Free actions of free groups on countable structures and property (T)
aa r X i v : . [ m a t h . G R ] S e p FREE ACTIONS OF FREE GROUPS ON COUNTABLESTRUCTURES AND PROPERTY (T)
DAVID M. EVANS AND TODOR TSANKOV
Abstract.
We show that if G is a non-archimedean, Roelcke precom-pact, Polish group, then G has Kazhdan’s property (T). Moreover, if G has a smallest open subgroup of finite index, then G has a finite Kazh-dan set. Examples of such G include automorphism groups of countable ω -categorical structures, that is, the closed, oligomorphic permutationgroups on a countable set. The proof uses work of the second author onthe unitary representations of such groups, together with a separationresult for infinite permutation groups. The latter allows the constructionof a non-abelian free subgroup of G acting freely in all infinite transitivepermutation representations of G . Introduction
Main results.
A topological group G is non-archimedean if it has abase of open neighbourhoods of the identity consisting of subgroups. Thesymmetric group Sym( X ) on a set X , consisting of the group of all permu-tations of X , equipped with the topology of pointwise convergence, is anexample of such a group: pointwise stabilizers of finite sets form a base ofopen neighbourhoods of the identity. It is well-known that a Polish group G is non-archimedean if and only if it is isomorphic to a closed subgroupof Sym( X ) for some countable X . Moreover, such groups are exactly auto-morphism groups of first-order structures on X .A group G ≤ Sym( X ) is said to be oligomorphic (in its action on X )if G has only finitely many orbits on X n , for all n ∈ N (where the actionon X n is the diagonal action). Such groups have been extensively studiedfrom the point of view of infinite permutation groups, combinatorics, modeltheory, and topological dynamics (see, for example, the references [4], [11]and [9]). They arise as automorphism groups of ω -categorical structures andmodel-theoretic methods produce a wide variety of examples of these.In [13], the second author studied the unitary representations of oligo-morphic permutation groups, showing that they are completely reducibleand giving a description of the irreducible representations. Many of the re-sults of [13] hold under a weaker (and more intrinsic) assumption than thatof being an oligomorphic group: that of Roelcke precompactness (see Defini-tion 1.5 here). For G ≤ Sym( X ) this means that whenever Y is a union offinitely many G -orbits, then G acts oligomorphically on Y (see Lemma 1.6). Mathematics Subject Classification.
Primary 22A25, 20B27; Secondary 03C15.The second author was partially supported by the ANR grant GrupoLoco (ANR-11-JS01-008).
We note that in the interesting cases, Roelcke precompact groups are notlocally compact: more precisely, a Roelcke precompact topological group islocally compact iff it is compact.Using this description of the unitary representations, the paper [13] showsthat Kazhdan’s property (T) holds for a natural class of closed, oligomorphicpermutation groups G ≤ Sym( X ) ([13, Theorem 6.6]; for the definition ofproperty (T), see Definition 4.2 here). Furthermore, [13, Theorem 6.7] givessome examples — including the automorphism groups of the rational order-ing ( Q , ≤ ) and the random graph — of such groups with strong property(T), where the Kazhdan set can be taken to be finite. In the latter case,the proof proceeds by finding a non-abelian free subgroup of G which actsfreely on X , an idea that goes back to Bekka [1]. We use a similar methodhere, combined with techniques from permutation group theory, to provethe following general result. Theorem 1.1.
Suppose that G is a non-archimedean, Roelcke precompact,Polish group and G ◦ is the intersection of the open subgroups of finite indexin G . Then G and G ◦ have Kazhdan’s property (T) and G ◦ has a finiteKazhdan set. While so far property (T) has found most of its applications in the realmof locally compact groups, we note that there are some interesting conse-quences in our setting as well. Combining Theorem 1.1 with the results ofGlasner and Weiss [5], we obtain the following.
Corollary 1.2.
Let G be a non-archimedean, Roelcke precompact groupand G y X a continuous action on a compact Hausdorff space X . Thenthe simplex of G -invariant measures on X is a Bauer simplex , i.e., the setof its extreme points is closed.
The extreme points of the simplex of invariant measures are exactly theergodic measures (a measure µ is ergodic if every µ -invariant measurableset A ⊆ X is null or co-null; a set A is µ -invariant if for every g ∈ G , µ ( A △ gA ) = 0). While Glasner and Weiss only state their theorem forlocally compact groups, the proof works equally well in general; see thebook of Phelps [12] for the general version of the ergodic decompositiontheorem needed in the proof.We also note that while amenable locally compact groups with property(T) must be compact, this is not necessarily true in our setting. Indeed,there are a number of automorphism groups of ω -categorical structures thatare amenable: for example, this is true if the structure has the so-called Hrushovski property , see [8] for more details. Amenability is relevant forCorollary 1.2 because it ensures that the simplex of invariant measures isnon-empty for any G -flow X .Apart from the description of the representations of non-archimedean,Roelcke precompact, Polish groups given in [13] (see Theorem 4.1 here), themain ingredient in the proof is the following, which is the main contributionof the current paper. Theorem 1.3.
Suppose G is a non-archimedean, Roelcke precompact Polishgroup and G ◦ is the intersection of the open subgroups of finite index in G .Suppose G ◦ = { G } . Then there exist f, g ∈ G ◦ which generate a (non-abelian) free subgroup F of G ◦ with the property that if H ≤ G is open andof infinite index, then F acts freely on the coset space G/H . Here, recall that a group F acting on a set Y is acting freely if for allnon-identity g ∈ F and y ∈ Y we have gy = y . Thus each F -orbit on Y isregular.Theorems 1.1 and 1.3 answer Questions (1) (for non-archimedean groups)and (2) at the end of [13].Recall that an action of a group G on a discrete space X is called amenable if there is a G -invariant finitely additive measure on X . As afurther corollary to Theorem 1.3, we note: Corollary 1.4.
Suppose G is a non-archimedean, Roelcke precompact Pol-ish group which is acting continuously on the discrete space X . Suppose theaction of G on X is amenable. Then X contains a finite orbit. The proof of Theorem 1.3 rests ultimately on Neumann’s Lemma (seeLemma 2.1), a very general result about separating finite sets in an infinitepermutation group.The required consequences of this for closed, Roelcke precompact per-mutation groups are given in Section 2. These results can be deduced from‘folklore’ results in model theory, but we provide proofs of them in the lan-guage of permutation groups. Theorem 1.3 is proved in Section 3 along withCorollary 1.4. Section 4 discusses Kazhdan’s property (T) and contains theproof of Theorem 1.1.
Notation.
Our notation for permutation groups is fairly standard. Groupsact on the left. If G is a group acting on X and A ⊆ X , then G A is thepointwise stabilizer { g ∈ G : ga = a for all a ∈ A } . If A = { a } is a singleton,we denote this by G a . If G is the automorphism group of some structure M with domain X , then we write G = Aut( M ) and use the alternative notationAut( M/A ) for G A . We do not usually distinguish notationally between astructure and its underlying set. Acknowledgements.
Both Authors thank Dugald Macpherson for help-ful discussions about some of the material in this paper. The paper wascompleted while the Authors were participating in the trimester programme‘Universality and Homogeneity’ at the Hausdorff Institute for Mathematics,Bonn. We are also grateful to the anonymous referee for carefully readingthe paper and making useful suggestions.1.2.
Background.Definition 1.5.
The topological group G is called Roelcke precompact iffor every open neighbourhood U of the identity, there is a finite set E suchthat G = U EU . DAVID M. EVANS AND TODOR TSANKOV If G is non-archimedean, so G ≤ Sym( X ) for some X , then U in Defini-tion 1.5 can be taken to be an open subgroup and the condition for Roelckeprecompactness says that there are only finitely many double cosets of U in G . In fact, if we take ¯ a ∈ X n and U = G ¯ a its stabilizer, then the conditionsays that there are finitely many G -orbits on Y , where Y is the G -orbit on X n containing ¯ a . Recall that a group G is said to act oligomorphically on aset Y if G has finitely many orbits on Y n for all n ∈ N . The following is arestatement of [13, Theorem 2.4]: Lemma 1.6.
Suppose G ≤ Sym( X ) . Then G is Roelcke precompact if andonly if whenever Y is a union of finitely many G -orbits on X , then G actsoligomorphically on Y . If G is acting on X as in the above, then we say that G is locally oligo-morphic on X . Note that in this case, if A ⊆ X is finite, then its pointwisestabilizer G A is also locally oligomorphic on X (this follows easily fromRoelcke precompactness).We summarise the above as: Corollary 1.7.
A topological group G is non-archimedean, Polish and Roel-cke precompact if and only if it can be represented as a closed, locally oligo-morphic subgroup of Sym( X ) , for countable X . Closed subgroups of Sym( X ) are precisely automorphism groups of first-order structures on X . Indeed, if G ≤ Sym( X ), we consider the canonicalstructure which has a relation for each G -orbit on X n , for all n ∈ N . Then G is a closed subgroup of Sym( X ) if and only if it is the full automor-phism group of this canonical structure. If X is countable, then, by theRyll-Nardzewski Theorem, G is oligomorphic if and only if this canonicalstructure is ω -categorical. (The book [4] is a convenient reference for thismaterial.)The following fact is an easy consequence of Roelcke precompactness;see [13, Corollary 2.5] for a proof. Lemma 1.8.
Suppose X is countable and G ≤ Sym( X ) is locally oligomor-phic. Then G has only countably many open subgroups. We also note that for such G , there is a ‘universal’ choice for the set X . Lemma 1.9.
Suppose G is a non-archimedean Polish group with countablymany open subgroups. Then there is a faithful action of G on a countable set X = X ( G ) with the property that for every open subgroup U ≤ G , there is a ∈ X such that U = G a . Moreover, G is closed in Sym( X ) in this action.Proof. Let ( U i : i ∈ I ) be a system of representatives for the set of conjugacyclasses of open subgroups of G . Let X ( G ) be the disjoint union of the (left)coset spaces G/U i . So X ( G ) is countable and if U ≤ G is open there is aunique i ∈ I such that U is conjugate to U i , so U = gU i g − for some g ∈ G .Then U is the stabilizer of the coset a = gU i ∈ X ( G ).It remains to prove that G is a closed subgroup of Sym( X ( G )). As G isPolish, it suffices to show that the original topology of G is the same as the one inherited from Sym( X ( G )). Indeed, if { g n } is a sequence in G convergingto 1 in Sym( X ( G )), then it eventually enters every open subgroup of G ,which, as G is non-archimedean, means that g n → G . (cid:3) Remarks 1.10.
It is worth noting that if M is a countable ω -categoricalstructure and G = Aut( M ), then the action of G on X ( G ) (as in Lemma 1.9)is essentially that of G on M eq .We finally observe that our setting is slightly more general than theclassical one of oligomorphic groups (and ω -categorical structures). Locallyoligomorphic groups can be represented as inverse limits of oligomorphicones (which, however, are not necessarily closed in Sym( X )). Examples canbe obtained by taking the disjoint union of countably many ω -categoricalstructures (in disjoint languages); then the automorphism group of thisstructure is the direct product of the automorphism groups of the individualstructures and is locally oligomorphic. Perhaps more interestingly, considerthe abelian group M which is a direct sum of countably many copies of thePr¨ufer p -group Z ( p ∞ ) (complex roots of 1 of order a power of p ) for someprime p . It can be checked that Aut( M ) acts locally oligomorphically on M . Another example is given by the ℵ -partite random graph (with namedparts). Further interesting structures can be constructed from inverse limitsof finite covers of ω -categorical structures.2. Algebraic closure and Neumann’s Lemma
The following result is sometimes called
Neumann’s Lemma (cf. Corol-lary 4.2.2 of [6]). It is equivalent to a well known result of B. H. Neumannon covering groups by cosets; an independent, combinatorial proof can befound in [3], or [4, 2.16].
Lemma 2.1.
Suppose G is a group acting on a set X and all G -orbits on X are infinite. Suppose A, B ⊆ X are finite. Then there is some g ∈ G with gA ∩ B = ∅ . The result has the following model-theoretic consequence, which can beregarded as ‘folklore’.
Lemma 2.2.
Suppose M is a countable, saturated first-order structure and A, B, C ⊆ M are algebraic closures of some finite subsets of M with B ⊆ C .Then there is g ∈ Aut(
M/B ) such that g ( C ) ∩ A = B ∩ A . Here, the algebraic closure in M of a set E is the union of the finite E -definable subsets of M . Saturation means that if E is finite and S isa family of E -definable subsets of M with the finite intersection property,then T S = ∅ . It implies that the algebraic closure of a finite E ⊆ M is the union of the finite Aut( M/E )-orbits (where Aut(
M/E ) denotes thepointwise stabilizer of E in the automorphism group Aut( M )).We give an analogous result for locally oligomorphic permutation groups. DAVID M. EVANS AND TODOR TSANKOV
Definition 2.3.
Suppose G ≤ Sym( X ) is locally oligomorphic on X . If E ⊆ X is finite, the algebraic closure acl( E ) of E in X is the union of thefinite G E -orbits.Note that, with this notation, if Y is a G -orbit on X , then acl( E ) ∩ Y isfinite. Lemma 2.4.
Suppose M is countable and G ≤ Sym( M ) is closed andlocally oligomorphic on M . If E ⊆ M is finite, then G acl( E ) is also locallyoligomorphic on M and has no finite orbits on M \ acl( E ) .Proof. Let B = acl( E ). Note that B is the union of a chain E = E ⊆ E ⊆ E ⊆ · · · of finite G E -invariant sets. Claim 1:
Suppose ¯ a ∈ M n is a finite tuple and let A i be the G E i -orbitcontaining ¯ a . Then there is N ∈ N such that A i = A N for all i ≥ N .Indeed, note that G E i is a normal subgroup of finite index in G E . Itfollows that { gA i : g ∈ G E } is a G E -invariant partition of A (with finitelymany parts). As G E has finitely many orbits on A , there are only finitelymany possibilities for such a partition, so as A i ⊇ A i +1 , they must be equalfor sufficiently large i . ( ✷ Claim . )It is worth noting that the N here depends only on the G E -orbit con-taining ¯ a , not the particular representative ¯ a . As G B = T i G E i , it followsthat in each G E -orbit, the G B -orbits coincide with the G E N -orbits, and as G E N has finite index in G E , there are only finitely many of them, showingthat G B is locally oligomorphic and that G B · ¯ a is infinite whenever G E · ¯ a is. If a ∈ M \ acl( E ), then by the Claim, G acl( E ) · a = G E N · a for some N ; as G E N has finite index in G E and G E · a is infinite, this means that G acl( E ) · a is also infinite. (cid:3) Lemma 2.5.
Suppose M is countable and G ≤ Sym( M ) is closed andlocally oligomorphic on M . Suppose A, B, C are algebraic closures in M ofsome finite subsets of M and B ⊆ C . Then there is g ∈ G B such that g ( C ) ∩ A = B ∩ A .Proof. Let ¯ a, ¯ c be finite tuples with A = acl(¯ a ) and C = acl(¯ c ) (wherealgebraic closure is from the action of G on M , of course). Consider theaction of H = G B on M . By Lemma 2.4, H is locally oligomorphic on M and has no finite orbits on M \ B . Let S be the H -orbit containing ¯ c andlet S , . . . , S k be the H ¯ a -orbits on S (note that there are finitely many ofthese, as H is oligomorphic on the union of the H -orbits which contain theelements of the tuples ¯ a, ¯ c ).Write M \ B as the union of a chain X ⊆ X ⊆ X ⊆ · · · of subsetseach of which is a finite union of H -orbits. Recall that the intersection of A, C with each X i is finite. So by Lemma 2.1, for each j ∈ N , there is some¯ c ′ ∈ S such that ( X j ∩ acl(¯ c ′ )) ∩ ( X j ∩ acl(¯ a )) = ∅ . Thus, for each j ∈ N there is some i ≤ k such that this holds for all ¯ c ′ ∈ S i .So there is some i ≤ k such that this holds for all j and for all ¯ c ′ ∈ S i . Inparticular, there is ¯ c ′ ∈ S such that acl(¯ c ′ ) ∩ acl(¯ a ) ⊆ B , as required. (cid:3) Remarks 2.6.
The result can be derived from the model-theoretic state-ment Lemma 2.2, though some care is required as the canonical structurefor a locally oligomorphic G ≤ Sym( X ) is not saturated if there are in-finitely many G -orbits on X . However, if we regard it as a multi-sortedstructure (with a sort for each of the G -orbits on X ), then it is saturated(as a multi-sorted structure) and Lemma 2.2 also holds in this context. Nev-ertheless, it seems worthwhile to offer a direct proof which does not use themodel-theoretic terminology. Definition 2.7.
Suppose G is a topological group. Then G ◦ denotes theintersection of the open subgroups of finite index in G . Lemma 2.8.
Suppose G is a non-archimedean, Roelcke precompact Polishgroup. Consider G as a closed subgroup of Sym( X ( G )) (as in Lemma 1.9).Then G ◦ = G acl( ∅ ) and it is Roelcke precompact. Moreover ( G ◦ ) ◦ = G ◦ .Proof. By definition, an element of X ( G ) is in acl( ∅ ) if and only if its stabi-lizer is open and of finite index. Moreover, any such subgroup is the stabilizerof some point of X ( G ), so the first statement is immediate from Lemma 2.4.For the second statement, suppose U ⊆ G ◦ is open and of finite index.The topology on G ◦ is the subspace topology so there is some e ∈ X ( G )such that G e ∩ G ◦ ≤ U ≤ G ◦ . Let D be the U -orbit on X ( G ) containing e ; let C be the G ◦ -orbit and E the G -orbit. Note that by Claim 1 in theproof of Lemma 2.4, C is equal to the G X -orbit containing e , for somefinite, G -invariant X ⊆ acl( ∅ ). Thus { gC : g ∈ G } is a finite partition of E . So as U is of finite index in G ◦ , we have that { gD : g ∈ G } is also afinite partition of E . Let V be the setwise stabilizer of D in G . So this isan open subgroup of finite index in G and V ∩ G ◦ = U . (To see this, let v ∈ V ∩ G ◦ . Then there is u ∈ U such that v · e = u · e , so u − v ∈ G e .Therefore v ∈ uG e ∩ G ◦ = u ( G e ∩ G ◦ ) ⊆ U .) But V ≥ G ◦ , so U = G ◦ . (cid:3) Lemma 2.9.
Let G be a non-archimedean, Roelcke precompact, Polish groupand π : G → K be a continuous homomorphism to a compact Polish group.Then π ( G ) is closed in K . In particular, G/G ◦ is a compact, profinite group.Proof. Without loss of generality, we may assume that π ( G ) is dense in K .Let V ≤ G be an open subgroup. As G is Roelcke precompact, there is afinite F ⊆ G such that V F V = G . Using that K is compact, we obtain K = π ( V ) π ( F ) π ( V ) = π ( V ) π ( F ) π ( V ) . So K is the disjoint union of finitely many double cosets of the compactgroup π ( V ), which are closed and, therefore, open. In particular, π ( V ) isopen.The open subgroup V ≤ G was arbitrary, so for all V , π ( V ) is somewheredense. As G is Polish, this implies that π ( G ) is non-meager in K (see, forexample, [8, Proposition 3.2]). As π ( G ) is also a Borel subgroup of K , itmust be open and closed [7, 9.11], implying that π is surjective. DAVID M. EVANS AND TODOR TSANKOV
For the last statement, let K = lim ←− G/N , where the inverse limit istaken over all finite index, open, normal subgroups of G directed by reverseinclusion. (Note that, as G is Roelcke precompact, it has only countablymany open subgroups, so K is Polish.) Then there is a natural injectivehomomorphism π : G/G ◦ → K with dense image. By the main statementof the lemma, π is also surjective, and, therefore, a topological group iso-morphism. (cid:3) Constructing automorphisms without fixed points
Theorem 3.1.
Suppose M is a countable set and G ≤ Sym( M ) is closedand locally oligomorphic. Suppose G ◦ = 1 . Then there exist elements f, g ∈ G ◦ generating a non-abelian free subgroup F of G ◦ which acts freely on M \ acl( ∅ ) . Suppose that M and G are as in the theorem. Note that by Lemma 2.4, G ◦ is closed and locally oligomorphic on M \ acl( ∅ ). Also, by Lemma 2.8,( G ◦ ) ◦ = G ◦ . So for the rest of the proof we may assume without loss ofgenerality that G = G ◦ and acl( ∅ ) = ∅ . Note that the assumption that G ◦ = 1 means that some G ◦ -orbit is infinite.We will regard M as a first order structure with automorphism group G (for example, by giving it its canonical structure).Let A = { acl( E ) : E ⊆ M finite } be the set of algebraic closures offinite subsets of M . By a partial automorphism of M we mean a bijection P → Q between elements of A which extends to an element of G . We build f, g in the theorem by a back-and-forth argument as the union of a chainof partial automorphisms, ϕ : A → B and γ : A ′ → B ′ (approximating f, g respectively). It will suffice to show how to extend the domain of one of ϕ, γ in the ‘forth’ step (by symmetry, the argument for extending images in the‘back’ step will be the same).Consider F = h a, b i , the free group on generators a, b . The non-identityelements of F can be thought of as reduced words ω ( a, b ) in a, b : so ex-pressions ω ( a, b ) = c c · · · c r where c i ∈ { a, b, a − , b − } and c i = c − i +1 forall i < r . If ϕ, γ are partial automorphisms of M then by ω ( ϕ, γ ) we meanthe composition obtained by substituting ϕ for a and γ for b in ω ( a, b ). Werefer to this as a reduced word in ϕ, γ . This is a bijection between subsets of M which extends to an element of G (of course, it could be empty); a fixedpoint of this is an element x ∈ M such that ω ( ϕ, γ ) x is defined and equalto x .Theorem 3.1 follows from the following: Proposition 3.2.
Suppose M is as above and ϕ : A → B , γ : A ′ → B ′ arepartial automorphisms of M such that no reduced word in ϕ, γ has a fixedpoint. Suppose A ⊆ C ∈ A . Then there is an extension ˜ ϕ : C → D of ϕ toa partial automorphism with domain C such that no reduced word in ˜ ϕ, γ has a fixed point.Proof. First we show that we can choose D (and ˜ ϕ ) so that D ∩ ( C ∪ B ∪ A ′ ∪ B ′ ) = B . As ϕ extends to an element of G , there is some partial automorphism ϕ ′ : C → D ′ extending ϕ . So B ⊆ D ′ and by Lemma 2.5there is g ∈ G B with gD ′ ∩ ( C ∪ B ∪ A ′ ∪ B ′ ) = B . Let D = gD ′ and˜ ϕ : C → D be the composition g ◦ ϕ ′ . This has the required property.Now we show that this choice of ˜ ϕ works.We first note the following: Observation:
If ˜ ϕ − z is defined and z ∈ C ∪ B ∪ A ′ ∪ B ′ , then z ∈ ( C ∪ B ∪ A ′ ∪ B ′ ) ∩ D = B . So z ∈ B and therefore ϕ − z is defined and˜ ϕ − z = ϕ − z ∈ A . Similarly, if ˜ ϕw is defined and ˜ ϕw ∈ C ∪ B ∪ A ′ ∪ B ′ then˜ ϕw ∈ ( C ∪ B ∪ A ′ ∪ B ′ ) ∩ D = B , so w ∈ A , ϕw is defined and ϕw = ˜ ϕw .Now suppose π π · · · π r is a reduced word in ˜ ϕ, γ and x, y ∈ M are suchthat( ∗ ) π π · · · π r x = y. So π i ∈ { ˜ ϕ, ˜ ϕ − , γ, γ − } and π i +1 = π − i . We show that most of the terms˜ ϕ , ˜ ϕ − in this equation can be replaced by ϕ , ϕ − without changing thevalidity of the equation. Claim:
Suppose π i = ˜ ϕ − and i < r . Then we can replace π i by π ′ i = ϕ − in ( ∗ ). Similarly, if π j = ˜ ϕ and j > π j by π ′ j = ϕ in( ∗ ).To see this, note that as the word is reduced, π i +1 is equal to γ, γ − , or˜ ϕ − (as we will want to repeat this argument we also consider the possibilitythat it is ϕ − ). If z = π i +1 . . . π r x then z is in the image of π i +1 , so z ∈ A ′ ∪ B ′ ∪ C . By the observation, it follows that ϕ − z is defined (and equalto ˜ ϕ − z ), so we can replace π i by ϕ − as required. Similarly π j − is equalto γ , γ − or ˜ ϕ (or ϕ ). So w = π j +1 . . . π r x is such that ˜ ϕw is defined and inthe set A ′ ∪ B ′ ∪ C . So by the observation, ϕw is defined (and equal to ˜ ϕw )and we can make the required replacement. ( ✷ Claim )Now make all of the replacements allowed by the Claim. The only pos-sible ˜ ϕ − remaining is if π r = ˜ ϕ − and the only possible ˜ ϕ remaining is if π = ˜ ϕ .Thus, after the replacements we have( ∗∗ ) ˜ ϕ s β ˜ ϕ − t x = y, where s, t ∈ { , } and β is a (possibly trivial) reduced word in ϕ, γ . Sup-pose, for a contradiction, that x = y .We consider various cases. If β is trivial, then exactly one of s, t is 1and (possibly after rearranging ( ∗∗ )) we have ˜ ϕx = x . Then x ∈ C ∩ D , so x ∈ B and ϕx is defined. Thus ϕx = x , contradicting the assumption on ϕ, γ . Suppose now that β is non-trivial. If s = t = 1, then we can rearrange( ∗∗ ) to obtain βz = z where z = ˜ ϕ − x . As β is a non-trivial reduced wordin ϕ, γ , this is a contradiction. We also have a contradiction if s = t = 0.For the remaining cases, by rearranging ( ∗∗ ) if necessary, we can assume s = 1 and t = 0, that is, ˜ ϕβx = x . Then x ∈ D ∩ ( A ∪ B ∪ A ′ ∪ B ′ ), so x ∈ B , βx ∈ A and ϕ ( βx ) is defined, with ϕβx = x . But ϕβ is a non-trivial reduced word in γ, ϕ (it comes from the same word as π . . . π r ) so we havea contradiction. (cid:3) Proof of Theorem 3.1.
Recall that we are assuming (without loss of gener-ality) that acl( ∅ ) = ∅ and M is infinite (the latter from the assumption that G ◦ = 1). We build chains of partial automorphisms ϕ ⊆ ϕ ⊆ ϕ ⊆ · · · and γ ⊆ γ ⊆ γ ⊆ · · · such that f = S i ϕ i and g = S i γ i are automorphisms. At each stage we useProposition 3.2 to extend the domain or image of one of ϕ i , γ i so that f, g willbe automorphisms. We can start off with ϕ , γ so that no reduced word in ϕ , γ has any fixed points. To do this, we just ensure (using Lemma 2.5) thatthe domains and images of φ , γ are all disjoint. Then by the Proposition,the same will be true of all the ϕ i , γ i , and therefore of f, g . So no reducedword in f, g has any fixed points: in particular, every reduced word in f, g is not the identity so f, g freely generate a free group whose non-identityelements have no fixed points on M . (cid:3) We can now prove Theorem 1.3 from the introduction.
Proof of Theorem 1.3:
Consider G as acting as a closed, locally oligomorphicsubgroup of Sym( X ( G )), where X ( G ) is as in Lemma 1.9. Let f, g be asgiven by Theorem 3.1 with M = X ( G ). Suppose H ≤ G is open and ofinfinite index and 1 = k ∈ F . By construction of X ( G ), there is an injective G -morphism from the coset space G/H to X ( G ). As H is of infinite indexin G , clearly the image of this is in M \ acl( ∅ ). It follows that k has no fixedpoints on G/H , as required. ✷ We now obtain Corollary 1.4 from the introduction.
Proof of Corollary 1.4.
Suppose G is as in the statement of the Corollary. If G ◦ = 1, then by Lemmas 2.4 and 2.8, every orbit of G is finite. Otherwise, byTheorem 1.3 there is a non-abelian free subgroup F of G as in Theorem 1.3.If all orbits of X are infinite, then F acts freely on X . But this is impossibleif the action of G (and therefore of F ) on X is amenable. (cid:3) As a further application of Theorem 1.3, we note the following.
Corollary 3.3.
Suppose G is a non-archimedean, Roelcke precompact Pol-ish group. Then G is not equal to the union of its open subgroups of infiniteindex.Proof. Open subgroups of infinite index are exactly the stabilizers of ele-ments of X ( G ) \ acl( ∅ ). So the statement follows once we know that someelement of G fixes no element of X ( G ) \ acl( ∅ ). But we just showed that thereis a free group of rank 2 with this property, which is more than enough. (cid:3) Note that very little is used in the proof of Theorem 3.1 apart from Neu-mann’s Lemma, in the form of Lemma 2.5. As the corresponding result holdsfor countable, saturated structures (Lemma 2.2) or, more generally, count-able multi-sorted structures which are saturated as multi-sorted structures,the proof of Theorem 3.1 also gives the following.
Corollary 3.4.
Suppose M is a countable saturated structure with someinfinite sort. Then there exist f, g ∈ Aut( M/ acl( ∅ )) such that F = h f, g i isthe free group on f, g and every non-identity element of F fixes no elementsof M eq \ acl eq ( ∅ ) . ✷ Example 3.5.
We give an example of a countable (non-saturated) structurewith a rich automorphism group for which the above corollary fails. Let L be a language with countably many binary relation symbols ( E i : i ∈ N ).Consider the class C of finite L -structures A where each E i is an equivalencerelation on A and only finitely many of the E i are not the universal relation A on A . Then C has countably many isomorphism types and it is easy tocheck that it is a Fra¨ıss´e amalgamation class. Let M be the Fra¨ıss´e limit.Then M is a countable, homogeneous L -structure.As M is constructed as the union of a chain of finite structures in C ,if a, b ∈ M then there exists n such that E n ( a, b ). In particular, M is notsaturated. If g ∈ Aut( M ) and a ∈ M , then E n ( a, ga ) for some n , therefore g fixes the E n -class which contains a . It is easy to see that the E n -classes arenon-algebraic elements of M eq , therefore every automorphism of M fixes anon-algebraic element of M eq . In particular, Aut( M ) is the union of properopen subgroups of infinite index.Note that, of course, Neumann’s Lemma fails for M eq in this example:acl eq ( a ) contains the E n -equivalence classes of a and we have just observedthat there is no automorphism which moves this to a set disjoint from it(over acl eq ( ∅ )).4. Property (T) for non-archimedean, Polish, Roelckeprecompact groups
The book [2] is a convenient reference for the background to this section.A unitary representation of a topological group G is a homomorphism π : G → U ( H ) to the unitary group of some Hilbert space H which isstrongly continuous, meaning that for every ξ ∈ H the map G → H givenby g π ( g ) ξ is continuous. If H is an open subgroup of G , consider theaction of G on the coset space Y = G/H . Then Y is a discrete space and theaction of G on Y is continuous. It is easy to check that the correspondingaction of G on ℓ ( Y ) gives a unitary representation of G (called the quasi-regular representation λ G/H ).From [13], we have the following:
Theorem 4.1.
Suppose G is a non-archimedean, Roelcke precompact, Pol-ish group. Then every unitary representation of G is a direct sum of irre-ducible unitary representations. Moreover, every irreducible unitary repre-sentation of G is a subrepresentation of ℓ ( G/H ) , for some open subgroup H of G .Proof. The first statement is part of [13, Theorem 4.2]. The rest of the proofis similar to that of [13, Proposition 6.2]. If π : G → U ( H ) is an irreducibleunitary representation of G , then by [13, Theorem 4.2], π is isomorphic toan induced representation Ind GK ( σ ) for some open subgroup K of G andirreducible representation σ of K which factors through a finite quotient K/H of K . In particular, π is a subrepresentation of Ind GK (Ind KH (1 H )) (where1 H is the trivial representation of H ), which is the same thing as ℓ ( Y ) with Y = G/H . (cid:3) We now recall the definition of property (T) for topological groups.
Definition 4.2.
Suppose G is a topological group, Q ⊆ G and ε >
0. If π : G → U ( H ) is a unitary representation of G , we say that a non-zerovector ξ ∈ H is ( Q, ε ) -invariant (for π ) if sup x ∈ Q k π ( x ) ξ − ξ k < ε k ξ k .We say that ( Q, ε ) is a
Kazhdan pair if for every unitary representation π of G , if π has a ( Q, ε )-invariant vector, then it has a (non-zero) invari-ant vector. We say that G has Kazhdan’s property (T) (respectively, strongproperty (T) ) if there is a Kazhdan pair (
Q, ε ) with Q compact (respectively,finite).The following fact will be useful (see Proposition 1.7.6 and Remark 1.7.9in [2]). Lemma 4.3.
Let G be a completely metrizable group and N ✁ G a closednormal subgroup. If both N and G/N have property (T), then so does G . In [13, Theorem 6.6], it was shown that automorphism groups of certain ω -categorical structures (those without algebraicity and with weak elimi-nation of imaginaries) have property (T). Furthermore, [13, Theorem 6.7]gave some examples of ω -categorical structures M whose automorphismgroups G have strong property (T). In the latter case, the proof proceededby exhibiting a free action of a non-abelian free group on the structure. Weuse the same idea, together with Theorem 1.3 to prove the second part ofTheorem 1.1: if X is countable and G ≤ Sym( X ) is closed and Roelckeprecompact, then G ◦ has strong property (T). This generalises the resultsin [13]. Proof of Theorem 1.1.
Suppose G is a non-archimedean, Roelcke precom-pact Polish group. By Lemma 2.9, the quotient group G/G ◦ is compact andtherefore has property (T). In view of Lemma 4.3, to prove the theorem,it remains to show that G ◦ has property (T). By Theorem 1.3, there existsa set Q = { f , f } ⊆ G ◦ which generates a non-abelian free subgroup F of G ◦ with the property that if H is a proper, open subgroup of G ◦ , then F acts freely on the coset space G ◦ /H . Following an argument similar to the one in [1], we show that ( Q, p − √
3) is a Kazhdan pair for G ◦ . ByTheorem 4.1, it suffices to show that for any proper open subgroup H ≤ G ◦ and all ξ ∈ ℓ ( G ◦ /H ),(1) max i =1 , k π ( f i ) · ξ − ξ k ≥ q − √ k ξ k . By Theorem 3.1, the restriction of π to F is a direct sum of copies of theleft-regular representation of F and Kesten’s theorem [10] tells us that k π ( f ) + π ( f − ) + π ( f ) + π ( f − ) k = 2 √ . A simple calculation using the Cauchy–Schwartz inequality (see [1, pp. 515–516] for details) yields X i =1 k π ( f i ) · ξ − ξ k ≥ − √ , thus proving (1). (cid:3) References [1] M. B. Bekka,
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Unitary representations of oligomorphic groups , Geom. Funct. Anal. (2012), no. 2, 528–555. Department of Mathematics, Imperial College London, London SW7 2AZ,UK.
E-mail address : [email protected] Universit´e Paris 7, UFR de Math´ematiques, Case 7012, 75205 Paris cedex13, France.
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