j ∈ J D h j − ,j ) is not a union of towerbases in A . All such sets J resp. J give rise to the partitions α ( A ) resp. α ( A )of { , . . . , n } and there is a 1-1 correspondence between their atoms. An atom J ∈ α ( J ) corresponds to a collection of towers of which the union of roofs is mappedon the union of floors of a collection of towers corresponding to J ∈ α ( A ) via ϕ . Denote by h J := min { h j | j ∈ J } resp. h J := min { h j | j ∈ J } the minimalheight of a tower associated with an atom J ∈ α ( A ) resp. J ∈ α ( A ). For any i ∈ { , . . . , n } define J ( i ) ∈ α ( A ) resp. J ( i ) ∈ α ( A ) to be the atoms containing i .Let γ ∈ T ( ϕ ). There exists a Kakutani-Rokhlin partition A = { D k,i | ≤ k ≤ h i − , i ∈ { , . . . , n }} such that A refines the finite partitions { X γi } i ∈ I and { ϕ f i ( X γi ) } i ∈ I given in Proposition 3.1.3 and such that | f i | ≤ h A for all i ∈ I , this means f γ isconstant on atoms of A and k f γ k ∞ ≤ h A . For such a Kakutani-Rokhlin partitionthe pairs U ( A ) admit the following set:(i) U in ⊆ U ( A ) the set of pairs ( k, i ), such that γ ( D k,i ) ⊂ D ( i ).(ii) U top ⊆ U ( A ) the set of pairs ( k, i ), such that f γ | D k,i + k ≥ h i .(iii) U bot ⊆ U ( A ) the set of pairs ( k, i ), such that f γ | D k,i + k < See Subsection 1.2.2. .1.2 TOPOLOGICAL FULL GROUPS IN TERMS OF KAKUTANI-ROKHLINPARTITIONS When γ is applied, the atoms corresponding to pairs in U in stay in the same tower,atoms corresponding to pairs in U top move through the roof of the containing towerand atoms corresponding to pairs in U bot move through the floor of the containingtower. By the assumption | f i | ≤ h A for all i ∈ I , atoms corresponding to pairs in U top must lie at within a distance of h A from the top of the containing tower andanalogously atoms corresponding to pairs in U bot must lie at within a distance of h A from the floor of the containing tower. Definition 3.1.10 ([BK00], Definition 2.1) . (i) Let J ∈ α ( A ) and J ∈ α ( A ).Define for r ∈ { , . . . , h J − } resp. r ∈ { , . . . , h J − } the sets: F ( r, J ) := G j ∈ J D h j − h J + r,j and F ( r , J ) := G j ∈ J D r ,j (ii) A homeomorphism γ ∈ T ( ϕ ) satisfies the level condition (L) if the followingconditions hold:(a) if ( k, i ) ∈ U top and D k,i ⊆ X γl , then F ( h J ( i ) − h i + k, J ( i )) ⊆ X γl .(b) if ( k, i ) ∈ U bot and D k,i ⊆ X γl , then F ( k, J ( i )) ⊆ X γl .The level condition (L) amounts to the fact that when an atom D k,i is moved throughthe roof (resp. floor) of the containing tower D ( i ) by γ , then D h j − h i + k,j is movedthrough the roof of D ( j ) for all j ∈ J ( i ) (resp. D k,j is moved through the floor of D ( j ) for all j ∈ J ( i ) by γ ). Note, that if U in = U ( A ) holds for a homeomorphism γ ∈ T ( ϕ ) i.e. γ ( D ( i )) = D ( i ) for every tower D ( i ), then γ obviously satisfies thelevel condition. Definition 3.1.11 ([BK00], § . Let (
X, ϕ ) be a minimalCantor system and let A = { D k,i | ≤ k ≤ h i − , i ∈ { , . . . , n }} be a Kakutani-Rokhlin partition. Denote by T ( A ) the set of homeomorphisms γ ∈ T ( ϕ ) such that A refines the finite partitions { X γi } i ∈ I and { ϕ f i ( X γi ) } i ∈ I and | f i | ≤ h A holds for all i ∈ I and γ satisfies the level condition (L) with respect to A . Remark 3.1.12.
By definition the set T ( A ) is finite. Definition 3.1.13.
Denote by P ( A ) the subset of homeomorphisms γ ∈ T ( A ) suchthat U in = U ( A ) holds. In case of a nested sequence {A n } n ∈ N of Kakutani-Rokhlinpartitions satisfying property (H), we call the elements of P ( A n ) n -permutations .Every n -permutation p admits a decomposition p = p . . . p m such that the supportof every component p i lies within a single tower of the partition A n . Remark 3.1.14.
The set P ( A ) is a finite subgroup of T ( ϕ ) and isomorphic to L i ∈{ ,...,n } S h i , where S h i denotes the symmetric group of degree h i . In the caseof a nested sequence {A n } n ∈ N of Kakutani-Rokhlin partitions satisfying property(H), it holds that P ( A n ) ≤ P ( A n +1 ) and S n ∈ N P ( A n ) is a proper subgroup of T ( ϕ ). Every finite group embedds into S n ∈ N P ( A n ) and thus into T ( ϕ ). Note that P ( A ) ∼ = L i ∈{ ,...,n } A h i , where A h i denotes the alternating group of degree h i . 61 .1. BASIC STRUCTURE AND SIGNIFICANCE IN CANTOR DYNAMICS It holds that given a sufficient nested sequence {A n } n ∈ N of Kakutani-Rokhlin par-titions every element of the topological full group can be described as a homeomor-phism that permutes atoms of A n for some n ∈ N : Theorem 3.1.15 ([BK00], Theorem 2.2) . Let ( X, ϕ ) be a minimal Cantor systemand let {A n } n ∈ N be a nested sequence of Kakutani-Rokhlin partitions satisfying prop-erty (H). Then T ( ϕ ) = S n ∈ N T ( A n ) . In [Med07], Theorem 3.1.15 was generalized to aperiodic Cantor systems. Besidesproving the above theorems, [BK00] reflects them in the context of different examplesof Cantor systems and is devoted to study topological properties of F ( ϕ ) and T ( ϕ )– specifically, a topology on Homeo(X) is introduced with respect to which T ( ϕ ) isdense in F ( ϕ ).In the following we review the stronger statement from [GM14] in more detail. Itelucidates the stucture of topological full groups of minimal Cantor systems in that itshows that their structure is close to a union of permutational wreath products of Z .Note that T ( ϕ ) always admits embeddings of such permutational wreath products: Proposition 3.1.16.
Let ( X, ϕ ) be a minimal Cantor system. Then for every n ∈ N the group Z o n S n := Z n (cid:111) S n embeds into T ( ϕ ) .Proof. Let n ∈ N . By assumption there exists a tower of height n i.e. a clopensubset U ⊂ X such that U, ϕ ( U ) , . . . , ϕ n − ( U ) are pairwise disjoint. Then the groupgenerated by the induced transformation ϕ U and the group S n of permutations ofthis tower is isomorphic to Z o n S n .The stronger result is obtained by strengthening the assumptions on the approxi-mating nested sequence {A n } n ∈ N of Kakutani-Rokhlin partitions:Let { m n } n ∈ N be a fixed sequence of positive integers such that m n → ∞ for n → ∞ .By restriction to a subsequence, we can assume that(i) h A n ≥ m n + 2Any homeomorphism of a compact metric space is uniformly continuous by the Heine-Cantor theorem , which implies that in the construction of {A n } n ∈ N the base B ( A n ) can be made sufficiently small such that we can assume:(ii) diam( ϕ k ( B ( A n ))) ≤ n for − m n − ≤ i ≤ m n This condition allows to get rid of some of the more extensive vocabulary leading upto Theorem 3.1.15, because the Lebesgue number lemma then automatically assuresproperty(L) for γ ∈ T ( ϕ ) with respect to A n for every n big enough. Definition 3.1.17 ([GM14], Definition 4.5) . Let (
X, ϕ ) be a minimal Cantor systemand let {A n } n ∈ N be a nested sequence of Kakutani-Rohklin partitions that satisfies(H) and let n ∈ N . Every continuous map from a compact metric space to a metric space is uniformly continuous. .1.2 TOPOLOGICAL FULL GROUPS IN TERMS OF KAKUTANI-ROKHLINPARTITIONS (i) Let D n ( i ) be a tower. The set of atoms { D nk,i | ≤ k ≤ h ni − } contained in D n ( i ) can be regarded as a cyclic group of order h ni endowed with a metric d ni by associating D nk,i with e kπi/h ni and d ni correponds to the restriction of theintrinsic metric on the unit circle normalized by h ni / π .(ii) Let p i and p j be n -permutations with supp( p i ) ⊆ D n ( i ) and supp( p j ) ⊆ D n ( j )such that d nr ( D nk,r , p r ( D nk,r )) ≤ m n for all 0 ≤ k ≤ h nr and r ∈ { i, j } i.e. the action of p r on an atom in D n ( r ) can be uniquely interpreted asa rotation in positive or negative direction. Let k ∈ {− m n , . . . , , . . . , m n } .Then the notation p i ( k ) = p j ( k ) signifies, that in the case of 0 ≤ k ≤ m n ,the action of p i moves the atom D nk,i the same way as the action p j moves theatom D nk,j when interpreted as rotation in positive or negative direction and inthe case of − m n ≤ k ≤
0, the action of p i moves the atom D nh ni −| k | ,i the sameway as the action p j moves the atom D nh nj −| k | ,j when interpreted as rotation inpositive or negative direction.(iii) Define for 0 ≤ k ≤ h A n the sets U ( k ) := G i D nh i − k − ,i L ( k ) := G i D nk,i (iv) Let r ∈ N . A homeomorphism γ ∈ T ( ϕ ) is called an n-rotation with rotationnumber at most r if there exist a pair of subsets S u , S l ⊆ { , . . . , m n } called supportive sets and families of integers { l i } i ∈ S u and { k j } j ∈ S l with | l i | ≤ r , | k j | ≤ r such that: γ = Y i ∈ S u ( ϕ U ( i ) ) l i × Y i ∈ S l ( ϕ L ( j ) ) k j Remark 3.1.18. (i) The set U ( k ) consists of all atoms which are k levels beneaththe top in their containing tower and the set L ( k ) consists of atoms which are k levels above the bottom of their containing tower.(ii) Note, that this is well-defined by assumption (i) on the nested sequence ofKakutani-Rokhlin partitions made in the beginning of this subsection, as itguarantees for U ( k ) ∩ L ( l ) = ∅ for every 0 ≤ k ≤ h A n and 0 ≤ l ≤ h A n . Lemma 3.1.19 ([BM08], Proposition 2.1.(3)) . Let ( X, ϕ ) be a minimal Cantorsystem and let A be a clopen subset of X . Then ϕ − A ϕ is periodic i.e. every orbit isfinite.Proof. Note that ϕ − A | A = ϕ t ϕ − ,A ( x ) ( x ) and A = F n ∈ N F n − i =0 ϕ i ( A ∩ t − ϕ,A ( n )). It followsimmediatly from the definitions that for every n ∈ N and every x ∈ F n − i =0 ϕ i ( A ∩ t − ϕ,A ( n )) we have ( ϕ − A ϕ ) n ( x ).We sketch the more powerful result from [GM14]: 63 .1. BASIC STRUCTURE AND SIGNIFICANCE IN CANTOR DYNAMICS Theorem 3.1.20 ([GM14], Theorem 4.7) . Let ( X, ϕ ) be a minimal Cantor systemand let A n be a nested sequence of Kakutani-Rokhlin partitions satisfying property(H). Let γ ∈ T ( ϕ ) .(i) There exists an n ∈ Z , n > , such that for every n ≥ n there exists adecomposition γ = p γ r γ such that r γ is an n -rotation with rotation number atmost and P γ is an n -permutation with decomposition p γ = p γ, . . . p γ,i n (seeDefinition 3.1.11) satisfying the following properties:(a) The supportive sets S u , S l associated with the n -rotation r γ are containedin { , . . . , n − } .(b) d ni ( p γ,i ( D nk,i ) , D nk,i ) ≤ n for i ∈ { , . . . , i n } and ≤ k ≤ h ni − (c) p γ,i ( k ) = p γ,j ( k ) for all k ∈ [ − m n , m n ] and i, j ∈ { , . . . , i n } .(ii) If γ = p γ r γ is another decomposition of this kind, then p γ = p γ and r γ = r γ .(iii) For every finite subset F ⊆ T ( ϕ ) , there exists an n > such that for every n ≥ n and every pair of elements γ , γ ∈ F with γ = γ the decompositionsatisfies p γ = p γ .Sketch of Proof. (i) As mentioned in the beginning of this subsection, we can findan n big enough such that the cocylce f γ is constant on the atoms of A n andsuch that k f γ k ∞ ≤ n . Furthermore, choose n big enough such that n − is aLebesgue number for the partition { X γi } i ∈ Z induced by γ . Then assumption(ii) on the nested sequence of Kakutani-Rokhlin partitions in the beginning ofthis subsection assures that f γ is constant on ϕ k ( B ( A n )) for − m n − ≤ i ≤ m n for all n ≥ n . Let n ≥ n . Define R γ ( x ) := ϕ U ( h ni − − k ) ( x ) , if x ∈ D nk,i and f γ ( x ) + k ≥ h ni ϕ − L ( k ) ( x ) , if x ∈ D nk,i and f γ ( x ) + k < x, elseThis is a well defined n -rotation with rotation number at most 1 satisfyingproperty (a). Since f γ is constant on ϕ k ( B ( A n )) for − m n − ≤ i ≤ m n for n ≥ n , we can produce an n -permutation p γ from γ , by tweaking γ in such away that if an atom D nk,i is moved over the top of its containing tower (resp.over the bottom of its containing tower) to some atom D nl,j , it is moved to D nl,i (resp. D nh ni − h nj + l,i ) instead. This is well defined and the obtained permutation p γ = p γ, . . . p γ,i n satisfies the required properties (b) and (c). The proof of γ = p γ r γ is straightforward, one just has to keep track of atoms under therespective homeomorphisms.(ii) Let γ = p γ r γ be another decomposition of the described kind. Then p − γ p γ = r γ r γ . The left-hand side of this equation is an n -permutation, the right-handside an n -rotation. By definition, both sides must be the identity and hence p γ = p γ and r γ = r γ .64 .1.3 TOPOLOGICAL FULL GROUPS IN C*-ALGEBRAIC TERMS (iii) Let γ , γ ∈ T ( ϕ ) with γ = γ with corresponding decompositions γ = p γ r γ and γ = p γ r γ . There exists a clopen set C ⊂ X such that γ ( x ) = γ ( x )for all x in C . Property (i) of the deomposition implies that for every n largeenough the supports of r γ and r γ are contained in F j ∈{ ,...,k } U ( j ) t L ( j ) forsome fixed k ∈ N . By definition of A n for every n large enough there existsa subset C ⊂ C with C ∩ supp( r γ ) = ∅ = C ∩ supp( r γ ) and consequently, p γ | C = p γ | C .For odometer systems above situation is the most lucid in that their topological fullgroups in fact are unions of permutational wreath products: Proposition 3.1.21 ([Mat13b], Proof of Proposition 2.1) . Let ( X, ϕ ) be an odome-ter system of type a = ( a n ) n ∈ N . Then the topological full group is of the form T ( ϕ ) ∼ = [ n ∈ N Z o a n S a n (cid:16) : ∼ = [ n ∈ N Z a n (cid:111) S a n (cid:17) Proof.
Let {A n } n ∈ N = { D nk } n ∈ N , ≤ k ≤ a n − be the sequence of Kakutani-Rokhlin par-titions with property (H) given in Example 1.2.12. Denote by Γ n the group ofelements γ ∈ T ( ϕ ) of which the cocycle f γ is constant on atoms of A n . The family { Γ n } n ∈ N is a nested sequence of subgroups with S n ∈ N Γ n = T ( ϕ ). Since for every n ∈ N the partition A n consists of only one tower of height a n every γ ∈ Γ n inducesa well-defined permutation σ γ ∈ S a n . The induced map π n : Γ n → S a n is a grouphomomorphism and has a section π − n defined by π − n ( σ )( x ) = ϕ σ ( k ) − k ( x ) for σ ∈ S a n and x ∈ D nk . Let γ ∈ ker( π ). Then for all k ∈ Z /a n Z it holds that γ ( D nk ) = D nk andthus γ | D nk = ϕ m γ,k · a n for some m γ,z ∈ Z . Then ker( π n ) → Z a n : γ → ( m γ,k ) k ∈ Z /a n Z isan isomorphisms and therefore Γ n ∼ = Z a n (cid:111) S a n . The first glances on topological full groups of minimal Cantor systems were froma perspective informed by topological dynamics and operator algebras. The signif-icance of full groups resp. topological full groups in topological dynamics is, thatunder sufficient conditions they determine the orbit equivalence class resp. flip con-jugacy class of a topological dynamical system. One appearance of T ( ϕ ) can betraced back to [Put89]:Let ( X, ϕ ) be a minimal Cantor system and let Y be a closed subset of X . Thenby Theorem 1.2.27 the C*-subalgebra A Y of C ( X ) (cid:111) ϕ Z generated by C ( X ) and uC ( X \ Y ) is an AF C*-algebra, by which it lent itself to the analysis on ideals in thewake of Theorem 1.2.6 by Stratila and Voiculescu. The maximal abelian subalgebrawith associated conditional expectation is given by C ( X ) and the restriction ofthe conditional expectation E from Remark 1.1.30(iii) to A Y . Fix the followingnotations:UN( C ( X ) , C ( X ) (cid:111) ϕ Z ) := { v ∈ U( C ( X ) (cid:111) ϕ Z ) | vC ( X ) v ∗ = C ( X ) } UN( C ( X ) , A Y ) := { v ∈ U( A Y ) | vC ( X ) v ∗ = C ( X ) } .1. BASIC STRUCTURE AND SIGNIFICANCE IN CANTOR DYNAMICS Both sets can be considered as groups of ∗ -automorphism of C ( X ) by their ad-joint action – the latter obviously being a subgroup of the former. Denote therespective quotients by U( C ( X )) of this groups by Γ (resp. Γ Y ). The groupUN( C ( X ) , A Y ) plays the role of the locally finite unitary subgroup U from The-orem 1.2.6, while Γ Y represents the corresponding group of homeomorphisms Γ U .By Lemma 5.1 in [Put89], every v ∈ UN( C ( X ) , C ( X ) (cid:111) ϕ Z ) can be written uniquelyas v = f P n ∈ Z u n p n , where f ∈ U( C ( X )) and { p n } n ∈ Z is a collection of pairwiseorthogonal projections p n ∈ C ( X ) with p n = 0 for almost all n such that X = X n ∈ Z p n = X n ∈ Z ϕ − n ( p n )holds. The projection p n is given by the absolute value of E n ( v ) (Remark 1.1.30(iii)).The group Γ is isomorphic to the group of invertible elements in the monoid C ( X, Z )where the monoid operation is given by( f · f )( x ) := f ( x ) + f ( ϕ − f ( x ) ( x ))This, however, is just the topological full group T ( ϕ ). Every v ∈ UN( C ( X ) , C ( X ) (cid:111) ϕ Z ) admits an element Φ( v ) ∈ T ( ϕ ) defined by Φ( v )( x ) := ϕ n ( x ) for all x ∈ supp( p n ). Proposition 3.1.22 ([Put89], Theorem 5.2) . Let ( X, ϕ ) be a minimal Cantor sys-tem. Then −→ U( C ( X )) ι −→ UN( C ( X ) , C ( X ) (cid:111) ϕ Z ) Φ −→ T ( ϕ ) −→ is an exact sequence which is split via the section γ v γ := P n ∈ Z u n X γn , where { X γn } n ∈ Z is the partition defined in Proposition 3.1.3. The above considerations are embedded in the groupoid setting, where they are anaspect of Renault’s topological version of Feldman-Moore. The following propositionis a special case of [Ren08], Proposition 4.8 (by putting twists aside):
Lemma 3.1.23 ([Mat12], Lemma 5.5) . Let G be a Hausdorff, locally compact, ef-fective, ´etale groupoid. For every partial isometry v ∈ C ∗ r ( G ) such that vv ∗ , v ∗ v ∈ C ( G (0) ) and vC ( G (0) ) v ∗ = vv ∗ C ( G (0) ) there exists a compact open slice V ∈ B o,k G such that V = { g ∈ G| v ( g ) = 0 } . The assignement v τ V induces a homomorphism Φ : UN( C ( G (0) ) , C ∗ r ( G )) → T ( G )such that uf u ∗ = f ◦ Φ( u ) for u ∈ UN( C ( G (0) ) , C ∗ r ( G ), in particular the followingholds: Proposition 3.1.24 ([Mat12], Proposition 5.6) . Let G be an effective, ´etale Cantorgroupoid. Then the following sequence is exact and splits −→ U( C ( G (0) )) ι −→ UN( C ( G (0) ) , C ∗ r ( G )) Φ −→ T ( G ) −→ Remark 3.1.25.
A canonical section of Φ is provided by mapping B ∈ T ( G ) onto B . Note that then B f ∗ B = f ◦ α B for all f ∈ C ( G (0) ).66 .1.4 LOCALLY FINITE SUBGROUPS Definition 3.1.26 ([GPS99], § . Let (
X, ϕ ) be a minimal Cantor system and let x ∈ X . Define T ( ϕ ) { x } to be the stabilizer T ( ϕ ) Orb + ϕ ( x ) = { γ ∈ T ( G ) | γ (Orb + ϕ ( x )) =Orb + ϕ ( x ) } . Remark 3.1.27.
It follows immediately from this definition that ϕ · T ( ϕ ) { x } · ϕ − = T ( ϕ ) { ϕ ( x ) } .Subgroups of this form can be characterized in terms of approximation by Kakutani-Rokhlin partitions and are necessarily locally finite as a corollary of Theorem 3.1.20: Corollary 3.1.28.
Let ( X, ϕ ) be a minimal Cantor system. Let x ∈ X and {A n } n ∈ N be a nested sequence of Kakutani-Rokhlin partitions around x satisfying property (H).Then the following hold:(i) T ( ϕ ) { x } = [ n ∈ N P ( A n ) : ∼ = [ n ∈ N M i ∈{ ,...,i n } S h ni . (ii) The group T ( ϕ ) { x } is locally finite and in consequence amenable.(iii) For every x, y ∈ X the groups T ( ϕ ) { x } and T ( ϕ ) { y } are isomorphic.Proof. (i) Let γ ∈ T ( ϕ ) { x } . Assume r γ is non-trivial, then by the definition of n -rotations, there exist points in Orb − ϕ ( x ) which are mapped to Orb + ϕ ( x ) by γ .Conversely, any γ ∈ P ( A n ) preserves towers and thus forward and backward orbitsof x ∈ X .(ii) and (iii) are immediate consequences of (i).(iv) follows from (i), Remark 3.1.27 and minimality. Corollary 3.1.29 ([GM14], Proposition 2.12 & [dC13], FAIT 2.2.4) . Let ( X, ϕ ) bea minimal Cantor system. Then the group T ( ϕ ) { x } satisfies no group law. It followsthat the groups T ( ϕ ) and T ( ϕ ) satisfy no group law.Proof. Assume there exists a sufficient non-empty word w ∈ F k for some k ∈ N .Then by Corollary 3.1.28(i) every finite symmetric group satisfies the group law w .Hence by Caley’s Theorem every finite group and in consequence every product offinite groups satisfies the law w . But since residually finite groups are preciselythe groups which embed into direct products of finite groups and free groups areresidually finite, this is a contradiction. Corollary 3.1.30 ([GM14], Proposition 5.2) . Let ( X, ϕ ) be a minimal Cantor sys-tem. The group T ( ϕ ) { x } is a maximal locally finite subgroup for every x ∈ X . Let w ∈ F k for some k ∈ N be a non-empty word. A group G satisfies the group law w if everyset of elements g , . . . , g k ∈ G evaluates under w as w ( g , . . . , g k ) = 1 ∈ G . See e.g. Corollary 2.2.6 and Theorem 2.3.1 of [CSC10]. .1. BASIC STRUCTURE AND SIGNIFICANCE IN CANTOR DYNAMICS Proof.
Let γ ∈ T ( ϕ ) \ T ( ϕ ) { x } . Then by Corollary 3.1.28, the decomposition of γ for a sufficient sequence of Kakutani-Rokhlin partition must always contain a non-trivial n -rotation r γ . Then r γ ∈ h T ( ϕ ) , γ i is of infinite order and thus, any subgroupof T ( ϕ ) that contains T ( ϕ ) { x } and γ cannot be locally finite.These locally finite subgroups had alread appeared implicitely in [Kri80] and in[Put89]: The subgroup T ( ϕ ) { x } is just the group Γ Y in the case of Y = { x } for x ∈ X considered by Putnam. For every x ∈ X the system ( X, T ( ϕ ) { x } ) is a min-imal AF-system (see Definition 1.2.24). Since every AF-system is conjugate to aBratteli-Vershik system ( X, ϕ ) conversely for every minimal AF-system Γ there ex-ists an x ∈ X such that Γ ∼ = T ( ϕ ) { x } . A more general result is [JM13], Lemma 4.1 &Lemma 4.2 where local finiteness for more general stabilizer subgroups is shown. Ma-tui generalizes Corollary 3.1.28 to AF-groupoids in [Mat06], § T ( G ) of the AF-groupoid G associated to aBratteli diagram ( V, E ) as in Example 2.6.17 is the increasing union of direct sumsof symmetric groups G k := L v ∈ V k S |P ,v | that act by permutation of cylinder sets inthe space of infinite paths. Conversely, if T ( G ) of an ´etale Cantor groupoid is locallyfinite, the groupoid G is AF.The commutator subgroups of the locally finite subgroups T ( ϕ ) { x } are simple groups: Corollary 3.1.31.
Let ( X, ϕ ) be a minimal Cantor system. Let x ∈ X and let {A n } n ∈ N be a nested sequence of Kakutani-Rokhlin partitions around x satisfyingproperty (H). Then T ( ϕ ) x } ∼ = [ n ∈ N M i ∈{ ,...,i n } A h ni where A h ni denotes the alternating group of degree h ni and furthermore: T ( ϕ ) ab { x } := T ( ϕ ) { x } (cid:30) T ( ϕ ) x } = lim −→ n ( Z / Z ) i n . Remark 3.1.32.
We note that for minimal AF-systems the following hold:lim −→ ( Z / Z ) i n ∼ = K ( C ( X ) (cid:111) T ( ϕ ) { x } ) ⊗ Z / Z Theorem 1.2.28 ∼ = K ( C ( X ) (cid:111) ϕ Z ) ⊗ Z / Z By this we have T ( ϕ ) ab { x } ∼ = K ( C ( X ) (cid:111) ϕ Z ) ⊗ Z / Z . Proposition 3.1.33.
Let ( X, ϕ ) be a minimal Cantor system and let x ∈ X . Then T ( ϕ ) x } is a simple group.Proof. Fix a nested sequence {A n } n ∈ N of Kakutani-Rokhlin partitions around x satisfying property (H). Let H be a non-trivial normal subgroup of T ( ϕ ) x } . Let γ be a non-trivial element of the subgroup H n := H ∩ L i ∈{ ,...,i n } A h ni . by Re-mark 1.2.15, there exists an l > n such that every summand of γ viewed as elementin L i ∈{ ,...,i k } A h ki for k ≥ l is non-trivial. But since every factor A h ki is simple, any See [GPS04] and [Mat06], § .1.5 A LEMMA BY GLASNER AND WEISS proper normal subgroup of L i ∈{ ,...,i k } A h ki is of the form L i ∈ I A h ki for some propersubset I ⊆ { , . . . , i k } . This means the normal closure of H n in L i ∈ I A h ki is thewhole group for all k large enough and thus H = T ( ϕ ) x } . In this Subsection, we give a swift sketch of a lemma in [GW95]. One purpose of[GW95] was to do the classification of minimal Cantor systems by the associateddimension groups obtained in [GPS95] without relying on C*-algebra theory. Inparticular, it uses the full group and the topological full group to construct properequivalences between minimal Cantor systems given isomorphisms of the associateddimension groups – thus [GW95] foreshadows the results in [GPS99].
Lemma 3.1.34 ([GW95], Lemma 2.5) . Let ( X, ϕ ) be a minimal Cantor system andlet A, B be clopen subsets of X such that µ ( B ) < µ ( A ) for every µ ∈ M ϕ . Thenthere exists an element γ ∈ T ( ϕ ) such that γ ( B ) ⊂ A , γ = id and γ | X \ ( B ∪ γ ( B )) = id .Sketch of Proof. If there exists a Kakutani-Rokhlin partition A that refines P := { B \ A, A \ B, A ∩ B, X \ ( A ∪ B ) } such that the number of atoms contained in A is greater than the number of atoms ofcontained in B , a sufficient homeomorphism γ ∈ T ( ϕ ) can easily be given by movingatoms of A . By the assumptions, R A − B d µ > µ ∈ M ϕ . Thenthere exists a c > { R A − B d µ | µ ∈ M ϕ } > c , because otherwisethere exists a sequence of measures for which the above integral goes to 0. By thecompactness of M ϕ in the weak*-topology, a cluster point µ of this sequence exists.But µ satisfies µ ( A ) = µ ( B ), which is a contradiction. In consequence, there existsan n ∈ N such that every n ≥ n satisfies n − · n − X i =0 ( A − B )( ϕ i ( x )) ≥ c for all x ∈ X . Assume otherwise the existence of an increasing sequence { n k } k ∈ N ofnaturals and a sequence of points { x k } k ∈ N such that n − k · n k − X i =0 ( A − B )( ϕ i ( x k )) ≤ c. Then a weak* cluster point ˜ µ of n − k · n k − X i =0 ϕ i ( δ x k )satisfies R A − B d˜ µ ≤ c which is a contradiction. Let A be Kakutani-Rokhlinpartition that refines P such that h A > n . Then the above inequality implies that A contains more atoms than B . 69 .1. BASIC STRUCTURE AND SIGNIFICANCE IN CANTOR DYNAMICS We note that Lemma 3.1.34 is crucial for the proof of Theorem 3.1.53. Furthermore,it implies the following lemma, which is required to prove simplicity of T ( ϕ ) (seeSubsection 3.2.2): Lemma 3.1.35 ([Mat06], Lemma 4.7) . Let ( X, ϕ ) be a minimal Cantor system andlet x ∈ X . Then every clopen neighbouhood U of x contains a clopen neighbourhood V of x , such that [ V ] is -divisible in K ( X, ϕ ) . Proof.
Let V be a clopen neighbourhood of x such that 2 µ ( V ) < µ ( U ) for all µ ∈ M ( ϕ ). Then µ ( V ) < µ ( U \ V ) for all µ ∈ M ( ϕ ). Let γ be the homeomorphismobtained in Lemma 3.1.34 for the pair V and U \ V . We have to show [ γ ( V ) − V ] =0, because then [ V ] = [ γ ( V ) ], hence [ V ∪ γ ( V ) ] = 2[ V ]. Then C k := V ∩ f − γ ( k )for k ∈ Z is a partition of V and we can write γ ( V ) − V = X k ∈ Z (cid:16) γ ( C k ) − C k (cid:17) = X k ∈ Z (cid:16) ϕ k ( C k ) − C k (cid:17) Every summand can be written as a telescoping sum of which every summand istrivial in K ( X, ϕ ): Write ϕ k ( C k ) − C k = k − X i =0 ( ϕ i ( C k ) ◦ ϕ − − ϕ i ( C k ) )for all k > ϕ k ( C k ) − C k = | k |− X i =0 ( ϕ k + i ( C k ) − ϕ k + i ( C k ) ◦ ϕ − )for all k < The paper [GPS99] introduces another important subgroup by the means of the index map . For minimal Cantor systems the index map measures the transfer frombackward orbits to forward orbits. As the terminology suggests one motivation forthe index map lies in operator index theory.
Definition 3.1.36 ([GPS99], Definition 5.1 & 5.2) . Let (
X, ϕ ) be a minimal Cantorsystem with a nested sequence of Kakutani-Rohklin partitions {A n } n ∈ N that satisfies(H) and let x ∈ X .(i) Let γ ∈ T ( ϕ ). Define κ x ( γ ) := |{ y ∈ Orb − ϕ ( x ) | γ ( y ) ∈ Orb + ϕ ( x ) }| and define λ x ( γ ) := |{ y ∈ Orb + ϕ ( x ) | γ ( y ) ∈ Orb − ϕ ( x ) }| . There exists a class [ f ] ∈ K ( X, ϕ ) such that 2[ f ] = [ V ]. .1.6 THE INDEX MAP (ii) Let a ∈ N , b ∈ Z and let m ∈ N such that h A m > | b | + 2 a . Define thehomeomorphism σ k,l ∈ T ( ϕ ) by: σ k,l ( y ) := ϕ − | b |− a ( y ) for y ∈ D mk,i if | b | + 1 ≤ k ≤ | b | + aϕ | b | + a ( y ) for y ∈ D mk,i if h mi − | b | − a + 1 ≤ k ≤ h mi − | b | y else Remark 3.1.37. κ x ( γ ) and λ x ( γ ) are necessarily finite. The homeomorphisms σ k,l are elements in T ( ϕ ) of finite order. Lemma 3.1.38 ([GPS99], Lemma 5.3.) . Let ( X, ϕ ) be a minimal Cantor systemand x ∈ X . Then the following holds: T ( ϕ ) = a k ∈ N ,l ∈ Z T ( ϕ ) { x } ϕ l σ k,l T ( ϕ ) { x } . Proof.
Let β x : T ( ϕ ) → N × N be defined by β x ( γ ) := (cid:16) κ x ( γ ) , λ x ( γ ) (cid:17) . Varying overall possible pairs k, l lets β x ( ϕ l σ k,l ) vary over all of N × N , thus β x is surjective. Let γ ∈ T ( ϕ ) such that p = κ x ( γ ) , λ x ( γ ) = q . If p ≥ q , then β x ( γ ) = β x ( ϕ p − q σ q,p − q ) andif p < q , then β x ( γ ) = β x ( ϕ p − q σ p,q − p ). By applying a homeomorphism γ in T ( ϕ ) { x } ,we can bring elements in Orb ϕ ( x ) into a position such that applying ϕ p − q σ q,p − q resp. ϕ p − q σ p,q − p maps orbit points into the same half-orbits as γ , hence there existsa homeomorphism γ in T ( ϕ ) { x } such that γ = γ ϕ l σ k,l γ for sufficient k, l . Proposition 3.1.39 ([GPS99], Proposition 5.4. & 5.5.) . Let ( X, ϕ ) be a mini-mal Cantor system and let µ be a ϕ -invariant probability measure on X . Then Hom( T ( ϕ ) , Z ) is isomorphic to Z and I µ ( γ ) := R X f γ d µ defines the unique grouphomomorphism I µ : T ( ϕ ) → Z with I µ ( ϕ ) = 1 .Proof. By Lemma 3.1.38 elements of the topological full group can be written asproducts of elements of finite order and multiples of ϕ , thus homeomorphisms inHom( T ( ϕ ) , Z ) only depend on the image of ϕ , hence Hom( T ( ϕ ) , Z ) ∼ = Z . Let γ , γ ∈ T ( ϕ ). By Lemma 3.1.4, ϕ -invariance of the measure and linearity of the integral,we have I µ ( γ γ ) = I µ ( γ ) + I µ ( γ ), thus I µ defines a group homomorphism withIm( I µ ) ⊆ R and I µ ( ϕ ) = 1. Elements of finite order must necessarily be containedin ker( I µ ), thus Im( I µ ) = Z . Definition 3.1.40.
The map I = I µ described in Proposition 3.1.39 is called the index map .One can find a covariant representation ρ of C ∗ ( X, ϕ ) on H = ‘ ( Z ) (the left regularrepresentation) together with a projection P on H , such that I ( γ ) = dim ker( P ρ ( v γ ) P ) − dim ker(( P ρ ( v γ ) P ) ∗ ) = κ x ( γ ) − λ x ( γ )Thus I is the manifestation of a Fredholm index. The motivation of this lies in non-commutative differential geometry. The cyclic cohomology HC ∗ ( A ) of a C*-algebra71 .1. BASIC STRUCTURE AND SIGNIFICANCE IN CANTOR DYNAMICS A is a non-commutative version of de Rham homology introduced by Connes. Itadmits a bilinear pairing h , i : HC ( C ( X ) (cid:111) ϕ Z ) × K ( C ( X ) (cid:111) ϕ Z ) → C . Every γ ∈ T ( ϕ ) induces a canonical unitary v γ ∈ U ( C ∗ ( X, ϕ )) by Proposition 3.1.22thus further inducing a class [ v γ ] ∈ K ( C ( X ) (cid:111) ϕ Z ). Every ϕ -invariant measure µ induces a class of one-cocycles [ c µ ] ∈ HC ( C ( X ) (cid:111) ϕ Z ), the pairing induces amap π µ : T ( ϕ ) → C . It can be shown that π µ ( γ ) = I ( γ ). A direct proof of I ( γ ) = κ x ( γ ) − λ x ( γ ) follows from Theorem 3.1.20: Corollary 3.1.41 ([GM14], Lemma 5.3) . Let ( X, ϕ ) be a minimal Cantor system.Then Im( I ) = Z and I = κ x − λ x .Proof. Let γ ∈ T ( ϕ ) and let {A n } n ∈ N be a sufficient nested sequence of Kakutani-Rokhlin partitions. Let γ = p γ r γ be a decomposition into an n -permutation p γ and an n -rotation r γ with rotation number at most 1. Then I ( γ ) = I ( r γ ). ByLemma 3.1.19, I ( ϕ − A ) = I ( ϕ − A ϕϕ − ) = I ( ϕ − ) = − A ⊆ X .Thus by definition of r γ we have I ( r γ ) = | U | − | L | , where U is the set of numbers k ∈ { , . . . , m n } such that γ moves U ( k ) = F i D nh i − k − ,i over the roof of the partition,hence | U | = κ x ( γ ) and L is the set of numbers k ∈ { , . . . , m n } such that γ moves L ( k ) = F i D nk,i below the base of the partition, hence | L | = λ x ( γ ). Definition 3.1.42 ([GPS99], Definition 5.7) . Let (
X, ϕ ) be a minimal Cantor sys-tem. Denote by T ( ϕ ) the kernel of I . Remark 3.1.43.
We have T ( ϕ ) { x } ≤ T ( ϕ ) for all x ∈ X , and since commutatorsin T ( ϕ ) vanish under the index map, we have T ( ϕ ) ≤ T ( ϕ ) . Proposition 3.1.44 ([Mat06], Lemma 4.1) . Let ( X, ϕ ) be a minimal Cantor systemand let x, y ∈ X with Orb ϕ ( x ) = Orb ϕ ( y ) . Then T ( ϕ ) = T ( ϕ ) { x } · T ( ϕ ) { y } .Proof. Let γ ∈ T ( ϕ ) . As 0 = I ( γ ) = κ x ( γ ) − λ x ( γ ), there exists a bijection b between the finite sets: A = { n ∈ N | γ ( ϕ n ( x )) ∈ Orb − ϕ ( x ) } , B = { n ∈ N | γ ( ϕ − n ( x )) ∈ Orb + ϕ ( x ) } . Let m = max( A ∪ B ). Let U be a clopen set containing x such that U does notcontain ϕ i ( x ) for i ∈ {− m, . . . , m } and V = [ n ∈ A ϕ − n ( U ) ∪ [ n ∈ B ϕ n ( U )does not contain ϕ i ( y ) for i ∈ { , . . . , m − } . Then ˜ γ ∈ T ( ϕ ) y defined by˜ γ ( z ) := ϕ − b ( n ) − n , z ∈ ϕ n ( U ) for n ∈ Aϕ n − b − ( n ) , z ∈ ϕ − n ( U ) for n ∈ Bz, z ∈ X \ V satisfies γ ˜ γ ∈ T ( ϕ ) { x } , thus γ = ( γ ˜ γ )˜ γ − gives the desired factorization.This has immediate consequences:72 .1.7 ISOMORPHISM THEOREMS Corollary 3.1.45.
Let ( X, ϕ ) be a minimal Cantor system. Then the followinghold:(i) T ( ϕ ) ab0 is an elementary abelian -group.(ii) T ( ϕ ) = T ( ϕ ) (iii) T ( ϕ ) ab = Z × T ( ϕ ) ab0 Proof.
By Corollary 3.1.28(i) and Proposition 3.1.44 the group T ( ϕ ) is generatedby its involutions implying (i). By Proposition 3.1.44 and Remark 3.1.27, we have ϕ · T ( ϕ ) · ϕ − = T ( ϕ ) , which implies T ( ϕ ) / T ( ϕ ) = T ( ϕ ) / T ( ϕ ) × T ( ϕ ) ab0 = Z × T ( ϕ ) ab0 The group Z × T ( ϕ ) ab0 is abelian, implying (ii) which in turn implies (iii).The notion of the index map generalizes to the setting of groupoids: Definition 3.1.46 ([Nek17], Definition 2.4) . Let G be an ´etale Cantor groupoid.The index map is the homomorphism I : T ( G ) → H ( G ), which maps B ∈ T ( G ) ontoto the equivalence class of its characteristic function [ B ]. Remark 3.1.47.
The index map is a homomorphism as an immediate consequenceof Proposition 2.6.4.For the transformations groupoids associated to a minimal Cantor system, the dy-namical homology is isomorphic to the K-theory of the associated crossed productand above map genereralizes the index map of Definition 3.1.40.
Definition 3.1.48.
The kernel of the index map I is a subgroup of T ( G ) denotedby T ( G ) .By Proposition 2.6.4 we immediately have: Corollary 3.1.49.
Let G be an ´etale Cantor groupoid and let B ∈ B o,k G such that s ( B ) ∩ r ( B ) = ∅ . Then T B ∈ T ( G ) . The chief purpose of [GPS99] was the classification of minimal Cantor systems bythe means of full groups and their subgroups.
Definition 3.1.50 ([GPS99], Definition 4.1, 4.10., 5.1. & 5.2.) . [ B ] is by definition contained in ker δ = ker( s ∗ − r ∗ ). .1. BASIC STRUCTURE AND SIGNIFICANCE IN CANTOR DYNAMICS (i) Let ( X , G ) and ( X , G ) be topological dynamical systems. An isomorphism α : G → G is called spatial if there exists a homeomorphism ϕ : X → X such that α ( g ) = ϕ ◦ g ◦ ϕ − for all g ∈ G .(ii) Let ( X, ϕ ) be a Cantor system. Let A ⊆ X and let G ≤ Homeo( X ). Denote by G A the subgroup { g ∈ G | supp( g ) ⊆ A } . If A is a clopen subset the resultinggroup is called a local subgroup of G .A crucial ingredient is the abundance of certain involutions: Lemma 3.1.51 ([GPS99], Lemma 3.3) . Let ( X, ϕ ) be a Cantor system and let agroup G ≤ Homeo( X ) either be F ( ϕ ) resp. T ( ϕ ) resp. T ( ϕ ) x for some x ∈ X . Let U, V be clopen subsets of X . Then the following are equivalent:(i) [ U ] = [ V ] in K ( X, ϕ ) resp. K ( X, ϕ ) / Inf( K ( X, ϕ )) resp. K ( X, T ( ϕ ) x ) .(ii) There exists an element g ∈ G such that g ( U ) = V .(iii) There exists an element g ∈ G such that g ( U ) = V , g = Id and g | ( U ∪ V ) C =Id ( U ∪ V ) C .If the action of G on X is minimal, for every clopen set U ⊆ X and for every x ∈ U there exists a g ∈ G such that g = Id , g ( x ) = x and g | U C = Id U C . Remark 3.1.52.
The only non-trivial implication is ( i ) ⇒ ( ii ). In the case of G = F ( ϕ ) it follows by ”patching together” elements in T ( ϕ ) that arise as inLemma 3.1.34. In the other cases it follows from the Bratteli-Vershik represen-tations of the respective systems. Theorem 3.1.53 ([GPS99], Theorem 4.2) . Let ( X , ϕ ) and ( X , ϕ ) be minimalCantor systems and let G i either be F ( ϕ i ) , T ( ϕ i ) , or T ( ϕ i ) x i for i ∈ { , } and x i ∈ X i . Then any group isomorphism α : G → G is spatial. By Stone duality homeomorphisms between the Cantor spaces X and X are in 1-1correspondence with Boolean isomorphisms between the associated Boolean algebrasof clopen sets CO( X ) and CO( X ), thus it is enough to construct an Booleanisomorphism a : CO( X ) → CO( X ) such that α ( g ) a = ag for all g ∈ G . Thekey steps are that local subgroups characterize clopen sets and that local subgroupshave an algebraic characterization. Then for every U ∈ CO( X ) the group α (( G ) U )proves to be a local subgroup of Γ and thus corresponds to a clopen subset a ( U ) ⊆ CO( X ) setting up a bijection a : CO( X ) → CO( X ). The proof is accomplishedby showing a preserves intersections and satisfies α ( g ) a = ag for all g ∈ G . Thistheorem is in the vein of Dye’s work, parts of the algebraic characterization oflocal subgroups are directly transferred from [Dye63]. Combining Theorem 3.1.53with the classification of Cantor systems obtained in [GPS95] (see Theorem 1.2.30resp. Theorem 1.2.32 resp. Remark 1.2.31) shows that F ( ϕ ), T ( ϕ ) and T ( ϕ ) { x } areinvariants for the respective types of equivalence: These groups are termed of class F in [GPS99], we reserve this terminology for a more generaldefinition by Matui. .1.7 ISOMORPHISM THEOREMS Corollary 3.1.54 ([GPS99], Corollary 4.6) . Let ( X , ϕ ) and ( X , ϕ ) be minimalCantor systems. The following are equivalent:(i) The systems ( X , ϕ ) and ( X , ϕ ) are orbit equivalent.(ii) The groups F ( ϕ ) and F ( ϕ ) are isomorphic. Corollary 3.1.55 ([GPS99], Corollary 4.4) . Let ( X , ϕ ) and ( X , ϕ ) be minimalCantor systems. The following are equivalent:(i) The systems ( X , ϕ ) and ( X , ϕ ) are flip conjugate.(ii) The groups T ( ϕ ) and T ( ϕ ) are isomorphic. Corollary 3.1.56 ([GPS99], Corollary 4.11) . Let ( X , ϕ ) and ( X , ϕ ) be minimalCantor systems. The following are equivalent:(i) The systems ( X , ϕ ) and ( X , ϕ ) are strong orbit equivalent.(ii) The groups T ( ϕ ) x and T ( ϕ ) x are isomorphic for all x ∈ X , x ∈ X . [GPS99] concludes by showing that isomorphisms between index map kernels arespatial, again by the algebraic characterization of local subgroups, which impliesthat T ( ϕ ) is a complete invariant for flip-conjugacy. Corollary 3.1.57 ([GPS99], Corollary 5.18) . Let ( X , ϕ ) and ( X , ϕ ) be minimalCantor systems. The following are equivalent:(i) The systems ( X , ϕ ) and ( X , ϕ ) are flip conjugate.(ii) The groups T ( ϕ ) and T ( ϕ ) are isomorphic. In [BM08] the authors found the following adaption of Theorem 3.1.53 by a differentapproach:
Theorem 3.1.58 ([BM08], Theorem 4.2) . Let ( X , ϕ ) and ( X , ϕ ) be minimalCantor systems and let Γ i either be T ( ϕ i ) , T ( ϕ i ) or T ( ϕ i ) for i ∈ { , } and x i ∈ X i . Then any group isomorphism α : Γ → Γ is spatial. As a corollary the above list extends by:
Corollary 3.1.59.
Let ( X , ϕ ) and ( X , ϕ ) be minimal Cantor systems. Thefollowing are equivalent:(i) The systems ( X , ϕ ) and ( X , ϕ ) are flip conjugate.(ii) The groups T ( ϕ ) and T ( ϕ ) are isomorphic. The above results permit to use dynamical properties invariant under the respec-tive notions of equivalence to distinguish between groups of the respective type upto isomorphism, in particular, invariants of flip conjugacy e.g. topological entropyor ergodicity, allow to distinguish topological full groups up to isomorphy. By Re-mark 1.1.20(ii), Example 1.1.21 and Corollaries 3.1.55 & 3.1.57 & 3.1.59 we have:
Proposition 3.1.60.
There exists an uncountable family of pairwise non-ismorphictopological full groups of minimal subshifts. .2. RESULTS BY MATUI In this section we review Matui’s contributions to the study of topological full groupsand their properties. Subsection 3.2.1 gives a outline of Matui’s spatial realizationtheorem on ´etale Cantor groupoids from [Mat15]. Subsection 3.2.2 recollects theresults on topological full groups of minimal Cantor systems from [Mat06]. In Sub-section 3.2.3 we turn to generalizations in the context of ´etale Cantor groupoids.Subsection 3.2.4 finishes the section with Matui’s proof of exponential growth of thetopological full groups of minimal subshifts.
The results in the wake of [Dye63] are instance of a larger principle: The reconstruc-tion of structures from their groups of transformations. The proof of Theorem 3.1.53shows that isomorphisms between certain automorphism groups of the Boolean al-gebras CO( X ) and CO( X ) are generated by isomorphisms between CO( X ) andCO( X ). In [BM08] the spatial realization of isomorphisms for the groups T ( ϕ ), T ( ϕ ) and T ( ϕ ) of a minimal Cantor system ( X, ϕ ) is demonstrated by followingthe proof of Theorem 384D in [Fre11]. The same method was applied to the re-spective groups in the context of general minimal topological dynamical systemsover Cantor spaces (see [Med11]). The flip conjugacy of Cantor minimal systems isequivalent to isomorphy of the corresponding transformation groupoids ([Mat16b],Theorem 8.1), thus the groups T ( G ), T ( G ) and T ( G ) are complete isomorphisminvariants for transformation groupoids. We take a look at Matui’s proof of spatialrealization modeled after Theorem 384D in [Fre11]: Definition 3.2.1 ([Mat15], Definition 3.1) . Let X be a Cantor space. A subgroup G ≤ Homeo( X ) is said to be of class F if it satisfies the following properties:(F1) For every g ∈ G , g = 1 implies that supp( g ) is clopen.(F2) For every clopen set U ⊆ X and for every x ∈ U there exists a g ∈ G \ { } such that g = 1 and x ∈ supp( g ) ⊆ U .(F3) For every g ∈ G \ { } with g = 1 and for every non-empty clopen set U ⊂ supp( g ) there exists a g ∈ G \{ } such that supp( g ) ⊂ U ∪ g ( U ) and g | supp( g ) = g | supp( g ) .(F4) For every non-empty clopen subset U ⊆ X there exists a g ∈ G with supp( g ) ⊆ U and g = 1.Property (F2) immediately allows to encode the order structure on regular closedsubsets of X by local subgroups: A closed subset A of a topological space X is called regular if A = A ◦ .2.1 ISOMORPHISMS THEOREMS II Lemma 3.2.2 ([Mat15], Lemma 3.2) . Let X be a Cantor space and let G ≤ Homeo( X ) be a group of class F. Then for a pair of regular closed subsets A, B ⊆ X the following are equivalent:(i) A ⊆ B (ii) G A ⊆ G B Next, local subgroups over supports of involutions are algebraically characterized:
Definition 3.2.3 ([Mat15], p.7) . Let X be a Cantor space, let G ≤ Homeo( X ) bea group of class F and let τ ∈ G be a non-trivial involution. Define:(i) C τ := { g ∈ G | τ gτ = g } (ii) U τ := { g ∈ C τ | g = 1 , gcgc − = cgc − g ∀ c ∈ C τ } (iii) S τ := { g ∈ G | g ∈ G, ugu − = g ∀ u ∈ U τ } (iv) W τ := { g ∈ G | sgs − = g ∀ s ∈ S τ } Lemma 3.2.4 ([Mat15], Lemma 3.3) . Let X be a Cantor space, let G ≤ Homeo( X ) be a group of class F and let τ ∈ G be a non-trivial involution. Then W τ = G supp( τ ) . Lemma 3.2.2 and Lemma 3.2.4 then imply:
Lemma 3.2.5 ([Mat15], Lemma 3.4) . Let X , X be Cantor spaces, let G ≤ Homeo( X ) and G ≤ Homeo( X ) be groups of class F and let Φ : G → G bean isomorphism of groups. Let τ, σ ∈ G be involutions. Then the following equiva-lences hold:(i) supp( τ ) ⊆ supp( σ ) ⇔ supp(Φ( τ )) ⊆ supp(Φ( σ )) (ii) supp( τ ) ∩ supp( σ ) = ∅ ⇔ supp(Φ( τ )) ∩ supp(Φ( σ )) = ∅ This allows to construct a spatial homeomorphism:
Theorem 3.2.6 ([Mat15], Theorem 3.5) . For i ∈ { , } let X i be Cantor spaces andlet G i ≤ Homeo( X i ) be groups of class F. Let α : G → G be an isomorphism andlet x ∈ X . Fix the sets T ( x ) := { g ∈ G | x ∈ supp( g ) , g = 1 } and P ( x ) := \ g ∈ T ( x ) supp( α ( g )) . Then P ( x ) is a singleton for every x ∈ X and the map ϕ : X → X given by ϕ ( x ) := P ( x ) for x ∈ X is a homeomorphism such that α ( g ) = ϕ ◦ g ◦ ϕ − for all g ∈ G . .2. RESULTS BY MATUI Matui proceeds by showing:
Proposition 3.2.7 ([Mat15], Proposition 3.6) . Let G be a minimal, effective, ´etaleCantor groupoid. Then every subgroup G ≤ T ( G ) with T ( G ) ≤ G is a subgroup of Homeo( G (0) ) of class F.Proof. It is sufficient to show that T ( G ) is of class F.(F1) This holds by definition.The properties (F2), (F3) and (F4) follow from minimality by Corollary 3.1.9.(F2) Let U ⊆ G (0) be non-empty and clopen and let x ∈ U . Then there exist non-empty, compact, open bisections W, V ∈ B o,k G such that s ( W ) ∪ r ( W ) ⊆ U , x ∈ s ( W ), s ( W ) ∩ r ( W ) = ∅ , s ( V ) ∪ r ( V ) ⊆ s ( W ), x ∈ s ( V ) and s ( V ) ∩ r ( V ) = ∅ . The element[ τ W , τ V ] is the required involution.(F3) Let γ be a non-trivial involution in T ( G ) and let U ⊆ supp( γ ) be non-emptyand clopen. There exists a non-empty, compact, open bisection W ∈ B o,k G with s ( W ) ∪ r ( W ) ⊆ U such that s ( W ), r ( W ), γ ( s ( W )) and γ ( r ( W )) are pairwise disjoint.Define ˜ γ ∈ T ( G ) by ˜ γ ( x ) := γ ( x ) , if x ∈ s ( W ) ∪ γ ( s ( W )) x, elseThe element [˜ γ, [ γ, τ W ]] is sufficient.(F4) Let U ⊆ G (0) be non-empty and clopen. Then there exist non-empty, compact,open bisections W, V ∈ B o,k G such that s ( W ) ∪ r ( W ) ⊆ U , s ( V ) ∪ r ( V ) ⊆ U , the sets s ( W ), r ( W ) and r ( V ) are pairwise disjoint and s ( V ) ⊆ s ( W ). Then the element[( τ ( W s ( V )) ) − , τ V ] has order three. Proposition 3.2.8 ([Mat15], Proposition 3.8) . For i ∈ { , } let G i be a minimal,effective, ´etale Cantor groupoid and let G i be subgroups of T ( G i ) with T ( G i ) ≤ G i .Then every spatial isomorphism α : G → G induces an isomorphism of groupoidsbetween G and G .Sketch of Proof. Let h : G (0)1 → G (0)2 such that α : G → G is given by γ h ◦ γ ◦ h − . By minimality every g ∈ G is contained in an open, compact slice B ∈ B o,k G such that the associated element α B ∈ T ( G ) is in T ( G ) (This is verified similarilyas the proof of (F4) in Proposition 3.2.7). For every g ∈ G the element g :=( s | h ◦ B ◦ h − ) − ( h ( s ( g ))) ∈ G is well defined and the map induced by g → g is agroupoid isomorphisms between G and G .Theorem 3.2.6, Proposition 3.2.7 and Proposition 3.2.8 then imply: Theorem 3.2.9 ([Mat15], Theorem 3.10) . Let G , G be effective, minimal, Haus-dorff, ´etale Cantor groupoids. The following are equivalent:(i) G ∼ = G .2.2 SIMPLICITY AND FINITE GENERATION (ii) T ( G ) ∼ = T ( G ) (iii) T ( G ) ∼ = T ( G ) (iv) T ( G ) ∼ = T ( G ) Later on it was noticed that spatial realization follows directly from reconstructionresults of Matatyahu Rubin:
Definition 3.2.10 ([Mat16a], § . Let X be a topological space. A subgroup G ⊆ Homeo( X ) is called locally dense if for every x ∈ X and every open set x ∈ U ⊆ X the set { f ( x ) | f ∈ G, f | X \ U = id X \ U } is somewhere dense.The following theorem is a direct consequence of Corollary 3.5 in [Rub89]: Theorem 3.2.11 ([Mat16a], Theorem 5.3) . Let X , X be locally compact, Haus-dorff topological spaces containing no isolated points. Let G ⊆ Homeo( X ) , G ⊆ Homeo( X ) be isomorphic, locally dense groups of homeomorphisms. Then for everyisomorphism ϕ : G → G , there exists a unique homeomorphism φ : X → X suchthat ϕ ( g ) = φ ◦ g ◦ φ − . In [Mat06], the first paper primarily concerned with algebraic properties of T ( ϕ ),Matui showed that T ( ϕ ) is simple by characterization of the abelianization of T ( ϕ )in terms of K ( C ∗ ( X, ϕ )) and in the case of a subshift finitely generated. Matuiapplies C*-algebraic methods, in that he uses a surjective homomorphism T ( ϕ ) → K ( C ∗ ( X, ϕ )) ⊗ ( Z / Z ) which is a modification of the mod map ([GPS99], Definition2.10) – thus the name signature map . To set up the signature map, Matui provesa structural lemma on T ( ϕ ) describing it in terms of the locally finite subgroups T ( ϕ ) { x } .In Remark 3.1.32, we mentioned the isomorphism T ( ϕ ) ab { x } ∼ = K ( C ∗ ( X, ϕ )) ⊗ ( Z / Z ).The preomposition with the quotient homomorphism induces a homomorphismsgn x : T ( ϕ ) { x } → K ( C ∗ ( X, ϕ )) ⊗ ( Z / Z ) . The group homomorphism sgn on T ( ϕ ) then is obtained by patching together themaps { sgn x } x ∈ X via the factorization obtained in Proposition 3.1.44. The homo-morphism sgn x can be expressed in the following way:Let γ ∈ T ( ϕ ) { x } . For every x ∈ X , define g γ ( x ) to be the smallest positive integer k such that γ k ( x ) = x . This sets up a continuous function g γ : X → N . Let {A n } n ∈ N be a nested sequence of Kakutani-Rokhlin partitions such that the cocycle f γ isconstant on atoms, then the function g γ is constant on atoms. Hence for every k ∈ N with g − γ ( k ) non-empty and there exists a union D k = S i ∈ I D i of atoms of thepartition {A n } n ∈ N such that the sets D k , γ ( D k ) , . . . , γ k − ( D k ) are pairwise disjointand their union is g − γ ( k ). Define a function G γ ∈ C ( X, Z ) by G γ = P k ∈ N D k .This function allows to characterize sgn x : 79 .2. RESULTS BY MATUI Lemma 3.2.12 ([Mat06], Lemma 3.7) . Let ( X, ϕ ) be a minimal Cantor system andlet γ ∈ T ( ϕ ) { x } . Then sgn x ( γ ) = [ G γ ] + 2 K ( C ∗ ( X, ϕ )) ⊗ ( Z / Z ) . This characterization implies sgn x | T ( ϕ ) { x } ∩ T ( ϕ ) { y } = sgn y | T ( ϕ ) { x } ∩ T ( ϕ ) { y } for all x, y ∈ X . It holds that the map sgn x is surjective on intersecions T ( ϕ ) { x } ∩ T ( ϕ ) { y } ([Mat06],Lemma 4.3). Furthermore, let γ ∈ T ( ϕ ) . By Proposition 3.1.44 for any x, y ∈ X with Orb ϕ ( x ) = Orb ϕ ( y ) there exist γ x ∈ T ( ϕ ) { x } and γ y ∈ T ( ϕ ) { y } with γ = γ x γ y .Define sgn( γ ) = sgn x ( γ x ) + sgn y ( γ y ). By Lemma 4.4 in [Mat06], sgn( γ ) neitherdepends on the choice of x, y ∈ X nor on the choice of factors γ x , γ y . Thus oneobtains a well-defined surjective map sgn : T ( ϕ ) → K ( C ∗ ( X, ϕ )) ⊗ ( Z / Z ), whichis in fact a group homomorphism ([Mat06], Proposition 4.6). Equipped with thishomomorphism, Matui characterizes the abelianization of T ( ϕ ) : Theorem 3.2.13 ([Mat06], Theorem 4.8.) . Let ( X, ϕ ) be a minimal Cantor system.Then the signature map satisfies ker(sgn) = T ( ϕ ) giving rise to the isomorphism T ( ϕ ) ab0 ∼ = K ( C ∗ ( X, ϕ )) ⊗ ( Z / Z ) .Sketch of Proof. The inclusion T ( ϕ ) ⊆ ker(sgn) is obvious. Assume γ ∈ ker(sgn).Let x, y ∈ X with Orb ϕ ( x ) = Orb ϕ ( y ). Then by Lemma 3.1.35, we can choose U in the proof of Proposition 3.1.44 such that [ U ] is 2-divisible and thus byLemma 3.2.12, the element ˜ γ ∈ T ( ϕ ) satisfies sgn(˜ γ ) ∈ ker(sgn y ) = T ( ϕ ) y } . Inconsequence, γ ˜ γ ∈ ker(sgn x ) = T ( ϕ ) x } holds and therefore γ is a product of com-mutators. Corollary 3.2.14.
Let ( X, ϕ ) be a minimal Cantor system. Then we have T ( ϕ ) ab = Z ⊕ ( K ( C ∗ ( X, ϕ )) ⊗ ( Z / Z )) . Furthermore, Matui proves with the help of the signature map:
Theorem 3.2.15 ([Mat06], Theorem 4.9.) . Let ( X, ϕ ) be a minimal Cantor system.Then T ( ϕ ) is a simple group.Proof. Let H be a non-trivial normal subgroup of T ( ϕ ) = T ( ϕ ) . Let γ ∈ H bea non-trivial element and define l := max { f γ ( x ) | x ∈ X } . Fix an arbitrary x ∈ X .Then there exists a y ∈ X with Orb ϕ ( x ) = Orb ϕ ( y ) such that y = γ ( y ) = ϕ ( y ) andfurthermore a clopen neighbourhood U of x such that U ∩ ϕ ( U ) = ∅ , such that U doesnot contain the elements ϕ ( y ) , γ ( y ) , ϕ − γ ( y ) and ϕ k ( x ) for − l − ≤ k ≤ l +1 and, byLemma 3.1.35, such that U is 2-divisible in K ( X, ϕ ). Then the homeomorhpism τ U defined by τ U ( x ) := ϕ ( x ) , if x ∈ Uϕ − ( x ) , if x ∈ ϕ ( U ) x, elseis by definition contained in T ( ϕ ) { x } and by the description of sgn x in Lemma 3.7 in[Mat06] we have τ u ∈ ker(sgn x ). Thus the non-trivial homeomorphism τ U ◦ γ ◦ τ U ◦ γ − is contained in T ( ϕ ) x } ∩ H . But since T ( ϕ ) x } is simple by Proposition 3.1.33, itholds that T ( ϕ ) x } ⊆ H for all x ∈ X and thus H = T ( ϕ ) .80 .2.3 GENERALIZATIONS Theorem 3.2.13 and Theorem 3.2.15 combined imply:
Corollary 3.2.16 ([Mat06], Corollary 4.10.) . Let ( X, ϕ ) be a minimal Cantor sys-tem. The group T ( ϕ ) is simple if and only if K ( C ∗ ( X, ϕ )) is 2-divisible. In [BM08] the authors obtained a different proof of simplicity of T ( ϕ ) by moreapproachable techniques. Just as Matui they rely on the Lemma of Glasner andWeiss (see Lemma 3.1.34).Matui demonstrated that in the case of minimal subshifts the group T ( ϕ ) is finitelygenerated, that the groups T ( ϕ ), T ( ϕ ) and T ( ϕ ) cannot be finitely presented andgave a sufficient and necessary condition for the finite generation of T ( ϕ ) and T ( ϕ ) .The proofs will be omited, since, firstly, in Subsection 3.3.3 a generalization ofTheorem 3.2.17 will be given and, secondly, finite presentation for minimal subshiftsfollows as a corollary from results discussed in Subsection 3.5.1. Theorem 3.2.17 ([Mat06], Theorem 5.4.) . Let ( X, ϕ ) be a minimal Cantor system.The group T ( ϕ ) is finitely generated if and only if ( X, ϕ ) a minimal subshift. Combining Theorem 3.2.17 with Theorem 3.2.13 gives:
Corollary 3.2.18 ([Mat06], Corollary 5.5.) . Let ( X, ϕ ) be a minimal Cantor system.The following are equivalent:(i) ( X, ϕ ) is a minimal subshift and K ( C ∗ ( X, ϕ )) ⊗ ( Z / Z ) is finite.(ii) T ( ϕ ) is finitely generated.(iii) T ( ϕ ) is finitely generated. Matui’s proof of finite presentation follows by contradiction relying on results onminimal subshifts:
Theorem 3.2.19 ([Mat06], Theorem 5.7) . Let ( X, ϕ ) be a minimal subshift. Then T ( ϕ ) cannot be finitely presented. Using the facts that a finite index subgroup H of a group G is finitely presented ifand only if the group G is finitely presented and that finite presentability is stableunder extensions (see Chapter V of [dlH00]), gives the following corollary: Corollary 3.2.20 ([Mat06], Corollary 5.8) . Let ( X, ϕ ) be a minimal subshift. Thenthe groups T ( ϕ ) and T ( ϕ ) cannot be finitely presented. By Proposition 2.6.20, the transformation group of free Z n actions on a Cantorspace are almost finite. Thus the following simplicity results of Matui in particulargeneralize Theorem 3.2.15: 81 .2. RESULTS BY MATUI Theorem 3.2.21 ([Mat13b], Theorem 4.7) . Let G be an almost finite, minimal,effective, ´etale Cantor groupoid. Any non-trivial subgroup of T ( G ) normalized by T ( G ) contains T ( G ) . Thus T ( G ) is simple. Theorem 3.2.22 ([Mat13b], Theorem 4.16) . Let G be a purely infinite, minimal,effective, ´etale Cantor groupoid. Any non-trivial subgroup of T ( G ) normalized by T ( G ) contains T ( G ) . Thus T ( G ) is simple. Both of these simplicity theorems hinge on a generalization of Lemma 3.1.34. Inthe almost finite case this is represented by Lemma 6.7 in [Mat12]. In the purelyinfinite case it comes for free by the structure of the groupoid by the following lemma– which we state because of its use in Subsection 3.6.2:
Proposition 3.2.23 ([Mat15], Proposition 4.11.) . Let G be an ´etale Cantorgroupoid. The following are equivalent:(i) G is purely infinite and minimal.(ii) For every pair of clopen sets A, B ⊂ G (0) with B = ∅ , there exists a compact,open slice U ∈ B o,k G with s ( U ) = A and r ( U ) ⊂ B .(iii) For every pair of clopen sets A, B ⊂ G (0) with A = G (0) and B = ∅ , there existsan S ∈ T ( G ) such that r ( SA ) ⊂ B .Proof. (i) ⇒ (ii): Since B is properly infinite, there exist compact open slices U, V satisfying s ( U ) = s ( V ) = B , r ( U ) ∪ r ( V ) ⊂ B and r ( U ) ∩ r ( V ) = ∅ . The family { V n } n ∈ N of compact, open slices defined inductively by V := U and V n +1 = V V n satisfies s ( V n ) = B , r ( V n ) ⊂ B and r ( V n ) ∩ r ( V m ) = ∅ for n = m . Since G isminimal there exists a finite family of slices { W i } i ∈ I as chosen in Lemma 3.1.7.Then U = S i ∈ I V i W i is the required slice. (ii) ⇒ (iii): B \ A is not empty : By (ii) there exists a compact open slice U with s ( U ) = A and r ( U ) ⊂ B \ A . The compact, open slice S := U ∪ U − ∪ ( G (0) \ ( s ( U ) ∪ r ( U )) is sufficient. B \ A is empty : We have B ⊂ A . By the previous case, there exist S , S ∈ T ( G )with r ( S A ) ⊂ G (0) \ A and r ( S ( G (0) \ A )) ⊂ B . The element S := S S is sufficient. (iii) ⇒ (i): Minimality is immediate, as by this property { Su | S ∈ T ( G ) } is dense in G (0) for every u ∈ G (0) . Let A be a non-empty clopen subset of G (0) . Let B , B , B ⊆ A be non-empty clopen and pairwise disjoint. Let S , S ∈ T ( G ) such that r ( S ( A \ B )) ⊂ B and r ( S ( B ∪ B )) ⊂ B . Then U = B ∪ S ( A \ B ) and V = S U arecompact open slices with s ( U ) = s ( V ) = A , r ( U ) ⊂ B ∪ B and r ( V ) ⊂ B , thus A is properly infinite.In light of above simplicity results and of Subsection 3.2.2, it makes sense to studythe abelization of T ( G ). Matui formulates the following conjecture, which he calls AH-conjecture :82 .2.3 GENERALIZATIONS
Conjecture 3.2.24 ([Mat16a], Conjecture 2.9) . Let G be a minimal, effective, ´etaleCantor groupoid. Then the following sequence is exact H ( G ) ⊗ ( Z / Z ) j −→ T ( G ) ab I −→ H ( G ) −→ i.e. T ( G ) ab is an extension of H ( G ) by a quotient of H ( G ) ⊗ ( Z / Z ) . Remark 3.2.25 ([Mat16a], Remark 2.10) . At first Matui considered a strongerversion where the sequence is short exact. While it is true for some classes ofgroupoids, there are counterexamples highlighted by Volodymyr V. Nekrashevych.This stronger version is termed the strong AH-conjecture .Since T ( G ) ≤ T ( G ) , the homomorphism I induced by the index map I is well-defined. Of course we need to clarify how the homomorphism j comes about: Bythe description of H ( G ) ⊗ ( Z / Z ) given in Proposition 2.6.5, it is sufficient todefine the images j ( U ) for clopen subsets U ⊂ G (0) . By minimality Lemma 3.1.7applies and there exists a finite family { B i } i ∈ I of compact open slices such that r ( B i ) ⊂ G (0) \ U , U = S i s ( B i ) and s ( B i ) ∩ s ( B j ) = ∅ for i = j . Define j ( U ) tobe the class [ Q i ∈ I T B i ] ∈ T ( G ) ab . Verifying the AH-conjecture requires to show thatthis map is even well-defined and that the arising sequence is exact, in particular,that the index map I is surjective. Matui was able to show surjectivity of I in thepurely infinite and in the almost finite case: Theorem 3.2.26 ([Mat12], Theorem 7.5 & [Mat15], Theorem 5.2) . Let G be aneffective, ´etale Cantor groupoid. If G is either almost finite or purely finite, theindex map I : T ( G ) → H ( G ) is surjective. With regards to Conjecture 3.2.24 Matui obtained:
Theorem 3.2.27.
The AH-conjecture is satisfied by ´etale Cantor groupoids whichare:(i) ([Mat16a], Theorem 3.6) principal, minimal and almost finite.(ii) ([Mat16a], Theorem 5.8) a finite product of SFT-groupoids.(iii) (Corollary 3.2.14) the transformation groupoid of a minimal Cantor system. Assume that j is well-defined, then Corollary 3.1.49 implies Im j ⊆ ker ¯ I . Oneobtains ker ¯ I ⊆ Im j in particular if T ( ϕ ) is generated by transpositions of the form T B . In the case of minimal Cantor systems this is represented by Proposition 3.1.44.In [Mat12], Matui considered the subgroup of elementary homeomorphisms in T ( G )i.e. the subgroup of homeomorphisms γ ∈ T ( G ) of finite order such that { x ∈G (0) | γ n ( x ) = x } is clopen for all n ∈ N and related this subgroup with the indexmap kernel T ( G ) in the almost finite, principal case: For transformation groupoids associated to minimal Z n -actions on Cantor spaces with n > .2. RESULTS BY MATUI Theorem 3.2.28 ([Mat12], Theorem 7.13) . Let G be an almost finite, effective,´etale Cantor groupoid. Then every element of T ( G ) can be written as product offour elements in T ( G ) of finite order. In principal, ´etale Cantor groupoids an element in T ( G ) is elementary if and only if it is of finite order. In [Mat13b], Matui shows that if T ( ϕ ) is finitely generated, the group ( X, ϕ ) nec-essarily has exponential growth. This follows from the fact that minimal Cantorsystems which are not odometers – these in particular encompasses all minimal sub-shifts – are precisely those of which T ( ϕ ) contains a copy of the lamplighter group L . Definition 3.2.29.
The semi-direct product L := ( L Z Z / Z ) (cid:111) Z where Z actson the copies of Z / Z by the shift is called the lamplighter group . A standardpresentation of the lamplighter group is L = D x, y | x , [ x, y n xy − n ] n ∈ N E . Proposition 3.2.30 ([dlH00] VII.A.1) . Any finitely generated group that containsa free sub-semigroup on 2 generators has exponential growth.
Remark 3.2.31.
As the subsemigroup in L with above presentation generated by y and xy is free, L has exponential growth.The following is immediate from the structure of topological full groups of odometersystems given by Proposition 3.1.21: Corollary 3.2.32 ([Mat13b], Proposition 2.1) . Let ( X, ϕ ) be a minimal Cantor sys-tem. If ( X, ϕ ) is an odometer, then every finitely generated subgroup of T ( ϕ ) is vir-tually abelian and thus has polynomial growth and contains no free sub-semigroups. This implies that for any odometer (
X, ϕ ), the groups T ( ϕ ) and T ( ϕ ) can notcontain copies of the lamplighter group L . Lemma 3.2.33 ([Mat13b], Proposition 2.3) . Let ( X, ϕ ) be a minimal Cantor sys-tem. If it is not an odometer, then there exists a clopen subset C ⊂ X such that forevery finite subset F of Z the following holds for all x ∈ X : f C,F ( x ) := X k ∈ F C ◦ ϕ k ( x ) ≡ See Appendix A.1 for definition of the growth rate of a finitely generated group. .2.4 ON GROWTH Proof.
By Lemma 1.1.22, there exists continuous function g : X → { , } Z with g ( X ) infinite that satisfies g ◦ ϕ = σ ◦ g . Define C to be the set of elements x suchthat g ( x ) = 1. By definition, we have g ( x ) k = C ( ϕ k ( x )) Let F be a finite subsetof Z such that f C,F is 0 (mod 2) everywhere and let l = min F and m = max F .Let x, y ∈ X : g ( x ) l − − g ( y ) l − ≡ X k ∈ F \ l g ( x ) k − − g ( y ) k − (mod 2)Thus if g ( x ) k = g ( y ) k for all l ≤ k ≤ m , then g ( x ) l − = g ( y ) l − . By iteration, itfollows that every value g ( x ) k with k ∈ Z depends only on the values of g ( x ) k for l ≤ k ≤ m contradicting the assumption of infinity of g ( X ). Theorem 3.2.34 ([Mat13b], Theorem 2.4) . Let ( X, ϕ ) be a minimal Cantor system.If it is not an odometer, then the group T ( ϕ ) contains a copy of the lamlighter group L .Proof. Assume (
X, ϕ ) is not an odometer. Let U ⊂ X be a non-empty clopensubset such that U, ϕ ( U ) , ϕ ( U ) , ϕ ( U ) are pairwise disjoint. For any non-emptyclopen subset V ⊂ U define a homeomorphism τ V ∈ T ( ϕ ) by: τ V ( x ) := ϕ ( x ) , if x ∈ Vϕ − ( x ) , if x ∈ ϕ ( V ) x, elseNote, that all homeomorphisms of this kind pairwise commute and that for anyfinite family of non-empty clopen V , . . . , V n ⊂ U the homeomorphism τ V ◦ · · · ◦ τ V n is the identity if and only if P ni =1 V i ( x ) = 0 (mod 2) for all x ∈ X . Define ahomeomorphism r = ϕ U ◦ ϕ ◦ ϕ U ◦ ϕ − ∈ T ( ϕ ), where ϕ U is the induced transformationassociated with U (Definition 3.1.17(iii)). It holds, that supp( r ) ⊂ U ∪ ϕ ( U ) and r ◦ τ V ◦ r − = τ ϕ U ( V ) . The derivative ( U, ϕ U | U ) of ( X, ϕ ) over U can not be anodometer and thus by Lemma 3.2.33 there exists a non-empty clopen subset C suchthat for every finite subset F ⊂ Z f C,F ( x ) := X k ∈ F C ◦ ϕ kU ( x ) ≡ x ∈ U . It holds that supp( ϕ C ) ⊂ U ∪ ϕ ( U ). The group generated by r and τ C is isomorphic to the lamplighter group, because r k ◦ τ C ◦ r − k = τ ϕ kU ( C ) for any k ∈ Z and P k ∈ F ϕ kU ( C ) = 0 for every finite F ⊂ Z , it holds that Q k ∈ F r k ◦ τ C ◦ r − k isnot the identity for every finite F ⊂ Z . Since supp( r ) ∩ supp( ϕ ◦ r ◦ ϕ − ) = ∅ andsupp( τ C ) ∩ supp( ϕ ◦ τ C ◦ ϕ − ) = ∅ , the group generated by [ r − , ϕ − ] and [ τ − C , ϕ − ]is contained in T ( ϕ ) and isomorphic to the lamplighter group L . Corollary 3.2.35 ([Mat13b], Corollary 2.5) . Let ( X, ϕ ) be a minimal Cantor sys-tem. If T ( ϕ ) is finitely generated, it has exponential growth. .2. RESULTS BY MATUI Proof.
By Theorem 3.2.17, the group T ( ϕ ) associated with a minimal Cantor system( X, ϕ ) is finitely generated if and only if ( X, ϕ ) is conjugate to a minimal subshift,thus it can not be an odometer and must contain a copy of L . By Remark 3.2.31,the growth of the lamplighter group is exponential and thus T ( ϕ ) has exponentialgrowth. For the definition of Higman-Thompson groups see Appendix A.7, of Cuntz-Kriegeralgebras Definition B.1.11 and for SFT-groupoids Subsection 2.6.4. In [Mat15] Ma-tui mentions the fact that under the representation of Cuntz algebras as groupoidC*-algebras – see Theorem 2.6.27 – the Higman-Thompson groups turn out to betopological full groups of SFT-groupoids. For a more precise description of corre-spondences see [MM17].
Definition 3.2.36.
Let n ∈ N and let { A i,j } i,j ∈{ ,...,n } ∈ M n × n ( Z / Z ) be a matrixof which every row and column is non-zero and which is not a permutation matrix.Let { S i } i ∈{ ,...,n } be a collection of non-zero partial isometries over some Hilbertspace H that generate a faithful representation of the Cuntz-Krieger algebra O A .Let w = ( w , . . . , w k ) ∈ { , . . . , n } k be a finite word of length k ∈ N . Denote by S w the product S w . . . S w k . Remark 3.2.37.
Note that S w = 0 if and only if A w i ,w i +1 = 1 for all i ∈ { , . . . , n − } .Matui’s observation is based on a result obtained in [Bir04] and in [Nek04]: TheHigman-Thompson group G n, has a faithful representation as unitary subgroup ofthe Cuntz algebra O n . Theorem 3.2.38 ([Nek04], Proposition 9.6) . Let n ∈ N \ { } . Then the map π : G n, → O n given for every g ∈ G n, represented as a table of cofinite bases in X A ∗ n by g = x w . . . x w k x u . . . x u k ! S u S ∗ w + · · · + S u k S ∗ w k is a faithful unitary representation. Moreover G n, is isomorphic to the group gen-erated by unitary elements S ∈ O n which are of the form S = S u S ∗ w + · · · + S u k S ∗ w k for some k ∈ N and w i , u i ∈ A ∗ n . Let G A be an SFT-groupoid. As in Subsection 2.6.4 an SFT-groupoid G A comeswith the data of a matrix A that satisfies (Inp), an associated finite directed graphΓ A = ( V, E ) with induced edge shift σ A on the space X A of one-ended infinitedirected paths and G A is the equivalence groupoid induced by the tail equivalence ofpaths. Under the identification of E with { , . . . , n } the condition of Remark 3.2.37is equivalent to ω = ( ω , . . . , ω k ) ∈ E ∗ being a finite directed path in Γ A . Definition 3.2.39.
Denote by D ˜ A the subalgebra of O ˜ A generated by elements ofthe form S ω S ∗ ω for ω ∈ E ∗ .86 .2.5 HIGMAN-THOMPSON GROUPS AS TOPOLOGICAL FULL GROUPS The C*-subalgebra D ˜ A is commutative and isomorphic to C ( X A ) via the assignment S ω S ∗ ω C ω , in fact, it is a Cartan subalgebra of O ˜ A isomorphic C ( G (0) A ) under theisomorphism of Theorem 2.6.27. Lemma 3.2.40.
Let G A be SFT-groupoid. Denote by G A the group of all home-omorphisms h ∈ Homeo( X A ) such that there exists a pair of continuous maps k h , l h : X A → Z ≥ such that σ k h ( x ) A ( x ) = σ l h ( x ) A ( h ( x )) . Then G A ∼ = T ( G A ) . For inverse fibers of k h and l h there exist finite clopen partitions into cylinder sets andby this every B ∈ T ( G A ) decomposes as compact open slice into a union F i ∈ I B ω i ,υ i of compact open slices given by a finite collection of pairs of finite directed paths( ω i = ( ω i, , . . . , ω i,l i ) , υ i = ( υ i, , . . . , υ i,m i ) i ∈ I which satisfy r ( ω i,l i ) = r ( υ i,m i ), suchthat X A = F i ∈ I C ω i = F i ∈ I C υ i . With this description the isomorphism of unitarynormalizers induced by the isomorphism of Theorem 2.6.27 is provided by the as-signment B → P i ∈ I S ω i S ∗ υ i . In the case of a full shift of order n the image preciselyamounts to the copy of G n, in O n described in Theorem 3.2.38. More generally –see § G n,r turn out to correspond to T ( G A n,r ) for thematrix A n,r := . . . . . . n . . . Matui thus considered topological full groups T ( G A ) associated to irreducible one-sided shifts of finite type ( X A , σ A ) as generalized Higman-Thompson groups andshowed properties of the groups G n,r generalize to this context: Theorem 3.2.41.
Let G A be an SFT-groupoid. Then the following hold:(i) The group T ( G A ) is not amenable.(ii) ([Mat15], Theorem 6.7) The group T ( G A ) satisfies the Haagerup property. (iii) ([Mat15], Theorem 6.21) The group T ( G A ) is of type F ∞ . Remark 3.2.42.
Theorem 3.2.41(i) is a consequence of being purely infinite andminimal – see Lemma 2.6.24 and Remark 3.5.16(ii). See § The
Haagerup property is a notion weaker than amenability and similarily is an opposite toProperty(T). It is of significance in connection with the
Baum-Connes conjecture . [Geo08] Let G be a group. A CW-complex X is called an Eilenberg-MacLane complex of type K ( G, π ( X ) ∼ = G and π k ( X ) is trivial for all k = 1. Let n ∈ N . A group is said to be of type F n if there exists an Eilenberg-MacLane complex of type K ( G,
1) with finite n -skeleton. It is saidto be of type F ∞ if it is of type F k for all k ∈ N . It holds that a group G is finitely generated (resp.finitely presented) if and only if it is of type F (resp. of type F ), which is why this propertiesare termed finiteness properties . .3. ON A PAIR OF SUBGROUPS Nekrashevych defined a pair of subgroups of T ( G ) denoted by S ( G ) and A ( G ) interms of slices. These have subsequently been termed the symmetric- and alternat-ing full groups. In the effective case S ( G ) is just Matui’s subgroup of elementaryelements mentioned at the end of Subsection 3.2.3. The subgroups S ( G ) (resp. A ( G )) are in the vein of the description of the locally finite subgroups T ( ϕ ) { x } (resp. T ( ϕ ) x } ) in terms of permutations (resp. even permutations) of atoms in an as-sociated nested sequence of Kakutani-Rokhlin partitions (see Corollary 3.1.28 &3.1.31). Similar to the situation of Proposition 3.1.44, these new subgroups areclosely related to the subgroups T ( G ) (resp. T ( G ) ), in fact, they are equal for cer-tain classes of groupoids. Subsection 3.3.1 gives the definition of S ( G ) and A ( G ). InSubsection 3.3.2 we recount Nekrashevych’s proof of simplicity of A ( G ) for minimal,effective, ´etale Cantor groupoid and in Subsection 3.3.3 his proof of finite generationof A ( G ) for minimal, expansive, ´etale Cantor groupoids. Definition 3.3.1 ([Nek17], Definition 3.1) . Let G be an ´etale Cantor groupoid andlet d be a positive integer. A family of compact, open slices M = { M i,j } ≤ i,j ≤ d withthe properties(i) M j,k M i,j = M i,k for all 1 ≤ i, j, k ≤ d (ii) M i,i ⊆ G (0) for all 1 ≤ i ≤ d (iii) M i,i ∩ M j,j = ∅ if i = j for all 1 ≤ i, j ≤ d is called multisection of degree d . The set S i M i,i is called the domain of M . Let U be a clopen subset contained in some component M i ,i of the domain. Then M| U := { M i,j r ( M i ,i U ) } di,j =1 is a multisection of degree d called the restriction of M to U .The following lemma assures us to have enough multisections: Lemma 3.3.2 ([Nek17], Lemma 3.1) . Let G be an ´etale Cantor groupoid. Let { u , . . . , u d } be a finite set of elements in G (0) which is contained in a single G -orbitand in an open neighbourhood U . There exists a multisection M = { M i,j } ≤ i,j ≤ d ofdegree d , such that u i ∈ M i,i , x j = r ( M i,j u i ) and U contains the domain of M .Proof. Since all u i are contained in one G -orbit, there exist elements g , g . . . g d such that s ( g i ) = u and r ( g i ) = u i , in particular set g = u . Since G is an´etale Cantor groupoid, for every i ∈ { , . . . , d } there exists a compact, open slice B i ∈ B o,k G such that g i ∈ B i . Since G (0) is a Cantor space, there exists a clopenneighbourhood u ∈ W with W ⊆ s ( B i ) ∩ U such that the sets r ( B i W ) are disjointand B W = W (by assuming W ⊆ s ( B )). Then M i,j := ( B j W )( B i W ) − gives thedesired multisection.88 .3.1 SYMMETRIC- AND ALTERNATING FULL GROUPS Definition 3.3.3 ([Nek17], Defintion 3.2) . Let G be an ´etale Cantor groupoid andlet M be a multisection of degree d with domain U . There is an embedding of thesymmetric group S d of degree d into T ( G ) given by π
7→ M π := S di =1 M i,π ( i ) ∪ ( G (0) \ U ) and accordingly an embedding of the alternating group A d of degree d . Denotethe image of the symmetric group under this embedding by S ( M ) and by A ( M )the image of A d . The subgroup of T ( G ) generated by the union of the subgroups S ( M ) for all multisections M of degree d is denoted by S d ( G ), the subgroup of T ( G )generated by the union of the subgroups A ( M ) for all multisections M of degree d is denoted by A d ( G ). The subgroup S ( G ) := S ( G ) is called the symmetric fullgroup and the subgroup A ( G ) := A ( G ) the alternating full group . Remark 3.3.4. (i) The subgroups S d ( G ) and A d ( G ) are normal, since conju-gates of multisections of degree d are multisections of degree d . It holds, that S d ( G ) ≥ S d +1 ( G ) and A d ( G ) ≥ A d +1 ( G ), since for a multisection M of degree d + 1 the groups S ( M ) resp. A ( M ) are generated by all the subgroups S ( M )resp. A ( M ) for all multisections M of degree d in M .(ii) Since every transposition in S ( G ) is in T ( G ) , it is immediate that S ( G ) ≤ T ( G ) . Since 3-cycles are generated by commutators, it holds that A ( G ) ≤ T ( G ) . Proposition 3.3.5 ([Nek17], Proposition 3.6 & Corollary 3.7) . Let G be an ´etaleCantor groupoid. If all G -orbits have n or more points, S ( G ) = S d ( G ) for all ≤ d ≤ n and A ( G ) = A d ( G ) for all ≤ d ≤ n . Thus, if all G -orbits are infinite, S ( G ) = S d ( G ) for all ≤ d and A ( G ) = A d ( G ) for all ≤ d .Proof. By Remark 3.3.4(i) it holds that S d ( G ) ≥ S d +1 ( G ). It remains to show S d ( G ) ≤ S d +1 ( G ) for 2 ≤ d ≤ n −
1. Let M be a multisection of degree d . Let { x , x , . . . x d } be a set of points with x i ∈ M i,i contained in the same G -orbit.By assumption there exists an additional point x d +1 in the same G -orbit. Let U be a clopen neighbourhood of { x , x , . . . , x d +1 } containing the domain of M .Lemma 3.3.2 allows to construct a multisection M of degree d + 1 with M i,j ⊆ M i,j for i, j ∈ , , . . . , d . Since the domain of M is compact, it is possible to applythis construction to get a finite family M k of multisections of degree d + 1 withdisjoint domains such that S k M ki,j ⊇ M i,j . This implies S ( M ) ⊆ S d +1 ( G ). Theproof for A ( G ) is analogous and the second statement is a simple corollary of thefirst statement.By [Nek17] Theorem 3.2.11 can be used to extend the list in Theorem 3.2.9 by:(v) S ( G ) ∼ = S ( G )(vi) A ( G ) ∼ = A ( G ) 89 .3. ON A PAIR OF SUBGROUPS The purpose of this subsection is to recount the proof of simplicity of A ( G ) forminimal, effective, ´etale Cantor groupoids obtained in [Nek15]. As a consequenceone obtains a verification of the AH-conjecture for minimal, almost finite, principal,´etale Cantor groupoids. Lemma 3.3.6 ([Nek17], Lemma 3.3) . Let A d be the alternating group of degree d .If d ≥ , it holds that h ˜ A ∪ ˜ A i = A d × A d × A d , where ˜ A := { ( π, π, | π ∈ A d } and ˜ A := { (1 , π, π ) | π ∈ A d } .Proof. Since [ ˜ A , ˜ A ] = { (1 , [ A d , A d ] , } holds and because the assumption d ≥ A d , A d ] = A d , it follows that { (1 , A d , } ⊆ h ˜ A ∪ ˜ A i and consequently { ( A d , , } , { ( A d , , } ⊆ h ˜ A ∪ ˜ A i Proposition 3.3.7 ([Nek17], Proposition 3.2) . Let G be an ´etale Cantor groupoidand M be a multisection of degree d ≥ . If there is a collection of multisections M k for k = 1 , , . . . , n for some n ∈ N such that S nk =1 M ki,j = M i,j , then A ( M ) ⊆h S nk =1 A ( M k ) i .Proof. Let n = 2. The families I := { M i,j ∩ M i,j } di,j =1 , D := { M i,j \ M i,j } di,j =1 and D := { M i,j \ M i,j } di,j =1 are multisections. The domains of I and D i are invariantunder elements of A ( M i ) and the restriction of an element M iπ to the domain of I (resp. D i ) equals I π (resp. D iπ ) for every π ∈ A d . Restrictions of elementsin A ( M ) to the domain of D are trivial and so are restrictions of elements in A ( M ) to the domain of D . Thus, we can apply Lemma 3.3.6 and A ( I ) , A ( D i ) ≤h A ( M ) ∪ A ( M ) i holds as desired. The other cases follow by induction. Theorem 3.3.8 ([Nek17], Theorem 4.1) . Let G be a minimal, effective, ´etale Cantorgroupoid. Any non-trivial subgroups of T ( G ) normalized by A ( G ) must contain A ( G ) .Thus A ( G ) is simple.Proof. Let H ≤ T ( G ) be a non-trivial subgroup normalized by A ( G ). Let B ∈ H ⊂ B o,k G be a non-trivial element. By effectiveness, there exists an element g ∈ G with g ∈ B and such that s ( g ) = r ( g ). Let U be a clopen neighbourhood of s ( g )such that U ∩ BU = ∅ . Let M be a multisection of degree ≥ U – this exists by Lemma 3.3.2 – and let A , A ∈ A ( M ) ⊆ A ( G ).Then B M B − is a multisection of which the domain is contained in BU and suchthat BA i B − ∈ A ( B M B − ) for i ∈ { , } . The computation of [ B − , A ] on thesets ( U ∪ BU ) C , U and BU shows [[ B − , A ] A ] = [ A , A ]. The assumption of H being normalized by A ( G ) implies [[ B − , A ] A ] ∈ H , hence [ A ( M ) , A ( M )] ⊆ H .Since A ( G ) ≤ T ( G ) by Remark 3.3.4(ii), it follows A ( M ) = [ A ( M ) , A ( M )] ⊆ H .3.2 SIMPLICITY OF THE ALTERNATING FULL GROUP and in consequence A ( G| U ) ≤ H , where G| U is the restriction of G to U as de-fined in Definition 2.1.5. The proof is completed by showing that the conjucateclosure of A ( G| U ) in A ( G ) is A ( G ). By Proposition 3.3.5 it is sufficient to showthat for any multisection M of degree 5, it holds that A ( M ) is contained in theconjugate closure of A ( G| U ) in A ( G ). To this end let M be a multisection of de-gree 5 and let x ∈ F , be arbitrary. The points x i := r ( F ,i x ) are containedin one G -orbit. By the assumption of minimality, there exist elements ρ i , σ i ∈ G for all i ∈ { , . . . , } with r ( ρ i ) = s ( σ i ) = x i such r ( σ i ) ∈ U and since all or-bits are infinite the elements x , . . . x , s ( ρ ) , . . . , s ( ρ ) , r ( σ ) , . . . , r ( σ ) can be as-sumed pairwise different. By Lemma 3.3.2 there exist families of compact, openslices { R i } i =1 and { S i } i =1 such that ρ i ∈ R i , σ i ∈ S i , r ( R i ) ⊆ U and the sets s ( R ) , . . . , s ( R ) , r ( R ) , . . . , r ( R ) , r ( S ) , . . . , r ( S ) are pairwise disjoint, and thereexists a multisection M := { M i,j } i,j =1 such that M i,j ⊆ M i,j and M i,i = r ( R i ) = s ( S i ). Then the collection U := { S j M i,j S − i } i,j =1 forms a multisection of which thedomain is contained in U . The sets C i := R i ∪ S i ∪ ( S i R i ) − are compact, open slicessuch that s ( C i ) = r ( C i ) and the sets s ( C ) , . . . , s ( C ) are pairwise disjoint. The com-pact, open slice A := S i =1 C i ∪ (cid:16) G (0) \ S i =1 s ( C i ) (cid:17) is an element of T ( G ) and in par-ticular of A ( G ) such that U = A M A − . Thus A ( M ) = A − A ( U ) A ≤ A − A ( G| U ) A .Since the domain of M is compact, there exists a finite collection M k of restrictionsof M constructed in the same way as M and satisfying the same properties. Ap-plying Proposition 3.3.7 shows A ( M ) ≤ h A ( G| U ) i A ( G ) and the proof is finished.Theorem 3.3.8, Theorem 3.2.21 and Theorem 3.2.22 imply: Corollary 3.3.9.
Let G be an minimal, effective, ´etale Cantor groupoid. If G iseither almost finite or purely finite, then A ( G ) = T ( G ) holds. Corollary 3.3.9 allowed Nekrashevych to proof Conjecture 3.2.24 in the case of min-imal, almost finite, principal, ´etale Cantor groupoids: For principal, ´etale Cantorgroupoids, every element γ ∈ T ( G ) is of finite order if and only if γ ∈ S ( G ) holds,thus Theorem 3.2.28 and Remark 3.3.4(ii) imply S ( G ) = T ( G ) . If G is minimal andalmost finite, then A ( G ) = T ( G ) by Corollary 3.3.9. This implies the assignement of j ( s ( S ) ) as the class of S ∪ S − ∪ ( G (0) \ ( s ( S ) ∪ r ( S )) in T ( G ) ab from Conjecture 3.2.24has its range in S ( G ) / A ( G ) and one even has: Theorem 3.3.10 ([Nek15], Theorem 7.2) . Let G be a minimal, almost finite, prin-cipal, ´etale Cantor groupoid. Then j is a well-defined grouphomomorphism and j ( H ( G ) ⊗ ( Z / Z )) = S ( G ) / A ( G ) . The following theorem, a direct proof of which can be found in Theorem 3.6 from[Mat16a], then follows from Theorem 3.2.26 and Theorem 3.3.10:
Theorem 3.3.11 ([Nek15], Proposition 7.5) . Let G be a minimal, almost finite,effective, ´etale Cantor groupoid. Then the following sequence is exact: H ( G ) ⊗ ( Z / Z ) j −→ T ( G ) ab I −→ H ( G ) −→ A ( G| U ) naturally embeddeds into A ( G ) by extending with units outside of U . .3. ON A PAIR OF SUBGROUPS In Subsection 2.6.2 expansive groupoids where descibed as generalizations of trans-formation groupoids associated with shifts. Nekrachevych obtained for these a gen-eralization of Theorem 3.2.17.
Lemma 3.3.12 ([Nek17], Lemma 3.5) . Let A d be the alternating group of degree d .Let X , X be sets with ≤ | X i | ≤ ∞ and | X ∩ X | = 1 . Then the conjugate closureof [ A X , A X ] in h A X ∪ A X i equals A X ∪ X .Proof. Denote by { s } := X ∩ X . The group A X ∪ X is generated by 3-cycles.If the 3 elements permutated by a 3-cycle are all contained in one of the sets X i ,then the corresponding 3-cycle is trivially in A X ∪ X . A cycle containing s is alsocontained in A X ∪ X , since ( x , s, x ) with x ∈ X , x ∈ X is [( x , s, x ) , ( x , s, x )]for x ∈ X , x ∈ X . A cycle ( x , x , x ) with x , x ∈ X \ { s } and x ∈ X \ { s } is just the conjugate of the cycle ( x , x , s ) ∈ A X by the cycle ( x , x , s ). Thus h A X ∪ A X i = A X ∪ X . Cycles of the form ( x , s, x ) with x ∈ X , x ∈ X are in [ A X , A X ] and by conjugation with elements in A X ∪ X all 3-cycles can beobtained. Lemma 3.3.13 ([Nek17], Proposition 3.4) . Let G be an ´etale Cantor groupoid and B , B be multisections with degrees d resp. d greater than such that the in-tersection of their domains equals I := B , ∩ B , . Denote by ˜ B the multisectionwith domain equal to the domain of B | I ∪ B | I and its collection of slices given bycompositions of slices in B | I ∪ B | I . Then A ( ˜ B ) ≤ h A ( B ) ∪ A ( B ) i .Proof. The elements of [ A ( B ) , A ( B )] act trivially on ( B , ∪ B , ) \ I are preciselythe elements in [ A ( B | I ) , A ( B | I )] and accordingly are contained in A ( ˜ B ). Similarilyconjugating elements in A ( ˜ B ) by elements in vA ( B ) ∪ A ( B ) is the same as conjugat-ing by the corresponding elements in A ( B | I ) ∪ A ( B | I ). This means the conjugateclosure of [ A ( B ) , A ( B )] in h A ( B ) ∪ A ( B ) i is contained in A ( ˜ B ) and by applyingLemma 3.3.12 equality follows.We will use subdivisions to ease the navigation through Nekrashevych’s proof offinite generation: Theorem 3.3.14 ([Nek17], § . Let G be an expansive, ´etale Cantor groupoid suchthat every G -orbit contains or more elements. Then A ( G ) is finitely generated.Proof. Fix for the duration of the proof a metric d on G (0) .(i) By assumption for every u ∈ G (0) there exist elements g , g , g , g ∈ G with s ( g i ) = u and such that u, r ( g ) , r ( g ) , r ( g ) , r ( g ) are pairwise different, bywhich there exist open, compact slices G , G , G , G with identical sourcesuch that g i ∈ G i and the sets s ( G ), r ( G ), r ( G ), r ( G ) and r ( G ) arepairwise disjoint. Hence, there exists an ε u > ε u -neighbourhood of u have G -orbits containing at least 5 elements whichhave a distance of at least ε u from each other. Since G (0) is compact, there92 .3.3 FINITE GENERATION OF ALTERNATING FULL GROUPS exists a countable family of elements in G (0) with neighbourhoods chosen asabove such that G (0) is covered. Hence there exists an ε > all units have G -orbits containing at least 5 elements which have a distance of atleast ε from each other. Thus any finite cover of G (0) by disjoint clopen setswith components of diameter less than ε produces a finite, clopen partition P := { P i } of G (0) such that every G -orbit intersects at least 5 components of P .Let S be a compact, open generating set of G and let P be a clopen partition of G (0) as chosen in (i).(ii) We can assume S to be symmetric – by taking the union with its inverse. Inaddition by Lemma 2.6.14 we can assume S to be contained in G \
Iso( G ).This implies d ( s ( g ) , r ( g )) = 0 for all g ∈ S and since S is compact thereexists an ε > d (cid:16) s ( g ) , r ( g ) (cid:17) ≥ ε for all g ∈ S . In consequence s ( g ) and r ( g ) lie in different components of P for all g ∈ S by choosing asufficient refinement. Moreover it can be assumed that | Sg | ≥ g ∈ G (0) just by taking the union with sufficient open, compact slices contained in G \
Iso( G ). This implies that for all g ∈ S there exist elements g , g , g , g ∈ S such that s ( g i ) = g and the units g, r ( g ) , r ( g ) , r ( g ) , r ( g ) are contained inpairwise different components of P . Since G was assumed to be expansive,by Remark 2.6.9 there exists an expansive cover S of S and by choosing asufficient refinement it can be assumed that s ( F ) and r ( F ) are contained indifferent components of P for all F ∈ S . Denote by T the set of elements g ∈ S k =1 S k for which s ( g ) and r ( g ) are contained in different components of P . For every g ∈ T and every V ∈ S k =1 S k there exists a multisection M ofdegree 5 with g ∈ M , ⊆ V and all M i,i are contained in pairwise differentcomponents of P . The set T is compact, hence there exists a finite collection {M k } of multisections choosen as M such that S k M k , covers T . Let A denotethe group h S k A ( M k ) i . The proof is accomplished by showing A ( G ) = A .(iii) Let g ∈ G be such that s ( g ) and r ( g ) sit in different components of P . Thenfor every open, compact slice B with g ∈ B there exists a multisection N ofdegree 5, such that A ( N ) ⊆ A , g ∈ N , ⊆ B and the sets N i,i are containedin pairwise different components of P : Since S is an expansive cover of thegenerating set S , the proof reduces to the case where B is a finite productof open compact slices B B . . . B n with B i ∈ S . This allows to proof thestatement by induction on n . If n ≤ M k in (ii). Suppose the statement is true for products of length n . Let B = B B . . . B n +1 . Then g can be uniquely written as a product g = g g . . . g n +1 with g i ∈ B i . By the assumptions on S there exist anelement g ∈ S such that s ( g ) = s ( g ) and r ( g ) , s ( g ) , r ( g ) are contained inpairwise different components of P . Then there exists an open, compact slice B with g ∈ B . Denote g := g g g and B := B B B . It holds that g ∈ B B B . . . B n +1 ⊆ B . By the induction hypothesis there exists a multisection N of degree 5 with N i,i contained in pairwise different components of P such93 .3. ON A PAIR OF SUBGROUPS that g g . . . g n +1 ∈ N , ⊆ B B . . . B n +1 , A ( N ) ≤ A and we can additionalyassume, that N , and r ( g ) are contained in pairwise different components of P . Denote by N the induced multisection of degree 3 on { N i,j } i,j =1 . Since B is by definition contained in S and g is contained in T , there exists an M j (as defined in (ii)) with g ∈ M , ⊆ G . Choosing a sufficient submultisectionof M j gives a multisection M of degree 3 with g ∈ M , ⊆ G and such thatthe only intersecting components of the domains of N and M are N , and M , . Applying Lemma 3.3.13 to N and M gives the desired multisection.(iv) Let g ∈ G \ Iso( G ) be such that s ( g ) and r ( g ) are contained in the samecomponent of P . Then for every open, compact slice B with g ∈ B there existsa multisection N of degree 5, such that A ( N ) ⊆ A , g ∈ N , ⊆ B and thesets N , ∪ N , , N , , N , , N , are contained in pairwise different componentsof P : To this end let P ∈ P such that s ( g ) , r ( g ) ∈ P . There exists a g ∈ G with s ( g ) = s ( g ) and g / ∈ P , hence by applying the result obtained in (iii)there exists a multisection N of degree 3 with A ( N ) ≤ A , g ∈ N , and thecomponents of its domain contained in pairwise different components of P .Similarily replacing g by g g − gives a multisection N of degree 3 with thecorresponding properties. It can be assumed that N , and N , lie in differentcomponents of P . Lemma 3.3.13 applied to N and N verifies the statement.(v) Let g , g ∈ G be such that s ( g ) = s ( g ) and s ( g ) , r ( g ) , r ( g ) pairwise dif-ferent. Let N be a multisection of degree 5 given by (iii) or (iv) for g –whichever applies. It can be assumed that r ( g ) is not in N , or N , . Thereare at most 3 components of P intersecting { s ( g ) , r ( g ) , r ( g ) } , hence thereexists a k ∈ , , r ( g ) / ∈ N k,k . Consequently r ( g ) is not contained inthe domain of the submultisection ˜ N of degree 3 induced by N , , N , , N k,k .Let N be a multisection of degree 5 given by (iii) or (iv) for g . It can beassumed the component of the domain containing r ( g ) does not intersect thedomain of ˜ N . There exists an l ∈ , , N l,l is contained in a com-ponent of P which does not contain r ( g ) and does not intersect the domain of˜ N . Denote by ˜ N the submultisection of degree 3 induced by N , , N , , N l,l .The multisection arising from the application of Lemma 3.3.13 to ˜ N and˜ N contains a sufficient submultisection ˜ N of degree 3 such that g , g arecontained in respective slices contained in ˜ N and A ( ˜ N ) ≤ A .(vi) The multisection ˜ N that has just been constructed in (v) shows that for everymultisection M of degree 3 and any element g in its domain M there exists arestriction N to some subset containing g such that A ( N ) ≤ A . By compact-ness of the domain of M there exists a finite family {N k } of multisections ofdegree 3 such that their domains are a finite cover of the domain of M and A ( N k ) ≤ A . Proposition 3.3.7 then implies A ( G ) = A ( G ) = A .94 .4 In terms of non-commutative Stone duality In [Law16] and [Law17] Lawson undertook a translation of Matui’s isomorphismtheorems on ´etale groupoids under non-commutative Stone duality to the world ofinverse semigroups and gave a purely algebraic proof. Subsection 3.4.1 contains re-fined versions of non-commutative Stone duality which include the groupoids Matuiconsidered and Subsection 3.4.2 deals with the isomorphism theorem. All proofswill be omitted and we only give a suggestive description of the correspondences.
The sufficient duality is in vein of Stone’s classical duality between the category ofBoolean algebras and the category of Stone spaces. Definition 3.4.1 ([Law16]) . (i) An ´etale groupoid G is called a Boolean if itssubspace of units G (0) is a Stone space.(ii) An inverse monoid S is called Boolean if its idempotents E ( S ) form a Booleanalgebra with the induced order. Theorem 3.4.2 ([Law16], Theorem 3.4) . There is a duality of categories betweenthe category of Boolean inverse ∧ -monoids is dually equivalent to the category ofHausdorff Boolean groupoids. Remark 3.4.3.
Since a proper filter of a Boolean inverse ∧ -monoid is prime ifand only if it is an ultrafilter ([Law16], Lemma 3.2), on the level of objects inthe passing from a Boolean inverse ∧ -monoid S to a Boolean groupoid G( S ) thecompletely prime filters can be replaced by ultrafilters and the obtained groupoid istopologized by taking the sets V s as a basis. In the other direction one associates toa Hausdorff Boolean groupoid G the inverse monoid of compact, open slices B o,k G .We do not dwell upon what the respective correct definitions for morphisms are.Under this duality the countable Boolean inverse ∧ -monoids translate to secondcountable Hausdorff Boolean groupoids. Definition 3.4.4 ([Law16], p. 6) . Let S be a Boolean inverse monoid.(i) An ideal I of S is called ∨ -closed , if for all a, b ∈ I existence of a ∨ b implies a ∨ b ∈ I .(ii) It is said to be 0 -simplifying , if it contains no non-trivial ∨ -closed ideals.The ∨ -closed ideals in a Boolean inverse ∧ -monoid S correspond to unions of G( S )-orbits (see [Law12], § Note that as in the “commutative” setting the discovery of the refined version predates andmotivates the general, frame theoretic version. .4. IN TERMS OF NON-COMMUTATIVE STONE DUALITY Theorem 3.4.5 ([Law16], Corollary 4.8) . Let S be a Boolean inverse ∧ -monoid.The groupoid of ultrafilters G( S ) is minimal if and only if S is -simplifying. In Lemma 3.1.7 we saw that for ´etale groupoids minimality corresponds to theexistence of a certain family of slices which translates as follows:
Lemma 3.4.6.
Let S be a Boolean inverse monoid and let e, f ∈ E ( S ) with e, f = 0 .If S is -simplifying, there exists a finite set { s , . . . , s k } ⊂ S such that e = W ki =1 d ( s i ) and r ( s i ) ≤ f for all i ∈ { , . . . , k } . The algebraic characterization of effectiveness is established through the followingcorrespondence:
Lemma 3.4.7 ([Law16], Lemma 4.9) . Let S be a Boolean inverse ∧ -monoid and G( S ) be the associated groupoid of ultrafilters. For every s ∈ S the following areequivalent:(i) a ∈ Z ( E ( S )) (ii) V a ⊆ Iso(G( S ))The following definition is a specialized version of a notion from classical inversesemigroup theory (see [Law98]): Definition 3.4.8 ([Law98], p.139f) . Let S be an inverse semigroup.(i) The centralizer Z ( E ( S )) of idempotents in S is the set of elements s ∈ S suchthat se = es for all e ∈ E ( S ).(ii) It is said to be fundamental , if E ( S ) = Z ( E ( S )) holds.The following follows from the characterization in Lemma 3.4.7: Theorem 3.4.9 ([Law16], Theorem 4.10) . Let S be a Boolean inverse ∧ -monoid.The groupoid of ultrafilters G( S ) is effective if and only if S is fundamental. We thus have a class of Boolean inverse ∧ -monoids for which one can state thealgebraic counterpart to the isomorphism theorem: Definition 3.4.10 ([Law17], p.381f) . (i) A Boolean inverse monoid is said to be simple if it is 0-simplifying and fundamental.(ii) A Tarski monoid S is a countable Boolean inverse ∧ -monoid S of which thesemilattice of idempotents E ( S ) is a countable, atomless Boolean algebra. See Remark 1.1.15(ii). .4.2 TRANSLATING THE RECONSTRUCTION The effective, minimal, Hausdorff, ´etale Cantor groupoids considered by Matui thuscorrespond under non-commutative Stone duality to simple Tarski monoids. Asmentioned in Remark 3.1.6 (ii) the topological full group of an ´etale Cantor groupoid G is by definition the unit group of the inverse monoid of compact open slices andthus Theorem 3.2.9 translates under non-commutative Stone duality to: Theorem 3.4.11 ([Law17], Theorem 2.10) . Let S and S be simple Tarski monoids.Then the following are equivalent:(i) S ∼ = S (ii) U ( S ) ∼ = U ( S ) Definition 3.4.12 ([Law17], p.382) . Let S be a Boolean inverse monoid with asso-ciated Boolean algebra of idempotents E ( S ).(i) Denote by X ( S ) the Stone space dual to E ( S ).(ii) Let e ∈ E ( S ). Denote by U e the set of ultrafilters of E ( S ) that contain e .The elements of X ( S ) are given by ultrafilters of E ( S ) and the open sets of X ( S )are of the form U e for some e ∈ E ( S ). The group of units U ( S ) acts on E ( S ) by e geg − . The following is no surprise considering Theorem 3.4.9: Proposition 3.4.13 ([Law17], Proposition 3.1) . Let S be a Boolean inverse monoid.Then S is fundamental if and only if the action of U ( S ) on E ( S ) is faithful. The algebraic proof of Theorem 3.4.11 is a consequence of the following proposition:
Proposition 3.4.14 ([Law17], Lemma 3.10) . Let S and S be fundamental Booleaninverse monoids and let G ≤ U ( S ) and G ≤ U ( S ) be subgroups such that S i =( G ↓ i ) ∨ for i ∈ { , } . Assume there exists a group isomorphism α : G → G andan isomorphism of Boolean algebras γ : E ( S ) → E ( S ) such that for all g ∈ G and e ∈ E ( S ) the following is satisfied γ ( geg − ) = α ( g ) γ ( e ) α ( g − ) . Then there exists a unique isomorphism
Θ : S → S that extends α and γ . In light of the requirement S i = ( G ↓ i ) ∨ Lawson introduces the notion of piecewisefactorizability: This is the closure of G ↑ i with respect to the join operation. .4. IN TERMS OF NON-COMMUTATIVE STONE DUALITY Definition 3.4.15 ([Law17], p.391) . Let S be a Boolean inverse monoid and let G ≤ U ( S ). Then S is said to be piecewise factorizable with respect to G if for every s ∈ S there exist finite collections { s , . . . , s k } ⊂ S and { g , . . . , g k } ⊂ G with s i ≤ g i for all i ∈ { , . . . , k } such that s is of the form s = W ki =1 g i d ( s i ).A crucial ingredient in the proofs of spatial realization modeled after the proofof Theorem 384D in [Fre11] e.g. Matui’s proof, is an “abundance of involutions ”– see Definition 3.2.1. Lawson demonstrates that infinitesimals in S give rise toinvolutions: Definition 3.4.16 ([Law17], p.383f) . Let S be an inverse monoid.(i) An element s ∈ S is called an infinitesimal if s = 0 and s = 0.(ii) A pair ( s, t ) of infinitesimals s, t ∈ S is called a 2 -infinitesimal of S if d ( s ) = r ( t ) and st is an infinitesimal.An element s of an inverse monoid S satisfies s = 0 if and only if r ( s ) d ( s ) = 0. Lemma 3.4.17 ([Law17], Lemma 2.5 & 2.7) . Let S be a Boolean inverse monoid.(i) Let s ∈ S be an infinitesimal. Then g := s ∨ s − ∨¬ ( d ( s ) ∨ r ( s )) is an involutionin U ( S ) with s ≤ g . (ii) Let ( s, t ) be a -infinitesimal of S . Then the element g := s ∨ t ∨ ( st ) − ∨ ¬ ( r ( s ) ∨ d ( s ) ∨ d ( t )) is an element of U ( S ) of order . Definition 3.4.18 ([Law17], p.383f) . Let S be a Boolean inverse monoid.(i) Every involution g ∈ U ( S ) that is constructed as in Lemma 3.4.17(i) is called a special involution and the subgroup of U ( S ) generated by the set of all specialinvolutions is denoted by S ( S ).(ii) Every g ∈ U ( S ) of order 3 that is constructed as in Lemma 3.4.17(ii) is calleda special -cycle and the subgroup of U ( S ) generated by the set of all special3-cycles is denoted by A ( S ).In the groupoid picture for a Tarski monoid the above construction of special involu-tions resp. special 3-cycles corresponds precisely to the construction of involutionsresp. 3-cycles from multisections as in Definition 3.3.3 and thus this groups areprecisely the groups S ( G ) and A ( G ) defined by Nekrashevych in the Tarski monoidpicture. The symbol ¬ is the negation for the Boolean algebra E ( S ). .4.2 TRANSLATING THE RECONSTRUCTION Proposition 3.4.19 ([Law17], § . Let S be a simple Tarski monoid. Then S ispiecewise factorizable with respect to S ( S ) and in consequence S = ( S ( S ) ↓ ) ∨ . Another important aspect in the proof of the isomorphism theorem in the groupoidsetting are the supports of partial homeomorphisms. The inverse monoid counter-part comes in the following shape:
Definition 3.4.20 ([Law17], p.393) . Let S be a Boolean inverse ∧ -monoid. The support operator σ : S → E ( S ) is given by σ : s
7→ ¬ ( s ∧ s − s .This definition indeed corresponds to the support in the topological sense in that byProposition 3.16 of [Law17] we have for every g ∈ U ( S ) in a fundamental Booleaninverse ∧ -monoid S U σ ( g ) = { F ∈ X ( S ) | gF g − = F } . This allows furthermore to give a translation of local subgroups of U ( S ): Definition 3.4.21 ([Law17], p.396) . Let S be a Tarski monoid. Let G ≤ U ( S ) suchthat G contains S ( S ). Let e ∈ E ( S ). The local subgroup G e is given by G e = { g ∈ G | σ ( g ) ≤ e } Analogous to the groupoid setting the order on E ( S ) corresponds precisely to thecontainment of local subgroups. In [Mat15] Matui introduces groups of class Fby a set of axioms, demonstrates that ismorphisms of such groups induce spatialhomeomorphisms and shows that subgroups of T ( G ) that contain T ( G ) have class F(– see Subsection 3.2.1). The approach in [Law17] is a little different in that class Fis translated to a list of axioms on Booleam inverse ∧ -monoids (and the associatedgroups of special involutions resp. 3-cycles). Definition 3.4.22 ([Law17], § . Let S be an Boolean inverse ∧ -monoid. It issaid to be of class F if it satisfies the following properties:(F1) For every e ∈ E ( S ) with e = 0 there exists a finite set of special involutions τ , . . . , τ k ∈ S ( S ) such that e = W ki =1 σ ( τ i ).(F2) For every involution g ∈ U ( S ) and e ∈ E ( S ) with e = 0 and e ≤ σ ( g )there exists a special involution τ ∈ S ( S ) such that σ ( τ ) ≤ d ( ge ) ∨ r ( ge ) and σ ( τ ) ≤ ∧ gτ .(F3) For every e ∈ E ( S ) with e = 0 there exists a special 3-cycle c such that σ ( c ) ≤ e .Considering Lemma 3.4.17 the existence of special involutions resp. 3-cycles followsfrom the existence of infinitesimals resp. 2-infinitesimals – this is guaranteed by theproperty of being 0-simplifying: See Definition 3.1.50(ii). .4. IN TERMS OF NON-COMMUTATIVE STONE DUALITY Lemma 3.4.23 ([Law17], Lemma 3.5) . Let S be a -simplifying Tarski monoid. Let F ⊆ E ( S ) be an ultrafilter and let e ∈ F . Then there exists an infinitesimal s ∈ S with d ( s ) ∈ F and s ∈ eSe . Remark 3.4.24.
As for Lemma 3.4.23: Note that submonoids of the form eSe correspond to restriction groupoids. Analogous to the construction of an involutionin T ( G ) following Lemma 3.1.7 the existence of a sufficient infinitesimal followshere from Lemma 3.4.6. Furthermore Lemma 3.4.23 induces the existence of 2-infinitesimals.With the help of the existence of sufficent infinitesimals and the construction ofinvolutions from infintesimals resp. of 3-cycles from 2-infinitesimals Lawson shows: Proposition 3.4.25 ([Law17], Proposition 3.20) . Simple Tarski monoids are ofclass F.
Let S be a simple Tarski monoid and let G ≤ U ( S ) such that G contains S ( S ).Lawson shows that local subgroups of the form G σ ( τ ) where τ is an involution in G can be described algebraically analogous to Definition 3.2.3 and Lemma 3.2.4([Law17], Theorem 3.23). Furthermore analogous to Lemma 3.2.5 the followingholds: Lemma 3.4.26 ([Law17], Lemma 3.24) . Let S and S be simple Tarski monoidsand for i ∈ { , } let G i ≤ U ( S i ) be such that S ( S i ) ≤ G i . Let α : G → G be anisomorphism and let s, t ∈ G be involutions. Then the following hold:(i) σ ( s ) ≤ σ ( t ) ⇔ σ ( α ( s )) ≤ σ ( α ( t )) (ii) σ ( s ) σ ( t ) = 0 ⇔ σ ( α ( s )) σ ( α ( t )) = 0The analogue of Theorem 3.2.6 is given by: Theorem 3.4.27 ([Law17], Proposition 3.6) . Let S , S be simple Tarski monoidsand for i ∈ { , } let G i ≤ U ( S i ) be such that S ( S i ) ≤ G i . Let α : G → G bean isomorphism. Then there exists a homeomorphism ϕ : X ( S ) → X ( S ) such that ϕ ( gF g − ) = α ( g ) ϕ ( F ) α ( g − ) . The function ϕ is given by ϕ ( F ) := ( { σ ( τ ) | τ ∈ G, τ = 1 , σ ( t ) ∈ F } ) ↑ . The home-omorphism ϕ of Stone spaces corresponds to an isomorphism of Boolean algebras γ : E ( S ) → E ( S ) with the following properties:(i) γ ( σ ( t )) = σ ( α ( t )) for all involutions in G (ii) γ ( geg − ) = α ( g ) γ ( e ) α ( g − ) for all g ∈ G and e ∈ E ( S )Putting together Proposition 3.4.14, Proposition 3.4.19 and Theorem 3.4.27 con-cludes the proof of Theorem 3.4.11. It needs thus to be verified that ϕ ( F ) is an ultrafilter of E ( S ), that the arising map is indeeda homeomorphism and that ϕ satisfies the required property. .5 Significance for geometric group theory We have already talked about combinatorial aspects and growth of topological fullgroups and their finitely generated subgroups in preceding subsections, however,it is now that we push forward to the aspects that constitute the peculiarity oftopological full groups in the context of group theory. Subsection 3.5.1 starts withapproximation properties as corollaries of the factorization result by Grigorchukand Medynets reviewed in Subsection 3.1.2. In Subsection 3.5.2 we discuss theamenability of T ( ϕ ) for minimal subshifts. In Subsection 3.5.3 we take a quick lookat [Mat14a] where the existence of topological full groups with the Liouville propertywas demonstrated. We conclude the section with a Subsection 3.5.4 on topologicalfull groups as a supply of new examples of groups with intermediate growth. Some of the main corollaries of Theorem 3.1.20 (Theorem 4.7 of [GM14]) have notyet been mentioned: Topological full groups of minimal Cantor systems are LEF-groups and cannot be finitely presented. Theorem 3.5.1 ([GM14], Theorem 5.1) . Let ( X, ϕ ) be a minimal Cantor system.Then T ( ϕ ) is a LEF-group.Proof. Let F ⊂ T ( ϕ ) be a finite subset. There exists an n ∈ N , such that thefollowing properties hold:(i) For every n ≥ n and every pair of elements γ , γ ∈ ( F ∪ { id } ) with γ = γ the decomposition satisfies P γ = P γ .(ii) There exists an m ∈ N such that for every γ ∈ F and n ≥ n the supportiveset of the n -rotation R γ is contained in { , . . . , m } .(iii) P − γ,i ( k ) = P − γ,j ( k ) for all γ ∈ F , n ≥ n , k ∈ {− m, . . . , , . . . , m } and i, j ∈{ , . . . , i n } .Property (i) follows from Theorem 3.1.20(iii), property (ii) follows from property(a) in Theorem 3.1.20(i) and property (iii) follows from property (c) in Theo-rem 3.1.20(i). Fix an n ≥ n . Then by (i)-(iii) the homeomorphism P − γ R γ P γ is an n -rotation for every γ , γ ∈ F . Consider the finite group P ( A n ) of n -permutations.The map α : ( F ∪ { id } ) → P ( A n ) defined by α : γ P γ is well-defined by unique-ness of the decomposition (see Theorem 3.1.20(ii)). It is injective by property (i).Since P − γ R γ P γ is an n -rotation for every γ , γ ∈ F , we have: α ( γ γ ) = α ( P γ R γ P γ R γ ) = α ( P γ ( P γ P − γ ) R γ P γ R γ ) = α (( P γ P γ )( P − γ R γ P γ R γ )) = P γ P γ See Appendix A.5. .5. SIGNIFICANCE FOR GEOMETRIC GROUP THEORY and thus α is multiplicative on F . Let ˜ α be the extension of α to all of T ( ϕ ) definedby ˜ α ( γ ) = id for γ ∈ T ( ϕ ) \ ( F ∪ { id } ) . Then ˜ α is the required map for theLEF-property.The LEF-property entails some immediate consequences : Corollary 3.5.2 ([GM14], Corollary 2.7) . Let ( X, ϕ ) be a minimal Cantor system.Then its topological full group T ( ϕ ) is(i) an LEA-group i.e. locally embeddable into the class of amenable groups.(ii) sofic. In Theorem 3.2.15, T ( ϕ ) of a minimal Cantor system is simple. By Corollary A.5.5there exists no infinite, finitely presented, simple LEF-group thus Theorem 3.5.1implies: Corollary 3.5.3.
Let ( X, ϕ ) be a minimal Cantor system. Then T ( ϕ ) is neverfinitely presented. In the first version of [GM14] it was conjectured that topological full groups of min-imal Cantor systems are amenable. This was confirmed by Juschenko and Monodin [JM13]. Together with the results of Matui this produces an uncountable fam-ily of pairwise non-isomorphic, finitely generated, infinite, simple, amenable groups.Examples of such groups were previously unknown!Amenability of T ( ϕ ) is obtained via an embedding into the wobbling group W ( Z ): Definition 3.5.4.
Let (
X, d ) be a metric space. The wobbling group W ( X ) is thegroup of all bijections f of X such that sup { d ( x, f ( x )) | x ∈ X } < ∞ holds.While the wobbling group W ( Z ) itself is not amenable, it allows for the followingamenability criterion: Theorem 3.5.5.
Let G be a group that admits a morphism ι : G → W ( Z ) . If thestabilizer subgroup G N of the action induced by ι and the natural action of W ( Z ) isamenable, then the group G is amenable. See [CSC10], Chapter 7 for terms not defined here. For the definition of amenability of groups and their actions and characterizations see Ap-pendix A.4. One can show there is a copy of Z / Z ∗ Z / Z ∗ Z / Z in W ( Z ). .5.2 AMENABILITY OF TOPOLOGICAL FULL GROUPS The demonstration of Theorem 3.5.5 requires some set-up: Let X be a set. Denote by P f ( X ) the set of finite subsets of X . Note, that P f ( X ) is a discrete abelian groupwith the symmetric difference ∆ as the operation. Note that P f ( X ) is naturallyisomorphic to L X Z / Z and is subgroup of the group of all subsets of X with ∆as operation, which is naturally isomorphic to ( Z / Z ) X . The dual group (cid:92) P f ( X )of characters of P f ( X ) is isomorphic to ( Z / Z ) X and the canonical duality pairing (cid:92) P f ( X ) × P f ( X ) → C computes for z = { z x } x ∈ X ∈ ( Z / Z ) X and A ∈ P f ( X ) as h z, A i = exp(i π X x ∈ A z x )The normalized Haar measure λ on (cid:92) P f ( X ) corresponds to the symmetric Bernoullimeasure on ( Z / Z ) X and gives rise to the Hilbert space L ( (cid:92) P f ( X ) , λ ). The action ofa group G on a set X naturally induces an action of G on L X Z / Z by permutationof summands producing the semi-direct product P f ( X ) (cid:111) G ∼ = ( Z / Z ) o X G which istermed the permutational wreath product or lamplighter group of Z / Z and G . Thepermutational wreath product admits action on P f ( X ) by ( A, g ) · B := g ( A )∆ B forall A, B ∈ P f ( X ) and g ∈ G . The proof of Theorem 3.5.5 requires two fundamentalingredients, Theorem A.4.12 and the fact that the action P f ( Z ) (cid:111) W ( Z ) (cid:121) P f ( Z )is amenable. This fact is the technical core of [JM13] and motivated the followingdefinition: Definition 3.5.6 ([JMMdlS18], Definition 1.1) . Let G be a group that acts on aset X via α : G (cid:121) X . The action is called extensively amenable if there exists a G -invariant mean m on P f ( X ) such that µ ( { A ∈ P f ( X ) | B ⊆ A } ) = 1 for every B ∈ P f ( X ) . This definition is justified by the following lemma:
Lemma 3.5.7 ([JNdlS16], Lemma 2.7) . Let G be a group that acts transitively ona set X and let y ∈ X . Then the following are equivalent:(i) The action of G on X is extensively amenable.(ii) There exists a net of unit vectors { f n } of in L ( (cid:92) P f ( X ) , λ ) such that k g · f n − f n k → and k f n | {{ z x } x ∈ X | z y =0 } k → .(iii) The action of P f ( X ) (cid:111) G on P f ( X ) is amenable. By definition ∅ ∈ P f ( X ). This is a wide generalization of the lamplighter group L from Definition 3.2.29. Its name comes from the fact that it is stable under a specific notion of extension of actions([JMMdlS18], Proposition 2.4 & Corollary 2.5). .5. SIGNIFICANCE FOR GEOMETRIC GROUP THEORY
The most intricate step in the proof of amenability for T ( ϕ ) from [JM13] is thefollowing result: Theorem 3.5.8 ([JM13], Theorem 2.1) . The action of the wobbling group W ( Z ) on Z satisfies condition (ii) from Lemma 3.5.7. Extensive amenability of group actions is a property considerably stronger thanamenability of group actions:
Lemma 3.5.9 ([JMMdlS18], Lemma 2.1) . Let G be a group that acts on a set X .If G is amenable, the action is extensively amenable and if the action is extensivelyamenable and X is non-empty the action is amenable.Sketch of Proof. The action of a group G on a set X induces an action of G on the set M of means on P f ( X ) such that µ ( { A ∈ P f ( X ) | B ⊆ A } ) = 1 for every B ∈ P f ( X ).The set of M can be shown to be a non-empty, compact, convex subset of ‘ ∞ ( X ) ∗ .Since G is amenable, the action of G on M is amenable i.e. it has a fixed point µ , which guarantees extensive amenability. For the second statement, by extensiveamenability there exists a G -invariant mean ˜ µ on P f ( X ) \ ∅ producing a G -invariantmean on X as follows: f ∈ ‘ ∞ ( X ) Z P f ( X ) \∅ | A | − X x ∈ A f ( x ) d˜ µ ( A ) . Extensive amenability passes over to actions of subgroups on subgroup orbits:
Lemma 3.5.10.
Let a group G act extensively amenable on a set X . Then for everysubgroup H ≤ G and every H -orbit Y the induced action is extensively amenable.Proof. By Lemma 3.5.7 there exists an P f ( X ) (cid:111) G -invariant mean m on P f ( X ).Let H be a subgroup and Y an H -orbit. The map α : P f ( X ) → P f ( Y ) given by A A ∩ Y is a P f ( Y ) (cid:111) H -equivariant map, since for all A ∈ P f ( X ) , h ∈ HB ∈ P f ( Y ) we have( B, h ) · ( A ∩ Y ) = B ∆ h ( A ∩ Y ) = ( B ∩ Y )∆( h ( A ) ∩ Y ) = (( B, h ) · A ) ∩ Y. The pushforward α ∗ m is then a P f ( Y ) (cid:111) H -invariant mean on P f ( Y ).Ideas of [JM13] where developed further in [JNdlS16], in particular a sufficient con-dition for condition (ii) from Lemma 3.5.7 was given in terms of stochastics onSchreier graphs : Definition 3.5.11 ([JNdlS16], § . Let a group G with finite, symmetric gener-ating set act transitively on a set X . A function f ∈ L ( (cid:92) P f ( X ) , λ ) is said to be a product of independent variables or p.i.r.-function if there exists a family { f x } x ∈ X offunctions in R ( Z / Z ) such that f ( z ) = Q x ∈ X f x ( z x ) for every z = { z x } x ∈ X ∈ ( Z / Z ) X . See Appendix A.3. .5.2 AMENABILITY OF TOPOLOGICAL FULL GROUPS
Theorem 3.5.12 ([JNdlS16], Theorem 2.8) . Let a group G with finite, symmetricgenerating set act transitively on a set X and let y ∈ X . Then the following areequivalent:(i) There exists a sequence of p.i.r.-functions { f n } ∈ L ( (cid:92) P f ( X ) , λ ) satisfying k g · f n − f n k → and k f n | {{ z x } x ∈ X | z y =0 } k → .(ii) The Schreier graph Γ( X, G, S ) is recurrent. Assuming Theorem 3.5.8 we take a quick look at a proof of Theorem 3.5.5 followingthe path of [dC13], § Proof. (Theorem 3.5.5) First we note that we may assume G ≤ W ( Z ): Sinceker( ι ) ≤ G N is amenable by assumption and the class of amenable groups is sta-ble under extensions and quotients, the group G is amenable if and only if ι ( G )is amenable. The first step of the proof is to show that G A ∆ N is amenable for all A ∈ P f ( Z ). The stabilizer G k + N is amenable for all k ∈ Z by the following argument:Assume G k + N is not amenable for some k ∈ Z . Since the action of W ( Z ) on Z isextensively amenable, the transitive action of G k + N on the orbit G k + N · j is exten-sively amenable for all j ∈ Z by Lemma 3.5.10 and thus amenable by Lemma 3.5.7.By Theorem A.4.12 thus ( G k + N ) i is non-amenable for some and hence all i ∈ Z .But since ( G k + N ) k ≤ G k +1+ N and ( G k + N ) k − ≤ G k − N , the containing groups arenon-amenable. By iteration of this argument G j + N is non-amenable for all j ∈ Z ,which is a contradiction to the assumption that G N is amenable. Let A ∈ P f ( Z ).Then there exists a unique k ∈ Z such that A ∆ N = σ ( k + N ) for a finitely supportedpermutation σ ∈ S ( Z ). Since G A ∆ N = σG k + N σ − , the stabilizer G A ∆ N is amenable.Let Φ : G → P f ( Z ) (cid:111) W ( Z ) be the embedding given by g ( g ( N )∆ N , g ). Theaction of G on P f ( Z ) induced by Φ is amenable, since the action of P f ( Z ) (cid:111) W ( Z ) isamenable by Theorem 3.5.8. Since for all A ∈ P f ( Z ) the point stabilizer G A of theaction induced by Φ is precisely the stabilizer G A ∆ N of the wobbling group action –which has been shown to be amenable for all A ∈ P f ( Z ) – thus G must be amenableby Theorem A.4.12. Corollary 3.5.13.
Let ( X, ϕ ) be a minimal subshift. Then T ( ϕ ) is amenable.Proof. Let (
X, ϕ ) be a minimal Cantor system. The group T ( ϕ ) injects into W ( Z )by its action on orbits i.e. since cocycles of elements in T ( ϕ ) are bounded, everypoint x ∈ X induces a map ι x : T ( ϕ ) → W ( Z ) defined by ι x ( γ )( k ) = f γ ( ϕ k ( x )) + k This map is a group homomorphism as for γ , γ ∈ T ( ϕ ) by Lemma 3.1.4 we have: ι x ( γ γ )( k ) = f γ γ ( ϕ k ( x )) + k = f γ ( γ ϕ k ( x )) + f γ ( ϕ k ( x )) + k == f γ ( ϕ f γ ( ϕ k ( x ))+ k ( x )) + ι x ( γ ) = ι x ( γ ) ι x ( γ )( k ) This is well-defined since g ∈ W ( Z ) implies g ( N )∆ N ∈ P f ( Z ). .5. SIGNIFICANCE FOR GEOMETRIC GROUP THEORY By minimality every γ ∈ T ( ϕ ) is uniquely determined by f γ | Orb ϕ ( x ) , hence this ho-momorphism is injective. This implies the group ι x ( T ( ϕ ) x ) = ι x ( T ( ϕ )) N is amenableby Corollary 3.1.28(ii), thus Theorem 3.5.5 applies. Remark 3.5.14.
By Corollary 2.4 in [Cho80] every finitely generated, simple, el-ementary amenable group is finite, thus by Theorem 3.2.17 and Theorem 3.2.15the topological full group of a minimal subshift is an amenable, non-elementaryamenable group.Putting together Proposition 3.1.60, and Corollary 3.5.13 implies:
Corollary 3.5.15.
There exists an uncountable family of pairwise non-isomorphic,infinite, finitely generated, simple, amenable groups.
Remark 3.5.16. (i) Corollary 3.5.13 does not generalize to minimal Z d -actionson a Cantor space. In [EM13] an example of a Z -action was given, where T ( Z ) contains a copy of the free group on two generators.(ii) In [Mat15], Proposition 4.10 it is demonstrated, that the topological full group T ( G ) of a Hausdorff, effective, ´etale Cantor groupoid G in which G (0) is properlyinfinite contains a subgroup isomorphic to the free product ( Z / Z ) ∗ ( Z / Z )and, hence, is not amenable.Extensive amenable actions G (cid:121) X were further studied (still implicitly) for moregeneral X in [JdlS15]. In [JNdlS16] a powerful amenability criterion was developedencompassing Theorem 3.5.5 which applies to a diverse class of amenable, non-elementary amenable groups: Theorem 3.5.17 ([JNdlS16], Theorem 1.1) . Let X be a topological space. Let G ≤ Homeo( X ) be finitely generated and let S be a generating set. Let H be asubgroupoid of the groupoid of germs Germ( X, Homeo( X )) such that the followingconditions hold:(i) For every g ∈ G the set of x ∈ X such that the germ [( g, x )] / ∈ H is finite.(ii) For every s ∈ S the set x ∈ X such that [( s, x )] = H is finite.(iii) The action G (cid:121) X is extensively amenable.(iv) For every x ∈ Germ(
X, G ) (0) the isotropy group Germ(
G, X ) | x is amenable.(v) The group T ( H ) is amenable.Then the group G is amenable. Extensive amenability was explicitely introduced and studied in [JMMdlS18]. Inparticular it was shown that extensive amenability of an action G (cid:121) X inducesamenability of the action F ( X ) (cid:111) G (cid:121) F ( X ) i.e. P f ( − ) is replaced by a functor F of certain type.106 .5.3 LIOUVILLE PROPERTY AND QUANTIFYING AMENABILITY The findings of [JM13] required further analysis of topological full groups of minimalsubshifts. Both papers [Mat14a] and [Mat14b] by Nicol´as Matte Bon are concernedwith topological full groups. In [Mat14a] Matte Bon obtains an upper bound forrandom walk entropy of shifts with bounded complexity: Theorem 3.5.18 ([Mat14a], Theorem 1.2) . Let ( X, ϕ ) be a subshift with no isolatedperiodic points of which the complexity ρ satisfies lim n →∞ log nn ! ρ ( n ) = 0 Then for every finitely supported symmetric probability measure µ on T ( ϕ ) thereexists a C > such that for every n ≥ the following holds: H ( µ ∗ n ) ≤ Cρ ( d C q n log n e ) log n This estimate implies in particular that T ( ϕ ) has zero random walk entropy thusimplying by the characterization in Theorem A.6.4: Corollary 3.5.19.
Let ( X, ϕ ) be a subshift as chosen in Theorem 3.5.18. Then T ( ϕ ) has the Liouville property. Subshifts that satisfy the criteria of Theorem 3.5.18 do exist – see Example 1.1.21(i).Since the Lioville property implies amenability, this produces a proof of amenabilityfor such shifts independent of [JM13]. It is of note that for this proof minimality ofthe subshift is not required. Combined with the isomorphism theorems this implies:
Theorem 3.5.20 ([Mat14a], Theorem 1.1) . There exists an uncountable family ofpairwise non-isomorphic, infinite, finitely generated, simple groups with the Liouvilleproperty.
The upper bound of Theorem 3.5.18 produces a lower bound for µ ∗ n ( e ) i.e thereturn probability of the random walk, and in consequence upper bounds of Følnerfunctions of finitely generated subgroups of T ( ϕ ): Corollary 3.5.21 ([Mat14a], Corollary 1.7) . Let ( X, ϕ ) be a subshift as chosen inTheorem 3.5.18.(i) For every finitely supported symmetric probability measure µ on T ( ϕ ) thereexists a C > such that for every n ≥ the following holds: µ ∗ n ( e ) ≥ C exp( − Cρ ( d C q n log n e ) log n ) For the definition of ρ see Definition 1.1.19, for the definition of the entropy H see Defini-tion A.6.3(ii). .5. SIGNIFICANCE FOR GEOMETRIC GROUP THEORY (ii) If in addition there exists a C > and a α ∈ [0 , ) such that ρ ( n ) ≤ C n α , then for every finitely generated subgroup G ≤ T ( ϕ ) , every symmetric finitegenerating set S of G and for every ε > , there exists a C > such that forevery n ≥ the following holds: Fol
G,S ( n ) ≤ C exp( Cn α − α + ε ) In 1980 the general Burnside problem had been resolved long ago as a consequence ofthe Golod-Shafarevych theorem. The mathematical world was still looking for simpleconstructions of infinite, periodic, finitely generated group. A striking example wasgiven in [Gri80] by Grigorchuk as a group of Lebesgue measure preserving intervaltransformations now called the (first) Grigorchuk group . Today a representationin terms of automorphisms of an infinite, binary, rooted tree is more common. In[Gri83] Grigorchuk demonstrated that his group has intermediate growth resolving alongstanding question of John Milnor. As such it provided the first example of a non-elementary amenable group and initiated the study of branch groups and automatagroups. Over the years more examples of groups with intermediate growth havebeen found in particular non-residually finite such groups. However, for a long timethere were no examples of a simple, infinite, finitely generated group of intermediategrowth.In [Mat14b] Matte Bon shows that Grigorchuk groups embed into topological fullgroups of shifts, which by [JM13] produces a new proof of amenability of such groups. Proposition 3.5.22 ([Mat14b], Proposition 3.1) . Let ( X, ϕ ) be a minimal Cantorsystem. Then for every ω ∈ { , , } N + which is eventually constant the Grigorchukgroup G ω embedds into T ( ϕ ) . This is an immediate consequence of Proposition 3.1.16 and Proposition A.2.2. Ofmuch more interest is the question for the counterparts with intermediate growth:
Theorem 3.5.23 ([Mat14b], Proposition 3.7) . Let ω ∈ { , , } N + such that it isnot eventually constant. There exists a minimal subshift ( X ω , ϕ ω ) such that theGrigorchuk group G ω embedds into T ( ϕ ω ) . We do not dwell on the nature of the associated shift and the embedding for now, butimmediatly turn to more general ideas by Nekrashevych: In [Nek18] Nekrashevychshowed that topological full groups of fragmentations of dihedral group actions pro-duce examples of infinite, finitely generated, simple, periodic groups of intermediategrowth. In this subsection we take a quick look at [Nek18]. We dispense with proofs This is satisfied for example by Sturmian shifts. See Appendix A.2 for the definition of Grigorchuk groups. .5.4 NEW GROUPS OF BURNSIDE TYPE AND INTERMEDIATEGROWTH except for a sketch of the embedding of fragmentations of dihedral groups into topo-logical full groups of Cantor systems. For definitions of graphs associated to actionssee Appendix A.3.The minimality of an action of a finitely generated group on a compact metrizablespace enables to study the action via the study of orbital Schreier graphs. In partic-ular it entails that orbital Schreier graphs are repetitive i.e. for every finite subgraphΣ a copy can be found in bounded distance – depending on the size of Σ – of everyvertex:
Proposition 3.5.24 ([Nek18], Proposition 2.5) . Let a group G generated by a finitesymmetric set S act minimally on a compact metrizable space X by homeomor-phisms. Then for every r ∈ N , there exists an R ( r ) ∈ N such that for every G -regular point x ∈ X and every y ∈ X , there exists a vertex z ∈ Γ( y, G, S ) such that d ( z, y ) ≤ R ( r ) and such that the balls B x ( r ) ⊂ Γ( x, G, S ) and B z ( r ) ⊂ Γ( y, G, S ) are isomorphic as rooted graphs.Proof. Let r ∈ N and let x be a G -regular point. Then the ball B x ( r ) ⊂ Γ( x, G, S )is determined by the values of the function δ x : B ( r ) × B ( r ) → { , } given by δ x ( g, h ) := , for g ( x ) = h ( x ) in B x ( r )0 , elseSince x was assumed to be G -regular there exists a open neighbourhood U x of x such that for all y ∈ U x we have δ x = δ y and thus the balls B x ( r ) ⊂ Γ( x, G, S ) and B y ( r ) ⊂ Γ( y, G, S ) are isomorphic as rooted labelled graphs. By minimality the G -translates of U x form an open cover of X . By compactness there exists a finitecollection g , . . . , g n such that the family { g i ( U x ) } i ∈{ ,...,n } covers X . Let R x ( r ) :=max i ∈{ ,...,n } L G,S ( g i ). Then for every y ∈ X there exists an i ∈ { , . . . , n } such that g − i ( y ) ∈ U x . Then the balls B x ( r ) ⊂ Γ( x, G, S ) and B g − i ( x ) ( r ) ⊂ Γ( x, G, S ) are iso-morphic as rooted labelled graphs and d ( x, g − i ( x )) ≤ R x ( r ). To get rid of the depen-dency on the choice of the G -regular point x , note that the set of isomorphism classesof balls in orbital graphs of the form B x ( r ) where x is some arbitrary G -regular point x is finite. Then any complete set of representatives { B x i ( r ) } i ∈ I provides an estimateindependent of the choice of a G -regular point by R ( r ) = max i ∈ I R x i ( r ). Definition 3.5.25 ([Nek18], Definition 2.6) . Let a group G generated by a finitesymmetric set S act minimally on a compact metrizable space X by homeomor-phisms. Then the action is called linearly repetitive if there exists a K ∈ R > suchthat R ( r ) < Kr .Nekrashevych introduces a new class of groups by fragmentations of actions of di-hedral groups on Cantor spaces. This class in particular includes the Grigorchukgroups: Definition 3.5.26 ([[Nek18], § . Let h be a homeomorphism on a Cantor space X with period 2. 109 .5. SIGNIFICANCE FOR GEOMETRIC GROUP THEORY (i) A finite group G of homeomorhisms of X is called a fragmentation of h if forevery g ∈ G and x ∈ X either h ( x ) = x or h ( x ) = g ( x ) hold, and if for every x ∈ X there exists a g ∈ G such that h ( x ) = g ( x ).Let G be a fragmentation of h .(ii) Let g ∈ G . Denote by E g, the set of fixed points of g and by E g,h the set ofelements x ∈ X such that g ( x ) = h ( x ).(iii) Let h be such that its set of fixed points has empty interior. Denote by P G the family of subsets of X of the form T g ∈ G ( E g,i g ) ◦ where i g ∈ { , h } for all g ∈ G . The elements of P G are called the pieces of the fragmentation G .(iv) The infinite dihedral group D ∞ is given by the presentation D ∞ := h a, b | a , b i .(v) Let the infinite dihedral group D ∞ := h a, b | a , b i act on a Cantor set byhomeomorphisms. A fragmentation of the dihedral group h a, b i is a group G ofhomeomorphisms generated by A ∪ B where A resp. B are fragmentations of a resp. b . Example 3.5.27 ([Nek18], Example 3.3) . The first Grigorchuk group is a fragmen-tation of of a dihedral group acting on a Cantor space.By an elaborate analysis on the graphs associated to the action Nekrashevych ob-tains:
Theorem 3.5.28 ([Nek18], Theorem 4.1) . Let G be the fragmentation of a minimalaction of the dihedral group D ∞ on a Cantor space X such that there exists a purelynon-Hausdorff singularity. Then the group G is periodic. Remark 3.5.29.
By Lemma 3.2 in [Nek18] for every fixed point x of a non-free,minimal action of a dihedral group on a Cantor space, there exists a fragmentation G such that x is a purely non-Hausdorff singularity.For the refined case of linearly repetive actions Nekrashevych obtains an upperbound for the growth rate. Since any infinite, finitely generated, periodic group isnot virtually nilpotent, which by Gromov’s theorem (Theorem A.1.6) implies thatit has superpolynomial growth: Theorem 3.5.30 ([Nek18], Theorem 6.6) . Let G be the fragmentation of a minimalaction of the dihedral group D ∞ on a Cantor space X such that the orbital Schreiergraphs of G are linearly repetitive and there exists a purely non-Hausdorff singularity.Then G has intermediate growth. This assures that for all g ∈ G we have ( E g, ) ◦ ∩ ( E g,h ) ◦ = ∅ . The nilpotent finite-index subgroup would be periodic and finitely generated. Being nilpotentand periodic is equivalent to being locally finite and nilpotent. Locally finite, finitely generatedgroups are necessarily finite, which is a contradiction. .5.4 NEW GROUPS OF BURNSIDE TYPE AND INTERMEDIATEGROWTH
The following was already observed in the case of Grigorchuk groups by Matte Bonin [Mat14b]:
Proposition 3.5.31 ([Nek18], Proposition 5.1) . Every fragmentation G of a mini-mal action of the dihedral group D ∞ on a Cantor space X embeds into the topologicalfull group of a minimal subshift.Sketch of Proof. Let x ∈ X be a G -regular point. Then the orbital Schreier graphΓ( x, h a, b i ) is a two-ended infinite path. If two vertices v , v of the orbital Schreiergraph Γ( x, h a, b i ) are connected by an edge labelled by “ a ” (resp. “ b ”), then for every g ∈ A and i g ∈ { , a } (resp. g ∈ B and i g ∈ { , b } ) we have v ∈ ( E g,i g ) ◦ if andonly if v ∈ ( E g,i g ) ◦ , hence they lie in the same piece P ∈ P A (resp. P ∈ P B ). Byassociating adjacent edges in the orbital Schreier graph Γ( x, h a, b i ) with successiveintegers this gives rise to a sequence ω x ∈ ( P A ∪ P B ) Z . Define W ⊆ ( P A ∪ P B ) Z to be the subset of seqences ω ∈ ( P A ∪ P B ) Z such that every finite substring of ω is a substring of ω x . Then W is a closed shift-invariant subset of ( P A ∪ P B ) Z andProposition 3.5.24 implies that ( W , σ ) is minimal. Every g ∈ A (resp. g ∈ B )induces an element ι ( g ) ∈ T ( σ ) by defining for ω ∈ W : ι ( g )( ω ) := σ ( ω ) , if g | ω = a | ω (resp. g | ω = b | ω ) σ − ( ω ) , if g | ω − = a | ω − (resp. g | ω − = b | ω − ) ω, elseThus Corollary 3.5.13 implies: Corollary 3.5.32.
Every fragmentation G of a minimal action of the dihedral group D ∞ on a Cantor space X is amenable. In the case of expansive actions of dihedral groups the following propositions hold:
Proposition 3.5.33 ([Nek18], Proposition 5.2) . Let G be the fragmentation of anaction of the dihedral group D ∞ on a Cantor space. If the action of D ∞ is expansive,then the action of G is expansive. The following is a consequence of Theorem 3.3.14:
Proposition 3.5.34.
Let G = h A ∪ B i be the fragmentation of an expansive, min-imal action of the dihedral group D ∞ on a Cantor space and let G be its associatedtransformation groupoid. Then there exists a fragmentation ˜ G of this action suchthat A ( G ) ≤ ˜ G and the groups of germs G x /G ( x ) and ˜ G x / ˜ G ( x ) are isomorphic for all x ∈ X . They are connected in Γ( x, G ) by an edge for every element g ∈ A (resp. g ∈ B ) such that g | P = a (resp. g | P = b ). Hence orbital Schreier graphs of fragmentations of dihedral groups areone- or two-ended infinite paths “fattened up” by loops and multiple edges. .6. IRREDUCIBILITY OF KOOPMAN REPRESENTATIONS Combining the results on alternating full groups with Theorem 3.5.30, Proposi-tion 3.5.33 and Proposition 3.5.34 implies: Theorem 3.5.35 ([Nek18], Theorem 1.2) . Let G be the fragmentation of an ex-pansive, minimal action of the dihedral group D ∞ on a Cantor space such that theorbital Schreier graphs of G are linearly repetitive and there exists a purely non-Hausdorff singularity and let G be the transformation groupoid associated with theaction of G . Then the group A ( G ) is a infinite, finitely generated, simple, periodicgroup of intermediate growth. Let (
X, µ ) be a measure space such that µ is a finite measure. Let G be a locallycompact, second countable group acting ergodically and measure class preservingon ( X, µ ). The following question was put forward i.a. by Vershik:
Question ([Ver11], Problem 4) . In which of the above settings is the associatedKoopman representation irreducible? The list of actions for which this question has been answered affirmatively is quiteshort. Artem Dudko provides in [Dud18] a unified strategy of proving irreducibility for thenatural actions of Higman-Thompson groups as groups of piecewise linear homeo-morphisms and for the natural actions of weakly branch groups on tree boundaries.
Definition 3.6.1 ([Dud18], § . (i) Let ( X, µ ) be a probability space and let G be a locally compact, second countable group acting measure class preservingon ( X, µ ). This action is called measure contracting if for every measurablesubset A and M, (cid:15) > g ∈ G such that: (a) µ (supp( g ) \ A ) < (cid:15) (b) µ ( { x ∈ A | r d µ ( gx )d µ ( x ) < M − } ) > µ ( A ) − (cid:15) See Subsection 3.3.2 and Subsection 3.3.3. See Definition 1.1.10. In the measure preserving case the space of constant functions C · X provides a closed invariantsubspace thus one restricts to ( C · X ) ⊥ . For a list see [Dud18]. In this subsection let supp( g ) denote the set { x ∈ X | gx = x } . .6.1 MEASURE CONTRACTING ACTIONS (ii) Let ( X, µ ) be a measure space. Every function f ∈ L ∞ ( X, µ ) constitutesa multiplication operator T f ∈ B (L ( X, µ )) by T f ( g )( x ) = f ( x ) g ( x ) for all g ∈ L ( X, µ ). Denote by L ∞ the von Neumann algebra generated by { T f | f ∈ L ∞ ( X, µ ) } in B (L ( X, µ )).
Remark 3.6.2.
Let (
X, µ ) be a finite measure space. Then the von Neumannalgebra L ∞ defined in Definition 3.6.1(ii) is a masa in B (L ( X, µ )) – see e.g. Propo-sition III.1.5.16 of [Bla06].
Lemma 3.6.3 ([Dud18], Theorem 3) . Let G be a locally compact, second countablegroup acting ergodically and measure class preserving on a standard Borel space ( X, µ ) . Let A be the von-Neumann-algebra generated by M κ ∪ L ∞ in B (L ( X, µ )) .Then A = B (L ( X, µ )) .Proof. If we show A ⊆ C · Id, we are done, because then A = B (L ( X, µ )) andby the von Neumann bicommutant theorem A = B (L ( X, µ )). Accordingly, let A ∈ A . By Remark B.5.2(ii), A ⊆ ( L ∞ ) and A ⊆ M κ hold. By Remark 3.6.2 theabelian von-Neumann-algebra L ∞ is maximal, which is equivalent to ( L ∞ ) = L ∞ .Thus A is the multiplication operator T f associated with a function f ∈ L ∞ ( X, µ ).Since T f = A ∈ M κ , we have s d g ∗ µ d µ ( x ) f ( g − x ) h ( g − x ) = κ ( g ) T f ( h )( x ) == T f ( κ ( g ) h )( x ) = f ( x ) s d g ∗ µ d µ ( x ) h ( g − x )for all g ∈ G and h ∈ L ( X, µ ), thus f is G -invariant almost everywhere. Byergodicity this implies f must be constant almost everywhere and thus the associatedoperator satisfies T f ∈ C · Id – see [Gla03], Theorem 3.10.
Theorem 3.6.4 ([Dud18], Theorem 4) . Let ( X, µ ) be a probability space and let G be a locally compact, second countable group acting measure class preserving,ergodically and measure contracting on ( X, µ ) . Then L ∞ = M κ holds.Proof. Let A be a measurable subset of X . Then H A := { f ∈ L ( X, µ ) | supp( f ) ⊂ X \ A } is a subspace of L ( X, µ ). Denote by P A the orthogonal projection unto thissubspace. Since the action of G is measure contracting, we can find a sequence ofelements ( g Am ) m ∈ N satisfying conditions 3.6.1(i) and 3.6.1(ii) for M = m and (cid:15) = m − .Set A m = { x ∈ A | ( d µ ( g Am x )d µ ( x ) ) / < m − } , B m := supp( g am ) \ A m (3.1)By definition, A m ⊆ supp( g Am ) holds and we have: µ ( A \ A m ) = µ ( A ) − µ ( A m ) < µ ( A ) − ( µ ( A ) − m − ) = m − µ ( B m ) = µ (supp( g am ) \ A m ) ≤ µ (supp( g am ) \ A ) + µ ( A \ A m ) < m − (3.2) See Remark B.5.2(i). .6. IRREDUCIBILITY OF KOOPMAN REPRESENTATIONS
In the following we varify, that for all f , f ∈ L ( X, µ ) we have h κ ( g Am ) f , f i −→h P A f , f i for m → ∞ i.e. P A is in the weak operator closure of κ ( G ) and thus in M κ .Since L ( X, µ ) ∩ L ∞ ( X, µ ) is dense in L ( X, µ ), we may assume f , f ∈ L ∞ ( X, µ ).Let therefore k f i k ∞ = M i and let us fix the following abbrevations: I Am ( x ) := vuut d µ ( g Am x )d µ ( x ) f ( g Am x ) f ( x ) , II ( x ) := f ( x ) f ( x )Then we have: |h κ (( g Am ) − ) f , f i − h P A f , f i| = | Z X I Am ( x ) d x − Z X \ A II ( x ) d x | Note that I Am and II coincide on X \ (supp( g Am ) ∪ A ), thus this expression is equalto: | Z A m I Am ( x ) d x + Z B m I Am ( x ) d x + Z A \ supp( g Am ) II ( x ) d x − Z supp( g Am ) \ A II ( x ) d x | (3 . < m − Z A m | f ( g Am x ) f ( x ) | d x + | Z B m I Am ( x ) d x | + Z A supp( g Am ) | II ( x ) | d x ≤≤ m − M M + | Z B m I Am ( x ) d x | + Z A supp( g Am ) M M d x The first summand is obtained, since k f k ∞ = k f ◦ g Am k ∞ by measure class preser-vance and it obviously goes to 0 for m → ∞ . Since µ ( A supp( g Am )) ≤ µ ( A \ A m ) + µ ( B m ) (3 . < m − holds, the third summand can be bounded by 3 m − M M . By Cauchy-Schwartzthe second summand can be bounded by: q k κ (( g Am ) − ) f k k P X \ B m f k By unitarity of κ this is just q k f k k P X \ B m f k which tends to 0 for m → ∞ because: k P X \ B m f k ≤ Z B m | f ( x ) | d x < m − M We have established P A ∈ M κ for every measurable subset A . Since by assumption( X, µ ) is a probability space, µ is in particular finite and thus h{ A | A ⊆ X open }i is dense in L ∞ ( X, µ ) with respect to the L -Norm. This means that h{ P A | A ⊆ X open }i is dense in L ∞ with respect to the strong operator topology. This implies L ∞ = M κ . Theorem 3.6.5 ([Dud18], Theorem 4) . Let ( X, µ ) be a probability space and let G be .6.2 TOPOLOGICAL FULL GROUPS WITH IRREDUCIBLE KOOPMANREPRESENTATION a locally compact, second countable group acting measure class preserving, ergodicallyand measure contracting on ( X, µ ) . Then the associated Koopman representation κ is irreducible.Proof. Theorem 3.6.4 and Lemma 3.6.3 imply M κ = B (L ( X, µ )). But then κ ( G ) = κ ( G ) = M κ = B (L ( X, µ )) = C · Id. It follows that κ is irreducible, since anyprojection P H on a κ ( G )-invariant subspace H in B (L ( X, µ )) is contained in κ ( G ) and thus P must be either 0 or Id. We finish the main body of this text with a quick note on the irreducibility of Koop-man representations associated to the natural action of the topological full group T ( G ) of a purely infinite, minimal, effective, ´etale Cantor groupoid G on the space G (0) . Dudko applied the notion of measure contraction to show that natural actionsof Higman-Thompson groups F n,r and G n,r on ([0 , r ] , λ ) admit irreducible Koopmanrepresentations. The arguments for ergodicity and measure contraction relies onthe particular setting e.g. on the fact that the action is by piecewise linear homeo-morphisms. Contrasting to this “rigid” setting, in the groupoid setting the measurecontraction comes almost for free from the structure of the groupoid and the abun-dance of sufficient elements in T ( G ). By Proposition 2.3.11 and Proposition 2.3.12locally compact, ´etale groupoids always admit quasi-invariant, ergodic measuresand it is easy to see that such measures are quasi-invariant, ergodic measures withrespect to the natural action by the topological full group: Corollary 3.6.6.
Let G be an effective, ´etale Cantor groupoid and let µ be a quasi-invariant measure on G (0) . Then µ is quasi-invariant with respect to the naturalaction of T ( G ) on G (0) .Proof. This follows immediately from Proposition 2.4.21.
Corollary 3.6.7.
Let G be an ´etale Cantor groupoid and let µ be a quasi-invariant,ergodic probability measure on G (0) . Then the natural action of T ( G ) on G (0) isergodic with respect to µ .Proof. Let U ⊆ G (0) such that r ( BU ) = U for all B ∈ T ( G ) ⊂ B o,k G . Then U isnecessarily almost invariant and hence conull or null. Lemma 3.6.8.
Let G be an ´etale Cantor groupoid and let µ be a quasi-invariantprobability measure on G (0) . If G is minimal, the measure µ is strictly positive andhence for every clopen subset U of G (0) and every ε > there exists a clopen subset V ⊂ U with < µ ( V ) < ε . Every non-empty open subset of G (0) has positive measure. .6. IRREDUCIBILITY OF KOOPMAN REPRESENTATIONS Proof.
Assume there exists an open subset O ⊂ G (0) with µ ( O ) = 0. Let U ⊂ O be a clopen subset. By minimality and Lemma 3.1.7 there exists a finite familyof compact, open slices { B i } i ∈ I such that r ( B i ) ⊂ U , G (0) \ U = S i s ( B i ). Then S i s ( B i ) ∪ U is a finite, open cover of G (0) . Then by quasi-invariance of µ andProposition 2.4.21 this is a cover of G (0) by nullsets hence by subadditivity µ ( G (0) ) ≤ P i ∈ I µ ( S i ) + µ ( U ) = 0, which is a contradiction. The second statement follows thenimmediately from additivity. Theorem 3.6.9.
Let G be a purely infinite, minimal, ´etale Cantor groupoid and let µ be a quasi-invariant, ergodic probability measure on G (0) . Then the natural actionof T ( G ) on G (0) is measure contracting.Proof. Let U ⊆ G (0) be clopen and let M, (cid:15) >
0. By Lemma 3.6.8 for every ε > B ⊂ U such that 0 < µ ( B ) < ε . Fix the notation A := U \ B . Since G is purely infinite and minimal there exists by Proposition 3.2.23an open compact slice S with s ( S ) = A and r ( S ) ⊆ B . Denote by f S ( x ) := d µ ( T S x )d µ ( x ) Then we have µ ( { x ∈ U | q f S ( x ) ≥ M − } ) ≤ µ ( { x ∈ A | q f S ( x ) ≥ M − } ) + ε == µ ( { x ∈ A | ≤ M f S ( x ) } ) + ε = Z { x ∈ A | ≤ M f S ( x ) } d µ + ε ≤ M Z A f S ( x ) d µ + ε = M µ ( r ( SA )) + ε = M µ ( B ) + ε ≤ M ε + ε When chosing ε ≤ ε M this quantity is smaller or equal to ε , moreover supp( T S ) ⊆ U holds. Since measurable subsets of G (0) can be approximated by clopen subsetsarbitrarily well, the action is measure contracting.Combining Theorem 3.6.9 with Theorem 3.6.5 thus implies: Corollary 3.6.10.
Let G be a purely infinite, minimal, ´etale Cantor groupoid andlet µ be a quasi-invariant, ergodic probability measure on G (0) . Then Koopman rep-resentation associated to the action of T ( G ) on G (0) is irreducible. ppendicesppendix ASome terms of geometric grouptheory A.1 Growth of groups
The contents of this section can be found in § § Definition A.1.1.
Let G be a group generated by a symmetric, finite subset S = S − .(i) The word length L G,S ( g ) of g ∈ G with respect to S is the smallest integer n ∈ Z such that there exist generators s , . . . , s n ∈ S such that g = s . . . s n .By convention L G,S (1) = 0.(ii) The word metric d G,S on G with respect to S is given by d S ( g, h ) := L S ( g − h )for g, h ∈ G .(iii) The growth function γ G,S : N → [0 , + ∞ ) of G with respect to S is given by γ G,S ( n ) := |{ g ∈ G |L G,S ( g ) ≤ n }| (iv) Let f, g : N → [0 , + ∞ ) be non-decreasing functions. We write f (cid:22) g if thereexist constants C, α > f ( n ) ≤ Cg ( αn ) for all n ∈ N . We write f ∼ g if f (cid:22) g and g (cid:22) f holds. Remark A.1.2. (i) For finite, symmetric generating sets S , S of a group G wehave γ G,S ∼ γ G,S , henceforth we omit the generating set from the notation.(ii) Every finite group G satisfies γ G ∼ G satisfies n (cid:22) γ G (cid:22) e n . It is easy to see that this is an equivalence relation. efinition A.1.3.
Let G be a finitely generated group. The ∼ -equivalence class of γ G is called the growth type of G . The group G is said to have(i) polynomial growth if β G ( n ) ∼ n α for some α > superpolynomial growth if ln β G ( n )ln n −→ n →∞ ∞ .(iii) exponential growth if β G ( n ) ∼ e n .(iv) subexponential growth if ln β G ( n ) n −→ n →∞ intermediate growth if it has superpolynomial and subexponential growth. Remark A.1.4.
One has the implications ( i ) ⇒ ( iv ) and ( iii ) ⇒ ( ii ), moreover ( i )and ( ii ) as well as ( iii ) and ( iv ) are mutually exclusive. Every finitely generatedgroup either has polynomial, exponential or intermediate growth. Example A.1.5.
Every finitely generated, free abelian group Z d has polynomialgrowth, every free group F d with d ≥ Theorem A.1.6 ([dlH00], Theorem VII.29) . A finitely generated group has polyno-mial growth if and only if it is virtually nilpotent i.e. it contains a nilpotent subgroupof finite index.
A.2 Grigorchuk groups
For along time the existence of finitely generated groups with intermediate growthwas conjectural. A first example was found by Rostislav I. Grigorchuk. He in-troduced a class of groups – later termed Grigorchuk groups – indexed by infinite { , , } -sequences: Definition A.2.1 ([Mat14b], § . Let T be a infinite, labelled, binary, rootedtree, such that vertices are labelled by the set { , } ∗ by labelling the root with ∅ and the children of a vertex v are iteratively labelled by 0 v and 1 v . Let Ω be theset { , , } N + endowed with the product topology and the natural surjective map σ : Ω → Ω given by the shift. Let (cid:15) denote the transposition of 0 and 1 and for i, j ∈ { , , } let the symbol (cid:15) i,j denote the identity permutation on 0 and 1 if i = j and let (cid:15) i,j = (cid:15) if i = j . Define a tree automorphism a by a ( xv ) := (cid:15) ( x ) v for all x ∈ { , } and v ∈ { , } ∗ . Let ω ∈ Ω. Define further tree automorphisms b ω , c ω , d ω recursively by b ω (0 xv ) := 0 (cid:15) ,ω ( x ) v ; b ω (1 v ) = 1 b σ ( ω ) vb ω (0 xv ) := 0 (cid:15) ,ω ( x ) v ; c ω (1 v ) = 1 c σ ( ω ) vb ω (0 xv ) := 0 (cid:15) ,ω ( x ) v ; d ω (1 v ) = 1 d σ ( ω ) v Tree automorphisms are graph automorphism that fixes the root. .3. GRAPHS OF ACTIONS for all x ∈ { , } and v ∈ { , } ∗ . The group generated by the tree automorphisms a, b ω , c ω , d ω is called the Grigorchuk group G ω . The first Grigorchuk group is theGrigorchuk group G ω where ω = 012012012 . . . Proposition A.2.2.
Let ω be sequence in { , , } N + which is eventually constanti.e. there exists an N ∈ N such that ω n = ω n +1 for all n ≥ N . Then there exists an n ∈ N such that the group G ω embedds into Z o n S n ∼ = Z n (cid:111) S n . Remark A.2.3.
In [Gri85] Grigorchuk actually proves that G ω embedds into D ∞ o n S n for some n ∈ N , this group however admits an embedding into Z o n S n . Corollary A.2.4.
Let ω be sequence in { , , } N + which is eventually constant.Then the group G ω is virtually abelian and in consequence has polynomial growth. The far more interesting Grigorchuk groups however are provided by the rest:
Theorem A.2.5 ([Gri85], Theorem 2.1 & Corollary 3.2) . Let ω be sequence in { , , } N + which is not eventually constant. Then the group G ω has intermediategrowth. Moreover G ω is periodic if and only if every element of { , , } appearsinfinite times in ω . A.3 Graphs of actions
The graphs in this section are oriented graphs, which are in general not simple.Except where noted the definitions can be found in [Nek18], § Definition A.3.1.
Let G be a group with finite, symmetric generating set S .(i) Let G act transitively on a set X . The Schreier graph Γ( X, G, S ) of thisaction is given by the following data:(a) The vertex set is given by X .(b) A pair ( x, y ) ∈ X × X is an edge if and only if there exists an s ∈ S suchthat y = sx .(ii) Let G act on a set X . For every x ∈ X the orbital Schreier graph Γ( x, G, S )is the Schreier graph of the induced action of G on the orbit G · x .(iii) The Schreier graph associated with the left-multiplication of G on itself is the Cayley graph Γ( G, S ). Remark A.3.2. (i) We omit the generating set from the notion if it is apparentfrom the context. It contains a finite index abelian subgroup. Transitivity of the action assures that the arising graph is connected. x, G, S ) is naturally isomorphic to the Schreiergraph Γ(
G/G x , G, S ) associated to the action of G on the set of cosets G/G x . Definition A.3.3.
Let a group G with finite, symmetric generating set S act on acompact metrizable space X by homeomorphisms. Let x ∈ X .(i) Let G ( x ) denote the subgroup of elements g ∈ G x , such that there exists anopen neighbourhood U x of x for which g | U x = Id U x .(ii) The group G x /G ( x ) is called the group of germs of the action at the point x .(iii) The graph of germs ˜Γ( x, G, S ) is the Schreier graph Γ( G/G ( x ) , G, S ) associatedto the action of G on the set of cosets G/G ( x ) . Remark A.3.4.
There is a natural graph covering π : ˜Γ( x, G, S ) (cid:16) Γ( x, G, S ) wherethe group of germs G x /G ( x ) acts as deck transformations. Definition A.3.5.
Let a group G with finite, symmetric generating set S act on acompact metrizable space X by homeomorphisms.(i) An element x ∈ X is called a G -regular point if G x = G ( x ) i.e. for every g ∈ G that leaves x invariant there exists a neighbourhood U x of x that is pointwisefixed by g .(ii) An element x ∈ X is called a singular point if it is not G -regular.Let x ∈ X be a singular point.(iii) It is called a Hausdorff singularity if for every g ∈ G x \ G ( x ) the element x isnot an accumulation point of the interior of the set of fixed points of g .(iv) It is called a non-Hausdorff singularity if there exists a g ∈ G x \ G ( x ) such that x is an accumulation point of the interior of the set of fixed points of g .(v) It is called a purely non-Hausdorff singularity if x is an accumulation point ofthe interior of the set of fixed points of g for every g ∈ G x .One aspect of graphs of actions is that they allow to study finitely generated groupsvia stochastics on graphs. Definition A.3.6 ([Woe00], § I.1.B & § I.1.C ) . (i) Let S be a countable set. A Markov chain with state space S arises from the data of an initial probabilitydistribution λ = ( λ s ) s ∈ S on S and a stochastic matrix P = { p ( s, t ) } s,t ∈ S assequence of random variables { X n } n ∈ N for which the following holds for all n ∈ N and s , . . . , s n ∈ S :P[ X = s , X = s , . . . , X n = s n ] = λ s n − Y i =0 p ( s i , s i +1 ) The group G ( x ) is indeed a normal subgroup of G x . .4. AMENABILITY (ii) Let Γ = ( V, E ) be locally finite graph. The simple random walk on Γ withstarting point v ∈ V is the Markov chain with state space V , initial probabilitydistribution δ v and stochastic matrix P = { p ( v, u ) } v,u ∈ V given by p ( v, u ) = |{ ( v, w ) ∈ E }| − , if ( v, u ) ∈ E , elseThe graph Γ = ( V, E ) is called recurrent if for every v ∈ V the simple randomwalk on Γ with starting point v ∈ V is recurrent i.e. ∞ X i =0 P[min { n ≥ | X n = v } = n ] = 1 . A.4 Amenability
For encompassing treatments on amenability of groups the reader be refered toChapter 4 of [CSC10] or [Jus15]. Amenability is a pivotal notion in geometric grouptheory. While historically amenability (or more precisely non-amenability) alreadyappeared intrinsically in Felix Hausdorff’s work on paradoxical decompositions ofthe sphere, it was John von Neumann who isolated the property which posed anobstruction to existence of paradoxical decompositions:
Definition A.4.1.
Let G be a discrete group. Then G is said to be amenable ifthere exists a finitely additive probability measure µ on G such that µ ( gA ) = µ ( A )for every A ⊆ G and g ∈ G . Remark A.4.2. (i) Already von Neumann knew, that finite groups and abeliangroups are amenable and that the class of amenable groups is closed under theoperations of forming subgroups, forming quotients and extensions.(ii) The class of amenable groups is closed under direct limits and thus everylocally finite group is amenable.(iii) The free group F is non-amenable and in consequence every discrete groupcontaining it. It was a long-standing conjecture if containment of F charac-terizes non-amenable groups. A counterexample was provided by AleksandrY. Olshanskii in 1980 by the construction of Tarski monsters .Mahlon M. Day formulated the following question: Is the class of amenable groupsthe smallest class of groups that contains all finite and abelian groups which is closedwith respect to subgroups, quotients and extension? This class has been termed theclass of elementary amenable groups and is denoted by EG. Fundamental in theresolution of this problem was Ching Chou’s proof of the following growth typedichotomy on finitely generated EG-groups – a strict generalization of the theoremrestricted to finitely generated solvable groups obtained by John Milnor and JosephA. Wolf:122 heorem A.4.3.
Every finitely generated, elementary amenable group, has eitherpolynomial or exponential growth.
Moreover one can establish the following fact:
Theorem A.4.4.
Finitely generated groups of subexponential growth are amenable.
By this the Grigorchuk groups of intermediate growth provided first counterexamplesto Day’s question. Since then many examples of non-elementary amenable, amenablegroups have been found. There is a number of characterizations of amenability:
Definition A.4.5. (i) Let X be a set. A mean on X is a linear functional m ∈ L ∞ ( X ) ∗ such that the following conditions hold:(a) m ( X ) = 1(b) m ( f ) ≥ f ∈ L ∞ ( X ) with f ≥ α : G (cid:121) X be an action of a discrete group G on a set X . A mean m on X is called G -invariant if m ( g · f )( x ) := m ( f ( g − · x )) = m ( f )( x ) for all f ∈ L ∞ ( X ), x ∈ X and g ∈ G i.e. m is a fixed point of the induced action of G .By an application of the Hahn-Banach theorem , there is a 1-1 correspondence be-tween means and finitely additive probability measures on a set producing the fol-lowing characterization:
Proposition A.4.6.
Let G be a discrete group. Then G is amenable if and only if G carries a G -invariant mean (with respect to left multiplication). Moreover the following hold:
Theorem A.4.7.
A discrete group G is amenable if and only if C ∗ ( G ) ∼ = C ∗ r ( G ) . Theorem A.4.8.
Let G be a finitely generated group with finite generating set S .The following are equivalent:(i) The group G is amenable.(ii) There exists a sequence { F i } i ∈ N of finite sets called Følner sets contained in G such that there exists a sequence { ε i } i ∈ N of positive reals with lim i →∞ ε i = 0 suchthat | gF i ∆ F i | ≤ ε i F for all g ∈ S and i ∈ N . Følner sets allow to quantify amenability of finitely generated groups by the growthof Følner functions:
Definition A.4.9.
Let G be a finitely generated amenable group with symmetricfinite generating set S .(i) Let F ⊆ G . Define ∂ S F := { g ∈ G |∃ s ∈ S : sg / ∈ F } . 123 .5. LEF GROUPS (ii) The Følner function
Fol
G,S : N → N is given by:Fol G,S ( n ) := min {| F | : F ⊆ G, | ∂ S F | ≤ n | F |} A fundamental tool to establish amenability of a group is to study its actions:
Definition A.4.10.
Let α : G (cid:121) X be an action of a discrete group G on a set X .The action is called amenable if X carries a G -invariant mean. Theorem A.4.11.
Every action of an amenable group is amenable.
The converse is not true in general, one needs to add a crucial condition:
Theorem A.4.12.
Let α : G (cid:121) X be an amenable action of a group G on a set X .If the point-stabilizer G x is amenable for every x ∈ X , then G is amenable. A.5 LEF groups
The LEF property, where “LEF” is an abbrevation for “ l ocally e mbeddable into f inite groups”, is, as the name suggests, an approximation property: Definition A.5.1 ([CSC10], Definition 7.1.3) . Let C be a class of groups. A group G is said to be locally embeddable into C if for every finite subset F ⊂ G there existsa group H F in the class C and a map φ : G → H F such that φ ( xy ) = φ ( x ) φ ( y ) for all x, y ∈ F and φ | F is injective. If C is the class of finite groups, we call such a groupa LEF-group .For countable groups the LEF-property translates as follows:
Theorem A.5.2 ([GV98], Corollary 1.3) . Let G be a countable group. The followingare equivalent:(i) G is a LEF-group(ii) There exists a sequence { F n } n ∈ N of finite groups and a sequence of maps { π n : G → F n } n ∈ N such that for all g, h ∈ G the following properties hold:(a) ( g = h ) ⇒ ( ∃ N ∈ N : ∀ n > N, π n ( g ) = π n ( h )) .(b) ∃ N ∈ N : ∀ n > N, π n ( gh ) = π n ( g ) π n ( h ) . Finitely generated LEF-groups can be characterized in the following way:
Proposition A.5.3 ([GV98], p.53) . A group with a fixed finite generating set { g , . . . , g n } has the LEF-property if and only if it is the limit of a sequence of finitegroups in the space of marked groups. { π n : G → F n } n ∈ N is made up of group homomorphism,condition (ii) results in the definition of residually finite groups and thus residuallyfinite groups form a subclass of LEF-groups. In particular free groups and profinitegroups are LEF-groups. In the inverse direction finite presentation is a sufficientcondition: Proposition A.5.4 ([GV98], p.58) . Every finitely presented LEF-group is residuallyfinite.
As a consequence we have:
Corollary A.5.5 ([GV98], Corollary 3) . There exists no infinite, finitely presented,simple LEF-group.
A.6 The Liouville property
The content of this section can be found in [Mat14a]. A poperty of finitely generatedgroups that lies “between” subexponential growth and amenability is the Liouvilleproperty.
Definition A.6.1.
Let G be a finitely generated group.(i) Let µ be a probability measure. A function f : G → R is said to be µ -harmonic if f ( g ) = P h ∈ G f ( gh ) µ ( h ) for every g ∈ G . (ii) Let µ be a probability measure on G . The pair ( G, µ ) is said to have the
Liouville property if every µ -harmonic function is constant on the subgroup h supp( µ ) i ≤ G .(iii) The group G is said to have the Liouville property if for every symmetric,finitely supported probability measure µ on G the pair ( G, µ ) has the Liouvilleproperty.
Theorem A.6.2.
Every finitely generated group of subexponential growth has theLiouville property and every finitely generated group with the Liouville property isamenable.
A fundamental criterion in proving the Liouville property is the study of randomwalks on groups:
Definition A.6.3.
Let G be an countable, infinite, discrete group and let µ be aprobability measure on G . A profinite group is a topological group that is inverse limit of a directed system of discretefinite groups. This condition is also written as f = f ∗ µ . The values of f are in some sense given by aweighted average. .7. HIGMAN-THOMPSON GROUPS (i) The right random walk on ( G, µ ) is the Markov chain with state space G andtransition probabilities p ( g, h ) := µ ( h − g ) for every g, h ∈ G .(ii) The entropy of the probability measure µ is the number H ( µ ) := − X g ∈ supp( µ ) µ ( g ) log µ ( g )(iii) The entropy of the random walk on ( G, µ ) is the number h ( G, µ ) := lim n →∞ n H ( µ ∗ n )The Liouville property can be characterized in terms of the random walk entropy: Theorem A.6.4.
Let G be an countable, infinite, discrete group and let µ be a proba-bility measure on G . The ( G, µ ) has the Liouville property if and only if h ( G, µ ) = 0 . A.7 Higman-Thompson groups
In 1965 Richard J. Thompson introduced a triple of infinite, finitely presented groups F ⊆ T ⊆ G . The group F , now called Thompson’s group , has been a historically firstcandidate for a possible counterexample to the von-Neumann conjecture i.e. for anon-amenable, finitely generated group that contains no free subgroup. It has sincegained notoriety for defying any proof of non-amenability or amenability. The groups T and G gave rise to first examples of infinite, finitely presented, simple groups.Later on manifestations of this groups arose in quite diverse contexts. The Higman-Thompson groups G n,r are generalizations of the group G = G , considered byHigman in [Hig74]. The Higman-Thompson group G n,r arises as the automorphismgroup of the free J´onsson–Tarski algebra of type n on r generators . We follow[Sco84], [Bro87] and [Par11]:Let n ∈ N \ { } and r ∈ N . Let A n be the alphabet given by { , . . . , n } and let A ∗ n denote the free monoid generated by A n i.e. the set of finite (possibly empty)words with cocatenation as operation. Let X r = { x i } i ∈{ ,...,r } be a finite set. Denoteby X r A ∗ n the set of finite words of the form x i w where x i ∈ X r and w ∈ A ∗ n . Thisstructure can be represented by a labelled forest F n,r consisting of r ordered, infinite,complete, n -ary, rooted trees { T i } i ∈{ ,...,r } . A tree T i is labelled by labelling its rootwith x i and recursively labelling the children of a node x i w by x i w , . . . , x i wn withthe order inherited from the natural order of { , . . . , n } . Let a, b ∈ X r A ∗ n . Define a ≤ b to mean there exists an w ∈ A ∗ n such that b = aw In the tree picture this isequivalent to b being a descendant of a . This induces a partial order on X r A ∗ n . Asubset S ⊆ X r A ∗ n is called independent if all elements in S are pairwise incomparablewith respect to ≤ . A non-empty subset V ⊆ X r A ∗ n is called a subspace of X r A ∗ n if itis closed with respect to right cocatenation by elements in A ∗ n . Let V be a subspaceof X r A ∗ n . It is called cofinite if | X r A ∗ n \ V | is finite. A subset B ⊆ V is called a basis of V if it is independent and V = B A ∗ n i.e. V is the upward closure of B with126espect to ≤ . Note that every subspace V ⊆ X r A ∗ n has a canonical basis consisitingof the elements which are minimal with respect to ≤ . A subset B ⊆ X r A ∗ n is a basis if it is basis of some subspace W of X r A ∗ n . A basis B is called cofinite if thesubspace B A ∗ n is cofinite. Note that a basis is cofinite if and only if it is finite andmaximal. Moreover every finite basis is contained in some cofinite basis. Let B bea basis and let b ∈ B . The set ( B \ { b } ) ∪ { b , . . . , bn } is a basis called a simpleexpansion of B . A basis C that is obtained by a finite chain of simple expansionsfrom B is called an expansion of B . In the tree picture a subspace V consist of afamily { T a } a ∈ A of infinite complete subtrees with roots a where A corresponds to thecanonical basis of V . Let V, U be subspaces of X r A ∗ n . A map α : V → U is calleda homomorphism of subspaces if α ( vw ) = α ( v ) w holds for all v ∈ V and for all w ∈ A ∗ n . In the tree picture it is a map between disjoint families of infinite completesubtrees α : F a ∈ A T a → F b ∈ B T b with respective indpendents subsets A, B ⊂ X r A ∗ n of roots such that the ordered rooted tree structure of trees T a in the domain ispreserved. If in addition a homomorphism of subspaces is bijective, it is calledan isomorphism of subspaces . An isomorphism of subspaces α : V → W betweencofinite subspaces V, W , is called a cofinite isomorphism . Let α : V → W be acofinite isomorphism. A cofinite isomorphism β : ˜ V → ˜ W is called an extension of α if V ⊆ ˜ V and β ( v ) = α ( v ) for every v ∈ V . A cofinite isomorphism which has nonon-trivial extensions is called maximal . Example A.7.1.
Let r, n = 2. Let α be the cofinite isomorphism given by T x i T x i and T x i T x i for i ∈ { , } . Then the cofinite isomorphism ˜ α given by T x T x and T x T x is an extension of α . Lemma A.7.2 ([Sco84], Lemma 1) . Let n ∈ N \ { } and r ∈ N . Every cofiniteisomorphism α of subspaces of X r A ∗ n has a unique maximal extension. Denote the unique maximal extension of a cofinite isomorphism α of subspaces of X r A ∗ n by α ∗ . Let α : V → W and β : ˜ V → ˜ W be cofinite isomorphisms of subspacesof X r A ∗ n . Then α (cid:12) β := ( β | W ∩ ˜ V ◦ α | α − ( W ∩ ˜ V ) ) ∗ is a well-defined cofinite isomorphism of subspaces of X r A ∗ n . Lemma A.7.3 ([Sco84], Lemma 2) . Let n ∈ N \ { } and r ∈ N . The set of maximalcofinite isomorphisms of subspaces of X r A ∗ n forms a group with respect to (cid:12) . Definition A.7.4.
The
Higman-Thompson group ( G n,r , (cid:12) ) is the group of maximalcofinite isomorphisms of subspaces of X r A ∗ n .It is immediate that the groups G n,r admit a natural action on a Cantor space bytheir natural action on the pathspace of the forest F n,r , furthermore cofinite basesallow for a description of G n,r by tables , which will be needed to state Theorem 3.2.38.Let B = { b , . . . , b N } and C = { c , . . . , c N } be cofinite bases of the same cardinality.Then any bijection β between those bases extends naturally to a cofinite isomorphism127 .7. HIGMAN-THOMPSON GROUPS of subspaces between B A ∗ n and C A ∗ n , thus inducing a unique element in G n,r denotedby b . . . b N β ( b ) . . . β ( b N ) ! Conversely, for every g ∈ G n,r there exists a N ∈ N and cofinite bases B = { b , . . . , b N } and C = { c , . . . , c N } such that g = b . . . b N c . . . c N ! If a table is obtained from another by permutation of columns, they are said to be equivalent . If a table is obtained from another by replacing a column b i c i ! with b i . . . b i nc i . . . c i n ! we call it a simple expansion of the initial table. If a table is obtained from anotherby a finite string of simple expansions it is called an expansion of the initial table. Lemma A.7.5.
Two tables have expansions that are equivalent if and only if theyinduce the same element of G n,r . Let g ∈ G n,r given by the table g = b . . . b N c . . . c N ! The natural homeomorphisms associated with g on the pathspace of the forest F n,r is given by mapping the open sets of infinite paths { b i w | w ∈ A N n } onto the open setsof infinite paths { c i w | w ∈ A N n } via g ( b i w ) = c i w for all i ∈ { , . . . , N } .The Higman-Thompson fall within a class of groups of piecewise linear intervaltransformations, the following description is essentially due to [Bro87]: Theorem A.7.6.
Let n ∈ N \ { } and r ∈ N . Denote by ˜ G n,r the group of right-continuous, piecewise linear bijections f : [0 , r ) → [0 , r ) with finitely many singulari-ties, such that all singularities of f are contained in Z [ n ] , such that f ( Z [ n ] ∩ [0 , r )) = Z [ n ] ∩ [0 , r ) and for every point x ∈ [0 , r ) where f is not singular, there exists a k ∈ Z such that f ( x ) = n k . Then ˜ G n,r ∼ = G n,r ppendix BC*-algebras B.1 Basic terms
The definitions of this section are taken from Chapter I of [Dav96] or Chapter 2 of[Mur90] except where noted.
Definition B.1.1. (i) A normed algebra is an algebra A endowed with a submul-tiplicative norm k · k (i.e. ∀ a, b ∈ A : k ab k ≤ k a kk b k ). If A has a multiplicativeunit 1, it is said to be unital . A complete normed algebra is called Banachalgebra .(ii) Let a be an element in a unital Banach algebra A . The spectrum of a is the set σ ( a ) := { λ ∈ C : λ − a is non-invertible } . The spectral radius of a , denotedby spr( a ), is defined as spr( a ) := sup λ ∈ σ ( a ) | λ | .In some contexts it is useful to have a unit around: Definition B.1.2.
Let A be a Banach algebra. The vectorspace A × C endowedwith the product ( a, λ )( b, µ ) = ( ab + λb + µa, λµ ) is a unital algebra denoted by A + with (0 ,
1) as unit and A as a maximal ideal via the embedding a ( a, A naturally extends to a complete norm on A + by k ( a, λ ) k = k a k + | λ | . TheBanach algebra A + is called the unitization of A .Banach ∗ -algebras (pronounced ”star-algebra”) are Banach algebras with an isomet-ric involution. C*-algebras are an important subclass: Definition B.1.3. (i) A ∗ -algebra is a C -algebra A with anti-involution ∗ : A → A called the adjoint , i.e. a map ∗ : a a ∗ which satisfies:(a) ∀ a, b ∈ A : ∀ z , z ∈ C : ( z a + z b ) ∗ = z a ∗ + z b ∗ (conjugate-linear)(b) ∀ a ∈ A : a ∗∗ = a (self-inverse)(c) ∀ a, b ∈ A : ( ab ) ∗ = b ∗ a ∗ (anti-multiplicative). .1. BASIC TERMS (ii) A ∗ -homomorphism is an algebra-homomorphism ϕ : A → B between ∗ -algebras A , B such that ϕ ( − ∗ ) = ϕ ( − ) ∗ .(iii) A Banach ∗ -algebra is a Banach algebra A which is a ∗ -algebra such that k a ∗ k = k a k for all a ∈ A .(iv) Let A be a unital Banach ∗ -algebra. An element a ∈ A is called(a) self-adjoint if a ∗ = a .(b) normal if a ∗ a = a ∗ .(c) an isometry if a ∗ a = 1 and a coisometry if aa ∗ = 1(d) a partial isometry if aa ∗ a = a . Denote by Par( A ) the set of all partialisometries in A .(e) unitary if it is an isometry and a coisometry.(f) positive , write a ≥
0, if it is self-adjoint and σ ( a ) ⊆ [0; ∞ [.(g) an idempotent if a = a .(h) a projection , if it is a self-adjoint idempotent. Denote by Proj( A ) the setof all projections of A .The self-adjoint elements in A can be endowed with an order relation by defin-ing a ≤ b if b − a ≥
0. Denote the set of all positive elements in A by A pos .(v) A C*-algebra is a Banach ∗ -algebra A such that k a ∗ a k = k a k for all a ∈ A .(vi) Let A be a C*-algebra. A C*-subalgebra of A is a norm-closed subalgebraclosed with respect to the adjoint. An ideal of A is a norm-closed two sidedideal of the underlying C -algebra.(vii) Let A be a unital C*-algebra and let a ∈ A . The C*-subalgebra generated by a , denoted by C ∗ ( a ), is the norm-closure of the linear span of I and productsof a and a ∗ .(viii) Let A be a C*-algebra. A net ( i λ ) λ ∈ Λ of elements in A is called approximateunit if:(a) i λ ≥ , k i λ k ≤ λ ∈ Λ(b) λ ≤ κ implies i λ ≤ i κ (c) lim λ ∈ Λ i λ a = lim λ ∈ Λ ai λ = a for all a ∈ A .(ix) Let A be a C*-algebra and let B be a C*-subalgebra of A . The normalizer of B in A is the set N ( B , A ) := { a ∈ A | a ∗ B a ⊆ B , a B a ∗ ⊆ B } .Let A be a C*-algebra, then the set { a ∈ A : a ≥ , k a k < } is a directed set withrespect to the order defined on the self-adjoint elements. Therefore we have:130 heorem B.1.4 ([Dav96], Theorem I.4.8) . Every C*-algebra has an approximateunit.
As with general Banach algebras it is often useful to have a unit around:
Theorem B.1.5 ([Mur90], Theorem 2.1.6.) . If A is a C*-algebra, the extended normmakes A + into a C*-algebra. After this verbiage we look at the first concrete examples of such algebras:
Example B.1.6. (i) The field C with its usual norm is a unital C*-algebra withconjugation as ∗ -morphism.(ii) For any locally compact Hausdorff space X the space C ( X ) of continuous C -valued functions vanishing at infinity endowed with the supremum normand f f as adjoint is a commutative C*-algebra. If X is compact, C ( X )contains X as a unit. If X is non-compact, C ( X ) is non-unital.(iii) Let H be a Hilbert space. Then the space B ( H ) of bounded linear operatorson H is a C*-algebra with respect to the operator norm and the ∗ -morphismgiven by the adjoint of operators. Definition B.1.7. (i) Let A be a commutative Banach algebra. A character of A is a non-zero algebra-homomorphism A → C . Denote by Ω( A ) the set of allcharacters of A .Let A be a Banach ∗ -algebra.(ii) A linear functional ϕ : A → C is called positive , if ϕ ( a ) ≥ a ∈ A . Positive linear functionals of norm 1 are called states . Denoted by S ( A ) the set of all states of A . A state is said to be pure if it is an extremepoint in S ( A ).(iii) Let A be a C*-algebra. A weight on A is a function φ : A pos → [0 , ∞ ] suchthat φ ( λa ) = λφ ( a ) for all a ∈ A pos and λ ∈ R ≥ and φ ( a + b ) = φ ( a ) + φ ( b )for all a, b ∈ A pos .Abstract C*-algebras become accessible through their representations: Definition B.1.8.
Let H be a Hilbert space and let A be a Banach ∗ -algebra. A ∗ -algebra-representation of A on H is a ∗ -homomorphism A → B ( H ). A ∗ -algebra-representation φ : A → B ( H ) is called A C -valued function on a locally compact space X is said to vanish at infinity if the set { x ∈ X : | f ( x ) | ≥ (cid:15) } is compact for every (cid:15) > A point x contained in a convex subset C of some K -vectorspace V is called extreme point if itcan not be represented as non-trivial convex combination i.e. if there exist λ ∈ [0; 1] and y, z ∈ K with x = λy + (1 − λ ) z , it follows that x = y or x = z ([Rud91], § .1. BASIC TERMS (i) trivial if φ ( A ) = 0(ii) topologically irreducible if φ ( A ) contains no proper closed invariant subspace(iii) algebraically irreducible if φ ( A ) contains no proper invariant subspace(iv) faithful if it is injective(v) non-degenerate if φ ( A ) H = H (vi) cyclic if it has a cyclic vector i.e. there exists an element x ∈ H such that φ ( A ) x is dense in H . Remark B.1.9. ([Dav96], Theorem I.9.3). In the case of C*-algebras topological-and algebraical irreducibility are equivalent and one just speaks of irreducible rep-resentations.Originally, C*-algebras were defined as norm-closed subalgebras of B ( H ) for someHilbert space H closed under taking adjoints ([Seg47], § B ( H ) for some Hilbert space H orin other words: Theorem B.1.10 ([Dav96], Theorem I.9.12) . Every C*-algebra A has an isometricfaithful representation. The Gelfand-Naimark theorem is a consequence of the Gelfand representation, itscommutative manifestation, which relates a commutative Banach algebra A withthe space C (Ω( A )) of continuous functions vanishing at infinity on its space ofcharacters. By Gelfand theory, every locally compact Hausdorff space X gives riseto the commutative C*-algebra C ( X ) and any commutative C*-algebra A gives riseto the locally compact Haudorff space of characters Ω( A ). Given the correct notion ofmorphisms this produces an equivalence of categories LoCoHaus ’ CommC ∗ op ,called Gelfand duality . This is why the theory of general C*-algebra theory hasbeen termed non-commutative topology. It is thus sometimes helpful to think ofC*-algebraical terms as generalized topological notions :topological space ←→ C*-algebraopen subsets ←→ idealsclosed subsets ←→ quotientsclopen subsets ←→ projectionscompact spaces ←→ unital C*-algebrasproper continous maps ←→ unital *-morphismsAlexandroff compactification ←→ unitization2nd countable ←→ separable This is a cut down version of the list in § Definition B.1.11 ([CK80], § . Let n ∈ N and let { A i,j } i,j ∈{ ,...,n } ∈ M n × n ( Z / Z )be a matrix of which every row and column is non-zero and which is not a permu-tation matrix. Let { S i } i ∈{ ,...,n } be a collection of non-zero partial isometries oversome Hilbert space H with associated support projections Q i = S ∗ i S i and rangeprojections P i = S i S ∗ i such that the following hold:(i) P i P j = 0 for all i, j ∈ { , . . . , n } with i = j .(ii) Q i = P j ∈{ ,...,n } A i,j P j for all i ∈ { , . . . , n } .Denote by C ∗ ( S , . . . , S n ) be the C*-subalgebra of B ( H ) generated by { S i } i ∈{ ,...,n } .The up to isomorphism unique C*-algebra O A := C ∗ ( S , . . . , S n ) is called the Cuntz-Krieger algebra of A . If every entry of A is 1, the arising C*-algebra only dependson n . It is denoted by O n and called the Cuntz algebra of order n . B.2 Group C*-algebras
For a more thourough treatment of group C*-algebras see Chapter VII of [Dav96].
Definition B.2.1.
Let G be a locally compact topological group.(i) Let H be a Hilbert space. Denote by U ( H ) the group of unitary elementsin B ( H ). A unitary representation of G on H is a group-homomorphism π : G → U ( H ) such that the map g π ( g ) · x is continuous for all x ∈ H . Such a representation π is called irreducible if π ( G ) commutes with no properprojection of H and it is called cyclic , if it has a cyclic vector.(ii) Let H , H be Hilbert spaces with unitary representations π : G → U ( H )and π : G → U ( H ) of G . The representations π , π are said to be unitarilyequivalent if there exists a Hilbert space ismorphism (i.e. a bijection preservingthe inner product) Φ : H → H such that φ ◦ π ( g ) = π ( g ) ◦ φ for all g ∈ G .Locally compact topological groups possess what is called a left resp. right Haarmeasure i.e. every such group G supports a left- resp. right-invariant, non-zero, regu-lar Borel measure µ G , which is unique up to a scalar. Let G be locally compact group.As customary in measure theory, every p ∈ [1; ∞ [ defines a seminorm on the space M ( G ) := { f : G → C | f is measurable w.r.t. µ G } by k f k p := ( R G | f | p d µ G ) /p . Tak-ing the quotient of the semi-normed vectorspaces L p ( G ) := { f ∈ M µ G : k f k p < ∞} by ker( k · k p ) gives – by abuse of notation – the normed vector spaces (L p ( G ) , k · k p ). It is continuous with respect to the strong operator topology. .2. GROUP C*-ALGEBRAS
The space L ( G ) can be given the structure of a Banach ∗ -algebra with approximateunit with the following adjoint and product: f ∗ ( g ) = ∆( g − ) f ( g − )( f ∗ f )( g ) := Z h ∈ G f ( h ) f ( h − g ) d µ G ( h )The function ∆ is the modular function , which is a continuous group-homomorphism∆ : G → R + , that satisfies µ G ( Sg ) = ∆( g ) µ G ( S ) for all measureable sets S . Note,that in general L ( G ) is not a C*-algebra. It is a fundamental feature of the theory,that there is a close relation between unitary representations of the underlying locallycompact group G and ∗ -representations of the convolution algebra L ( G ): Everyunitary representation π of G gives rise to a ∗ -algebra-representation ˜ π of L ( G ) byintegration: ˜ π ( f ) := Z g ∈ G f ( g ) π ( g )d µ G ( g )Conversely every non-degenerate ∗ -algebra-representation ˜ π of L ( G ) induces a uni-tary representation π of G by setting π ( g ) := SOT- lim λ ∈ Λ ˜ π (cid:16) ( I λ )( g − ( − ) (cid:17) where ( I λ ) λ ∈ Λ is an approximate unit of L ( G ) and SOT- lim denotes the limit withrespect to the strong operator topology. L ( G ) can be endowed with norm definedby: k f k C ∗ := sup {k π ( f ) k : π is a ∗ -algebra-representation of L ( G ) } This supremum is well defined, since k π ( f ) k ≤ k f k for all ∗ -algebra-respresentationsof L ( G ) as they are norm-decreasing. The completion of L ( G ) with respect to thisnorm is a C*-algebra, called the (maximal) group C*-algebra of G denoted by C ∗ ( G ).The left regular representation λ of G is the unitary representation on L ( G ) givenby ( λ ( g ) f )( x ) := f ( g − x ). The reduced group C*-algebra of G denoted by C ∗ r ( G )is the closure of ˜ λ (L ( G )) with respect to the operator norm or equivalently thecompletion of L ( G ) with respect to the norm k f k r := k ˜ λ ( f ) k . If G is a discretegroup, C c ( G ) densely embeddeds in L ( G ). In direct consequence C ∗ ( G ) has thefollowing universal property: Proposition B.2.2.
Let G be a discrete group and let u : G → B ( H ) be a unitaryrepresentation. Then there exists a unique ∗ -algebra-representation π u : C ∗ ( G ) → B ( H ) such that π u | G = u . There is a 1-1 correspondence between the unitary representations of G and non-degenerate representations of C ∗ ( G ). Since consequently the regular representationextends naturally to a surjective ∗ -algebra-morphism λ : C ∗ ( G ) → C ∗ r ( G ), the Operators λ ( g ) are unitary, since the Haar measure is translation invariant. C ∗ r ( G ) is a quotient of C ∗ ( G ). In this case it could bedefined alternatively by C ∗ r ( G ) := λ ( C ∗ ( G )). B.3 Hilbert bundles
The following is mostly adopted from [Bla06], § II.7.
Definition B.3.1.
Let X be a second countable, locally compact, Hausdorff spaceendowed with a probability measure µ . Let H := { H x } x ∈ X be a collection of Hilbertspaces indexed by X and let Σ ⊂ Q x ∈ X H x be a countable set of sections. The triple( H , µ, Σ) is called a measured Hilbert bundle if it satisfies the following properties:1. The set { σ x | σ ∈ Σ } is a dense subspace of H x for every x ∈ X .2. For every σ , σ ∈ Σ the map x
7→ h σ ( x ) , σ ( x ) i is µ -measurable.Measured Hilbert bundles induce a Hilbert space by integration: Definition B.3.2.
Let ( X, H = { H x } x ∈ X , µ ) be a measured Hilbert bundle.(i) A section ξ ∈ Q x ∈ X H x is called measurable if for every σ ∈ Σ the function X → C : x
7→ h ξ ( x ) , σ ( x ) i is µ -measurable. Denote by Γ( X, H , µ ) the set ofall measurable sections.(ii) The space Γ( X, H , µ ) carries a semi-norm defined by k σ k := (cid:16) Z x ∈ X k σ ( x ) k x d µ (cid:17) . The quotient of µ -integrable measurable sections by the space of null sectionsi.e. { σ ∈ Γ( X, H , µ ) : k σ k < ∞} . { σ ∈ Γ( X, H , µ ) : k σ k = 0 } is a Hilbert space with the product given by h σ , σ i := R x ∈ X h σ ( x ) , σ ( x ) i x d µ called the direct integral of ( X, H , µ ) and is denoted by R ⊕ X H x d µ . B.4 A hamfisted portrait of K-groups
Operator K-theory is the non-commutative generalization of topological K-theoryand is an indispensible tool in the theory of C*-algebras and Banach algebras. Itcomprises continuous, half exact, homotopy-invariant and stable functors K and K , which associate to a Banach algebra A abelian groups K ( A ) resp. K ( A ),in particular the functors satisfy K ( C ) = Z and K ( C ) = 0. For monographson operator K-theory see e.g. [WO93] or [Bla08], we content ourselves with anadumbration of K -groups in most handwavy terms:The invariant K is obtained by putting algebraic structure on classes of idem-potents, in the case of C*-algebras one can restrict to classes of projections. In135 .4. A HAMFISTED PORTRAIT OF K-GROUPS C*-algebras there are three types of equivalence relations between projections toconsider. For p, q ∈ Proj( A ) we get the following list of equivalence relations, whereevery entry implies the preceding equivalence relations:(i) The projections p, q are algebraically equivalent , write p ∼ q , if there existelements xy ∈ A such that p = xy and q = yx .(ii) The projections p, q are unitarily equivalent , write p ∼ u q , if there exist aunitary element u ∈ A + such that p = uqu ∗ .(iii) The projections p, q are homotopic , write p ∼ h q , if there exist a norm contin-uous path of projections from p to q .The sum of two projections p + q is a projection if and only if the projections p, q are orthogonal i.e. ef = 0. Addition of orthogonal projections behaves well withrespect to equivalence, but not for every pair of equivalence classes of projectionsdoes there exist an orthogonal pair of respective contained representants. The ideais to pass over to the C*-algebra M ∞ ( A ) = S n ∈ N M n ( A ) of infinite matrices withfinite non-zero entries in A . Let p, q ∈ Proj( A ), then the following implications hold: p ∼ q ⇒ p
00 0 ! ∼ u q
00 0 ! , p ∼ u q ⇒ p
00 0 ! ∼ h q
00 0 ! . This means in this setting all the defined equivalence relations coincide. Moreover,for any p ∈ Proj( A ) we have: ! p
00 0 ! ! = p ! By conjugating with a unitary, a projection can be pushed down diagonally, and sincethe matrices we consider can be arbitrarily large, for any pair p, q ∈ Proj( M ∞ ( A ))we can find a orthogonal pair of representants of the respective equivalence classesby pushing one of the involved projections far enough down the diagonal. Thusthe set V ( A ) := Proj( M ∞ ( A )) / ∼ becomes a commutative monoid with respect tothe direct sum. This construction produces a covariant functor from C*-algebrasto commutative monoids. Any *-algebra homomorphism α : A → B induces a well-defined homomorphism of commutative monoids V ( α ) : V ( A ) → V ( B ). As in theconstruction of the integers Z from the natural numbers N , the commutative monoid V ( A ) gives rise to an abelian group K ( A ) via the Grothendieck group constructionby considering equivalence classes of formal differences between elements in V ( A ).In particular, we have K ( C ) = Z . Again, the construction is functorial and comeswith a canonical map ˜ ι A : V ( A ) → K ( A ). Any α : A → B induces a group ho-momorphism K ( α ) : K ( A ) → K ( B ). This definition, however, is insufficient inthe case of non-unital C*-algebras – in general K is not half-exact, but it is theexact sequences of K-theory, that make it tick! To have a unified way of definitionone turns to A + (see Definition B.1.2): The C*-algebra A + comes with a canonical136urjective *-algebra homomorphism π : A + → C and by functoriality one can define K ( A ) := ker (cid:16) K ( π ) : K ( A + ) → Z (cid:17) . Again, K is functorial and there is a canonical map ι A : V ( A ) → K ( A ). If A isunital, much of theory becomes simpler e.g. we have K ( A ) = K ( A ). There arethree classes of elements in K ( A ): Elements arising out of projections in A , elementsarising out of adjoining a unit or elements arising out of the matrices over A + . The K -groups carry more information by keeping track of the commutative monoid V ( A ) and the projections steming from A : Fix the notations K ( A ) + := Im( ι A ) andΣ A is the set of elements in K ( A ) that come from elements in Proj( A ). Undersufficient conditions i.e. when A is unital and stably finite, the set K ( A ) + is apositive cone for K ( A ) making it an ordered abelian group. Furthermore, if 1 is theunit of A , then Σ A = { [ p ] ∈ K ( A ) + (cid:12)(cid:12)(cid:12) [ p ] ≤ [1] } , thus the scale is uniquely determinedby [1]. For such a C*-algebra A the triple ( K ( A ) , K ( A + , [1]) is an ordered abeliangroup with order unit e.g. this is the case, if A is an AF C*-algebra. The group K ( A ) is not defined via the projections but via the unitaries of a C*-algebra A ,moreover, it can be given in terms of K , in that K ( A ) is the K -group of the suspension of A . The fundamental feature of K - and K -groups is, that any shortsequence 0 −→ A ϕ −→ B ψ −→ C −→ cyclical 6-term exact sequence of K-groups of C*-algebras K ( A ) K ( B ) K ( C ) K ( C ) K ( B ) K ( A ) K ( ϕ ) K ( ψ ) δ δ K ( ψ ) K ( ϕ ) in which δ signifies the index map . This designation hints at the ties of K-theorywith index theory. B.5 Von Neumann algebras
Von Neumann algebras form a subclass of C*-algebras. Their theory is very distinctin that it is noncommutative measure theory. In this section we recall the terms weneed in Section 3.6. A useful reference on von Neumann algebras is Chapter 4 of[Mur90] or § I.9 and § III of [Bla06].
Definition B.5.1.
Let H be a Hilbert space.(i) Let S be a subset of B ( H ). The set S := { S ∈ B ( H ) | S S = SS ∀ S ∈ S} iscalled the commutant of S . Being only unital is not sufficient, the
Cuntz algebra (see Definition B.1.11) represents acounter-example. .5. VON NEUMANN ALGEBRAS (ii) A ∗ -subalgebra A of the bounded linear operators B ( H ) which is closed w.r.t.the strong operator topology and contains Id is called von Neumann algebraon H .(iii) Since intersections of von Neumann Algebras are again von Neumann algebras,we can define for any S ⊆ B ( H ) the von Neumann algebra M S generated by S as the intersection of all von Neumann algebras containing S .(iv) Let G be a countable group and let π : G → B ( H ) be a unitary representationof G . Denote by M π the von Neumann Algebra generated by the set ofoperators { π ( g ) | g ∈ G } .(v) An abelian von Neumann algebra A is called maximal if it is not containedin a strictly larger abelian von Neumann algebra. An abelian von Neumannalgebra A is maximal if and only if A = A . Such a von Neumann algebra isacronymically termed a masa .Measure spaces give rise to commutative von Neumann algebras – the theory of vonNeumann algebras is in this sense non-commutative measure theory. Remark B.5.2. (i) By the von Neumann bicommutant theorem a ∗ -subalgebra A of B ( H ) is a von-Neumann-algebra if and only if A = A .(ii) For any subsets S, S , S ⊂ B ( H ) we have S ⊆ S , S ⊆ S implies S ⊆ S and thus S = S .(iii) Let π : G → B ( H ) be a unitary representation of a discrete group G . Thenthe image is self-adjoint and contains id H and thus M π is just the doublecommutant π ( G ) . This is the coarsest topology on B ( H ) which makes f ( x,y ) : B ( H ) → C , T
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