aa r X i v : . [ m a t h . L O ] J a n LOGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG)
EDITOR: ANDR´E NIES
Contents
Part 1. Group theory and its connections to logic
21. Nies and Stephan: non-word automatic free nil-2 groups 22. Nies: open questions on classes of closed subgroups of S ∞
43. Ivanov and Majcher: amenable subgroups of S ( ω ) 84. Nies: Stone-type duality for totally disconnected locally compactgroups 115. Nies: Closed subgroups of S ( ω ) generated by their permutationsof finite support 226. Harrison-Trainor and Nies: Π r -pseudofinite groups 25 Part 2. Computability theory and randomness
Part 3. Set theory • rapidly announcing results and questions related to logic • putting up results and their proofs for further research • parking results for later use • getting feedback before submission to a journal • foster collaboration.Each year’s blog is posted on arXiv a 2-3 months after the year has ended.Logic Blog 2019 (Link: http://arxiv.org/abs/2003.03361 )Logic Blog 2018 (Link: http://arxiv.org/abs/1902.08725 )Logic Blog 2017 (Link: http://arxiv.org/abs/1804.05331 )Logic Blog 2016 (Link: http://arxiv.org/abs/1703.01573 )Logic Blog 2015 (Link: http://arxiv.org/abs/1602.04432 )Logic Blog 2014 (Link: http://arxiv.org/abs/1504.08163 )Logic Blog 2013 (Link: http://arxiv.org/abs/1403.5719 ) Logic Blog 2012 (Link: http://arxiv.org/abs/1302.3686 )Logic Blog 2011 (Link: http://arxiv.org/abs/1403.5721 )Logic Blog 2010 (Link: http://dx.doi.org/2292/9821 ) How does the Logic Blog work?Writing and editing.
The source files are in a shared dropbox. Ask Andr´e([email protected]) in order to gain access.
Citing.
Postings can be cited. An example of a citation is:H. Towsner,
Computability of Ergodic Convergence . In Andr´e Nies (editor),Logic Blog, 2012, Part 1, Section 1, available at http://arxiv.org/abs/1302.3686 .The logic blog, once it is on arXiv, produces citations e.g. on GoogleScholar.
Part Group theory and its connections to logic Nies and Stephan: non-word automatic free nil-2 groups
We follow up on a post on word automaticity of various nilpotent groupsby the same guys last year [10, Section 6]. To be word automatic meansthat given an appropriate encoding of the elements by strings, the domainand operations are recognizable by finite automata. Other terms in use forthis notion include FA-presentable, and even “automatic”.We supply a proof that examples of the last of three types that wereprovided there are not word-automatic. The first two types were shown tobe word automatic there. For an odd prime p , let L p be the free nilpotent-2exponent p group of infinite rank. In the notation of [10, Section 6] one has N ∼ = Q ; let y i,k ( i < k ) be free generators of N ; then L p is given by thefunction φ ( x i , x k ) = y i,k if i < k . So we have L p = h x i , y i,k | x pi , [ x i , x k ] y − i,k , [ y i,k , x r ]( i < k ) i thus the y i,k are actually redundant. This group in a sense generalises theHeisenberg group over the ring F p , which would be the case of two generators x , x . Theorem 1.1. L p is not word automatic.Proof. Let α, β denote strings in the alphabet of digits 0 , . . . , p −
1, whichare thought of as extended by 0’s if necessary. Let [ α ] = Q i< | α | x α i i . Eachelement of L p has a normal form[ α ] Q s For each finite set S ⊆ D , there is a (cid:22) -least string u = u S ∈ D such that (i) ∀ r [ r ∈ S − C ⇒ [ r, u ] S ] . (ii) ∀ v, w ∈ S ([ v, u ] = [ w, u ] → ∃ c ∈ C cv = w ) . To see this, let k be so large that only x i with i < k occur in the normalform of any element of S , and let u = x k .For (i) note that r contains some x i with i < k , so the normal form of[ r, u ] contains [ x i , x k ], while the normal form of an element of S does notcontain such commutators.For (ii) let v = [ α ] c with c central. Then the normal form of [ v, u ] endsin Q i For each n , the subgroup h u , . . . , u n i generated by u , . . . , u n is contained in V n +1 . This is checked by induction on n . For n = 0 we have h u i ⊆ V because w = v = 1 is allowed in (1.1). For the inductive step, note that by thenormal form (and freeness of L p ) each element y of h u , . . . , u n i has theform Q i ≤ n u α i i Q s ≤ n Q r The elements Cu i are linearly independent in G/C . (b) The elements u i,k are linearly independent in C . EDITOR: ANDR´E NIES We use induction on a bound n for the indices. For (a) note that u in +1 V n C for i < p , for otherwise v = u in +1 c ∈ V n for some central c , andclearly [ v, u n +1 ] = 1 contradicting the condition (i) in Claim 1.2. Therefore C h u n +1 i ∩ C h u , . . . , u n i = 0.For (b), inductively the u i,k , i < k ≤ n are a basis for a subspace T ≤ C .The linear map given by Cw [ w, u n +1 ] is 1-1 by (ii) of Claim 1.2. So, by(a), the [ u i , u n +1 ] are independent, generating a subspace T . The condition(i) of Claim 1.2 implies that T ∩ T = 0. This concludes the inductive stepand verifies the claim.We now obtain our contradiction. Given n , by Claim 1.3 we have Q i Given a Polish space X , let F ( X )denote the set of closed subsets of X . The Effros structure on X is the Borelspace consisting of F ( X ) together with the σ -algebra generated by the sets C U = { D ∈ F ( X ) : D ∩ U = ∅} ,for open U ⊆ X . Clearly it suffices to take all the sets U in a countablebasis h U i i i ∈ N of X . The inclusion relation on F ( X ) is Borel because for C, D ∈ F ( X ) we have C ⊆ D ↔ ∀ i ∈ N [ C ∩ U i = ∅ → D ∩ U i = ∅ ]. Representing elements of the Effros structure of S ( ω ) . For a Polishgroup G , we have a Borel actions G y F ( G ) by translation and by conjuga-tion. We will only consider the case that G = S ( ω ). In the following σ, τ, ρ OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 5 will denote injective maps on initial segments of the integers, that is, on tu-ples of integers without repetitions. Let [ σ ] denote the set of permutationsextending σ : [ σ ] = { f ∈ S ( ω ) : σ ≺ f } (this is often denoted N σ in the literature). The sets [ σ ] form a base forthe topology of pointwise convergence of S ( ω ). For f ∈ S ( ω ) let f ↾ n bethe initial segment of f of length n . Note that the [ f ↾ n ] form a basis ofneighbourhoods of f . Given σ, σ ′ let σ ′ ◦ σ be the composition as far as itis defined; for instance, (7 , , , , ◦ (3 , , 6) = (1 , σ − bethe inverse of σ as far as it is defined. Definition 2.1. For n ≥ 0, let τ n denote the function τ defined on { , . . . , n } such that τ ( i ) = i for each i ≤ n . Definition 2.2. For P ∈ F ( S ( ω )), by T P we denote the tree describing P asa closed set in the sense that [ T P ] ∩ S ( ω ) = P . Note that T P = { σ : P ∈ C [ σ ] } . Lemma 2.3. The closed subgroups of S ( ω ) form a Borel set U ( S ( ω )) in F ( S ( ω )) .Proof. D ∈ F ( S ( ω )) is a subgroup iff the following three conditions hold: • D ∈ C [(0 , ,...,n − for each n • D ∈ C [ σ ] → D ∈ C [ σ − ] for all σ • D ∈ C [ σ ] ∩ C [ τ ] → D ∈ C [ τ ◦ σ ] for all σ, τ .It now suffices to observe that all three conditions are Borel. (cid:3) Note that U ( S ( ω )) is a standard Borel space. The statement of the lemmaactually holds for each Polish group in place of S ( ω ).So much for the preliminaries. We now note any known results on thecomplexity of isomorphism. Let G always denote a closed subgroup of S ( ω )(another, and cumbersome, but persistent, term is non-Archimedean group).By ‘group’ we usually mean such a G . The class of all closed subgroups of S ( ω ) . It is not hard to verify that theisomorphism problem is analytic [25]. It is Borel above ( ≥ B ) graph iso-morphism GI. Nothing else appears to be known on its complexity. Thefollowing was asked in Kechris et al. [25]. Question 2.4. Is the isomorphism relation between closed subgroups of S ( ω ) analytic complete? The isomorphism problem for abelian Polish groups is known to be ana-lytic complete [11], but the groups used there are not non-Archimedean (areArchimedean?). Discrete groups. Discreteness is Borel because it is equivalent to saying thatthe neutral element is isolated. Note that G is discrete iff G is countable.The isomorphism relation for discrete groups is Borel equivalent to graphisomorphism. The upper bound is fairly standard, the lower bound is ob-tained by A. Mekler’s technique [31] encoding countable graph isomorphism(for a sufficiently rich class of graphs) into isomorphism of countable nil-2groups of exponent p , where p is some fixed odd prime. EDITOR: ANDR´E NIES Procountable groups. The following are equivalent (see e.g. Malicki [29, Lemma 1]):(i) G is procountable, i.e., an inverse limit of a chain of countable groups G n +1 → G n (ii) G is a closed subgroup of a Cartesian product of discrete groups(iii) there is nbhd base of the neutral element consisting of open normalsubgroups.It is also equivalent to ask that(iv) G has a compatible bi-invariant metric (such a metric will be neces-sarily complete because G is a Polish group).This class includes the abelian closed subgroups of S ( ω ) (see below), andof course the discrete groups. So graph isomorphism GI can be reduced toits isomorphism problem. Abelian groups. To be abelian is easily seen to be Borel because G is abelian iff ∀ σ, τ ∈ T G [ σ − τ − στ ≺ id ω ].(In fact any variety of groups is Borel, by a similar argument.) As a nonlo-cally compact example, consider Z ω . Also there is a universal abelian closedsubgroup of S ( ω ).Each abelian closed subgroup of S ( ω ) has an invariant metric and hence ispro-countable by Malicki [30, Lemma 2], also Malicki [29, Lemma 1]. Su Gao(On Automorphism Groups of Countable Structures, JSL, 1998) has provedthat if Aut( M ) is abelian (or merely solvable) for a countable structure M ,then the Scott sentence of M has no uncountable model. Topologically finitely generated groups. This property is analytical. Thesymmetric group S ( ω ) is f.g. because the group of permutations with finitesupport is dense in S ( ω ), and is contained in a 2-generated group, namely thegroup generated by the successor function on Z and the transposition (0 , Question 2.5. Is being topologically finitely generated Borel?Given k ≥ , is being k -generated Borel? We observe that among the compact groups, being k -generated is Borel. Forin a Borel way we can represent G as proj lim n ∈ N G n for discrete finite groups G n (with some unnamed projections G n +1 → G n ). Then G is k -generatediff each G n is k -generated: for the nontrivial implication, for each n let a n be a k -tuple of generators for G n . Now take a converging subsequence of asequence of pre-images of the a n in G n . The limit generates G .Being 1-generated (monothetic) is Borel, because as it is well known, sucha group is either discrete, or an inverse limit of cyclic groups (e.g. ( Z p , +))and hence compact. See e.g. Malicki [29, Lemma 5]. Hewitt and Ross intheir book have a more detailed structure theorem for such groups.Among the abelian groups, being f.g. is Borel because such a group is pro-countable. If it is k -generated, it has to be an inverse limit of k -generatedcountable abelian groups. An onto map Z s → Z s is of course a bijection. So G is k -generated iff G is of the form Z r × H where H is a product of k − r procyclic groups. This condition is Borel.Outside the abelian, it is easy to provide an example of a complicatedpro-countable 2-generated group. In the free group F ( a, b ) let v z = b − z ab z OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 7 ( z ∈ Z ). For k ∈ N + let N k ≤ F ( a, b ) be the normal subgroup generatedby commutators [ v , v r ] , r ≥ k . Then N > N > . . . and T k N k = { } .Let G be the inverse limit of the system h F ( a, b ) /N k i k ∈ N with the naturalprojections. Compactly generated groups. One says that G is compactly generated ifthere is a compact subset S that topologically generates G . Note that if G is locally compact, it has a compact open subgroup K (van Dantzig), so thecompact subset KS generates G algebraically. Fact 2.6. For locally compact groups G ≤ c S ( ω ) , being compactly generatedis Borel. To see this, note that if G is c.g. iff it is topologically generated by acompact open set C . Such a set C is given as a finite union of sets [ T G ] ∩ [ σ ]for strings on T G , and we can describe arithmetically whether a set [ T G ] ∩ [ σ ]is compact. So we have to express that there is C such that for each η on T G , there is a term t and finitely many σ i with [ σ i ] ∩ T G ⊆ C so that t applied to the σ i yields an extension of η . This is arithmetical. Question 2.7. Is being (topologically) compactly generated Borel?Oligomorphic groups. To be oligomorphic means that for each n there areonly finitely many n -orbits. Equivalently the orbit structure M G is ω -categorical. By Coquand’s work (elaborated in Ahlbrandt and Ziegler [2])oligomorphic groups G, H are isomorphic iff M G and M H are bi-interpretable.Nies, Schlicht and Tent [37] have proved that isomorphism of oligomorphicgroups is ≤ B E ∞ , the universal countable Borel equivalence relation. So it isway below graph isomorphism. In fact it is unknown to be nonsmooth. ByHarrington-Kechris-Louveau, nonsmoothness is equivalent to an affirmativeanswer to the following. Question 2.8. Is E ≤ B isomorphism of oligomorphic groups?Amenable groups (with A. Iwanow and B. Majcher). Recall that a Polishgroup G is amenable if each compact space it acts on has an invariantprobability measure. G is extremely amenable if each such action has afixed point (so the point mass on it is the required probability measure).For discrete groups, this is equivalent to the usual Folner condition.Discrete amenable group form a Borel subset. For, applying Kuratowski-Ryll-Nardzewski selectors the Folner condition can be presented in a Borelform.Nilpotent groups are amenable. Thus, Mekler’s result can be also ap-plied to the isomorphism relation of discrete amenable groups, making itequivalent to GI.Amenability (without the assumption of discreteness) is Borel by its char-acterisation due to Schneider and Thom [43]. The description is more lessin the style of Folner.For more detail, including extreme amenability, see Section 3. Maximal-closed groups. Fixing some bijection Q n ↔ ω , the group AGL( Q n )of affine linear transformations can be seen as a closed subgroup of S ( ω ).Kaplan and Simon [24] showed that it is a maximal closed subgroup (that EDITOR: ANDR´E NIES is also countable). Agarwal and Kompatscher [1] have provided continuummany maximal-closed groups that are not even algebraically isomorphic,using “Henson digraphs” that were introduced in a paper of Henson.Clearly being maximal-closed is Π . It is not known to be Borel. Recursion theoretic view. The Effros space is insufficient here. We needa more concise way to represent closed subgroups of S ( ω ). They are givenby trees without dead ends satisfying a certain Π condition.Let T be the tree of all pairs h σ, σ ′ i of the same length n such that σ ( i ) = k ↔ σ ′ ( k ) = i for each i, k < n . In other words, there is f ∈ S ( ω )such that σ ≺ f and σ ′ ≺ f − .If B is a subtree of T without dead ends, then for each h f, f ′ i ∈ [ B ], f isa permutation of ω with inverse f ′ . We can formulate as a Π condition on B that { f : h f, f − i ∈ [ B ] } is closed under inverses and product.If B is a computable tree we say that the group given by [ B ] is computable. Question 2.9. Are there two compact, computably isomorphic computablesubgroups of S ( ω ) such that no computable copies are conjugate via a com-putable permutation of S ( ω ) ?. Ivanov and Majcher: amenable subgroups of S ( ω )In this post we show that the properties of being amenable and extremelyamenable for Polish groups are Borel.Given a Polish space Y let F ( Y ) denote the set of closed subsets of Y .The Effros structure on F ( Y ) is the Borel space with respect to the σ -algebragenerated by the sets C U = { D ∈ F ( Y ) : D ∩ U = ∅} , for open U ⊆ Y . For various Y this space serves for analysis of Borelcomplexity of families of closed subsets (see [25] for some recent results).It is convenient to use the fact that there is a sequence of Kuratowski-Ryll-Nardzewski selectors (selectors, in brief) s n : F ( Y ) → Y , n ∈ ω ,which are Borel functions such that for every non-empty F ∈ F ( Y ) the set { s n ( F ); n ∈ ω } is dense in F .We consider S ( ω ) as a complete metric space by defining d ( g, h ) = X { − n | g ( n ) = h ( n ) or g − ( n ) = h − ( n ) } . Let S < ∞ denote the set of all bijections between finite substes of ω . Let S + < ∞ = { σ ∈ S < ∞ | dom [ σ ] is an initial segment of ω } . The family {N σ | σ ∈ S + < ∞ } is a basis of the Polish topology of S ( ω ).We mention here that the set U ( S ( ω )) of all closed subgroups of S ( ω ) isa Borel subset of F ( S ( ω )) (see Lemma 2.5 of [25]).Since S ( ω ) is a Polish group (in particular the multiplication is contin-uous) we may extend the set of selectors s n , n ∈ ω , by group words ofthe form w (¯ s ) which define Borel maps F ( S ( ω )) → S ( ω ) and respectively OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 9 U ( S ( ω )) → S ( ω ). In particular for any closed G ≤ S ( ω ) all w (¯ s )( G ) forma dense subgroup. Below for simplicity we will always assume that alreadyall s n ( G ), n ∈ ω , form a dense subgroup of G .3.1. Closed subgroups and amenability. In this section we apply thedescription of amenable topological groups found by F.M. Schneider and A.Thom in [44] in order to analyse amenability for closed subgroups of S ( ω ).Let G be a topological group, F , F ⊂ G are finite and U be an identityneighbourhood. Let R U be a binary relation defined as follows: R U = { ( x, y ) ∈ F × F : yx − ∈ U } . This relation defines a bipartite graph on ( F , F ). Let µ ( F , F , U ) = | F | − sup {| S | − | N R ( S ) | : S ⊆ F } , where N R ( S ) = { y ∈ F : ( ∃ x ∈ S )( x, y ) ∈ R U } . By Hall’s matching theo-rem this value is the matching number of the graph ( F , F , R U ). Theorem4.5 of [44] gives the following description of amenable topological groups.Let G be a Hausdorff topological group. The following are equivalent.(1) G is amenable.(2) For every θ ∈ (0 , E ⊆ G , and every identityneighbourhood U , there is a finite non-empty subset F ⊆ G such that ∀ g ∈ E ( µ ( F, gF, U ) ≥ θ | F | ) . (3) There exists θ ∈ (0 , 1) such that for every finite subset E ⊆ G , and everyidentity neighbourhood U , there is a finite non-empty subset F ⊆ G suchthat ∀ g ∈ E ( µ ( F, gF, U ) ≥ θ | F | ) . It is worth noting here that when an open neighbourhood V contains U the number µ ( F, gF, U ) does not exceed µ ( F, gF, V ). In particular in theformulation above we may consider neighbourhoods U from a fixed base ofidentity neighbourhoods. For example in the case of a closed G ≤ S ( ω ) wemay take all U in the form of stabilizers V [ n ] = { f ∈ G : f ( i ) = i for i < n } .It is also clear that we can restrict all θ by rational numbers. From now onwe work in this case. Theorem 3.1. The class of all amenable closed subgroups of S ( ω ) is Borel.Proof. Since the family of all closed subgroups of S ( ω ) is Borel it suffices toprove the following claim.CLAIM. For every basic open neighbourhood U of the unity, any rational θ ∈ (0 , 1) and any pair of tuples ¯ s and ¯ s ′ of selectors the family of all closed Z ⊆ S ( ω ) with the condition ∀ g ∈ ¯ s ( Z )( µ (¯ s ′ ( Z ) , g ¯ s ′ ( Z ) , U ) ≥ θ | ¯ s ′ ( Z ) | )is Borel.Indeed, let us denote the condition of the claim by F ø( U, θ, ¯ s, ¯ s ′ ). Thenhaving Borelness as above we see that the (countable) intersection by all U , θ and ¯ s of the families [ {{ G ≤ c S ( ω ) : G | = F ø( U, θ, ¯ s, ¯ s ′ ) } : ¯ s ′ is a tuple of selectors } is also Borel. Note that this family exactly consists of closed subgroups G having dense sabgroups satisfying condition (2) of Schneider-Thom’s the-orem. It is well-known that groups having dense amenable subgroups areamenable. In particular we see that the claim above implies the theorem.Let us prove the claim. For a closed Z ⊆ S ( ω ), g ∈ ¯ s ( Z ) and F = { f , . . . , f k } consisting of entries of ¯ s ′ ( Z ) to guarantee the inequality µ ( F, gF, U ) ≥ θ | F | we only need to demand that for every S ⊆ F the following inequalityholds: | S | − k + θ · k ≤ | N R ( S ) | , where N R ( S ) is defined with respect to ( F, gF ) and U . To satisfy thisinequality we will use the observation that when S ′ ⊆ gF and ρ is a function S ′ → S such that gf ( ρ ( gf )) − ∈ U for each gf ∈ S ′ then | S ′ | ≤ | N R ( S )) | .The following condition formalizes µ ( F, gF, U ) ≥ θ | F | : ^ S ⊆ F _ { ^ gf ∈ S ′ ( gf ( ρ ( gf )) − ∈ U ) : S ′ ⊆ gF , ρ : S ′ → S , | S | − k + θ · k ≤ | S ′ |} . By the choice of g and F we see that all closed Z ⊆ S ( ω ) satisfying it forma Borel family. (cid:3) Let U be the Urysohn space. By [47] every Polish group is realized as aclosed subgroup of Iso ( U ). Applying the proof given above to F ( Iso ( U ))we obtain the following corollary. Corollary 3.2. The class of all amenable closed subgroups of Iso ( U ) isBorel. Closed subgroups and extreme amenability. Let G be a topologi-cal group. The group G is said to be extremely amenable if every continuousaction of G on a non-empty compact Hausdorff space admits a fixed point.We begin by fixing a left-invariant metric d inducing the topology of S ( ω )(resp. Iso ( U )). Recall from ([42], Theorem 2.1.11) that G ≤ c S ( ω ) isextremely amenable if and only if the left-translation action of G on ( G, d )is finitely oscillation stable. From ([42], Theorem 1.1.18) and ([32], proof ofTheorem 3.1) this is equivalent to the following condition:For any ε > F ⊂ G there exists a finite K ⊆ G such that for any function c : K → { , } there exists i ∈ { , } and g ∈ G such that for any f ∈ F there exists k ∈ c − ( i ) with d ( gf, k ) < ε .We consider the case when G has a countable base of the topology. Bythe definition of extreme amenability if G has a dense subgroup which isextremely amenable, then G is extremely amenable too. Now it is easy tosee that when D ⊆ G is a countable dense subgroup of G then extremeamenability of G is equivalent to condition above for the elements taken in D .We now see that when G ∈ F ( S ( ω )) (resp. F ( Iso ( U ))), extreme amenabil-ity of G is equivalent to a countable conjunction of the folowing conditions. OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 11 Let ¯ s be a tuple of selectors and ε ∈ Q + . Then there is aselectors ¯ t such that for any function c : ¯ t → { , } thereexists i ∈ { , } and a selector s ′ such that for any s ∈ ¯ s there exists k ∈ c − ( i ) with d ( s ′ ( G ) s ( G ) , k ( G )) < ε .We see that extreme amenability is a Borel property.3.3. Comments. 1. The argument given in Section 3.2 is adapted fromthe proof of Theorem 1.3 in [32]. Originally [32] considers the followingsituation. Let G be a Polish group and Γ be a countable group. Let usconsider the Polish space Hom (Γ , G ) of all homomrphisms from Γ to G .By Theorem 3.1 in [32] the subset of all π ∈ Hom (Γ , G ) such that π (Γ) isextremely amenable is a G δ subset of Hom (Γ , G ). By Corollary 18 of [23]the set of all representations from Hom (Γ , G ) whose image is an amenablesubgroup of G is also G δ in Hom (Γ , G ).2. Let G n be the space of all n -generated (discrete) groups with distinguished n -tuples of generators ( G, ¯ g ) (so called marked groups ). This is a compactspace under so called Grigorchuk topology . In papers [3] and [48] descriptivecomplexity in G n of some versions of amenability is considered. The authorsof [3] show that amenability is Π . They ask if it is Π -complete.A group G is called elementarily amenable if it is in the smallest class ofgroups which contains all abelian and finite ones and is closed under quo-tients, subgroups, extensions corresponding to exact sequences 1 → K → G → H → ω (i.e.computably presented groups). Under this indexation computable groupscorrespond to groups with decidable word problem. It is easy to see thatFølner’s condition of amenability (or the Schneider-Thom’s condition of Sec-tion 3.1 in the case of discrete groups) define a Π subset of indices. On theother hand applying Theorem 3 of [21] it is easy to see that this property isΠ -hard (it is a Markov property). Similarly one easily obtains that extremeamenability is Π -complete.E-mail: Aleksander.Iwanowpolsl.pl4. Nies: Stone-type duality for totally disconnected locallycompact groups In this post all topological groups will be Polish, and they all have a basisof neighborhoods of 1 consisting of open subgroups. As is well-known, sucha group is topologically isomorphic to a closed subgroup of the symmetricgroup on N , denoted S ( ω ). A homeomorphic embedding into S ( ω ) is ob-tained for instance by letting the group act by left translation on the leftcosets of open subgroups in that basis of neighborhoods of 1.Nies, Schlicht and Tent [37] developed the notion of coarse groups forclosed subgroups of S ( ω ), which first appeared in [25]. The idea is to doalgebra with approximations of elements, rather than with the elementsthemselves. The approximations are all the cosets of open subgroups (leftor right cosets, this makes the same class). Open cosets form the domainof the coarse group, and the structure is equipped with the ternary relation AB ⊆ C . The authors in [37] apply the notion primarily for the class ofoligomorphic groups, but also the profinite groups. (Ivanov has pointed outthat in that case a closely related structure was studied much earlier byChatzidakis [5].)Here we give a different approach to coarse groups, which is particularlyintended for the setting of totally disconnected locally compact (t.d.l.c.)groups. General references for t.d.l.c. groups include Willis [49, 50].In [37] all open cosets of a topological group G were considered, but theanalysis was restricted to classes of groups G which have only countablymany open subgroups. This is e.g. the case for Roelcke precompact groups(for each open subgroup U there is a finite set F such that U F U = G ).Such groups are in a sense opposite to the t.d.l.c. groups: the intersection ofthose two ‘large” classes consists merely of the profinite groups. However, asuperclass of both has also been studied: locally Roelcke precompact groups.The coarse group M ( G ) of a t.d.l.c. group G consists of the compact opencosets of G .4.1. Inductive groupoids, and inverse semigroups. A category is smallif the objects form a set (rather than a proper class). Recall that a groupoidis a small category such that each morphism A has an inverse, denoted A − .A partially ordered groupoid is a groupoid with a partial order ⊑ on the setof morphisms (and therefore also on the objects, which are identified withtheir identity morphisms) where the functional and the order structure arecompatible. An inductive groupoid is a partially ordered groupoid such thatthe partial order ⊑ restricted to the set of neutral elements is a semilattice.See Lawson [27, Section 4.1].A semigroup is called regular if for each a there is b , called the inverse of a , such that aba = a and bab = b . Inductive groupoids closely correspondto inverse semigroups . These are regular semigroups where the idempotents(elements e such that ee = e ) commute. In particular, the (large) categoriesof inductive groupoids and of inverse semigroups are isomorphic.For instance, given an inductive groupoid, to define the semigroup oper-ation, simply let AB = ( A | V )( V | B ), where V is the meet of the rightdomain of A and the left domain of B , and | denotes restriction, given byaxiom A a ) below. See Lawson [27] for detail.The representation theorem due to Wagner and Preston (both 1954, in-dependently) realizes every inverse semigroup S as an inverse semigroup ofpartial bijections on S . An element a ∈ S becomes the partial bijection τ a : a − S → S given by t at . Clearly τ b ◦ τ a = τ ba . If S is a group this isjust the left Cayley representation. See Lawson [27, Section 4.1 and 4.5].4.2. The coarse groupoid of a topological group. Given a topologicalgroup for which the open subgroups form a nbhd basis of 1, an inductivegroupoid is obtained as follows. • Objects correspond to open subgroups. In the abstract setting theywill be called ∗ subgroups. We use letters U, V, W for them. • Morphisms correspond to open cosets. Abstractly they are called ∗ cosets. We use letters A, . . . , E to denote them. A : U → V meansthat A is a right coset of U and a left coset of V . In brief we often OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 13 write U A V for this. The usual notation in the theory of groupoidsis U = d ( A ) (for domain) and V = r ( A ) (for range).We will treat coarse groupoids axiomatically. We begin with the following. Notation and conventions. An object U will be identified with the neu-tral morphism 1 U . So there are only morphisms, and objects merely forma convenient manner of speaking. We write RC ( U ) and LC ( U ) for the setsof right, resp. left ∗ cosets of U . In formulas we also write U A to mean that A ∈ RC ( A ), and A U to mean that A ∈ LC ( U ).We axiomatically require the usual properties defining groupoids and par-tial orders. For ease of language we adjoin a least element 0 to the par-tial order. We require that in the partial order ⊑ on the objects (i.e., the ∗ subgroups), any two elements U, V have an infimum, denoted U ∧ V . By A ⊥ B denote that A ∧ B = 0 are incompatible. If M = M ( G ) then 0 isinterpreted as the empty set.We have the following axioms connecting the groupoid and partial order.(Keep in mind that we identify U and 1 U .) Axioms 4.1. (A1) If A ⊑ B then A − ⊑ B − .(A2) Let U ⊑ V .( ↓ ) If V B then A ⊑ B for some U A .( ↑ ) If U A then A ⊑ B for some V B .(A3) if AB and A ′ B ′ are defined and A ⊑ A ′ , B ⊑ B ′ , then AB ⊑ A ′ B ′ .(A4) If U A and V B and U ⊑ V , then either A ⊑ B or A ⊥ B .(A5) If A B then there is C ⊑ A such that C ⊥ B .Remarks:Note that Axioms (A1), (A2 ↓ ) and (A3) are the usual axioms of orderedgroupoids, OG1, OG3 and OG2 respectively in Lawson [27, Section 4.1],only the notation there is a bit different.We have A U iff U A − by the definitions, which implies that the axiomsmentioning right ∗ cosets also holds for left ∗ cosets. See e.g. [27, Section 4.1,Prop 3(6)] for a proof of the left coset version of (A2 ↓ ) which Lawson calls(OG3 ∗ ).Axiom (A2 ↑ ) doesn’t seem to occur in the ordered groupoids literature.Axiom (A4) is special to the applications to topological groups we have inmind here. It implies that different right ∗ cosets of the same ∗ subgroup aredisjoint. Axiom (A5) essentially says that the topology is Hausdorff. The axioms are satisfied for structures of suitable open cosets. Inthe following, G is a topological group as above with countably many opensubgroups, or G is a t.d.l.c. group. Let M ( G ) denote the coarse groupoid:the ∗ subgroups are the open subgroups in the former case, and the compactopen subgroups in the t.d.l.c. case. The morphisms are the (compact) opencosets. We have A : U → V if A is a right coset of U and a left coset of V . Recall that in brief we write U A V for this. It is easily seen that theaxioms above hold. To show that M ( G ) (with 0 interpreted as the emptyset) is a lower semilattice, suppose that x ∈ aU ∩ bV for subgroups U, V ,then xU = aU and xV = bV . Let W = U ∩ V . Then aU ∩ bV = xW . Claim 4.6 below shows that this argument works in the general axiomaticsetting.Note that ∃ A : U → V iff U and V are conjugate in G . In this case, thereis a ∈ G such that U a = A = aV . Some consequences of the axioms. First we check that the ordering relation of morphisms carries over to theirleft and right domains. Claim 4.2. Suppose A : U → U , B : V → V and A ⊑ B . Then U i ⊑ V i for i = 0 , . To verify this: by (A1) we have A − ⊑ B − . Then by (A3) and identifying U with 1 U , we have U = AA − ⊑ BB − = V . Similarly, we’ve got U ⊑ V . Claim 4.3. For each A ∈ M and each ∗ subgroup U , there are a ∗ subgroup V ⊑ U and aleft ∗ coset B of V such that B ⊑ A . A similar fact holds for right ∗ cosets. To see this, suppose that A is left ∗ coset of W . Let V = W ∧ U . ByAxiom (A2 ↓ ) there is B ⊑ A such that B is left ∗ coset of V , as required.The following holds more generally in ordered groupoids. Claim 4.4 ([27], Section 4.1, Prop 3(5)) . If C ⊑ AB then there are A ′ ⊑ A and B ′ ⊑ B such that C = A ′ B ′ . Next we show that each left ∗ coset of a ∗ subgroup V is given by the left ∗ cosets of a ∗ subgroup U it contains. (In a sense it is the “union” of thesecosets.) Claim 4.5. Suppose U ⊑ V . If B V = C V then there is A U ⊑ B such that A ⊥ C . To verify this, we may suppose that B C . By Axiom (A5) there a ∗ subgroup W and D W ⊑ B such that D ⊥ C . Let U ′ = W ∧ U . Let E U ′ ⊑ D by (A2 ↓ ). There is A U ⊒ E by (A2 ↑ ). Since A ⊥ B fails (becauseof E ) we have A ⊑ B by (A4). However, A ⊑ C would imply E ⊑ C andhence contradict D ⊥ C . So A ⊥ C by (A4) again. Claim 4.6. Suppose A U ∧ B V = 0 . Let W = U ∩ V . Then A ∧ B is theunique left ∗ coset of W contained in A and B . If C W ′ ⊑ A, B , then W ′ ⊑ W by Claim 4.2 and definition of W . So by(A2 ↑ ) there is D W such that C ⊑ D . Letting C = A ∧ B , we see that A ∧ B is a left ∗ coset of W . If any left ∗ coset of W is contained in A, B it equals C by (A4). Normal ∗ subgroups. Recall that we write LC ( U ) and RC ( U ) for thesets of left, resp. right, ∗ cosets of U . We say that ∗ subgroups U, V are conjugate if ∃ A : U → V , or in other words, RC ( U ) ∩ LC ( V ) = ∅ . In M ( G )this replicates the usual meaning of conjugacy. For the slightly nontrivialdirection, if A = U a = bV then a − U a = a − bV . This is a subgroup,so a − b ∈ V , and hence U a = V . The axioms of groupoids imply thatconjugacy is an equivalence relation. OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 15 Normal ∗ subgroups V are the ones only conjugate to themselves: for each B ∈ RC ( V ) we have B − V B = V . This is equivalent to V B = BV foreach such B , or equivalently defined by the condition LC ( V ) = RC ( V ). Incategory language, all morphisms with left domain V also have right domain V , and vice versa. So there is a natural group operation on RC ( V ).A rather trivial fact from group theory becomes more demanding in theaxiomatic setting of coarse groupoids. Proposition 4.7. Suppose that U is a ∗ subgroup such that RC ( U ) , or equiv-alently LC ( U ) , is finite. Then there is a normal ∗ subgroup N ⊑ U . In usual topological group theory, the argument is as follows. Since U isa subgroup of G of finite index, the conjugacy class of U is finite. Let N = T g ∈ G U g . This is a finite intersection and hence defines an open subgroup,and N h = S g U gh = N for each N . So N is normal. Proof. Let D be the “conjugacy class” of U , namely D = { B − U B : B ∈ RC ( U ) } .The hypothesis implies that D is finite, so let N be the meet of all itsmembers. Then N ⊑ U . We show that N is normal as required.Given C ∈ RC ( N ), we will show that C ∈ LC ( N ). We define a bijection f C : D → D by f C ( W ) = D − W D where D ∈ RC ( W ) and C ⊑ D ;note that D exists and is unique because N ⊑ W , so f C is well-defined.To verify that f C is bijection, since D is finite it suffices to show that f C is1-1. Suppose f C ( W ) = f C ( W ) =: V . Then D − W D = D − W D = V where C ⊑ D , D and D i ∈ RC ( W i ). Since D , D ∈ LC ( V ), and D , D are not disjoint, this implies D = D and hence W = W .We have N ′ := C − N C ⊑ f C ( W ) for each W ∈ D using (A1) and (A3).So, since f C is onto, N ′ ⊑ N .Since C ∈ LC ( N ′ ) and N ′ ⊑ N , there is D ∈ LC ( N ) such that C ⊑ D by (A2 ↑ ). Then C ′ := D − ∈ RC ( N ), so C ′ ⊑ D ′ for some D ′ ∈ LC ( N )by the argument above. Then D ⊑ E := ( D ′ ) − ∈ RC ( N ) by (A1). So C ⊑ E and both are in RC ( N ). Hence C = D = E by (A4). This showsthat LC ( N ) ⊆ RC ( N ), hence also RC ( N ) ⊆ LC ( N ) by taking inverses. So RC ( N ) = LC ( N ) as required. (cid:3) The filter group associated with a coarse groupoid. We’ve seen howto turn a topological group into a coarse groupoid. Now we go the oppositeway. This is adapted from [37]. An alternative, possibly easier way to do thisis to take the topological group of “left automorphisms” of M , as detailedin Prop. 4.18 below.Let M be a coarse groupoid. A filter on a p.o. is a proper subset that isdownward directed and upward closed. For ( M, ⊑ ), a filter is thus a subset x that is upward closed, and A ∧ B exists and is in x , for any A, B ∈ x . Definition 4.8. A full filter is a filter x on the partial order ( M, ⊑ ) suchthat for each ∗ subgroup U ∈ M , there is a left ∗ coset and a right ∗ coset in x . Note that these ∗ coset are unique by (A4). F ( M ) denotes the set offull filters. We use variables x, y, z for full filters. Claim 4.9. For each A there is a full filter x such that A ∈ x . This follows by iterated applications of Claim 4.3.We begin with the topology on F ( M ). Definition 4.10 (Topology on the set of full filters) . As in [37] we define a topology on F ( M ) by declaring as subbasic the opensets b A = { x ∈ F ( M ) : A ∈ x } where A ∈ M . These sets form a base since filters are directed.Suppose M is countable. The following improves [37, Prop 2.5] that F ( M )a totally disconnected Polish space. Proposition 4.11. There is a homeomorphic embedding taking F ( M ) to aclosed subset of Baire space.Proof. Let h U n i n ∈ N be a descending sequence of ∗ subgroups that is cofinal(every ∗ subgroups contains an U n ). Fix a bijection f : N → M . We definean injection ∆ from F ( M ) into Baire space ω ω .Suppose that x ∈ F ( M ). Let ∆( x )(0) be the left ∗ coset of U in x .Suppose ∆( x )(2 n ) is defined and a left ∗ coset of some U r . Let ∆( x )(2 n +1)be the k such that A = f ( k ) ∈ x , and A is a right ∗ coset of U m where m > r is chosen least possible. This exists by Claim 4.3, since the sequence h U n i iscofinal, and (A2).Similarly, suppose ∆( x )(2 n + 1) is defined and a right ∗ coset of some U r .Let ∆( x )(2 n + 2) be the k such that A = f ( k ) ∈ x , and A is a left ∗ coset of U m where m > r is chosen least possible.By the axioms, ∆ is injective because x is the filter generated by ∆( x ).One checks that it is a homeomorphism because full filters correspond tothe paths on the subtree of the strings given by the possible next choices ateach step. (cid:3) Next we define the group operations on F ( M ). For filters x, y we let x − = { A − : A ∈ x } xy = { C : ∃ A, B [ AB ⊑ C ] } Here writing AB implies that it is defined, i.e. ∃ U A U B . Claim 4.12. If x, y are full filters, then x − and z = xy are full filters. That x − is a full filter is straightforward. For the second statement,clearly z is upwards closed. We verify that z is downwards directed.Always let i = 0 , 1. Suppose C i ∈ z . Then there are A i ∈ x , B i ∈ y and U i such that A i U i B i and A i B i ⊑ C i .Let U = U ∧ U . By definition of full filters there are A ∈ x and B ∈ y such that A U B . Then C = AB ∈ z . By (A4) and since filters aredownwards directed, we have A ⊑ A i and B ⊑ B i . So by (A3) we have C ⊑ C i as required. OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 17 The neutral element e is the full filter consisting of all the ∗ subgroups. Itis clear from the groupoid axioms that ( F ( M ) , · ) is a group with this neutralelement, and the inverse operation above.That leaves continuity of the group operations. First, as in [37, Claim2.11] we need to check that the transfer from formal to semantic conceptsworks between ∗ cosets A and the corresponding actual open cosets b A . Notethat we are NOT claiming that these are the only open cosets; this is nottrue unless we require further axioms special to the particular class of groupswe are interested in. Claim 4.13. Let A, B, C ∈ M . (a) A ⊑ B ⇐⇒ b A ⊆ b B . (b) d B − = ( b B ) − . (c) If A · B is defined then [ A · B = b A b B . (d) b U is a subgroup of F ( M ) . (e) A ∈ LC ( U ) ⇐⇒ b A is a left coset of b U .Similarly for right cosets.Proof. (a) The implication ⇒ is upward closure of full filters. For the impli-cation ⇐ , suppose that A B . By (A5) there is C ⊑ A such that C ⊥ B .By Claim 4.9 let x be a full filter such that C ∈ x . Then A ∈ x and B x .(b) is immediate. For (c), ⊇ is by definition, and ⊆ follows from Claim 4.9.(d) is immediate using U U = U .(e) We follow [37]: ⇒ : take any x ∈ b A . We show that x b U = b A .For x b U ⊆ b A , let y ∈ b U . Since A ∈ LC ( U ), we have AU ⊑ A . So x · y ∈ b A b U ⊆ b A by (c).For b A ⊆ x b U , let y ∈ b A . To show that y ∈ x b U , or equivalently x − y ∈ b U ,note that we have x − y ∈ b A − b A = d A − b A ⊆ b U by (b) and (c). ⇐ : Suppose b A = x b V . There is B ∈ x such that B ∈ LC ( V ). By the forwardimplication, b B is a left coset of b V . Also x ∈ b A ∩ b B , so A ⊥ B fails. Since A, B ∈ LC ( V ) this implies A = B by (A4).The case of right cosets follows by taking inverses. (cid:3) Another transfer fact will be useful. Claim 4.14. The map x, y → x · y − is continuous on F ( M ) . This follows since the sets of the form b D form a basis. If xy − ∈ b D , i.e., D ∈ xy − , then by definition there are A ∈ x and B ∈ y such that AB − ⊑ D . Now, by Claim 4.13, b A b B − ⊆ b D as required. Claim 4.15. For any left coset x b V in F ( M ) , there is A ∈ LC ( V ) such that x b V = b A .Proof. Since x is a full filter, there is some left ∗ coset A of V in x . Weclaim that x b V = b A . We have x b V ⊆ b A b V = b V , since A ∈ x and b A is a leftcoset of b V by Claim 4.13. To see that b A ⊆ x b V , let y ∈ b A . Since x, y ∈ b A , x − y ∈ b A − b A = d A − b A ⊑ b V by Claim 4.13. Thus y ∈ x b V . (cid:3) By definition of the topology, the open subgroups of F ( M ) form a nbhdbase of 1. So if M is countable, F ( M ) is a non-Archimedean Polish group.The operation F recovers a topological group from its coset structurewhen that is countable. It also works in the t.d.l.c. setting where M ( G )denotes the compact open cosets. Proposition 4.16 (cf. [25], after Claim 3.6, and [37], Prop 2.13) . Suppose that G is a closed subgroup of S ( ω ) such that M ( G ) is countable.There is a natural group homeomorphism Φ : G ∼ = F ( M ( G )) given by g 7→ { A : A ∋ g } ,with inverse given by x g where T x = { g } . The inverse map simply sends a full filter x to the point it converges to.Note that x isn’t really a filter in the sense of topology, only on certain opensets, but that suffices for the convergence notion. Example 4.17. For an instructive example of a coarse groupoid, considerthe oligomorphic group G = Aut( Q , < ). The open subgroups of G are thestabilizers of finite sets. If U, V are stabilizers of sets of the same finitecardinality, there is a unique morphism A : U → V in the sense above,corresponding to the order-preserving bijection between the two sets. Thecoarse groupoid for Aut( Q , < ) is canonically isomorphic to the groupoidof finite order-preserving maps on Q , with the partial order being reverseextension. For compatible maps A, B , the meet A ∧ B is the union of thosemaps.A filter x corresponds to an arbitrary order-preserving map ψ on Q . Thefilter x contains a right coset of each open subgroup iff ψ is total, and aleft coset of each open subgroup iff ψ is onto. So the set of full filterscorresponds to Aut( Q ) as expected. (Incidentally, this example shows thatin Definition 4.8 we need both types of cosets, and that not every maximalfilter is full.) The filter group as an automorphism group. Let M be a coarsegroupoid. By M left we will denote the structure with domain M and the op-erations ∧ and ( r B ) B ∈ M where Ar B = AB in case r ( A ) = l ( B ), and Ar B = 0otherwise. We show that the left action of F ( M ) on M corresponds to theautomorphisms of this “rewrite” of M . (This is simlar to showing that agroup is isomorphic to the automorphism group of a Cayley graph given bya generating set, with edge relations labelled according to the generators.)Note that for each automorphism p of M left , and each A , we have r ( p ( A )) = r ( A ) . This is because where U = r ( A ), we have p ( A ) U = p ( AU ) = p ( A ). Also,note that p is determined by its restriction to the ∗ subgroups, because foreach right ∗ coset B of a ∗ subgroup U we have p ( B ) = p ( U ) B . Proposition 4.18. F ( M ) is topologically isomorphic to Aut( M left ) via acanonical isomorphism Θ .Proof. For x ∈ F ( M ), the left action U B A = x · B is given by A = CB where C U ∈ x . The isomorphism Θ : F ( M ) → Aut( M left ) maps x to its leftaction: OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 19 Θ( x )( A ) = x · A .Clearly Θ( x ) ∈ Aut( M left ), and Θ preserves the group operations.Let s M ∈ F ( M ) denote the full filter of ∗ subgroups, which is the neutralelement of F ( M ). We claim that the inverse Φ of Θ is given byΦ( p ) = p ( s M ).Clearly x = Φ( p ) is a filter. To show that x is a full filter, let U be a ∗ subgroup in M . Since p is an automorphism, firstly, we have p ( U ) U = p ( U ),so p ( U ) ∈ LC ( U ) and p ( U ) ∈ x . Secondly, there is B such that p ( B − ) = U .Then p ( B − B ) = U B = B . So B ∈ RC ( U ). Now V = B − B = r ( B ) is a ∗ subgroup. Since p ( V ) = B we have B ∈ x .We verify that Θ , Φ are inverses of each others. Φ(Θ( x )) = x because A U ∈ x ↔ xU = A ↔ Θ( x )( U ) = A .Θ(Φ( p )) = p because p ( U ) = A ↔ A ∈ Φ( p ) ↔ Φ( p ) U = A ↔ Θ(Φ( p ))( U ) = A .To show Θ and Φ are continuous at 1, note that if p = Θ( x ), then p ( U ) = U is equivalent to x ∈ b U . (cid:3) Profinite groups, and ∗ compact coarse groupoids. As mentioned,Chatzidakis [5] carried out the first research related to the application ofcoarse groupoids to profinite groups. In her version the coarse groupoid wasrestricted to normal open ∗ subgroups, which suffices in that case. (Thanksto the Ivanovs for pointing this out.)We say that a coarse groupoid M is ∗ compact if ∀ U [ LC ( U ) is finite].This is, of course, equivalent to requiring that ∀ U [ RC ( U ) is finite]. Clearly M ( G ) for profinite G has this property. By Prop. 4.7, ∗ compactness impliesthat each ∗ subgroup U contains a normal ∗ subgroup V . (This was requiredseparately in [37].) Proposition 4.19. Let M be a coarse groupoid. Then M is ∗ compact ⇔ F ( M ) is compact.Proof. ⇐ : Given U ∈ M , by Claim 4.13(d) b U is open in F ( M ). So it hasfinite index. By (e) of the same claim, this implies that LC ( U ) is finite. ⇒ : Using Prop. 4.7, let h N k i k ∈ N be a descending chain of normal ∗ subgroupssuch that ∀ U ∃ k [ N k ⊑ U ]. Let G k be the group induced by M on LC ( N k ).We define an onto map p k : G k +1 → G k as follows: given A ∈ LC ( N k +1 ),using (A2 ↑ ) let p k ( A ) = B where A ⊑ B ∈ LC ( N k ). Each p k is a homomor-phism by Axioms (A1, A2).Let G be the inverse limit: G = proj lim k ( G k , p k ). Thus G = ( { f ∈ Q k G k : ∀ k f ( k ) = p k ( f ( k + 1)) } , · ),which is closed and hence compact group subgroup of the Cartesian productof the G k . We claim that G ∼ = ( F ( M ) , · ) via the map Φ that sends f ∈ G to the filter in F ( M ) generated by the ∗ cosets f ( k ), namelyΦ( f ) = { C ∈ M : ∃ k f ( k ) ⊑ C } . It is clear that Φ is a monomorphism. To show Φ is onto, given a full filter x ∈ F ( M ), for each k there is f ( k ) = B k ∈ LC ( N k ) such that B k ∈ x . Then f ∈ G , and clearly Φ( f ) = x .Note the b N k form a base of nbhds of 1 in F ( M ). Since Φ − ( b N k ) = { f : f ( k ) = N k } and the letter sets form a base of nbhds of 1 in G , we getthat Φ is a homeomorphism. Thus F ( M ) is compact. (cid:3) Coarse groupoids versus diagrams, for profinite groups. We will characterize the coarse groupoids M ( G ) obtained from profinitegroups G by adding an axiom to the ∗ compactness condition.Consider profinite G = proj lim k ( G k , p k ) where the G k are finite groupsand each p k : G k +1 → G k is an epimorphism. We say that h G k , p k i is a diagram for G . By the proof of Prop. 4.19, each diagram for G can be seenas a coarse groupoid M with F ( M ) ∼ = G . So a coarse groupoid for G isnot unique. Intuitively, they may be open subgroups of F ( M ) that M ismissing. To avoid this we need another axiom. In the axiom to follow, “CC”stands for “completeness in case of compactness”. We will see that it impliesin the compact case that each open subgroup of F ( M ) has a “name”. Axiom CC. Let M be a ∗ compact coarse groupoid. Let N be a normal ∗ subgroup of M .If a set S ⊆ LC ( N ) is closed under products and inverses, then there is a ∗ subgroup U such that A ⊑ U ↔ A ∈ S , for each A ∈ LC ( N ) . Clearly M ( G ) for profinite G satisfies this axiom. The axiom implies thedual of Prop. 4.16 in the compact case: Proposition 4.20. Let M be a ∗ compact coarse groupoid satisfying Ax-iom CC. Then M ∼ = M ( F ( M )) via the map A b A .Proof. By Claim 4.13 it suffices to show that the map is onto.Firstly, let U be an open subgroup of F ( M ). By definition of the topologyand Prop. 4.7, there is a normal ∗ subgroup N in M such that b N ⊆ U . ByProp. 4.19, F ( M ) is compact, so U is the union of finitely many cosets of b N .By Claim 4.15 each such coset has the form b A for some A ∈ LC ( N ). Let S be the set of such A in LC ( N ). The set S is closed under product andinverses since U is a subgroup, using Claim 4.13. So there is a ∗ subgroup U as in Axiom CC. Clearly b U = U .Secondly, given a left coset B of an open subgroup U in F ( M ), byClaim 4.15 we have B = b B for some B ∈ LC ( U ) as required. (cid:3) T.d.l.c. groups and ∗ locally compact coarse groupoids. We say that a coarse groupoid M is ∗ locally compact if for each ∗ subgroup K ∈ M the coarse subgroupoid induced on { A : A ⊑ K } is ∗ compact. Notethat M ( G ) for t.d.l.c. G has this property: by van Dantzig’s theorem (thatevery t.d.l.c. group has an open compact subgroup) M ( G ) is non-empty,and by definition M ( G ) consists of compact open cosets.We call M weakly ∗ locally compact if for some ∗ subgroup K ∈ M theinductive subgroupoid on { A : A ⊑ K } is ∗ compact. Proposition 4.21. Let M be a coarse groupoid. Then M is weakly ∗ locally compact ⇔ F ( M ) is locally compact OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 21 Proof. ⇐ : By van Dantzig’s theorem, F ( M ) has a compact open sub-group L . Let K ∈ M be a ∗ subgroup such that b K ⊆ L . As above let M K be the coarse subgroupoid of M on { A : A ⊑ K } . Then F ( M K ) ∼ = b K .So F ( M K ) is compact. Hence M K is ∗ compact by Prop. 4.19. ⇒ . Let M be weakly ∗ locally compact via K . Then F ( M K ) ∼ = b K is compact. Since b K is an open subgroup of F ( M ) this makes F ( M ) locally compact. (cid:3) Example 4.22. (a) If G is countable discrete group, then M ( G ) consistsof the isomorphisms between finite subgroups.(b) Let G = ( Q p , +). The proper open subgroups are compact, and are allof the form U r = p r Z p for some r ∈ Z . In this abelian setting each morphismis an endomorphism. The group G r of endomorphisms A : U r → U r has thestructure of C p ∞ (the direct limit of the cyclic groups C p n with the canonicalembeddings). Let f ( x ) = px for x ∈ C p ∞ and view f as a map G r → G r +1 .Then for A ∈ LC ( U r ) , B ∈ LC ( U r +1 ), the ordering relation A ⊑ B isequivalent to f ( A ) = B . We see that the coarse groupoid is a bit like adiagram for a profinite group, but goes not only to closer approximations ofelements (backwards), but also to less close ones (forward).(c) G d = Aut( T d ) for d ≥ 2. This is the group of automorphism of the d -regular tree T d , defined as an undirected graph without a specified root, firststudied by Tits. It is known that each proper open subgroup is compact.Each compact (open or not) subgroup is contained in the stabilizers of avertex, or the stabilizers of an edge (which are compact open). See [12, p.12]. It would be interesting to describe more of the structure of M ( G d ).Recall that in the locally compact setting, the coarse groupoid M ( G ) hasas a domain the compact open cosets of G . We replace Axiom CC from thecompact setting by a variant that works in the more general setting. Axiom CLC. Let M be a ∗ locally compact coarse groupoid. Let N be a ∗ subgroup of a M .If a finite set S ⊆ LC ( N ) is closed under products and inverses, thenthere is a ∗ subgroup U such that A ⊑ U ↔ A ∈ S for each A ∈ LC ( N ) . Clearly, if G is t.d.l.c. then M ( G ) satisfies this axiom. We verify that theaxiom characterizes the ∗ locally compact coarse groupoids obtained in thisway. Proposition 4.23. Let L be a ∗ locally compact coarse groupoid satisfyingAxiom CLC. Then L ∼ = M ( F ( L )) via the map A b A .Proof. As in Prop. 4.20, by Claim 4.13 it suffices to show that the map A b A is onto.Firstly let U be a compact open subgroup of F ( L ). There is W ∈ L suchthat c W ⊆ U . Let L U = { A ∈ L : b A ⊑ U } .Clearly L U is a coarse subgroupoid of L . In L U , RC ( W ) is finite, so byProp. 4.7 there is N ⊑ W such that S := LC ( N ) = RC ( N ) in L U . Clearlythe hypothesis of Axiom CLC applies to S , so we get a ∗ subgroup U . Then b U = U . Secondly, given a left coset B of an open subgroup U in F ( L ), by Claim 4.15we have B = b B for some B ∈ LC ( U ) as above. (cid:3) M and F as functors. We will view closed subgroups of S ( ω ) (also callednon-Archimedean groups) as a category where the morphisms f : G → H arethe continuous epimorphisms. The kernel of f is a closed normal subgroup.If G is compact/locally compact then H has the same property. Covariant functor, for the locally compact case. The open mappingtheorem for Hausdorff topological groups states that a surjective continuoushomomorphism of a σ -compact group onto a Baire (e.g., a locally compact)group is an open mapping. (This is proved using Baire category.) Each(separable) t.d.l.c. group is σ -compact again by van Dantzig. So if G islocally compact and f : G → H is onto, then for each compact open A ⊆ G ,the image f ( A ) is open (and of course compact) in H .On t.d.l.c. groups with continuous epimorphisms we can now view the op-erator M as a covariant functor M + . If f : G → H then M + ( f ) : M ( G ) →M ( H ) is given by A f ( A ). Here we view coarse groupoids as a weak cate-gory C weak where the morphisms r : M → N preserve the groupoid structureand the partial order in the forward direction only. Contravariant functor. If f : G → H is an epimorphism of non-Archimedeangroups, then we have a map M − ( f ) : M ( H ) → M ( G ), where M − ( f )( A ) = f − ( A ). This clearly preserves the inductive groupoid structure. We wantto identify the right type of morphisms on coarse groupoids so that M isa contravariant functor with inverse F (suitably extended to morphisms)for the classes of groups of interest. Consider a map M − ( f ) above and let R ⊆ M ( G ) be its range. R consists of all open [compact] cosets of subgroupof G that contain the kernel of f . So, • R is closed upwards, and • R is closed under conjugation of subgroups. Thus if ∃ A [ U A V ] and U ∈ R then V ∈ R .So we have to take the strong category C str with objects the (countable)coarse groupoids and with morphisms q : N → M that preserve the groupoidand the meet semilattice structure (in particular, they preserve the orderingand hence have to be 1 − 1) and have range with the two properties above.For such an q : N → M we can define F ( q ) : F ( M ) → F ( N ) by x q − ( x ). Claim 4.24. Suppose q : N → M is a morphism in C str .Then F ( q ) : F ( M ) → F ( N ) is a continuous epimorphism. Nies: Closed subgroups of S ( ω ) generated by theirpermutations of finite support Let SF ( ω ) denote the groups of permutations of ω that have finite sup-port. We note that SF ( ω ) plays a role in S ( ω ) similar to the role that Q plays in R . Clearly SF ( ω ) is dense in S ( ω ). The inherited topology has asa basis of neighbourhoods of 1 the open subgroups SF ( ω ) ∩ U n of SF ( ω ),where U n ≤ o S ( ω ) is the pointwise stabilizer of { , . . . , n } . For example thesubgroup T algebraically generated by { (01)(2 n n + 1) : n ≥ } of SF ( ω )is not closed; its closure is T ∪ { (01) } . OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 23 Given a closed subgroup G of S ( ω ), it is of interest to find out how muchof G can be recovered from the closed subgroup G ∩ SF ( ω ) of SF ( ω ). B.Majcher-Iwanov [28] called G ∩ SF ( ω ) the finitary shadow of G . Often G ∩ SF ( ω ) will be trivial, for instance if G is torsion free.Note that a closed subgroup G of S ( ω ) is compact iff each G -orbit isfinite. Majcher–Iwanov [28] studied the distribution of finitary shadows ofcompact subgroups of S ( ω ) within the subgroup lattice of SF ( ω ), and madeconnections to cardinal characteristics.One can also start from a subgroup H of SF ( ω ) and study how it isrelated to its closure G = H in S ( ω ). (Closures will be taken in S ( ω ) unlessotherwise stated.)Firstly, for each open subgroup U of H , its closure U is open in G . For,suppose U n ∩ H ≤ U . Since U n is closed we have U n ∩ H = U n ∩ H . Hence U n ∩ G ≤ U , so U is open in G .Conversely, each open subgroup of G is given as the closure of the opensubgroup L ∩ H of H : Lemma 5.1. Let H ≤ SF ( ω ) and let G = H be the closure of H in S ( ω ) .Then L = L ∩ H for each open subgroup L of G .Proof. Since L is closed in G , we only need to verify the inclusion ⊆ .Let g ∈ L . Since L is open in G , there is k such that for each v , if gv − ∈ U k then v ∈ L . Now let n ≥ k be arbitrary. Since G = H , there is r ∈ H such that gr − ∈ U n . So r ∈ L . This shows g ∈ L ∩ H . (cid:3) Remark 5.2. To summarize, there is a natural isomorphisms between theopen subgroups L of G and the open subgroups U of H , given by L → L ∩ H ,with inverse U → U . The compact case. The textbook on permutation groups by Dixon andMortimer [7, Lemma 8.3D] contains a proof of the following equivalences fora group H ≤ SF ( ω ):(i) every H -orbit is finite (i.e., H is compact)(ii) H only has finite conjugacy classes (such H is called an FC-group)(iii) H is residually finite.The implication (i) → (ii) is pretty elementary: for each x ∈ H there is afinite H -invariant set ∆ such that the support of x is contained in ∆. Thenumber of conjugates of x is then bounded by the number of conjugates of x | ∆ (restriction to ∆) in S ∆ . (ii) → (iii) also easy, the remaining implication(iii) → (i) is harder. It was proved in [33, Thm. 2].It would be interesting to determine the complexity of the topologicalisomorphism relation for closed subgroups H of SF ( ω ) that are FC-groups.Since H = H ∩ SF ( ω ), where the closure is taken in S ( ω ) as usual, it is Borelreducible to topological isomorphism of profinite groups, which by Kechriset al. [25] is Borel isomorphic to countable graph isomorphism. However,the groups employed there to reduce graph isomorphism do not have a fini-tary shadow satisfying the conditions above. [25] uses an extension to thetopological setting of Mekler’s construction to code a countable graph intoa countable step 2 nilpotent group of exponent a fixed odd prime. Mekler’sgroups being FC would mean that the coded graph is co-locally finite (each vertex connected to cofinitely many vertices). But for technical reasons re-lated to the definability of the vertex set, Mekler’s method only can encodegraphs that are triangle and square free, and such graphs are of course notco-finitary. The oligomorphic case. Recall that a group H ≤ S ( ω ) is called oligo-morphic if for each r its action on ω has only finitely many r -orbits. It isopen whether E can be Borel reduced to topological isomorphism of closedoligomorphic subgroups of S ( ω ). See Section 2, or [37] for background.For H ≤ SF ( ω ), note that H is oligomorphic iff H is. The hope was thatone can reduce E to topological isomorphism of closed subgroups of SF ( ω ),which should be easier to control. To be useful, this actually assumed anaffirmative answer to the following question: Question 5.3. Is the following true?Let H , H ≤ c SF ( ω ) be oligomorphic. Then H , H are homeomorphic ⇔ H , H are homeomorphic. For the implication ⇒ , we note that by Remark 5.2 the structures of opencosets for H i and H i are isomorphic for i = 0 , 1. That is M ( H i ) ∼ = M ( H i ).By hypothesis M ( H ) ∼ = M ( H ). Since the H i are closed oligomorphicsubgroups of S ( ω ), M ( H ) ∼ = M ( H ) implies that they are topologicallyisomorphic by [37].However, the intended application doesn’t not work, because oligomorphicsubgroups H of SF ( ω ) are very restricted, and in particular there are onlycountably many up to isomorphism.Some examples of such groups H come to mind. First take permutationsof finite support preserving the evens. This group is topologically isomorphicto SF ( ω ) × SF ( ω ). More generally, take a finite power of SF ( ω ). Anothertype of example is letting H be the automorphism group of a countablyinfinite structure obtained taking a finite structure M , and have an equiva-lence relation E with a “copy” of M on each class. For instance, take oneunary function symbol f , and let M be a finite cycle given by f . If φ is thepermutation that is the union of all these cycles of fixed length, then H isthe centralizer of φ in SF ( ω ).Let R be the random graph and Q the random linear order. In contrast,Aut( R ) and Aut( Q ) have no nontrivial members of finite support at all (nothard to check). Even G = Aut( E ) where E is an equivalence relation withjust two infinite classes, is not the closure of G ∩ SF ( ω ) because an automor-phism of finite support cannot leave an equivalence class. More generally, if R is a definable relation and φ ( a ) = b where φ is an automorphism of finitesupport, then R ( a ) almost equals R ( b ).Towards a full classification of oligomorphic subgroups H of SF ( ω ), weuse some structure theory of subgroups of SF ( ω ) developed in the 1970s byPeter Neumann, e.g. [33], and in two short papers by Dan Segal, indepen-dently. Here I’m mostly using the notes on finitary permutation groups byChris Pinnock, available at chrispinnock.com/phdpublications/ . Unlessotherwise stated references to theorems etc refer to those notes.Let us begin by assuming that H ≤ SF ( ω ) is 1-transitive. The Jordan-Wielandt Theorem says for such a group H that if H is primitive then OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 25 H = SF ( ω ) or H equals the group of alternating permutations of finitesupport (which has index 2 in SF ( ω )). So we can assume otherwise. Let B be a block of imprimitivity. B has to be finite (Lemma 2.2 in Pinnock),simply because there is a permutation τ ∈ H moving B to a disjoint set τ ( B ), so that B is contained in the support of τ . The equivalence relation E with classes made up of the translates of B is H -invariant, and hence aunion of 2-orbits. Since there are only finitely many 2-orbits, there must bea maximal block of imprimitivity. Then H acts primitively on Ω /E and eachinduced permutation there has finite support. So it acts highly transitivelyon it. This is the second type of example above, the same finite model ineach equivalence class of E . Note that H is a subgroup of G ≀ R where R = SF ( ω ) or R is the alternating group (Thm 2.3).If H is not 1-transitive we are in the case of the first example above. Itis known that we get only finite products of 1-transitive groups, and a finitegroup.For a more detailed treatment, see Iwanow [22] who works in the moregeneral setting of automorphism groups of countable saturated structures.He uses the concept of cell, a permutation group that preserves an equiva-lence relation with all classes finite, and induces the full symmetric groupon the equivalence classes. A 2-cell is permutation group G ≤ c S ( ω ) suchthat there exists a partition ω into G - invariant classes Y i such that for anyinfinite Y i the group induced by G on Y i is a cell and is also induced by thepointwise stabilizer of the complement of Y i . Each oligomorphic 2-cell is afinite product of cells and a finite algebraic closure of the empty set. Theirnumber is countable as already mentioned. Each cell is a finite cover of apure set in the sense of Evans, Macpherson, Ivanov, Finite covers, 1997.These covers have been classified.6. Harrison-Trainor and Nies: Π r -pseudofinite groups The authors of this post worked at the (now defunct) Research CentreCoromandel in July. They started from some notes of Dan Segal (2014),which in turn are based on [20]. The main concept in these notes is this. Definition 6.1. A group G is called pseudofinite if each first-order sentencetrue in G also holds in some finite group.In the first two sections we present a simplified account of some materialin the Segal notes. Our account is somewhat more general than the notes,firstly as it takes into account the quantifier complexity of the sentences forwhich a group needs to be pseudofinite, and secondly because a lot of thiswork can be carried out in the setting of an arbitrary finitely axiomatisedvariety of algebraic structures.Throughout, let G be an algebraic structure in a language L with finitelymany function and constant symbols, and no relation symbols besides equal-ity. The Π and the Σ sentences are the quantifier free ones, thought of asdisjunctions of conjunctions of equalities and inequalities. Note that satis-faction of Π sentences is closed under taking substructures. Definition 6.2. (i) For r ∈ N we say that an L -structure G is Σ r -pseudofinite if G is a model of the Σ r -theory of finite L -structures. Equivalently, each Π r sentence that holds in G also holds in somefinite L -structure.(ii) Similarly, we define Π r -pseudofiniteness of an L -structure G . Definition 6.3. We say that an L -structure G has named generators if G is d -generated for some d ≥ 1, and the language L contains finitely manyconstant symbols c = ( c , . . . , c d ) naming such generators of G .In this case witnesses for outermost existential quantifiers be named byterms. So we have Fact 6.4. Let G be an L -structure with named generators. Let e G be thestructure with the constants naming the generators omitted. Let r ≥ . Thefollowing are equivalent.(i) e G is Π r +1 -pseudofinite(ii) G is Π r +1 -pseudofinite(iii) G is Σ r -pseudofinite.Proof. (ii) → (i) and (i) → (iii) are straightforward, and don’t rely on the factthat the c i name generators.For (iii) → (ii), suppose a sentence θ ≡ ∃ x ξ ( x ) holds in G , where x denotes( x , . . . , x m ) and ξ ( x ) is Π r . Since the c i name generators, we can pick L -terms t j without free variables as witnesses. The sentence ξ ( t , . . . , t m ) isΠ r and holds in G . So it holds in some finite L -structure. Hence θ holds inthat structure. (cid:3) The Σ -theory of finite groups equals the Σ -theory of the trivial group,which is decidable. Every group satisfies the Σ -theory of the trivial groupand hence is Σ -pseudofinite. So the notion Σ -pseudofiniteness really onlymakes sense for groups with named generators.In contrast, the Π -theory of finite groups is hereditarily undecidableby a result of Slobodskoi [46]. However, it has infinitely many decidablecompletions, e.g. the theory of ( Z n , +), for each n ≥ 1. To see this, let H be the ultraproduct of all cyclic groups of prime order. H is abelian, torsionfree, not f.g. and satisfies the theory of finite groups. Any subgroup of H satisfies the Π -theory of H . To see that H has infinite rank, let R i , i ∈ N be disjoint infinite sets of primes, let f i ( p ) = 1 for p ∈ R i and 0 otherwise.Then the [ f i ] are linearly independent over Z .6.1. Σ -embedding of G into an ultraproduct of witness structures. The following construction is based on Houcine and Point [20]. We fix somevariety V of L -structures, such as groups. Suppose G is as in Definition 6.3,and G is Σ -pseudofinite. Let h φ ′ i i i ∈ N be a list of the Π -sentences that holdin G , with φ ′ being the conjunction of the finitely many axioms for V . Let φ n = V i ≤ n φ ′ i . By hypothesis on G there is a finite L -structure E n (calleda witness structure) such that E n | = φ n . Since we are only consideringΠ -sentences, we may assume that each E n is generated by c E n .Let U be a free ultrafilter on N . Let E = Q n E n / U be the correspondingultraproduct. Define an embedding β : G → E by setting for each L -term sβ ( s ( c G )) = s ( c E ). OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 27 Note that β is well-defined and 1 − 1: for each terms s, t , if G | = s ( c ) = t ( c )then the sentence s ( c ) = t ( c ) equals φ ′ i for some i , and so E n | = s ( c ) = t ( c )for each n ≥ i . One argues similarly for the case G | = s ( c ) = t ( c ). So β is amonomorphism of L -structures. Fact 6.5. Suppose an L -structure G with named generators as in Defini-tion 6.3 is Σ -pseudofinite. The map β : G → E preserves satisfaction of Σ -formulas in both directions. Thus, for each p , . . . , p k ∈ G and eachquantifier free L -formula θ ( x , . . . , x k , ˜ y ) , G | = ∃ ˜ y θ ( p , . . . , p k , ˜ y ) ⇔ E | = ∃ ˜ y θ ( β ( p ) , . . . , β ( p k ) , ˜ y ) .Proof. We may incorporate the parameters into θ , so we may assume thatthere are none. Let φ ≡ ∃ ˜ y θ (˜ y ). For the nontrivial implication supposethat E | = φ . If G | = ¬ φ , then E n | = ¬ φ for almost all n , contradiction. So G | = φ . (cid:3) The number of factors needed for being in a verbal subgroup. In this section we only consider the case that G is a group. We will explainthe result of Segal that if G is Π -pseudofinite then the verbal subgroups G ( n ) and h G q i , n, q ≥ 1, are definable in G by a positive Σ formula.Below G satisfies Definition 6.3 where d denotes the number of generators.The language L consists of symbols for the group operations, the neutralelement, and constants c , . . . , c d naming the generators. Definition 6.6. Let d, r ≥ 1. Let w be a term with free variables x , . . . , x k in the language of group theory, identified with an element of the free group F k . We say that w is r -bounded for d if for each finite d -generated group H ,we have w ( H ) = w ∗ r ( H ).Here for any group H , by w ( H ) one denotes the usual verbal subgroup whichis generated by the values of w , and w ∗ r ( H ) is the set of products of up to r many values of w or their inverses.Nikolov and Segal [40, Thm 1.7], or [41, Thm. 1.2], show that this holdsfor the terms x q , where q ≥ 1, and iterated commutators such as [ x, y ] or[[ x, y ] , [ z, w ]]. For such terms, the result below (for pseudofinite groups) wasestablished in [20, Prop. 3.8], but an additional hypothesis was needed forthe case of terms x q . Proposition 6.7. Suppose a d -generated group G with named generators asin Definition 6.3 is Σ -pseudofinite w.r.t. the language L . Suppose furtherthat w is r -bounded for d . Then w ( G ) = w ∗ r ( G ) .By Fact 6.4 the hypothesis holds if G is Π -pseudofinite w.r.t. the languageof group theory. In this case w ( G ) is Σ -definable without parameters in thatlanguage.Proof. Write g i = c Gi and g = ( g , . . . , g k ). Suppose t ( g ) ∈ w ( G ) for someterm t . That is, for some m , there are k -tuples z j of elements of G and ǫ j = − , 1, for 1 ≤ j ≤ m , such that t ( g ) = Q ≤ j ≤ m w ( z j ) ǫ j . Then E satisfies the Σ -sentence ∃ z . . . ∃ z m [ t ( c ) = Q ≤ j ≤ m w ( z j ) ǫ j ], andhence the ultrafilter U contains the set S of those n such that E n satisfiesthis sentence. By hypothesis on w and since E n is d -generated by the c E n i ,for each n ∈ S there is a choice of η ℓ = − , 1, for 1 ≤ ℓ ≤ r , such that E n | = ∃ y . . . ∃ y m [ t ( c ) = Q ≤ j ≤ r w ( y j ) η j ].Since U is an ultrafilter, there must be one choice ( η ℓ ) ≤ ℓ ≤ r so that the n where this choice applies form a set in U . Hence for this choice we have E | = ∃ y . . . ∃ y m [ t ( c ) = Q ≤ j ≤ r w ( y j ) η j ].By Fact 6.5 this implies that t ( g ) ∈ w ∗ r ( G ) as required. (cid:3) As Houcine and Point point out [20, Lemma 2.11], parameter definablequotients and subgroups of pseudofinite groups are again pseudofinite. Also,finitely generated pseudofinite groups that are of fixed exponent, or soluble,are finite [20, Prop. 3.9]. So, G is an extension of a pseudofinite group, theverbal subgroup given by a term as above, by a finite group satisfying thelaw given by the term.6.3. Effectiveness considerations. We show that a variant of the basicconstruction above leading to Fact 6.5 is effective using the structure G asan oracle. Effective ultraproducts. Given a sequence h E n i of uniformly computablestructures for the same finite signature, and a non-principal ultrafilter U for the Boolean algebra of recursive sets, one can form the structure E = Q rec E n / U . It consists of ∼ U -equivalence classes of recursive functions f such that f ( n ) ∈ E n for each n . Here f ∼ U g denotes that functions f, g agree on a set in U . We interpret the relation and functions symbols on E as usual.A version of Los’ Theorem restricted to existential formulas holds. Fact 6.8. For each formula θ ( x , . . . , x m ) ≡ ∃ e y ξ ( x , . . . , x m , e y ) with ξ quan-tifier free, and each [ f ] , . . . , [ f m ] ∈ E , E | = θ ([ f ] , . . . [ f m ]) ⇔ R = { n : E n | = θ ( f ( n ) , . . . f m ( n )) } ∈ U .Proof. Note that the set R is computable. For the implication from leftto right, suppose [ h ] , . . . , [ h k ] are witnesses for E | = θ ([ f ] , . . . [ f m ]), forrecursive functions h , . . . , h k .This shows that R ∈ U by definition of theultraproduct.For the implication from right to left, note that when n is in the com-putable set R one can search for the witnesses w n in E n . So one can definecomputable witness functions h , . . . , h k for E | = θ ([ f ] , . . . [ f m ]), assigninga vacuous value in E n in case n R . (cid:3) To obtain an effective version of Fact 6.5, let ψ i be an effective list of allthe Π sentences in L , where ψ is the conjunction of the laws of V . Wemodify the construction of the E n as follows. Given n , look for the leaststage s and a finite L -structure, called E n , such that E n satisfies each ψ i , i ≤ n , that has not been shown to fail in G by stage s , in the sense that acounterexample has been found among the first s elements of G . OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 29 Now define the restricted ultraproduct E as above. The argument for thefact can be carried out as before: Suppose that E | = φ for an existentialsentence φ . If G | = ¬ φ , then E n | = ¬ φ for almost all n . This contradicts theweak version of Los theorem above.The argument of Prop 6.7 also works with this restricted ultraproduct.Note that the set S in the proof of Prop 6.7 is computable since the E n arefinite and given by strong indices. The set of n for which a particular choiceof η ℓ works is also computable.The upshot: if G is computable, then we have a canonical ultraproductversion E of G , with a Σ elementary embedding, and this version E is ina sense effective as well, except for the ultrafilter, which is necessarily highin a sense specified and proved in a 2020 preprint by Lempp, Miller, Nies andSoskova available at . Part Computability theory and randomness Greenberg, Nies and Turetsky: Characterising SJTreducibility The basic concepts. SJT- reducibility and its equivalents. SJT-reducibility was introducedin [35, Exercise 8.4.37]. Definition 7.1 (Main: SJT-reducibility) . For an oracle B , a B –c.e. trace isa u.c.e. in B sequence h T n i n ∈ N of finite sets. For a function h , such a traceis h -bounded if | T n | ≤ h ( n ) for each n . A set A is jump-traceable if there isa computably bounded ∅ –c.e. trace h T n i n ∈ N such that J A ( n ) is in T n if it isdefined.For sets A, B , we write A ≤ SJR B if for each order function h , there is a B –c.e., h -bounded trace for J A .This is transitive by argument similar to [34, Theorem 3.3].(Question: Is “strongly superlow” reducibility equivalent to ≤ SJR ? Thiswas also mentioned in [35, Exercise 8.4.37], there shown to be at least asstrong.) Proposition 7.2. For each K -trivial set B , there is a c.e. K -trivial set A such that A SJR B .Proof. For some fixed computable function h there is a functional Ψ and K -trivial set A such that Ψ A has no c.e. trace bounded by h , by a resultof [6]. (Also see [35, 8.5.1] where h ( n ) = 0 . n works.) Encoding Ψinto J , we get a fixed computable function g so that the statement holds for J and g instead of Ψ and h .Relativizing to B we can retain the same g , sofor each B there is a K -trivial in B set A such that J A ⊕ B does not have a B -c.e. trace bounded by g . In particular, A SJR B .If B is K -trivial, it is low for K , and hence A is also K -trivial. Finally,there is K -trivial c.e. set b A ≥ T A , so we can make A c.e. (cid:3) Definition 7.3. Let c be a cost function. For sets A, B , we write A | = B c if there is a B -computable enumeration of h A s i satisfying c . Definition 7.4 (Benign cost functions) . A cost function c is benign [16]if from a rational ǫ > 0, we can compute a bound on the length of anysequence n < s ≤ n < s ≤ · · · ≤ n ℓ < s ℓ such that c ( n i , s i ) ≥ ǫ for all i ≤ ℓ . For example, c Ω is benign, with the bound being 1 /ǫ .Conjecture that strengthens proposition above: For each benign c , foreach B | = c , there is a c.e. A | = c such that A SJR B .Conjecture for each noncomputable c.e. E there are c.e. ≤ SJR -incomparable A, B ≤ T E .Fact ≤ SJR is Σ on the K -trivials. Density might be easy on the K -trivialsusing this.7.1.2. Two relevant randomness notions. Definition 7.5 (Demuth randomness) . A Demuth test is a seqence h G m i m ∈ N of open subsets of 2 N such that λG m ≤ − m and there is an ω -c.a. function p such that G m = [ W p ( m ) ] ≺ . Since the function p is ω -c.a., there are acomputable approximation function p ( m, s ) and a computable bound b suchthat lim s p ( m, s ) = p ( m ) and the number of changes is bounded by b ( m ).One writes G m [ t ] for [ W p ( m,t ) ,t ] ≺ , the approximation of the m -th componentat stage t . One may assume that λG m [ t ] ≤ − m for each t by cutting offthe enumeration of G m when it attempts to exceed that measure. One saysthat Z is Demuth random if for each such test, one has Z G m for almostevery m . Definition 7.6 (Weak Demuth randomness) . A nested Demuth test is aDemuth test h G m i m ∈ N such that G m ⊇ G m +1 for each m . One says that Z is weakly Demuth random if Z T m G m for each nested Demuth test h G m i m ∈ N . Replacing G m [ t ] by T i ≤ m G i [ t ] (and noting that the number ofchanges remains computably bounded), one may assume that the approxi-mations are nested at each stage t , i.e., G m [ t ] ⊇ G m +1 [ t ] for each m .7.2. Equvalent characterizations of ≤ SJR .Theorem 7.7. The following are equivalent for K -trivial c.e. sets A, B . (a) A ≤ SJR B (b) A | = B c for every benign cost function c (c) A ≤ T B ⊕ Y for each ML-random set Y that is not weakly Demuthrandom (d) A ≤ T B ⊕ Y for each ML-random set Y ∈ C , where C is the class ofthe ω -c.a., superlow, or superhigh sets. We remark that the implication (b) ⇒ (a) was essentially obtained byGreenberg and Nies [16, Prop. 2.1]. They proved the following. Let A bea c.e., jump-traceable set, and let h be an order function. Then there is abenign cost function c such that if A obeys c , then J A has a c.e. trace whichis bounded by h . Suppose now that A | = B c . Apply the argument in theproof of [16, Prop. 2.1] to a B -computable enumeration of A showing this.Then the h -bounded trace h T n i for J A constructed there is c.e. relative to B .For C ⊆ MLR define C ⋄ . By the implication (a) ⇒ (c) we obtain: OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 31 Corollary 7.8. Let C be a nonempty class of ML-randoms that contains noweakly Demuth random. Then C ⋄ is downward closed under ≤ SJR . For instance, let C = { Ω R } for a co-infinite computable set R . This showsthat the subideals of K -trivials considered in [14, 15] are SJT-ideals.We will prove the implications of the theorem in a cycle, starting from(b). The implications (b) ⇒ (c) and (c) ⇒ (d) rely on literature results, orstandard methods from the literature. The implication (d) ⇒ (a) combinesmethods from [36] and [4]. The implication (a) ⇒ (b) uses the box promotionmethod; for background see [17, 18].7.2.1. Proof of (b) ⇒ (c). Hirschfeldt and Miller showed that for each nullΠ class H ⊆ N there is a cost function c such that A | = c and Y ∈ MLR ∩ H implies A ≤ T Y ; see [35, proof of 5.3.15] for a proof of thisotherwise unpublished result.Suppose that a ML-random set Y fails a nested Demuth test h G m i . Then Y ∈ H = T m G m which is a Π class. We will apply the method ofHirschfeldt and Miller but incorporating the additional oracle set B , andusing the particular representation of the Π null class by a nested Demuthtest to ensure that the cost function c is benign. Also see post on weakDemuth randomness by Kuˇcera and Nies in [8]Let p ( m ) and its approximation p ( m, t ) be as in Definition 7.6 of nestedDemuth tests. We define the benign cost function c as follows. Define for k ≤ t r ( k, t ) = min { m : ∃ s.k < s ≤ t [ p ( m, s − = p ( m, s )] } V k,t = [ k ≤ s ≤ t G r ( k,s ) [ s ] c ( k, t ) = λV k,t . Clearly r ( k − , t ) ≥ r ( k, t ) for k > 0, hence the V k,t are nested and c ( k, t ) is nonincreasing in k . Similarly, c ( k, t ) is nondecreasing in t . Notethat if p ( m, s ) has stopped changing by a stage k , then r ( k, t ) > m foreach t ≥ k , and hence V k,t is contained in the final version G m of the m -thcomponent. By the conventions in Definition 7.6 above, if c ( k, t ) > − m then p ( m, s − = p ( m, s ) for some s in the interval ( k, t ]. Since the numberof changes of p ( m, s ) is bounded computably in m , this shows that the costfunction c is indeed benign.Suppose now that A | = B c via a B -computable approximation h A s i . Toshow A ≤ T Y ⊕ B for each ML-random set Y ∈ T m G m , we enumeratea Solovay test S relative to B , i.e., a uniformly Σ sequence hS n i relativeto B such that P n λ S n < ∞ . At stage s , when x < s is least such that A s ( x ) = A s − ( x ), list V x,s in S . This yields a Solovay test relative B by thehypothesis that the approximation of A obeys c .Since B is low for ML-randomness, the set Y is ML-random relative to B . Hence Y passes the Solovay test S . So choose s such that Y V forany V listed in S after stage s . Given an input x ≥ s , using Y as anoracle compute t > x such that [ Y ↾ t ] ⊆ V x,t . We claim that A ( x ) = A t ( x ). Otherwise A s ( x ) = A s − ( x ) for some s > t , which would cause V x,s (or somesuperset V y,s , y < x ) to be listed in G , contrary to Y ∈ V x,t .7.2.2. Proof of (c) ⇒ (d). Note that each superlow set is ω -c.a. So it sufficesto show that no ω -c.a. set, and no superhigh set, is weakly Demuth random.For ω -c.a. sets this is immediate from the definitions. For superhigh sets,this is a result of Kuˇcera and Nies [26, Cor. 3.6].7.2.3. Proof of (d) ⇒ (a). For convenience we restate the implication to beshown: Let C be the class of superlow, or of superhigh sets. Suppose A and B are K -trivial c.e. sets. If A ≤ T Y ⊕ B for each Y ∈ C ∩ MLR , then A ≤ SJR B . We remark that for this implication, the hypothesis suffices that A is c.e.and superlow, and B is Demuth traceable as discussed below; in particular,this holds if B is c.e. and superlow.As mentioned, the proof of this implication combines methods from Nies [36]and Bienvenu et al. [4]. Below we will review and use some technical notionsfrom these articles. First we consider [4, Def. 1.7]. Given an oracle B , aDemuth BLR h B i test generalises a Demuth test h G m i m ∈ N in that G m ⊆ N is a Σ ( B ) class with λG m ≤ − m , and there is a function f taking m to anindex for G m such that f has a B -computable approximation g , with g ( m, t )having a number of changes that is computably bounded in m . (Here wewill only need the case that f is ω -c.a., in which case the only differenceto usual Demuth tests is that the versions of the components are uniformlyΣ ( B ), rather than Σ .)The authors in [4] defined that an oracle B is low for Demuth BLR testsif every Demuth BLR h B i test can be covered by a Demuth test, in the sensethat passing the Demuth test implies passing the Demuth BLR h B i test. In[4, Thm. 1.8] they characterized such oracles via a tracing condition calledDemuth traceability. For c.e. sets, this condition is equivalent to being jumptraceable, or again superlow [4, Prop. 4.3].The following lemma holds for any oracle B . We will prove it shortly. Lemma 7.9. For a given order function h and a superlow c.e. set A , onecan build a Demuth BLR h B i test ( H m ) m ∈ N such that, if A ≤ T Y ⊕ B forsome ML-random set Y passing this test, then the function J A has a B -c.e.trace with bound h . Nies [36] defined a class C ⊆ N to be Demuth test–compatible if eachDemuth test is passed by a member of C . Using some methods from [13], heshowed in [36, Section 4] that the superlow ML-random sets, as well as thesuperhigh ML-random sets, are Demuth test–compatible.If a class C is Demuth test–compatible and B is Demuth traceable, theneach Demuth BLR h B i -test is passed by a set in C . So the lemma suffices toestablish the implication (d) ⇒ (a) in question.To verify Lemma 7.9, let Φ be a Turing functional such that Φ(0 e Y ⊕ B ) = Φ e ( Y ⊕ B ) for each e, Y . We reduce the lemma to the following. Claim 7.10. There is a Demuth BLR h B i test ( S m ) m ∈ N such that, if A =Φ Y ⊕ B for some Y passing this test, then J A has a B -c.e. trace with bound h . OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 33 This claim suffices to obtain Lemma 7.9: let ( H m ) m ∈ N be the Demuth BLR h B i test obtained as in [36, Lemma 2.6] applied to the test ( S m ) m ∈ N . Thus, if aset Y passes ( H m ) m ∈ N , then 0 e Y passes ( S m ) m ∈ N for each e . By hypoth-esis of Lemma 7.9, A ≤ T Y ⊕ B for some Y passing ( H m ) m ∈ N , so we have A = Φ Y ⊕ Be for some e , and hence A = Φ(0 e Y ⊕ B ). Since 0 e Y passes( S m ) m ∈ N , we can conclude from the claim that J A has a B -c.e. trace withbound h .It remains to prove Claim 7.10. Write U e = [ W Be ] ≺ . For the durationof the proof of the claim, a sequence h G m i of open sets will be called an( A, B ) -special test if there is a Turing functional Γ such that G m is the finalversion of the sets G m [ t ] = U Γ A ( m,t ) over stages t , and there is a computablefunction g such that the number of changes of Γ A ( m, t ) is bounded by g ( m ).A set Y passes such a test in the usual sense of Demuth tests, namely, Y G m for almost all m .We first observe that since A is superlow and c.e., there is a Demuth BLR h B i test hS m i that covers h G m i in the sense that each Y passing hS m i passes h G m i . To see this, let Θ be a Turing functional such that Θ X ( m, i ) is the i -th value of Γ X ( m, i ), for each oracle X . Note that by [36, Lemma 2.7]there is a computable enumeration ( A s ) s ∈ N of A and a computable function f such that for at most f ( m, i ) times a computation Θ A s ( m, i ) is destroyed.At stage t , let S m [ t ] = U Θ At ( m,i ) where i is maximal such that the expression on the right is defined atstage t . Clearly the number of times a version S m [ t ] changes is boundedby P g ( m ) i =0 f ( m, i ). Thus, ( S m ) m ∈ N is a Demuth test. If an oracle Y is in G m then Y is in the final version S m [ t ], so the new test indeed covers h G m i .So to establish Claim 7.10 it suffices to build an ( A, B )-special test h G m i in place of hS m i . To do so, we mostly follow [36, proof of Thm 3.2]. For m ∈ N let I m = { x : 2 m ≤ h ( x ) < m +1 } .At stage t , let u be the maximum use of the computations J A ( x ) for x ∈ I m that exist. We enumerate into the current version G m [ t ] all oracles Z suchthat Φ Z ⊕ Bt (cid:23) A ↾ u , as long as the measure stays below 2 − m . Whenever anew computation J A ( x ) for x ∈ I m converges at a stage, we start a newversion of G m . Clearly, there will be at most I m many versions.More formally, there is a Turing functional Γ such that for each string α of length t , U Γ α ( m,t ) = { Z : ∀ x ∈ I m [ J αt ( x ) ↓ with use u ⇒ α ↾ u (cid:22) Φ Z ⊕ Bt ] } . Let G m [ t ] = U ( ≤ − m )Γ A ( m,t ) . Here for a Σ ( B ) set W and rational ǫ > 0, as in theCut-off Lemma [36, Lemma 2.1] by W ( ≤ ǫ ) one denotes the Σ ( B ) set givenby the enumeration capped by measure ǫ . By the uniformity of the Cut-offLemma, from m, t with the help of oracle A we can compute an index forthis effectively open class. Thus, the versions G m [ t ] define an ( A, B )-specialtest ( G m ) m ∈ N .The B -c.e. trace ( T x ) x ∈ N is defined as follows. At stage t , for each string α of length t such that y = J αt ( x ) is defined and the measure of the current approximation to the c.e. open set U Γ α ( m,t ) exceeds 2 − m , put y into T x .The idea is that, if y = J A ( x ), then this must happen for some α ≺ A ,otherwise Y can be put into G m because there is no cut-off.The verification is similar to [36, proof of Thm 3.2] mutatis mutandis . Weomit the proofs of the two claims that follow, which are similar to the claimsin the proof of the corresponding result [36, Thm 3.2]. Claim 1. ( T x ) x ∈ N is a B -c.e. trace such that for each x we have T x ≤ h ( x ) . Claim 2. For almost every x , if y = J A ( x ) is defined, then y ∈ T x . This completes the proof of Claim 7.10 and hence Lemma 7.9, and henceof the implication in question.7.2.4. Proof of (a) ⇒ (b). This implication works in the context of muchweaker assumptions on A and B . We state this separately. Proposition 7.11. If A ≤ SJR B and B is jump traceable, then for everybenign cost function c , we have A | = B c . First we need another lemma. Lemma 7.12. Suppose T is a finite tree, and v , . . . , v n − ∈ T are pairwisedistinct, such that each v i has at least 2 children in T . Then T has at least n + 1 leaves. We omit the proof, which is a simple induction on n . Proof of Proposition 7.11. Fix c a benign cost function and f an ω -c.a. func-tion with c ( f ( n )) < − n for all n . We also denote by f a computable approx-imation to f , so that lim s f ( n, s ) = f ( n ) for all n , and such that for each n , |{ f ( n, s ) : s ∈ ω }| ≤ g ( n ), where g is some total computable function. Wemay assume that f ( n, s ) is non-decreasing in both n and s . Fix h such that B is h -JT.First we employ standard tricks to assume we already have the tracesfor the partial functions we intend to build. We define sets I en and J en for e < n < ω : • For each e , the sets { I en : e < n < ω } partition ω [2 e ] such that | I en | = g ( n ) + X i Suppose first that ( U x ) x ∈ ω were an oracle-free c.e. k -trace of Φ A . We willhave a module for each n > e . Our module for n seeks to test A at variouslengths, in particular at length f ( n ). At stage s = n , or at stage s > n with f ( n, s ) = f ( n, s − f ( n, s ). It will also testvarious lengths as they are provided to it by the n + 1 module.For a length ℓ , it will first test A ↾ ℓ in an element of x ∈ I n – that is, wedefine Φ σ ( x ) = σ for all σ ∈ ℓ , and we monitor the strings enumerated into U x . This will narrow down the possibilities for A ↾ ℓ to a set of at most n strings. The module will then test each of those strings in J n (we will saymore about how testing is done in J n in a moment). If more than one ofthose strings were to pass this second test, we would promote ℓ , i.e. tell the n − A ↾ ℓ . We will arrange thatthe n module promotes at most n − n − g ( n − 1) + n − n = e + 1, the n module will still declare lengths to be promoted, eventhough there is no n − n module, and so it will continue on as if there were an n − U x ) x ∈ ω only traces Φ A with oracle B , so we will need to relyon ( V y ) y ∈ ω to approximate this. When we decide to test a length ℓ at x ∈ I n ,we will simultaneously define Ψ Y ( h e + 1 , x, i i ) for i < n and all oracles Y to be the i th element enumerated into U Yx , if such an element exists. Then A ↾ ℓ ∈ U x ⊆ ℓ ∩ [ i 1) + P i 0, fix strings σ i and σ i on the basis of which B decided to promote ℓ i .By construction, B decides ℓ i should be promoted before it decides ℓ i +1 should be, and thus σ i +10 ↾ ℓ i = σ i +11 ↾ ℓ i for i > 0. Clearly this also holds for i = 0. Now define the following sequence of sets: • Z n = { σ n , σ n } ; • For 0 < i < n , Z i = Z i +1 ∪ ( { σ i , σ i } \ { σ ↾ ℓ i : σ ∈ Z i +1 } ) . Note that each Z i is an antichain, and { σ i , σ i } ⊆ { σ ↾ ℓ i : σ ∈ Z i } by con-struction. Let T = { σ ↾ ℓ i : σ ∈ Z & i ≤ n } , which we think of as a tree.Note that the leaves of T are precisely Z .For i < n , let v i = σ i +10 ↾ ℓ i = σ i +11 ↾ ℓ i . Then the v i are pairwise distinctand each has at least 2 children in T (namely, σ i +10 and σ i +11 ↾ ℓ i ). Thus | Z | ≥ n + 1.Let D be the set of axes chosen for various σ ∈ Z , and define τ ∈ J n by τ ( d ) = (cid:26) d ∈ D, d D. Observe that τ ∈ J n ( d ) ⇐⇒ d ∈ D . Claim 7.13. For each σ ∈ Z , we make the definition Φ σ ( τ ) = σ .Proof. By construction, we will make this definition so long as we have notalready defined Φ σ ( τ ) to be something else. But our actions for any string σ ′ Z will never do this, as such σ ′ will have an axis d D , and so will notseek to make a definition at τ . And σ ′ ∈ Z \ { σ } will not do this, as theywill only seek to make a definition for Φ σ ′ ( τ ), and σ ′ and σ are incomparableas Z is an antichain. (cid:3) As each of the ℓ i is promoted, we have Z ⊆ U Bτ , contradicting | U Bτ | ≤ h ( τ ) = n .Thus the construction can proceed. The remainder of the argument isrelative to B .Fix b ℓ the longest length which B believes the e +1 module should promote.Nonuniformly fix A ↾ b ℓ . Let L ( n, s ) be the set of lengths being tested by the n module at stage s . At a stage s , define a partial sequence σ sn for e ≤ n ≤ s recursively: OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 37 • σ se = A ↾ b ℓ ; • Given σ sn , define σ sn +1 to be a string τ extending σ sn with | τ | =max L ( n + 1 , s ), and such that for each ℓ ∈ L ( n + 1 , s ), τ ↾ ℓ isconfirmed at n + 1 by stage s , if such a string τ exists. Claim 7.14. There is at most one possible choice for σ sn .Proof. For n = e , this is immediate.For n > e , suppose there were two distinct strings τ and τ which areappropriate to pick for σ sn . Fix ℓ ∈ L ( n, s ) least with τ ↾ ℓ = τ ↾ ℓ . Then τ ↾ ℓ , τ ↾ ℓ witness the promotion of ℓ at stage s , and ℓ > | σ sn − | , as τ and τ both extend σ sn − . This contradicts | σ sn − | = max L ( n − , s ) (or contradictsthe definition of b ℓ if n = e + 1). (cid:3) Claim 7.15. Let ℓ n = max S s L ( n, s ) for n > e , and ℓ e = b ℓ . Then A ↾ ℓ n =lim s σ sn for n ≥ e .Proof. Induction on n . The case n = e is immediate.For n > e , first observe that ℓ n − is either a length promoted by the n module (and so eventually an element of L ( n, s )) or is f ( n − , s ) ≤ f ( n, s )for some s , and so is bounded by an element of L ( n, s ). Thus ℓ n − ≤ ℓ n .Now fix s sufficiently large such that σ sm = A ↾ ℓ m for all m < n and s ≥ s , and such that L ( n, s ) = S s L ( n, s ). As Φ A is traced by ( U Bx ) x ∈ ω ,there is a stage s ≥ s such that each A ↾ ℓ for ℓ ∈ L ( n, s ) is confirmed at n . Then A ↾ ℓ n is a possible choice for σ sn for every s ≥ s , and thus is σ sn . (cid:3) Define a sequence of stages ( s t ) t ∈ ω as follows: • t = e . • Given s t , s t +1 is the least s > s t such that for every n with e < n ≤ t , σ sn exists.Define A t = σ s t t . Claim 7.16. ( A t ) t ∈ ω | = c Proof. Suppose e < n ≤ t and A t ( z ) = A t ( z + 1) for some z with c ( z, t ) ≥ − n . As c ( z, s t ) ≥ c ( z, t ), z < f ( n, s t ) ∈ L ( n, s t ). Thus σ s t n = σ s t +1 n . Fix m least with σ s t m = σ s t +1 m . Fix ℓ ∈ L ( m, s t ) least with σ s t m ↾ ℓ = σ s t +1 m ↾ ℓ . Ifno length less than ℓ and greater than max L ( m − , s t ) is promoted by the m module at a stage s ∈ ( s t , s t +1 ], then these witness the promotion of ℓ at stage s t +1 , and ℓ > max L ( m − , s t ), as both σ s t m and σ s t +1 m extend σ s t m − = σ s t +1 m − . So whenever there is such a z, n and t , there is a promotionby an m module for m ≤ n at a stage after s t .There can be at most P e The purpose of this post is to provide an example showing that the bound-edness hypothesis in [39, Prop. 24] is necessary. We briefly review some background and notation. Let A ∞ denote thetopological space of one-sided infinite sequences of symbols in an alphabet A . Randomness notions etc. carry over from the case that A = { , } . Ameasure ρ on A ∞ is called shift invariant if ρ ( G ) = ρ ( T − ( G )) for each open(and hence each measurable) set G . The empirical entropy of a measure ρ along Z ∈ A ∞ is given by the sequence of random variables h ρn ( Z ) = − n log | A | ρ [ Z ↾ n ] . A shift invariant measure ρ on A ∞ is called ergodic if every ρ integrable func-tion f with f ◦ T = f is constant ρ -almost surely. The following equivalentcondition can be easier to check: for any strings u, v ∈ A ∗ ,lim N N n − X k =0 ρ ([ u ] ∩ T − k [ v ]) = ρ [ u ] ρ [ v ] . For ergodic ρ , the entropy H ( ρ ) is defined as lim n H n ( ρ ), where H n ( ρ ) = − n X | w | = n ρ [ w ] log ρ [ w ] . Thus, H n ( ρ ) = E ρ h ρn is the expected value with respect to ρ . Recall thatby concavity of the logarithm function and subadditivity of the entropy H ( X, Y ) ≤ H ( X ) + H ( Y ), the limit exists and equals the infimum of thesequence. This limit is denoted H ( ρ ), the entropy of ρ . Theorem 8.1 (SMB theorem, e.g. [45]) . Let ρ be an ergodic measure onthe space A ∞ . For ρ -almost every Z we have lim n h ρn ( Z ) = H ( ρ ) . Proposition 8.2 ([39], Prop. 7.2) . Let ρ be a computable ergodic measureon the space A ∞ such that for some constant D , each h ρn is bounded aboveby D . Suppose the measure µ is Martin-L¨of a.c. with respect to ρ . Then lim n E µ | h ρn − H ( ρ ) | = 0 . We now given an example showing that the boundedness hypothesis onthe h ρn is necessary. In fact we provide a computable ergodic measure ρ such that some finite measure µ ≪ ρ makes the sequence E µ h ρn converge to ∞ . This condition µ ≪ ρ (every ρ null set is a µ null set) is stronger thanrequiring that µ is Martin-L¨of a.c. with respect to ρ . Example 8.3. There is an ergodic computable measure ρ (associated toa binary renewal process) and a computable measure µ ≪ ρ such thatlim n E µ h ρn = ∞ . (We can then normalise µ to become a probability measure,and still have the same conclusion.) Proof. Let k range over positive natural numbers. The real c = P k − k iscomputable. Let p k = 2 − k /c so that P p k = 1. Let b = P k k · p k which isalso computable.Let ρ be the measure associated with the corresponding binary renewalprocess, which is given by the conditions ρ [ Z = 1] = 1 /b and ρ (10 k ≺ Z | Z = 1) = p k . OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 39 Informally, the process has initial value 1 with probability 1 /b , and aftereach 1 with probability p k it takes k many 0s until it reaches the next 1. Seeagain e.g. [45, Ch. 1] where it is shown that ρ is ergodic. Write v k = 10 k ρ [ v k ] = p k /b .Define a function f in L ( ρ ) by f ( v k b Z ) = k − /p k and f ( X ) = 0 for any X not extending any v k . It is clear that f is L ( ρ )-computable, in the usualsense that there is an effective sequence of basic functions h f n i convergingeffectively to f : let f n ( X ) = f ( X ) in case v k ≺ X , k ≤ n , and f n ( X ) = 0otherwise. Define the measure µ by dµ = f dρ , i.e. µ ( A ) = R A f dρ . Thus µ [ v k ] = k − /b . Since ρ is computable and f is L ( ρ )-computable, µ iscomputable. Also note that µ (2 N ) = R f dρ is finite.For any n > 2, letting k = n − 2, we have E µ h ρn ≥ − n µ [ v k ] log ρ [ v k ] = − nk b ( k − bc ) ≥ k nb − O (1) . (cid:3) Nies and Tomamichel: the measure associated with aninfinite sequence of qubits For background and notation see the 2017 Logic Blog entry [9, Section 6].Recall that mathematically, a qubit is a unit vector in the Hilbert space C .We give a brief summary on “infinite sequences” of qubits. One considersthe C ∗ algebra M ∞ = lim n M n ( C ), an approximately finite (AF) C ∗ alge-bra. “Quantum Cantor space” consists of the state set S ( M ∞ ), which isa convex, compact, connected set with a shift operator, deleting the firstqubit.Given a finite sequence of qubits, “deleting” a particular one generallyresults in a statistical superposition of the remaining ones. This is is why S ( M ∞ ) consists of coherent sequences of density matrices µ = h µ n i n ∈ N where µ n is in M n ( C ) (density matrices formalise such superpositions), ratherthan just of sequences of unit vectors in ( C ) ⊗ n . To be coherent means that T ( µ n +1 ) = µ n where T is the partial trace operation deleting the last qubit.For more background on this, as well as an algorithmic notion of randomnessfor such states, see Nies and Scholz [38].We defined in [9, Section 6] what it means for a state µ on M ∞ to beqML-random with respect to a computable shift invariant state ρ .For each density matrix D ∈ M n its diagonal D is also a density matrix.This is because the operator A P x ∈{ , } n P x DP x is completely positiveand trace preserving (here as usual P x = | x ih x | is the projection onto thesubspace spanned by the basis vector given by x ). Clearly µ n = T ( µ n +1 ) foreach n because taking the partial trace means adding corresponding items onthe two quadratic 2 n × n components along the diagonal. So µ is a diagonalstate, and hence corresponds to a measure on Cantor space. Clearly if µ iscomputable then so is µ . Shift invariance is also preserved by this operation.Ergodicity of µ can be used as a test of the more complicated ergodicityfor µ . Fact 9.1. If µ is ergodic then so is µ . Proof. Suppose µ = αη + βν is a nontrivial convex decomposition of µ intoshift-invariant states. Then µ = αη + βν is a nontrivial convex decompositionof µ into shift-invariant states as well. (cid:3) Fact 9.2. Let ρ be a computable shift-invariant measure. If a state µ isqML-random wrt ρ then so is µ .Proof. Note that for each classical Σ set G we have µ ( G ) = µ ( G ), where onthe left hand side G is interpreted as an ascending sequence h p m i of clopenprojections p m ∈ M m , and then µ ( G ) = lim m µ ( p m ). But µ ( p m ) = Tr( µ ↾ m p m ) = Tr( µ ↾ m p m ) because p m is diagonal. (cid:3) We obtain a partial quantum version of Prop. 8.2. This answers onespecial case of Conjecture 6.3 in [9] (unfortunately the roles of µ and ρ areexchanged there). The boundedness hypothesis turned out to be necessaryby Example 8.3, but was not present in the statement of the conjecture backthen. Proposition 9.3. Let ρ be a computable ergodic measure on the space A ∞ such that for some constant D , each h ρn is bounded above by D . Write s = H ( ρ ) . Suppose the state µ is qML random with respect to ρ . Then lim n Tr( µ ↾ n | h ρn − sI n | ) = 0 . Here the function h ρn is viewed as defined on strings of length n , and inthe expression above we identify it with the corresponding diagonal matrixin M n . Proof. Let µ be the classical state (measure) such that µ ↾ n is the diagonalof µ ↾ n , as above. By the fact above we have µ ≪ ML ρ i.e., the non ρ -MLRbit sequences form a µ null set. NowTr( µ ↾ n | h ρn − sI n | ) = Tr( µ ↾ n | h ρn − sI n | ) = E µ | h ρn − s | .It now suffices to apply Prop. 8.2. (cid:3) If ρ is i.i.d. then the boundedness condition on the h ρn holds. This yieldsa new proof of [9, Thm. 6.4] (first turning the ergodic state ρ into a classicalstate by applying a fixed unitary “qubit-wise”, as before). Part Set theory Yu: perfect subsets of uncountable sets of reals We make some remarks on a recent result: Theorem 10.1 (Hamel, Horowitz, Shelah [19]) . Assume ZF + DC . If every uncountable Turing invariant set of reals has aperfect subset, then so has every uncountable set of reals. We obtained an improvement of the Theorem which was added in Sec-tion III of the most recent version of [19]. Theorem 10.2 (Yu [19]) . Assume ZF + AC ω . For any analytic countableequivalence relation E , if every uncountable E -invariant set of reals has aperfect subset, then so has every uncountable set of reals. OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 41 Remark 1 : Actually AC ω can be removed from Theorems 10.1 and 10.2.In the recursion theoretic proof of Theorem 10.1, the first use of AC ω is toprove that [ Q ] T ∩ A is uncountable. But this is clearly unnecessary, sinceotherwise Q ⊆ [[ Q T ] ∩ A ] T would be countable but without appealing AC ω due to the uniformity.The second use of AC ω is to prove that Q e,i ∩ A is uncountable for some e, i . But if Q e,i ∩ A is countable for all e, i , then the computation is uniformsince Q e,i ∩ A = Q e,i ∩ P is a countable closed set. AC ω can be removedfrom Theorem 10.2 for similar reasons. Remark 2 : Ironically we need AC ω to prove the conclusion for everycountable Borel equivalence relation since the Borelness implying Σ -nessrequires AC ω . But for most natural Borel countable equivalence relations,it seems AC ω is unnecessary. References [1] L. Agarwal and M. Kompatscher. Pairwise nonisomorphic maximal-closed subgroupsof Sym (N) via the classification of the reducts of the Henson digraphs. The Journalof Symbolic Logic , 83(2):395–415, 2018. 2[2] G. Ahlbrandt and M. Ziegler. Quasi finitely axiomatizable totally categorical theories. Annals of Pure and Applied Logic , 30(1):63–82, 1986. 2[3] M. Benli and B. Kaya. Descriptive complexity of subsets of the space of finitelygenerated groups. Arxiv: 1909.11163 , 2019. 3.3[4] L. Bienvenu, R. Downey, N. Greenberg, A. Nies, and D. Turetsky. Characterizinglowness for Demuth randomness. The Journal of Symbolic Logic , 79(2):526–569, 2014.7.2, 7.2.3[5] Z. Chatzidakis. Model theory of profinite groups having the Iwasawa property. IllinoisJournal of Mathematics , 42(1):70–96, 1998. 4, 4.2[6] P. Cholak, R. Downey, and N. Greenberg. Strongly jump-traceability I: the com-putably enumerable case. Adv. in Math. , 217:2045–2074, 2008. 7.1.1[7] J.D. Dixon and B. Mortimer. Permutation groups , volume 163. Springer Science &Business Media, 1996. 5[8] A. Nies (editor). Logic Blog 2011. Available at http://arxiv.org/abs/1403.5721 ,2011. 7.2.1[9] A. Nies (editor). Logic Blog 2017. Available at http://arxiv.org/abs/1804.05331 ,2017. 9, 9, 9[10] A. Nies (editor). Logic Blog 2019. Available at http://arxiv.org/abs/2003.03361 ,2019. 1[11] V. Ferenczi, A. Louveau, and C. Rosendal. The complexity of classifying separa-ble Banach spaces up to isomorphism. Journal of the London Mathematical Society ,79(2):323–345, 2009. 2[12] A. Fig`a-Talamanca and C. Nebbia. Harmonic Analysis and Representation Theoryfor Groups Acting on Homogenous Trees , volume 162. Cambridge University Press,1991. 4.22[13] N. Greenberg, D. Hirschfeldt, and A. Nies. Characterizing the strongly jump-traceablesets via randomness. Adv. Math. , 231(3-4):2252–2293, 2012. 7.2.3[14] N. Greenberg, J. S. Miller, and A. Nies. Computing from projections of randompoints. Journal of Mathematical Logic , page 1950014, 2019. 7.2[15] N. Greenberg, J. S. Miller, A. Nies, and D. Turetsky. Martin-l¨of reducibility and costfunctions. Arxiv: https://arxiv. org/abs/1707.00258 , 2017. 7.2[16] N. Greenberg and A. Nies. Benign cost functions and lowness properties. J. SymbolicLogic , 76:289–312, 2011. 7.4, 7.2[17] N. Greenberg and D. Turetsky. Strong jump-traceability and Demuth randomnesss. Proc. Lond. Math. Soc. , 108:738–779, 2014. 7.2 [18] N. Greenberg and D. Turetsky. Strong jump-traceability. Bulletin of Symbolic Logic ,24(2):147–164, 2018. 7.2[19] C. Hamel, H. Horowitz, and S. Shelah. Turing invariant sets and the perfect setproperty. preprint arXiv:1912.12558 , 2019. 10.1, 10, 10.2[20] A. O. Houcine and F. Point. Alternatives for pseudofinite groups. Journal of GroupTheory , 16(4):461–495, 2013. 6, 6.1, 6.2, 6.2[21] J. Chubb I. Bilanovic and A. Roven. Detecting properties from description of groups. Arch. Math. Log. , 59:293 – 312, 2020. 3.3[22] A Ivanov. Closed groups induced by finitary permutations and their actions on trees. Proceedings of the American Mathematical Society , 130(3):875–882, 2002. 5[23] A. Kaichouh. Amenability and ramsey theory in the metric setting. Fund. Math. ,231:19 – 38, 2015. 3.3[24] I. Kaplan and P. Simon. The affine and projective groups are maximal. Transactionsof the American Mathematical Society , 368(7):5229–5245, 2016. 2[25] A. Kechris, A. Nies, and K. Tent. The complexity of topological group isomorphism. The Journal of Symbolic Logic , 83(3):1190–1203, 2018. 2, 2, 3, 4, 4.16, 5[26] A. Kuˇcera and A. Nies. Demuth randomness and computational complexity. Ann.Pure Appl. Logic , 162:504–513, 2011. 7.2.2[27] M. Lawson. Inverse semigroups: the theory of partial symmetries . World Scientific,1998. 4.1, 4.2, 4.4[28] B. Majcher-Iwanow. Finitary shadows of compact subgroups of s ( ω ). Algebra Uni-versalis , 81(2):25, 2020. 5[29] M. Malicki. Abelian pro-countable groups and non-Borel orbit equivalence relations. Mathematical Logic Quarterly , 62(6):575–579, 2016. 2, 2, 2[30] Maciej Malicki. Abelian pro-countable groups and orbit equivalence relations. arXivpreprint arXiv:1405.0693 , 2014. Fundamentae. 2[31] A. Mekler. Stability of nilpotent groups of class 2 and prime exponent. The Journalof Symbolic Logic , 46(04):781–788, 1981. 2[32] J. Melleray and T. Tsankov. Generic representations of abelian groups and extremeamenability. Isr. J. Math. , 198:129 – 167, 2013. 3.2, 3.3[33] P. Neumann. The structure of finitary permutation groups. Archiv der Mathematik ,27(1):3–17, 1976. 5, 5[34] K.M. Ng. Beyond strong jump traceability. Proceedings of the London MathematicalSociety , 2010. To appear. 7.1.1[35] A. Nies. Computability and Randomness , volume 51 of Oxford Logic Guides . OxfordUniversity Press, Oxford, 2009. 444 pages. Paperback version 2011. 7.1.1, 7.1.1, 7.1.1,7.2.1[36] A. Nies. Computably enumerable sets below random sets. Ann. Pure Appl. Logic ,163(11):1596–1610, 2012. 7.2, 7.2.3, 7.2.3, 7.2.3[37] A. Nies, P. Schlicht, and K. Tent. Oligomorphic groups are essentially countable. arXiv preprint arXiv:1903.08436 , 2019. 2, 4, 4.2, 4.10, 4.2, 4.2, 4.2, 4.16, 4.2, 5, 5[38] A. Nies and V. Scholz. Martin-L¨of random quantum states. Journal of MathematicalPhysics , 60(9):092201, 2019. available at doi.org/10.1063/1.5094660. 9[39] A. Nies and F. Stephan. A weak randomness notion for measures. Available at https://arxiv.org/abs/1902.07871 , 2019. 8, 8.2[40] N. Nikolov and D. Segal. On finitely generated profinite groups. I. Strong completenessand uniform bounds. Ann. of Math. (2) , 165(1):171–238, 2007. 6.2[41] N. Nikolov and D. Segal. Generators and commutators in finite groups; abstractquotients of compact groups. Inventiones mathematicae , 190(3):513–602, 2012. 6.2[42] V. G. Pestov. Dynamics of infinite-dimensional groups. Inst. Mat. Pura. Apl. (IMPA),Rio de Janeiro, 2005) , 2005. 3.2[43] F.M. Schneider and A. Thom. Topological matchings and amenability. Fund. Math. ,238:167 – 200, 2017. 2[44] F.M. Schneider and A. Thom. On Folner sets in topological groups. Compositio Math-ematica , 154:1333–1361, 2018. 3.1[45] P. Shields. The Ergodic Theory of Discrete Sample Paths . Graduate Studies in Math-ematics 13. American Mathematical Society, 1996. 8.1, 8 OGIC BLOG 2020 (THE 10TH ANNIVERSARY BLOG) 43 [46] A. M. Slobodskoi. Unsolvability of the universal theory of finite groups. Algebra iLogika , 20:207–230, 1981. 6[47] V. Uspenskij. On the group of isometries of the Urysohn universal metric space. Comment. Math. Univ. Carolinae , 31:181 – 182, 1990. 3.1[48] Ph. Wesolek and J. Williams. Chain conditions, elementary amenable groups, anddescriptive set theory. Groups Geom. Dyn. , 11:649 – 684, 2017. 3.3[49] G. Willis. The structure of totally disconnected, locally compact groups. Mathema-tische Annalen , 300(1):341–363, 1994. 4[50] G. Willis. Computing the scale of an endomorphism of a totally disconnected locallycompact group.