aa r X i v : . [ m a t h . K T ] D ec K -THEORY AND ACTIONS ON EUCLIDEAN RETRACTS BARTELS, A.
Abstract.
This note surveys axiomatic results for the Farrell-Jones Conjec-ture in terms of actions on Euclidean retracts and applications of these toGL n ( Z ), relative hyperbolic groups and mapping class groups. Introduction
Motivated by surgery theory Hsiang [43] made a number of influential conjecturesabout the K -theory of integral group rings Z [ G ] for torsion free groups G . Theseconjectures often have direct implications for the classification theory of manifoldsof dimension ≥
5. A good example is the following. An h -cobordism is a com-pact manifold W that has two boundary components M and M such that bothinclusions M i → W are homotopy equivalences. The Whitehead group Wh( G ) isthe quotient of K ( Z [ G ]) by the subgroup generated by the canonical units ± g , g ∈ G . Associated to an h -cobordism is an invariant, the Whitehead torsion, inWh( G ), where G is the fundamental group of W . A consequence of the s -cobordismtheorem is that for dim W ≥
6, an h -cobordism W is trivial (i.e., isomorphic to aproduct M × [0 , G torsion free Wh( G ) = 0, and thus that in many cases h -cobordisms are products.The Borel conjecture asserts that closed aspherical manifolds are topologicallyrigid, i.e., that any homotopy equivalence to another closed manifold is homotopicto a homeomorphism. The last step in proofs of instances of this conjecture viasurgery theory uses a vanishing result for Wh( G ) to conclude that an h -cobordismis a product and that therefore the two boundary components are homeomorphic.Farrell-Jones [28] pioneered a method of using the geodesic flow on non-positivelycurved manifolds to study these conjectures. This created a beautiful connectionbetween K -theory and dynamics that led Farrell-Jones [30], among many otherresults, to a proof of the Borel Conjecture for closed Riemannian manifolds of non-positive curvature of dimension ≥
5. Moreover, Farrell-Jones [29] formulated (andproved many instances of) a conjecture about the structure of the algebraic K -theory (and L -theory) of group rings, even in the presence of torsion in the group.Roughly, the Farrell-Jones Conjecture states that the main building blocks for the K -theory of Z [ G ] is the K -theory of Z [ V ] where V varies of the family of virtu-ally cyclic subgroups of G . It implies a number of other conjectures, among themHsiang’s conjectures, the Borel Conjecture in dimension ≥
5, the Novikov Conjec-ture on the homotopy invariant of higher signatures, Kaplansky’s conjecture aboutidempotents in group rings, see [54] for a summary of these and other applications.My goal in this note is twofold. The first goal is to explain a condition formulatedin terms of existence of certain actions of G on Euclidean retracts that implies theFarrell-Jones Conjecture for G . This condition was developed in joint work withL¨uck and Reich [8, 12] where the connection between K -theory and dynamics has Date : December 2017.1991
Mathematics Subject Classification.
Key words and phrases.
Farrell-Jones Conjecture, K - and L -theory of group rings. been extended beyond the context of Riemannian manifolds to prove the Farrell-Jones Conjecture for hyperbolic and CAT(0)-groups. The second goal is to outlinehow this condition has been used in joint work with L¨uck, Reich and R¨uping andwith Bestvina to prove the Farrell-Jones Conjecture for GL n ( Z ) and mapping classgroups. A common difficulty for both families of groups is that their natural properactions (on the associated symmetric space, respectively on Teichm¨uller space)is not cocompact. In both cases the solution depends on a good understandingof the action away from cocompact subsets and an induction on a complexity ofthe groups. As a preparation for mapping class groups we also discuss relativelyhyperbolic groups.The Farrell-Jones Conjecture has a prominent relative, the Baum-Connes Con-jecture for topological K -theory of group C ∗ -algebras [15, 16]. The two conjecturesare formally very similar, but methods of proofs are different. In particular, theconditions discussed in Section 2 are not known to imply the Baum-Conjecture.The classes of groups for which the two conjectures are known differ. For exam-ple, by work of Kammeyer-L¨uck-R¨uping [44] all lattice in Lie groups satisfy theFarrell-Jones Conjecture; despite Lafforgues [51] positive results for many property T groups, the Baum-Connes Conjecture is still a challenge for SL ( Z ). Wegner [77]proved the Farrell-Jones Conjecture for all solvable groups, but the case of amenable(or just elementary amenable) groups is open; in contrast Higson-Kasparov [41]proved the Baum-Connes Conjecture for all a-T-menable groups, a class of groupsthat contains all amenable groups. On the other hand, hyperbolic groups satisfyboth conjectures. See Mineyev-Yu [62] and Lafforgue [52] for the Baum-ConnesConjecture and, as mentioned above, [8, 12] for the Farrell-Jones Conjecture. For amore comprehensive summary of the current status of the Farrell-Jones Conjecturethe reader is directed to [55, 70]. Acknowledgement.
It is a pleasure to thank my teachers, coauthors, and stu-dents for the many things they taught me. The work described here has beensupported by the SFB 878 in M¨unster.1.
The Formulation of the Farrell-Jones Conjecture
Classifying spaces for families.
A family F of subgroups of a group G is anon-empty collection of subgroups that is closed under conjugation and subgroups.Examples are the family Fin of finite subgroups and the family VCyc of virtuallycyclic subgroups (i.e., of subgroups containing a cyclic subgroup as a subgroupof finite index). For any family of subgroups of G there exists a G - CW -complex E F G with the following property: if E is any other G - CW -complex such that allisotropy groups of E belong to F , then there is a up to G -homotopy unique G -map E → E F G . This space is not unique, but it is unique up to G -homotopyequivalence. Informally one may think about E F G as a space that encodes thegroup G relative to all subgroups from F . Often there are interesting geometricmodels for this space, in particular for F = Fin. More information about thisspace can be found for example in [53]. An easy way to construct E F G is as theinfinite join ∗ ∞ i =0 ( ` F ∈F G/F ). If F is closed under supergroups of finite index(i.e., if F ∈ F is a subgroup of finite index in F ′ , then also F ′ ∈ F ), then thefull simplicial complex on ` F ∈F G/F is also a model for E F G ; we will denote thismodel later by ∆ F ( G ). The formulation of the conjecture.
The original formulation of the Farrell-Jones Conjecture [29] used homology with coefficients in stratified and twisted Ω-spectra. Here we use the equivalent [38] formulation developed by Davis-L¨uck [23]. -THEORY AND ACTIONS ON EUCLIDEAN RETRACTS 3
Given a ring R , Davis-L¨uck construct a homology theory X H G ∗ ( X ; K R ) for G -spaces with the property that H G ∗ ( G/H ; K R ) ∼ = K ∗ ( R [ H ]).Let F be a family of subgroups of the group G . Consider the projection map E F G → G/G to the one-point G -space G/G . It induces the F -assembly map α G F : H G ∗ ( E F ; K R ) → H G ∗ ( G/G ; K R ) ∼ = K ∗ ( R [ G ]) . Conjecture 1.1 (Farrell-Jones Conjecture) . For any group G and any ring R theassembly map α G VCyc is an isomorphism.
This version of the conjecture has been stated in [7]. The original formulationof Farrell and Jones [29] considered only the integral group ring Z [ G ]. Moreover,Farrell and Jones wrote that they regard this and related conjectures only as es-timates which best fit the know data at this time . However, the conjecture is stillopen and does still fit with all known data today. Transitivity principle.
Informally one can view the statement that the assemblymap α G F is an isomorphism for a group G and a ring R as the statement that K ∗ ( R [ G ]) can be assembled from K ∗ ( R [ F ]) for all F ∈ F (and group homology).If V is a family of subgroups of G that contains all subgroups from F , then onecan apply this slogan in two steps, for G relative to V and for each V ∈ V relativeto the F ∈ F with F ⊆ V . The implementation of this is the following transitivityprinciple. Theorem 1.2 ([29, 56]) . For V ∈ V set F V := { F | F ∈ F , F ⊆ V } . Assumethat for all V ∈ V the assembly map α V F V is an isomorphism. Then α G F is anisomorphism iff α G V is an isomorphism. Twisted coefficients.
Often it is beneficial to study more flexible generalizationsof Conjecture 1.1. Such a generalization is the Fibred Isomorphism Conjecture ofFarrell-Jones [29]. An alternative is the Farrell-Jones Conjecture with coefficientsin additive categories [14], here one allows additive categories with an action of agroup G instead of just a ring as coefficients. This version of the conjecture appliesin particular to twisted group rings. These generalizations of the Conjecture havebetter inheritance properties. Two of these inheritance property are stability underdirected colimits of groups, and stability under taking subgroups. For a summaryof the inheritance properties see [70, Thm. 27(2)]. Often proofs of cases of theFarrell-Jones Conjecture use these inheritance properties in inductions or to reduceto special cases. We will mean by the statement that G satisfies the Farrell-JonesConjecture relative to F , that the assembly map α G F is bijective for all additivecategories A with G -action. However, this as a technical point that can be safelyignored for the purpose of this note. Other theories.
The Farrell-Jones Conjecture for K -theory discussed so far hasan analog in L -theory, as it appears in surgery theory. For some of the applicationsmentioned before this is crucial. For example, the Borel Conjecture for a closedaspherical manifold M of dimensions ≥ M sat-isfies both the K -and L -theoretic Farrell-Jones Conjecture. However, proofs of theFarrell-Jones Conjecture in K - and L -theory are by now very parallel. Recently, thetechniques for the Farrell-Jones Conjecture in K - and L -theory have been extendedto also cover Waldhausen’s A -theory [25, 47, 74]. In particular, the conditions wewill discuss in Section 2 are now known to imply the Farrell-Jones Conjecture inall three theories. BARTELS, A. Actions on compact spaces
Amenable actions and exact groups.Definition 2.1 (Almost invariant maps) . Let X , E be G -spaces where E isequipped with a G -invariant metric d . We will say that a sequence of maps f n : X → E is almost G -equivariant if for any g ∈ G sup x ∈ X d ( f n ( gx ) , gf n ( x )) → n → ∞ . For a discrete group G we equip the space Prob( G ) of probability measures on G with the metric it inherits as subspace of l ( G ). This metric generates the topologyof point-wise convergence on Prob( G ). We recall the following definition. Definition 2.2.
An action of a group G on a compact space X is said to beamenable if there exists a sequence of almost equivariant maps X → Prob( G ).A group is amenable iff its action on the one point space is amenable. Groupsthat admit an amenable action on a compact Hausdorff space are said to be exact or boundary amenable . The class of exact groups contains all amenable groups,hyperbolic groups [1], and all linear groups [35]. Other prominent groups that areknown to be exact are mapping class groups [39, 48] and the group of outer auto-morphisms of free groups [18]. The Baum-Connes assembly map is split injectivefor all exact groups [40, 79]. This implies the Novikov conjecture for exact groups.This is an analytic result for the Novikov conjecture, in the sense that it has noknown proof that avoids the Baum-Connes Conjecture. There is no correspond-ing injectivity result for assembly maps in algebraic K -theory. For a survey aboutamenable actions and exact groups see [66]. Finite asymptotic dimension.
Results for assembly maps in algebraic K -theoryand L -theory often depend on a finite dimensional setting; the space of probabilitymeasures has to be replaced with a finite dimensional space. We write ∆( G ) for thefull simplicial complex with vertex set G and ∆ ( N ) ( G ) for its N -skeleton. The space∆( G ) can be viewed as the space of probability measures on G with finite support.We equip ∆( G ) with the l -metric; this is the metric it inherits from Prob( G ). Definition 2.3 ( N -amenable action) . We will say that an action of a group G ona compact space X is N -amenable if there exists a sequence of almost equivariantmaps X → ∆ ( N ) ( G ).The natural action of a countable group G on its Stone- ˇCech compactification βG is N -amenable iff the asymptotic dimension of G is at most N [37, Thm. 6.5].This condition (for any N ) also implies exactness and therefore the Novikov con-jecture [42]. For groups G of finite asymptotic dimension for which in addition theclassifying space BG can be realized as a finite CW -complex, there is an alterna-tive argument for the Novikov Conjecture [78] that has been translated to integralinjectivity results for assembly maps in algebraic K -theory and L -theory [3, 22].These injectivity results have seen far reaching generalizations to groups of finitedecomposition complexity [36, 45, 69]. N - F -amenable actions. Constructions of transfer maps in algebraic K -theoryand L -theory often depend on actions on spaces that are much nicer than βG . Agood class of spaces to use for the Farrell-Jones Conjecture are Euclidean retracts,i.e., compact spaces that can be embedded as a retract in some R n . Brouwer’sfixed point theorem implies that for an action of a group on an Euclidean retractany cyclic subgroup will have a fixed point. It is not difficult to check that thisobstructs the existence of almost equivariant maps to ∆ ( N ) (assuming G contains -THEORY AND ACTIONS ON EUCLIDEAN RETRACTS 5 an element of infinite order). Let F be a family of subgroups of G that is closedunder taking supergroups of finite index. Let S := ` F ∈F G/F be the set of allleft cosets to members of F . Let ∆ F ( G ) be the full simplicial complex on S and∆ ( N ) F ( G ) be its N -skeleton. We equip ∆ F ( G ) with the l -metric. Definition 2.4 ( N - F -amenable action) . We will say that an action of G on acompact space X is N - F -amenable if there exists a sequence of almost equivariantmaps X → ∆ ( N ) F ( G ). If an action is N - F -amenable for some N ∈ N , then we saythat it is finitely F -amenable. Remark . Let X be a G - CW -complex with isotropy groups in F and of dimension ≤ N . As ∆ F ( G ) is a model for E F G we obtain a cellular G -map f : X → ∆ ( N ) F ( G );this map is also continuous for the l -metric. In particular, the constant sequence f n ≡ f is almost equivariant. Therefore one can view N - F -amenability for G -spacesas a relaxation of the property of being a G - CW -complex with isotropy in F andof dimension ≤ N .This relaxation is necessary to obtain compact examples and reasonably small F :If a G - CW -complex is compact, then it has only finitely many cells. In particular,for each cell the isotropy group has finite index in G , so F would have to containsubgroups of finite index in G . Theorem 2.6 ([8, 12]) . Suppose that G admits a finitely F -amenable action on aEuclidean retract. Then G satisfies the Farrell-Jones Conjecture relative to F .Remark . The proof of Theorem 2.6 depends on methods from controlled topol-ogy/algebra that have a long history. An introduction to controlled algebra is givenin [67]; an introduction to the proof of Theorem 2.6 can be found in [4]. Here weonly sketch a very special case, where these methods are not needed.Assume that the Euclidean retract is a G - CW -complex X . As pointed outin Remark 2.5 this forces F to contain subgroups of finite index in G . As X iscontractible, the cellular chain complex of X provides a finite resolution C ∗ over Z [ G ] of the trivial G -module Z . Note that in each degree C k = L Z [ G/F i ] isa finite sum of permutation modules with F i ∈ F and F i of finite index in G .For a finitely generated projective R [ G ]-module P we obtain a finite resolution C ∗ ⊗ Z P of P . Each module in the resolution is a finite sum of modules of the form Z [ G/F ] ⊗ Z P with F ∈ F and of finite index in G . Here Z [ G/F ] ⊗ Z P is equippedwith the diagonal G -action and can be identified with the R [ G ]-module obtainedby first restricting P to an R [ F ]-module and then inducing back up from R [ F ] to R [ G ]. In particular, (cid:2) Z [ G/F ] ⊗ Z P (cid:3) ∈ K ( R [ G ]) is in the image of the assemblymap relative to the family F . It follows that [ P ] = P k ( − k [ C k ⊗ Z P ] is also inthe image. Therefore the assembly map H ( E F G ; K R ) → K ( R [ G ]) is surjective.(This argument did not use that F is closed under supergroups of finite index.) Example . Let G be a hyperbolic group. Its Rips complex can be compactifiedto a Euclidean retract [19]. The natural action of G on this compactification isfinitely VCyc-amenable [11].To obtain further examples of finitely F -amenable actions on Euclidean retracts,it is helpful to replace VCyc with a larger family of subgroups F . Groups that actacylindrically hyperbolic on a tree admit finitely F -amenable actions on Euclideanretracts where F is the family of subgroups that is generated by the virtually cyclicsubgroups and the isotropy groups for the original action on the tree [50]. Relativehyperbolic groups and mapping class groups are discussed in Section 4. Remark . A natural question is which groups admit finitely VCyc-amenableactions on Euclidean retracts. A necessary condition for an action to be finitely
BARTELS, A.
VCyc-amenable is that all isotropy groups of the action are virtually cyclic. There-fore, a related question is which groups admit actions on Euclidean retracts suchthat all isotropy groups are virtually cyclic. The only groups admitting such actionsthat I am aware of are hyperbolic groups. In fact, I do not even know whether ornot the group Z admits an action on a Euclidean retract (or on a disk) such thatall isotropy groups are virtually cyclic. There are actions of Z on disks withouta global fixed point. This is a consequence of Oliver’s analysis of actions of finitegroups on disks [65]. On the other hand, there are finitely generated groups forwhich all actions on Euclidean retracts have a global fixed point [2]. Homotopy actions.
There is a generalization of Theorem 2.6 using homotopyactions. In order to be applicable to higher K -theory these actions need to behomotopy coherent. The passage from strict actions to homotopy actions is alreadyvisible in the work of Farrell-Jones where it corresponds to the passage from theasymptotic transfer used for negatively curved manifolds [28] to the focal transferused for non-positively curved manifolds [30]. Definition 2.10 ([75, 76]) . A homotopy coherent action of a group G on a space X is a continuous map Γ : ∞ a j =0 (( G × [0 , j × G × X ) → X such thatΓ( g k , t k , . . . , t , g , x ) = Γ( g k , . . . , g j , Γ( g j − , . . . , x )) t j = 0Γ( g k , . . . , g j g j − , . . . , x ) t j = 1Γ( g k , . . . , t , g , x ) g = e, < k Γ( g k , . . . , t j +1 t j , . . . , g , x ) g j = e, ≤ j < k Γ( g k − , . . . , t , g , x ) g k = e, < kx g = e, k = 0Here Γ( g, − ) : X → X should be thought of the action of g on X , the mapΓ( g, − , h, − ) : [0 , × X → X is a homotopy from Γ( g, − ) ◦ Γ( h, − ) to Γ( gh, − ) andthe remaining data in Γ encodes higher coherences.In order to obtain sequences of almost equivariant maps for homotopy actions itis useful to also allow the homotopy action to vary. Definition 2.11 ( N - F -amenability for homotopy coherent actions) . A sequenceof homotopy coherent actions (Γ n , X n ) of a group G is said to be N - F -amenable ifthere exists a sequence of continuous maps f n : X n → ∆ ( N ) F ( G ) such that for all k and all g k , . . . , g ∈ G sup x ∈ X,t k ,...,t ∈ [0 , d ( f n (Γ( g k , t k , . . . , t , g , x ) , g k · · · g f n ( x )) → n → ∞ . Theorem 2.12 ([8, 76]) . Suppose that G admits a sequence of homotopy coherentactions on Euclidean retracts of uniformly bounded dimension that is finitely F -amenable. Then G satisfies the Farrell-Jones Conjecture relative to F .Remark . Groups satisfying the assumptions of Theorem 2.12 are said to behomotopy transfer reducible in [25]. The original formulations of Theorems 2.6and 2.12 were not in terms of almost equivariant maps, but in terms of certainopen covers of G × X .We recall here the formulation used for actions. (The formulation for homotopyactions is more cumbersome.) A subset U of a G -space is said to be an F -subset ifthere is F ∈ F such that gU = U for all g ∈ F and U ∩ gU = ∅ for all g ∈ G \ F . -THEORY AND ACTIONS ON EUCLIDEAN RETRACTS 7 A collection U of subsets is said to be G -invariant if gU ∈ U for all g ∈ G , U ∈ U .If no point is contained in more than N + 1 members of U , then U is said to be oforder (or dimension) ≤ N . A G -invariant cover by open F -subsets is said to be an F -cover.For a compact G -space X we equip now G × X with the diagonal G -action. For S ⊆ G finite an F -cover of G × X is said to be S -wide (in the G -direction) if ∀ ( g, x ) ∈ G × X ∃ U ∈ U such that gS ×{ x } ⊆ U. Then the action of G on X is N - F -amenable iff for any S ⊂ G finite there exists an S -wide F -cover U for G × X of dimension at most N [37, Prop. 4.5]. A translationfrom covers to maps is also used in [8, 12, 76]. From the point of view of covers (andbecause of the connection to the asymptotic dimension) it is natural to think of the N in N - F -amenable as a kind of dimension for the action of G on X , see [37, 72].A further difference between the formulations used above and in the referencesgiven is that the conditions on the topology of X are formulated differently, butcertainly Euclidean retracts satisfy the condition from [8]. Example . Theorem 2.12 applies to CAT(0)-groups where F = VCyc is thefamily of virtually cyclic subgroups [10, 76]. An application of Theorem 2.12 toGL n ( Z ) will be discussed in Section 4. Remark . There are interesting groups for whichone can deduce the Farrell-Jones Conjecture using Theorems 2.6 (or 2.12) andinheritance properties. However, it is not clear that these methods can accountfor all that is currently known. A third method, going back to work of Farrell-Hsiang [27], combines induction results for finite groups [24, 73] with controlledtopology/algebra. An axiomatization of this method is given in [9]. In impor-tant part of the proof of the Farrell-Jones Conjecture for solvable groups [77] is acombination of this method with Theorem 2.12.
Remark . The K -theory Novikov Conjecture concerns injec-tivity of assembly maps in algebraic K -theory, i.e., lower bounds for the algebraic K -theory of group rings. For the integral group ring of groups, that are only re-quired to satisfy a mild homological finiteness assumption, trace methods have beenused by B¨okstedt-Hsiang-Madsen [20] and L¨uck-Reich-Rognes-Varisco [57] to ob-tain rational injectivity results. The latter result in particular yields interestinglower bounds for Whitehead groups. For the group ring over the ring of Schattenclass operators Yu [80] proved rational injectivity of the Farrell-Jones assembly mapfor all groups. This is the only result I am aware of for the Farrell-Jones Conjecturethat applies to all groups! 3. Flow spaces
The construction of almost equivariant maps often uses the dynamic of a flowassociated to the situation.
Definition 3.1. A flow space for a group G is a metric space FS equipped with aflow Φ and an isometric G -action where the flow and the G -action commute. For α > δ > c, c ′ ∈ FS we write d fol ( c, c ′ ) < ( α, δ )to mean that there is t ∈ [ − α, α ] such that d (Φ t ( c ) , c ′ ) < δ . Example . Let G be the fundamental group of a Riemannian manifold M . Thenthe sphere bundle S ˜ M equipped with the geodesic flow is a flow space for thefundamental group of M . For manifolds of negative or non-positive curvature this BARTELS, A. flow space is at the heart of the connection between K -theory and dynamics usedto great effect by Farrell-Jones.This example has generalizations to hyperbolic groups and CAT(0)-groups. Forhyperbolic groups Mineyev’s symmetric join is a flow space [61]. Alternatively,it is possible to use a coarse flow space for hyperbolic groups, see Remark 3.7below. For groups acting on a CAT(0)-space a flow space has been constructedin [10]. It consists of all parametrized geodesics in the CAT(0)-space (technicallyall generalized geodesics) and the flow acts by shifting the parametrization.Almost equivariant maps often arise as compositions X ϕ −→ FS ψ −→ ∆ ( N ) F ( G ) , where the first map is almost equivariant in an ( α, δ )-sense, and the second map is G -equivariant and contracts ( α, δ )-distances to ε -distances. The following Lemmasummarizes this strategy. Lemma 3.3.
Let X be a G -space, where G is a countable group. Let N ∈ N .Assume that there exists a flow space FS satisfying the following two conditions.(A) For any finite subset S of G there is α > such that for any δ > there isa continuous map ϕ : X → FS such that for x ∈ X , g ∈ S we have d fol ( ϕ ( gx ) , gϕ ( x )) < ( α, δ ) . (B) For any α > , ε > there are δ > and a continuous G -map ψ : FS → ∆ ( N ) F ( G ) such that d fol ( c, c ′ ) < ( α, δ ) = ⇒ d ( ψ ( c ) , ψ ( c ′ )) < ε holds for all c, c ′ ∈ FS .Then the action of G on X is N - F -amenable.Proof. Let S ⊂ G be finite and ε >
0. We need to construct a map f : X → ∆ ( N ) F ( G )for which d ( f ( gx ) , gf ( x )) < ε for all x ∈ X , g ∈ S . Let α be as in (A) with respectto S . Choose now δ > G -map ψ : FS → ∆ ( N ) F ( G ) as in (B). Next choose ϕ : X → FS as in (A) with respect to this δ >
0. Then f := ψ ◦ ϕ has the requiredproperty. (cid:3) Remark ϕ ) . Maps ϕ : X → FS as in condition (A)in Lemma 3.3 can in negatively or non-positively curved situations often be con-structed using dynamic properties of the flow. We briefly illustrate this in a casealready considered by Farrell and Jones.Let G be the fundamental group of a closed Riemannian manifold of strict neg-ative sectional curvature M . Let ˜ M be its universal cover and S ∞ the sphere atinfinity for ˜ M . The action of G on ˜ M extends to S ∞ . For each x ∈ ˜ M there is acanonical identification between the unit tangent vectors at x and S ∞ : every unittangent vector v at x determines a geodesic ray c starting in x , the correspondingpoint ξ ∈ S ∞ is c ( ∞ ). One say that v points to ξ . The geodesic flow Φ t on S ˜ M has the following property. Suppose that v and v ′ are unit tangent vectors at x and x ′ pointing to the same point in S ∞ . Then d fol (Φ t ( v ) , Φ t ( v ′ )) < ( α, δ t ) where α depends only on d ( x, x ) and δ t → v, v ′ (still depending on d ( x, x )).This statement uses strict negative curvature. (For closed manifolds of non-positivesectional curvature the vector v ′ has to be chosen more carefully depending on v and t ; this necessitates the use of the focal transfer [30] respectively the use ofhomotopy coherent actions.)This contracting property of the geodesic flow can be translated into the con-struction of maps as in (A). Fix a point x ∈ ˜ M . Define ϕ : S ∞ → S ˜ M by sending -THEORY AND ACTIONS ON EUCLIDEAN RETRACTS 9 ξ to the unit tangent vector at x pointing to ξ . For t ≥ ϕ t ( ξ ) := Φ t ( ϕ ( ξ )).Using the contracting property of the geodesic flow is not difficult to check that forany g ∈ G there is α > α = d ( gx , x )) such that for any δ > t satisfying ∀ t ≥ t , ∀ ξ ∈ S ∞ d fol ( ϕ t ( gξ ) , gϕ t ( ξ )) < ( α, δ ) . Remark . Of course the space S ∞ used in Remark 3.4 is not contractible andtherefore not a Euclidean retract. But the compactification ˜ M ∪ S ∞ of ˜ M is adisk, in particular a Euclidean retract. As ˜ M has the homotopy type of a free G - CW -complex, there is even a G -equivariant map ˜ M → ∆ ( d ) ( G ) where d is thedimension of M . In particular, the action of G on ˜ M is d -amenable. It is notdifficult to combine the two statements to deduce that the action of G on ˜ M ∪ S ∞ is finitely F -amenable. This is best done via the translation to open covers of G × ( ˜ M ∪ S ∞ ) discussed in Remark 2.13, see for example [72].An important point in the formulation of condition (B) is the presence of δ > FS . If the action of G on FS is cocompact, then a version of theLebesgue Lemma guarantees the existence of some uniform δ >
0, i.e., it sufficesto construct ψ : FS → ∆ ( N ) F ( G ) such that d ( ψ ( c ) , ψ (Φ t ( c )) < ε for all t ∈ [ − α, α ], c ∈ FS . Remark FS ) . Maps ϕ as in condition (B) of Lemma 3.3 arebest constructed as maps associated to long thin covers of the flow space. Theselong thin covers are an alternative to the long thin cell structures employed byFarrell-Jones [28].An open cover U of FS is said to be an α -long cover for FS if for each c ∈ FS there is U ∈ U such that Φ [ − α,α ] ( c ) ⊆ U. It is said to be α -long and δ -thick if for each c ∈ FS there is U ∈ U containingthe δ -neighborhood of Φ [ − α,α ] ( c ). The construction of maps FS → ∆ ( N ) F ( G ) asin condition (B) of Lemma 3.3 amounts to finding for given α an F -cover U of FS of dimension at most N that is α -long and δ -thick for some δ > α ). For cocompact flow spaces such covers can be constructed in relativelygreat generality [11, 46]. Cocompactness is used to guarantee δ -thickness. Fornot cocompact flow spaces on can still find α -long covers, but without a uniformthickness, they do not provide the maps needed in (B). Remark . We outline the construction of the coarse flowspace from [5] for a hyperbolic group G . Let Γ be a Cayley graph for G . Thevertex set of Γ is G . Adding the Gromov boundary to G we obtain the compactspace G = G ∪ ∂G . Assume that Γ is δ -hyperbolic. The coarse flow space CF consists of all triples ( ξ − , v, ξ + ) with ξ ± ∈ G and v ∈ G such that there is somegeodesic from ξ − to ξ + in Γ that passes v within distance ≤ δ . Informally, v coarselybelongs to a geodesic from ξ − to ξ + . The coarse flow space is the disjoint union ofits coarse flow lines CF ξ − ,ξ + := { ξ − }× Γ ×{ ξ + } ∩ CF . The coarse flow lines are arequasi-isometric to R (with uniform constants depending on δ ).There are versions of the long thin covers from Remark 3.6 for CF . For α > U of bounded dimension that are α -long in the direction of thecoarse flow lines: for ( ξ − , v, ξ + ) ∈ CF there is U ∈ U such that { ξ − }× B α ( v ) ×{ ξ + }∩ CF ξ − ,ξ + ⊆ U .There is also a coarse version of the map ϕ t from Remark 3.4. To define it, fixa base point v ∈ G . For t ∈ N , ϕ t sends ξ ∈ ∂G to ( v , v, ξ ) where d ( v , v ) = t and v belongs to a geodesic from v to ξ . It is convenient to extend ϕ t to a map G × ∂G → CF , with ϕ t ( g, ξ ) := ( gv , v, ξ ) where now d ( v , v ) = t and v belongs to a geodesic from gv to ξ . Of course v is only coarsely well defined. Nevertheless, ϕ t can be used to pull long thin covers for CF back to G × ∂G . For S ⊆ G finitethere are then t > α > S -wide covers for G × ∂G . Theproof of this last statement uses a compactness argument and it is important atthis point that Γ is locally finite and that G acts cocompactly on Γ.4. Covers at infinity
The Farrell-Jones Conjecture for GL n ( Z ) . The group GL n ( Z ) is not a CAT(0)-group, but it has a proper isometric action on a CAT(0)-space, the symmetric space X := GL n ( R ) /O ( n ). Fix a base point x ∈ X . For R ≥ B R be the closed ballof radius R around x . This ball is a retract of X (via the radial projection alonggeodesics to x ) and inherits a homotopy coherent action Γ R from the action ofGL n ( Z ) on X . Let F be the family of subgroups generated by the virtually cyclicand the proper parabolic subgroups of GL n ( Z ). The key step in the proof of theFarrell-Jones Conjecture for GL n ( Z ) in [13] is, in the language of Section 2, thefollowing. Theorem 4.1.
The sequence of homotopy coherent actions ( B R , Γ R ) is finitely F -amenable. In particular GL n ( Z ) satisfies the Farrell-Jones Conjecture relative to F by The-orem 2.12. Using the transitivity principle 1.2 the Farrell-Jones Conjecture forGL n ( Z ) can then be proven by induction on n . The induction step uses inheri-tance properties of the Conjecture and that virtually poly-cyclic groups satisfy theConjecture.The verification of Theorem 4.1 follows the general strategy of Lemma 3.3 (ina variant for homotopy coherent actions). The additional difficulty in verifyingassumption (B) is that, as the action of GL n ( Z ) on the symmetric space is notcocompact, the action on the flow space is not cocompact either. The generalresults reviewed in Section 3 can still be used to construct for any α > α -long cover U for the flow space. However it is not clear that the resulting coveris δ -thick, for a δ > FS . The remedy for this short-coming is asecond collection of open subsets of FS . Its construction starts with an F -coverfor X at ∞ , meaning here, away from cocompact subsets. Points in the symmetricspace can be viewed as inner products on R n and moving towards ∞ correspondsto degeneration of inner products along direct summands W ⊂ Z n ⊂ R n . Thisin turn can be used to define horoballs in the symmetric space, one for each W ,forming the desired cover [33]. For each W the corresponding horoball is invariantfor the parabolic subgroup { g ∈ GL n ( Z ) | gW = W } , more precisely, the horoballsare F -subsets, but not VCyc-subsets. The precise properties of the cover at ∞ areas follows. Lemma 4.2 ([13, 33]) . For any α > there exists a collection U ∞ of open F -subsets of X of order ≤ n that is of Lebesgue number ≥ α at ∞ , i.e., there is K ⊂ X compact such that for any x ∈ X \ GL n ( Z ) · K there is U ∈ U ∞ containingthe α -ball B α ( x ) in X around x . This cover can be pulled back to the flow space where it provides a cover at ∞ for the flow space that is both (roughly) α -long and α -thick at ∞ . Then one is leftwith a cocompact subset of the flow space where the cover U constructed first is α -long and δ -thick.This argument for GL n ( Z ) has been generalized to GL n ( F ( t )) for finite fields F , and GL n ( Z [ S − ]), for S a finite set of primes [71] using suitable generalizationsof the above covers at ∞ . In this case the parabolic subgroups are slightly bigger, -THEORY AND ACTIONS ON EUCLIDEAN RETRACTS 11 in particular the induction step (on n ) here uses that the Farrell-Jones Conjectureholds for all solvable groups. Using inheritance properties and building on these re-sults the Farrell-Jones Conjecture has been verified for all subgroups of GL n ( Q ) [71]and all lattices in virtually connected Lie groups [44]. Relatively hyperbolic groups.
We use Bowditch’s characterization of relativelyhyperbolic groups [21]. A graph is fine if there are only finitely many embeddedloops of a given length containing a given edge. Let P be a collection of sub-groups of the countable group G . Then G is hyperbolic relative to P if G admitsa cocompact action on a fine hyperbolic graph Γ such that all edge stabilizers arefinite and all vertex stabilizers belong to P . The subgroups from P are said to beperipheral or parabolic. The requirement that Γ is fine encodes Farb’s BoundedCoset Penetration property [26]. Bowditch assigned a compact boundary ∆ to G as follows. As a set ∆ is the union of the Gromov boundary ∂ Γ with the set of allvertices of infinite valency in Γ. The topology is the observer topology; a sequence x n converges in this topology to x if given any finite set S of vertices (not including x ), for almost all n there is a geodesic from x n to x that misses S . (For generalhyperbolic graphs this topology is not Hausdorff, but for fine hyperbolic graphs itis.)The main result from [5] is that if G is hyperbolic relative to P , then G satisfiesthe Farrell-Jones Conjecture relative to the family of subgroups F generated byVCyc and P ( P needs to be closed under index two supergroups here for thisto include the L -theoretic version of the Farrell-Jones Conjecture). This result isobtained as an application of Theorem 2.6. The key step is the following. Theorem 4.3 ([5]) . The action of G on ∆ is finitely F -amenable. This is a direct consequence of Propositions 4.4 and 4.5 below, using the charac-terization of N - F -amenability from Remark 2.13 by the existence of S -wide coversof G × ∆.To outline the construction of these covers and to prepare for the mapping classgroup we introduce some notation. Pick a G -invariant proper metric on the set E of edges of Γ; this is possible as G and E are countable and the action of G on E has finite stabilizers. For each vertex v of Γ with infinite valency let E v be the setof edges incident to v . Write d v for the restriction of the metric to E v . For ξ ∈ ∆, ξ = v we define its projection π v ( ξ ) to E v as the set of all edges of Γ that are appearas initial edges of geodesics from v to ξ . This is a finite subset of E v (this dependsagain of fineness of Γ). Fix a vertex v of finite valence as a base point. For g ∈ G , ξ ∈ ∆ define their projection distance at v by d πv ( g, ξ ) := d v ( π v ( gv ) , π v ( ξ )) . For ξ = v , set d πv ( g, v ) := ∞ . (For relative hyperbolic groups a related quantityis often called an angle ; the terminology here is chosen to align better with thecase of the mapping class group.) If we vary g (in a finite set) and ξ (in an openneighborhood) then for fixed v the projection distance d πv ( g, ξ ) varies by a boundedamount. Useful is the following attraction property for projection distances: thereis Θ such that if d πv ( g, ξ ) ≥ Θ , then any geodesic from gv to ξ in Γ passes through v . Conversely, if some geodesic from gv to ξ misses v , then d πv ( g, ξ ) < Θ ′ for someuniform Θ ′ .Projection distances are used to control the failure of Γ to be locally finite.In particular, provided all projection distances are bounded by a constant Θ, avariation of the argument for hyperbolic groups (using a coarse flow space), can beadapted to provide S -long covers for the Θ-small part of G × ∆. The following is aprecise statement. Proposition 4.4.
There is N (depending only on G and ∆ ) such that for any Θ > and any S ⊆ G finite there exists a collection U of open VCyc -subsets of G × ∆ that is S -wide on the Θ -small part, i.e., if ( g, ξ ) ∈ G × ∆ satisfies d πv ( g, ξ ) ≤ Θ for all vertices v , then there is U ∈ U with gS ×{ ξ } ⊆ U . To deal with large projection distances an explicit construction can be used(similar to the case of GL n ( Z )). For ( g, ξ ) ∈ G × ∆ let V Θ ( g, ξ ) := { v | d πv ( g, ξ ) ≥ Θ } . As a consequence of the attraction property, for sufficiently large Θ, the set V Θ ( g, ξ )consists of vertices that belong to any geodesic from gv to ξ . In particular, it canbe linearly ordered by distance from gv .For a fixed vertex v and Θ > W ( v, Θ) ⊂ G × ∆ as the (interior of the)set of all pairs ( g, ξ ) for which v is minimal in V Θ ( g, ξ ), i.e., v is the vertex closestto gv for which d πv ( g, ξ ) ≥ Θ. Then W (Θ) := { W ( v, Θ) | v ∈ V } is a collection ofpairwise disjoint open P -subsets of G × ∆. Proposition 4.5.
Let S ⊂ G be finite. Then there are θ ′′ ≫ θ ′ ≫ θ ≫ such that W ( θ ) ∪ W ( θ ′ ) is a G -invariant collection of open P -subsets of order ≤ that is S -long on the θ ′′ -large part of G × ∆ : if d πv ( g, ξ ) ≥ θ ′′ for some vertex v , then thereis W ∈ W ( θ ) ∪ W ( θ ′ ) such that gS ×{ ξ } ⊂ W . A difficulty in working with the W ( v, Θ) is that it is for fixed Θ not possibleto control exactly how V Θ ( g, ξ ) varies with g and ξ . In particular whether or nota vertex v is minimal in V Θ ( g, ξ ) can change under small variation in g or ξ . Aconsequence of the attraction property that is useful for the proof of Proposition 4.5is the following: suppose there are vertices v and v with d πv i ( g, ξ ) > Θ ′ ≫ Θ, thenthe segment between v and v in the linear order of V Θ ( g, ξ ) is unchanged undersuitable variations of ( g, ξ ) depending on Θ ′ . Remark . A motivating example of relatively hyperbolic groups are fundamen-tal groups G of complete Riemannian manifolds M of pinched negative sectionalcurvature and finite volume. These are hyperbolic relative to their virtually finitelygenerated nilpotent subgroups [21, 26]. In this case we can work with the sphereat ∞ of the universal cover ˜ M of M . The splitting of G × S ∞ into a Θ-small partand a Θ-large part can be thought of as follows. Fix a base point x ∈ ˜ M . In-stead of a number Θ we choose a cocompact subset G · K of ˜ M . The small partof G × ∆ consists then of all pairs ( g, ξ ) for which the geodesic ray from gx to ξ iscontained in X ; the large part is the complement. Under this translation the coverfrom Proposition 4.5 can again be thought of as a cover at ∞ for ˜ M . Moreover,the vertices of infinite valency in Γ correspond to horoballs in ˜ M , and projectiondistances to time geodesic rays spend in horoballs.Note that the action of G on the graph Γ in the definition of relative hyperbolicitywe used is cocompact, but Γ is not a proper metric space. Conversely, in theabove example the action of G on ˜ M is no longer cocompact, but now ˜ M is aproper metric space. A similar trade off (cocompact action on non proper spaceversus non-cocompact action on proper space) is possible for all relatively hyperbolicgroups [34], assuming the parabolic subgroups are finitely generated. The mapping class group.
Let Σ be a closed orientable surface of genus g with afinite set P of p marked points. We will assume 6 g + 2 p − >
0. The mapping classgroup Mod(Σ) of Σ is the group of components of the group of orientation preservinghomeomorphisms of Σ that leave P invariant. Teichm¨uller space T is the space ofmarked complete hyperbolic structures of finite area on Σ \ P . The mapping classgroup acts on Teichm¨uller space by changing the marking. Thurston [32] defined -THEORY AND ACTIONS ON EUCLIDEAN RETRACTS 13 an equivariant compactification of Teichm¨uller space T . As a space T is a closeddisk, in particular it is an Euclidean retract. The boundary of the compactification PMF := T \ T is the space of projective measured foliations on Σ. The key stepin the proof of the Farrell-Jones Conjecture for Mod(Σ) is the following.
Theorem 4.7 ([6]) . Let F be the family of subgroups of Mod(Σ) that virtually fixa point in
PMF . The action of
Mod(Σ) on PMF is finitely F -amenable. From this it follows quickly that the action on T is finitely F -amenable as well,and applying Theorem 2.6 we obtain the Farrell-Jones Conjecture for Mod(Σ) rel-ative to F . Up to passing to finite index subgroups, the groups in F are centralextensions of products of mapping class groups of smaller complexity. Using thetransitivity principle and inheritance properties one then obtains the Farrell-JonesConjecture for Mod(Σ) by induction on the complexity of Σ. The only additionalinput in this case is that the Farrell-Jones Conjecture holds for finitely generatedfree abelian groups.The proof of Theorem 4.7 uses the characterization of N - F -amenability fromRemark 2.13 and provides suitable covers for Mod(Σ) ×PMF . Similar to the relativehyperbolic case the construction of these covers is done by splitting Mod(Σ) ×PMF into two parts. Here it is natural to refer to these parts as the thick part and thethin part. (The thick part corresponds to the Θ-small part in the relative hyperboliccase.)Teichm¨uller space has a natural filtration by cocompact subsets. For ε > ε -thick part T ≥ ε ⊆ T consist of all marked hyperbolic structures such thatall closed geodesics have length ≥ ε . The action of Mod(Σ) on T ≥ ε is cocom-pact [64]. Fix a base point x ∈ T . Given a pair ( g, ξ ) ∈ Mod(Σ) ×PMF there is aunique Teichm¨uller ray c g, that starts at g ( x ) and is “pointing towards ξ ” (tech-nically, the vertical foliation of the quadratic differential is ξ ). The ε -thick part ofMod(Σ) ×PMF is defined as the set of all pairs ( g, ξ ) for which the Teichm¨uller ray c g,ξ stays in T ≥ ε .An important tool in covering both the thick and the thin part is the complex ofcurves C (Σ). A celebrated result of Mazur-Minsky is that C (Σ) is hyperbolic [59].Klarreich [49] studied a coarse projection map π : T → C (Σ) and identified theGromov boundary ∂ C (Σ) of the curve complex. In particular, the projection maphas an extension π : PMF → C (Σ) ∪ ∂ C (Σ). (On the preimage of the Gromovboundary this extension is a continuous map; on the complement it is still only acoarse map.)Teichm¨uller space is not hyperbolic, but its thick part T ≥ ε has a number ofhyperbolic properties: The Masur criterion [58] implies that for ( g, ξ ) in the thickpart, c g,ξ ( t ) → ξ as t → ∞ . Moreover, the restriction of Klarreichs projectionmap π : PMF → C (Σ) ∪ ∂ C (Σ) to the space of all such ξ is injective. A result ofMinsky [63] is that geodesics c that stay in T ≥ ε are contracting. This is a propertythey share with geodesics in hyperbolic spaces: the nearest point projection T → c maps balls disjoint from c to uniformly bounded subsets. Teichm¨uller geodesics inthe thick part T ≥ ε project to quasi-geodesics in the curve complex with constantsdepending only on ε . All these properties eventually allow for the construction ofsuitable covers of any thick part using a coarse flow space and methods from thehyperbolic case. A precise statement is the following. Proposition 4.8.
There is d such that for any ε > and any S ⊂ Mod(Σ) finite there exists a
Mod(Σ) -invariant collection U of F -subsets of Mod(Σ) ×PMF of order ≤ d such for any ( g, ξ ) for which c g,ξ stays in T ≥ ε there is U ∈ U with gS ×{ ξ } ⊆ U . The action of the mapping class group on the curve complex C (Σ) does notexhibit the mapping class group as a relative hyperbolic group in the sense discussedbefore; the 1-skeleton of C (Σ) is not fine. Nevertheless, there is an importantreplacement for the projections to links used in the relatively hyperbolic case, thesubsurface projections of Masur-Minsky [60]. In this case the projections are notto links in the curve complex, but to curve complexes C ( Y ) of subsurfaces Y of Σ.(On the other hand, often links in the curve complex are exactly curve complexes ofsubsurfaces.) The theory is however much more sophisticated than in the relativelyhyperbolic case. Projections are not always defined; sometimes the projection isto points in the boundary of C ( Y ) and the projection distance is ∞ . Bestvina-Bromberg-Fujiwara [17] used subsurface projections to prove that the mappingclass group has finite asymptotic dimension. In their work the subsurfaces of Σare organized in a finite number N of families Y such that two subsurfaces in thesame family will always intersect in an interesting way. This has the effect that theprojections for subsurfaces in the same family interact in a controlled way with eachother. Each family Y of subsurfaces is organized in [17] in an associated simplicialcomplex, called the projection complex. The vertices of the projection complex arethe subsurfaces from Y . A perturbation of the projection distances can thoughtof as being measured along geodesics in the projection complex and now behavesvery similar as in the relative hyperbolic case, in particular the attraction propertyis satisfied in each projection complex. This allows the application of a variant ofthe construction from Proposition 4.5 for each projection complex that eventuallyyield the following. Proposition 4.9.
Let Y be any of the finitely many families of subsurfaces. Forany S ⊆ G finite there exists Θ > and a Mod(Σ) -invariant collection U of F -subsets of Mod(Σ) ×PMF of order ≤ such for any ( g, ξ ) for which there is Y ∈ Y with d πY ( g, ξ ) ≥ Θ there is U ∈ U with gS ×{ ξ } ⊆ U . The final piece, needed to combine Propositions 4.8 and 4.9 to a proof of Theo-rem 4.7, is a consequence of Rafi’s analysis of short curves in Σ along Teichm¨ullerrays [68]: for any ε > ε -short on c g,ξ (i.e.,if c g,ξ is not contained in T ≥ ε ) then there is a subsurface Y such that d πY ( g, ξ ) ≥ Θ. Remark . Farrell-Jones [31] proved topological rigidity results for fundamentalgroups of non-positively curved manifolds that are in addition A -regular. Thelatter condition bounds the curvature tensor and its covariant derivatives over themanifold. All torsion free discrete subgroups of GL n ( R ) are fundamental groups ofsuch manifolds. Similar to the examples discussed in this section a key difficultyin [31] is that the action of the fundamental group G of the manifold on the universalcover is not cocompact. The general strategy employed by Farrell-Jones seemshowever different, in particular, it does not involve an induction over some kindof complexity of G . The only groups that are considered in an intermediate stepare polycyclic groups, and the argument directly reduces from G to these and thenuses computations of K -and L -theory for polycyclic groups.This raises the following question: Can the family of subgroups in Theorem 4.1be replaced with the family of virtually polycyclic subgroups? Recall that thecover constructed for the flow space at ∞ is both α -long and α -thick, while only δ -thickness is needed. So it is plausible that there exist thinner covers at ∞ thatwork for the family of virtually polycyclic subgroups.For the mapping class group the family from Theorem 4.7 can not be chosento be significantly smaller; all isotropy groups for the action have to appear in -THEORY AND ACTIONS ON EUCLIDEAN RETRACTS 15 the family. But one might ask, whether there exist N - F -amenable actions (or N - F -amenable sequences of homotopy coherent actions) of mapping class groups onEuclidean retracts, where F is smaller than the family used in Theorem 4.7. References [1] S. Adams. Boundary amenability for word hyperbolic groups and an application to smoothdynamics of simple groups.
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