Dense locally finite subgroups of Automorphism Groups of Ultraextensive Spaces
Mahmood Etedadialiabadi, Su Gao, François Le Maître, Julien Melleray
aa r X i v : . [ m a t h . L O ] M a y DENSE LOCALLY FINITE SUBGROUPS OF AUTOMORPHISMGROUPS OF ULTRAEXTENSIVE SPACES
MAHMOOD ETEDADIALIABADI, SU GAO, FRANC¸ OIS LE MAˆITRE,AND JULIEN MELLERAY
Abstract.
We verify a conjecture of Vershik by showing that Hall’s universalcountable locally finite group can be embedded as a dense subgroup in theisometry group of the Urysohn space and in the automorphism group of therandom graph. In fact, we show the same for all automorphism groups ofknown infinite ultraextensive spaces. These include, in addition, the isometrygroup of the rational Urysohn space, the isometry group of the ultrametricUrysohn spaces, and the automorphism group of the universal K n -free graphfor all n ≥
3. Furthermore, we show that finite group actions on finite metricspaces or finite relational structures form a Fra¨ıss´e class, where Hall’s groupappears as the acting group of the Fra¨ıss´e limit. We also embed continuummany non-isomorphic countable universal locally finite groups into the isom-etry groups of various Urysohn spaces, and show that all dense countablesubgroups of these groups are mixed identity free (MIF). Finally, we give acharacterization of the isomorphism type of the isometry group of the Urysohn∆-metric spaces in terms of the distance value set ∆. Introduction
The concepts of ultraextensive metric spaces and ultraextensive relational struc-tures were introduced in [6] and [7], respectively, to capture some common prop-erties possessed by many Fra¨ıss´e limits. One of the main properties of an ultra-extensive space is that its automorphism group contains a countable dense locallyfinite subgroup. Known examples of ultraextensive spaces include the universalUrysohn space U , the universal rational Urysohn space QU , the universal ultra-metric Urysohn spaces, the random graph R , and the universal K n -free graphs H n (also known as Henson graphs). That the automorphism groups (or isometrygroups) of these spaces contain a countable dense locally finite subgroup was provedin Bhattacharjee–Macpherson [1], Pestov [21], Rosendal [23], and Siniora–Solecki[29]. In this paper we consider a concept of universal ∆-metric spaces that unifiesthe study of the Urysohn space U , the rational Urysohn space QU and the randomgraph R . Given a distance value set ∆, the universal ∆-metric space U ∆ is anultraextensive metric space. Mathematics Subject Classification.
Primary 03C13,03C55; Secondary 20E06,20E26.
Key words and phrases.
Hrushovski property, extension property for partial automorphisms(EPPA), partial isomorphism, HL-extension, HL-map, coherent, ultraextensive, ultrahomoge-neous, locally finite, Henson Graph, MIF, ∞ -MIF, omnigenous.The second author’s research was partially supported by the NSF grant DMS-1800323. Vershik [32] conjectured that Iso( U ), the isometry group of the Urysohn space,contains Hall’s universal countable locally finite group H as a dense subgroup. Hemade the same conjecture for the automorphism group of the countable randomgraph. Our first main result of this paper is to confirm Vershik’s conjecture. Infact, we establish this for all known examples of infinite ultraextensive spaces. Theorem 1.1.
The following groups contain Hall’s universal countable locally finitegroup H as a dense subgroup:(1) Iso( U ) , the isometry group of the Urysohn space;(2) Iso( QU ) , the isometry group of the rational Urysohn space;(3) Iso( U ∆ ) , the isometry group of the universal ∆ -metric space, for any dis-tance value set ∆ ;(4) Isometry groups of ultrametric Urysohn spaces;(5) Aut( R ) , the automorphism group of the random graph; and(6) Aut( H n ) , the automorphism group of the universal K n -free graph, for any n ≥ . In fact, we show that H appears canonically as a dense subgroup in these auto-morphism groups via the following theorems. Theorem 1.2.
Let K ∆ be the class of all structures ( X, G ) such that X is a finite ∆ -metric space, G is a finite group, and G acts on X by isometries. Then K ∆ is aFra¨ıss´e class. Letting ( X ∆ , H ∆ ) be the Fra¨ıss´e limit of K ∆ , then X ∆ is isometricto U ∆ , H ∆ is isomorphic to H , and H ∆ is dense in Iso( X ∆ ) . We note here that this result is related to some recent work of Doucha [5], seethe comments at the end of Section 3.
Theorem 1.3.
Let L be a finite relational language. Let T be a finite set of finite L -structures each of which is a Gaifman clique. Let K be the class of all pairs ( M, G ) such that M is a finite T -free L -structure, G is a finite group, and G acts on X byisomorphisms. Then K is a Fra¨ıss´e class. Letting ( N ∞ , H ∞ ) be the Fra¨ıss´e limit of K , then N ∞ is isomorphic to the universal T -free L -structure, H ∞ is isomorphicto H , and H ∞ is dense in Aut( N ∞ ) . In the proof of Theorem 1.2 we use a result of Rosendal [23] that characterizesthe RZ-property (after Ribes–Zalesski˘ı) by an extension property for finite metricspaces. Then, for Theorem 1.3 we need to use a concept of HL-property (afterHerwig–Lascar) and a characterization of the HL-property in the spirit of Rosendal’sresult, both developed in [7]. For both the RZ-property and the HL-property, we doneed to establish some new results about their closure under finite-index extensions.Next we turn to the problem of constructing many non-isomorphic dense locallyfinite subgroups of Iso( U ∆ ), the isometry group of the Urysohn ∆-metric spaces.We define a notion of omnigenous groups, which can be viewed as a natural gener-alization of Hall’s group. We then show that there are many such groups, and thatthey are all densely embeddable. Theorem 1.4.
There are continuum many non-isomorphic countable omnigenousgroups each of which is universal for countable locally finite groups.
UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 3
Theorem 1.5.
Every countable omnigenous group is embeddable into
Iso( U ∆ ) asa dense subgroup. In an effort to characterize all countable dense subgroups of Iso( U ∆ ), we considersome notions from the point of view of model theory and combinatorial grouptheory. In particular, we consider the property of being “mixed-identity free” (MIF)recently studied by Hull–Osin [15] and prove that any countable dense subgroup ofIso( U ∆ ), when | ∆ | ≥
2, must be MIF.
Theorem 1.6.
For any countable distance value set ∆ with | ∆ | ≥ , Iso( U ∆ ) aswell as any of its dense subgroup must be MIF. This theorem is false when | ∆ | = 1. In that case Iso( U ∆ ) is just the permutationgroup S ∞ and it is known that S ∞ is not MIF since it contains the non MIF groupof finitely supported permutations as a dense subgroup (cf. Theorem 5.9 of Hull–Osin [15]). Note that our result also provides an elementary proof of the fact thatthe group of finitely supported permutations cannot arise as a dense subgroup ofIso( U ∆ ) as soon as | ∆ | ≥ Remark . The theorem also yields a continuum family of universal countablelocally finite groups that are not embeddable as dense subgroups of Iso( U ∆ ), namelygroups of the form H ⊕ A , where A is a nontrivial abelian p -group. This can also beseen using the fact that Iso( U ∆ ) is always a topologically simple non abelian group(see Thm. 7.3), and no such group can contain a dense subgroup which decomposesas a nontrivial direct product.Furthermore, we introduce a new notion for locally finite groups which we call ∞ -MIF, and show that it is actually equivalent to being omnigenous.It would be very interesting to be able to distinguish topological groups of theform Iso( U ∆ ) by looking at their list of countable dense subgroups. As a first steptowards this, it is natural to ask which Iso( U ∆ ) can be densely embedded intoanother Iso( U Λ ). Indeed if so then Iso( U Λ ) will contain at least as many countabledense subgroups as Iso( U ∆ ). Our next result shows that these dense embeddingsonly occur in the obvious case, namely when Iso( U ∆ ) and Iso( U Λ ) are isomorphic,and provides a natural characterization in terms of the distance sets ∆ and Λ forthis to happen. This uses the following notion: we call ( d , d , d ) ∈ ∆ a ∆ -triangle if there is a metric space ( X, d ) with X = { x, y, z } such that d = d ( x, y ), d = d ( y, z ) and d = d ( z, x ). Theorem 1.8.
Let ∆ and Λ be countable distance value sets. Then the followingare equivalent:(i) There is a continuous homomorphism Iso( U ∆ ) → Iso( U Λ ) with dense range;(ii) Iso( U ∆ ) and Iso( U Λ ) are isomorphic as abstract groups;(iii) Iso( U ∆ ) and Iso( U Λ ) are isomorphic as topological groups;(iv) There exists a bijection θ : ∆ → Λ such that for any triple ( d , d , d ) ∈ ∆ , ( d , d , d ) is a ∆ -triangle iff ( θ ( d ) , θ ( d ) , θ ( d )) is a Λ -triangle. M. ETEDADIALIABADI, S. GAO, F. LE MAˆITRE, AND J. MELLERAY
The rest of the paper is organized as follows. In Section 2 we cover some pre-liminaries and verify that U ∆ is ultraextensive for any countable distance value set∆. In Section 3 we prove Theorem 1.2. In Section 4 we develop results about theHL-property of groups and prove Theorem 1.3. In Section 5 we study the notion ofomnigenous groups and prove Theorems 1.5 and 1.4. We apply these results alsoto the isometry groups of ultrametric Urysohn spaces. In Section 6 we study thenotions of discerning types, discerning structures, MIF groups, and ∞ -MIF groups.In Section 7 we prove that Iso( U ∆ ), as well as pointwise stabilizers on U ∆ , aretopologically simple; this is used to establish Theorem 1.8. Finally, in Section 8 wepose some open problems.2. Ultraextensive Metric Spaces
Basics of Fra¨ıss´e theory.
We briefly recall the basic concepts of Fra¨ıss´etheory. Throughout this paper let L be a countable language. Definition 2.1.
Let M be a countable L -structure. A partial automorphism of M is an isomorphism g : A → B , where A and B are finitely generated substructuresof M .The structure M is said to be ultrahomogeneous if every partial automorphismof M extends to an automorphism of M .In the cases considered in this paper, finitely generated substructures are alwaysfinite. For example, this happens when L is a relational language. Another casewe will consider is when L is the (finite) language of group theory, and M is acountable locally finite group. We will assume this property tacitly in all of ourdiscussions. Definition 2.2.
Let M be a countable L -structure. The age of M , denotedAge( M ), is the class of all finite substructures of M (considered up to isomor-phism).The age of any countable L -structure contains only countably many members upto isomorphism; also, any two members of the age embed in a third one (the jointembedding property ) and whenever A ∈ Age( M ) and B is a substructure of A thenalso B ∈ Age( M ) (the hereditary property ). Ages of ultrahomogeneous structuresare characterized by an additional condition. Definition 2.3.
Let K be a class of L -structures. We say that K has the amalga-mation property if, for any A, B, C ∈ K and any embedding β : A → B , γ : A → C ,there exists D ∈ K and embeddings β ′ : B → D and γ ′ : C → D such that β ′ ◦ β ( a ) = γ ′ ◦ γ ( a ) for all a ∈ A . K has the strong amalgamation property if in the above definition we have inaddition β ′ ( B ) ∩ γ ′ ( C ) = β ′ ◦ β ( A ). Theorem 2.4 (Fra¨ıss´e) . The age of any ultrahomogeneous L -structure satisfies theamalgamation property. Conversely, if K is a countable (up to isomorphism) classof finite L -structures which has the joint embedding, hereditary and amalgamation UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 5 properties then there exists a unique (up to isomorphism), ultrahomogeneous count-able L -structure M such that Age( M ) = K . A class K satisfying the assumptions of the theorem is called a Fra¨ıss´e class ,and the unique structure M above is called the Fra¨ıss´e limit of K . It is alsocharacterized by the statement that Age( M ) = K , and for any A ⊆ M , any B ∈ K and any embedding ϕ : A → B there exists an embedding ψ : B → M such that ψ ( ϕ ( a )) = a for all a ∈ A .2.2. ∆ -metric spaces.Definition 2.5. A distance value set is a nonempty subset ∆ of the open interval(0 , + ∞ ), such that ∀ x, y ∈ ∆ min( x + y, sup(∆)) ∈ ∆ . A ∆ -metric space is a metric space whose nonzero distances belong to ∆.In particular, when ∆ is bounded the definition implies that sup(∆) ∈ ∆. Thedefinition above is a particular case of what Conant [3] calls a distance monoid ,and our constructions could work for some more general distance monoids. Forsimplicity, we choose to work only in this more restricted case.In general, for a metric space ( M, d ), the isometry group Iso(
M, d ) is endowedwith the pointwise convergence topology, i.e. g n → g iff d ( g n ( x ) , g ( x )) → x ∈ M . When ( M, d ) is a separable complete metric space, Iso(
M, d ) becomes aPolish group, and we often write Iso( M ) instead of Iso( M, d ).In case (
M, d ) is countable , we use the following important convention. We willuse Iso( M ) to denote the group Iso( M, d ) but we will view it as a subset of thepermutation group Sym( M ) on M . As such it will become a closed subgroup ofSym( M ), and Iso( M ) will be endowed with the subspace topology of Sym( M ),which we will refer to as the permutation group topology . This again makes Iso( M )a Polish group. A basis of neighborhoods of 1 for this topology is given by pointwisestabilizers of finite tuples of elements of M .To apply Fra¨ıss´e theory, we will assume throughout this paper that ∆ is a count-able distance value set.Now any ∆-metric space ( M, d ) may be viewed as a first-order structure, in acountable relational language with a binary relational symbol R s for each element s of ∆, by declaring that M | = R s ( x, y ) iff d ( x, y ) = s. Since there should be no risk of confusion, we will be using the distance functioninstead of those binary relational symbols.
Lemma 2.6.
For any distance value set ∆, the class of finite ∆-metric spaces hasthe strong amalgamation property.
Proof.
Assume that
A, B, C are finite ∆-metric spaces and A is a subspace of both B and C . Let D denote the union of B and C , where both copies of A are identifiedand the values of the metric on B and C are imposed (and coincide on A ). We M. ETEDADIALIABADI, S. GAO, F. LE MAˆITRE, AND J. MELLERAY need to define d ( b, c ) for b ∈ B \ A and c ∈ C \ A . If A is empty, then we let m bethe maximum value taken by d on either B or C , and set d ( b, c ) = m for any b and c . If A is nonempty, then we set d ( b, c ) = min { min { d ( b, a ) + d ( a, c ) : a ∈ A } , sup(∆) } . Then D is a ∆-metric space. (cid:3) Thus if ∆ is a countable distance value set, then the class of finite ∆-metricspaces is a Fra¨ıss´e class. We denote by U ∆ the Fra¨ıss´e limit of this class, which isitself a countable ∆-metric space. We emphasize that G ∆ = Iso( U ∆ ) is endowedwith the permutation group topology.We will need the following characterization of U ∆ . The property in the proposi-tion is called the Urysohn property . Proposition 2.7.
Let ∆ be any countable distance value set. The space U ∆ isthe unique countable ∆-metric space X , up to isometry, that satisfies the followingproperty:Given any finite subset A of X and function f : A → ∆ satisfying | f ( a ) − f ( b ) | ≤ d ( a, b ) ≤ f ( a ) + f ( b ) , ∀ a, b ∈ A, there is an x ∈ X such that d ( x, a ) = f ( a ) for all a ∈ A .Functions f : A → ∆ as above are called Kat˘etov functions over A . We nowmention some well-known examples of spaces having the Urysohn property. Example 2.8. (1) ∆ is a singleton. For instance let ∆ = { } . Then U { } isa countable space with the discrete metric δ , where δ ( x, y ) = 1 iff x = y ,and G { } is isomorphic to Sym( N ) (also denoted S ∞ ). Here U { } can alsobe viewed as the complete graph K N , while Aut( K N ) = Sym( N ).(2) ∆ = { , , } . In this case U { , } is essentially the random graph R . In fact,if we define in R the metric d by d ( x, y ) = 1 iff there is an edge between x and y , then ( R , d ) is isometric with U { , } . In this case G { , } is isomorphicto Aut( R ).(3) ∆ = Q . Then U Q is the universal rational Urysohn space QU , and G Q =Iso( QU ).2.3. S-extensions.
We recall the notion of S-extension from [6].Let (
X, d X ) and ( Y, d Y ) be metric spaces. When there is no danger of confusion,we simply write X for ( X, d X ) and Y for ( Y, d Y ). We say that Y is an extension of X if ( X, d X ) is a subspace of ( Y, d Y ). Interchangeably, we use the same terminologywhen Y contains an isometric copy of X , i.e. when there is an (obvious) isometricembedding from X into Y .A partial isometry of X is an isometry between two finite subspaces of X . Theset of all partial isometries of X is denoted as P ( X ). P ( X ) is a groupoid with thecomposition ( p, q ) p ◦ q , where p ◦ q is only defined when rng( q ) = dom( p ), andthe inverse p p − . UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 7 If Y is an extension of X , then every partial isometry of X is also a partialisometry of Y . In symbols, we have P ( X ) ⊆ P ( Y ) if X ⊆ Y .If p, q ∈ P ( X ), we say that q extends p , and write p ⊆ q , if { ( x, p ( x )) : x ∈ dom( p ) } ⊆ { ( x, q ( x )) : x ∈ dom( q ) } . We let 1 X denote the identity isometry on X , i.e., 1 X ( x ) = x for all x ∈ X . Let P X denote the set of all p ∈ P ( X ) such that p X . We refer to elements of P X as nonidentity partial isometries of X . Note that if p ∈ P X then p − ∈ P X . Definition 2.9.
Let X be a metric space and P ⊆ P X such that P = P − . An S-extension of X with respect to P is a pair ( Y, φ ), where Y ⊇ X is an extensionof X , and φ : P → Iso( Y ) is such that φ ( p ) extends p for all p ∈ P . We also requirethat φ ( p − ) = φ ( p ) − for all p ∈ P . When P = P X we call ( Y, φ ) an
S-extension of X .The following strong notion of coherence was introduced by Solecki (cf. [23] and[29]). We use a terminology different from Solecki’s since we will have to deal witha weaker notion of coherence in the next subsection. Definition 2.10 (Solecki) . Let X be a metric space. An S-extension ( Y, φ ) of X is strongly coherent if for every triple ( p, q, r ) of partial isometries of X such that p ◦ q = r , we have φ ( p ) ◦ φ ( q ) = φ ( r ). Theorem 2.11 (Solecki [28] [23] [29]) . Let ∆ be any distance value set and X bea finite ∆ -metric space. Then, X has a finite, strongly coherent S-extension ( Y, φ ) where Y is a ∆ -metric space. The observation that finite, strongly coherent S-extensions can be constructedas ∆-metric spaces was explicit in Solecki’s unpublished notes [28] but follows im-plicitly from all proofs of existence of finite, strongly coherent S-extensions, e.g. inSiniora–Solecki [29] or Hubiˇcka–Koneˇcn`y–Neˇsetˇril [14].The following lemma highlights the importance of strongly coherent S-extensions.
Lemma 2.12.
Let X be a metric space and ( Y, φ ) be a strongly coherent S-extension of X . For every D ⊆ X , the map p φ ( p ) gives a group embeddingfrom Iso( D ) into Iso( Y ).2.4. Ultraextensive ∆ -metric spaces. We recall more notions from [6].For any metric space X and P ⊆ P X such that P = P − , we let F ( P ) denotethe free group generated by P , where for any p ∈ P , the inverse of p in F ( P ) is p − . If ( Y, φ ) is an S-extension of X with respect to P , then φ can be naturallyextended to a homomorphism from F ( P ) to Iso( Y ). We still use φ to denote thisgroup homomorphism, i.e., for any p , . . . , p n ∈ P , φ ( p · · · p n ) = φ ( p ) ◦ · · · ◦ φ ( p n ) . Definition 2.13.
Let X be a metric space and P ⊆ P X such that P = P − . AnS-extension ( Y, φ ) of X with respect to P is minimal if for any y ∈ Y there is g ∈ F ( P ) and x ∈ X such that y = φ ( g )( x ). M. ETEDADIALIABADI, S. GAO, F. LE MAˆITRE, AND J. MELLERAY
Definition 2.14.
Let X ⊆ X be metric spaces and ( Y i , φ i ) be an S-extension of X i for i = 1 ,
2. We say that ( Y , φ ) and ( Y , φ ) are coherent if(i) Y extends Y ,(ii) φ ( p ) extends φ ( p ) for all p ∈ P X ⊆ P X , and(iii) letting K i = φ i ( P X i ) ⊆ Iso( Y i ) for i = 1 ,
2, and letting κ : K → K be themap κ ( φ ( p )) = φ ( p ) for all p ∈ P X , then κ extends uniquely to a groupembedding from h K i into h K i .This notion of coherence is weaker than Solecki’s strong coherence, as witnessedby the following lemma. Lemma 2.15.
Let X ⊆ X be metric spaces, ( Y , φ ) be an S-extension of X ,and ( Y , ψ ) be a strongly coherent S-extension of X ∪ Y . Let φ : P X → Iso( Y )be defined as φ ( p ) = (cid:26) ψ ( φ ( p )) , if p ∈ P X , ψ ( p ) , if p ∈ P X \ P X .Then ( Y , φ ) and ( Y , φ ) are coherent. Proof.
From the definition of φ it is clear that for any p ∈ P X , p ⊆ φ ( p ) ⊆ ψ ( φ ( p )) = φ ( p ) . By Lemma 2.12, ψ gives a group embedding from Iso( Y ) to Iso( Y ). On theother hand, ψ coincides with the map φ ( p ) φ ( p ) for all p ∈ P X . Thus thismap extends uniquely to a group embedding from hP X i ≤ Iso( X ) into hP X i ≤ Iso( Y ). (cid:3) Definition 2.16.
A metric space U is ultraextensive if(i) U is ultrahomogeneous, i.e., there is a φ such that ( U, φ ) is an S-extensionof U ;(ii) Every finite X ⊆ U has a finite S-extension ( Y, φ ) where Y ⊆ U ;(iii) If X ⊆ X ⊆ U are finite and ( Y , φ ) is a finite minimal S-extension of X with Y ⊆ U , then there is a finite minimal S-extension ( Y , φ ) of X such that Y ⊆ U and ( Y , φ ) and ( Y , φ ) are coherent.One of the main properties of an ultraextensive metric space U is that Iso( U )always contains a countable dense locally finite subgroup when U is separable (The-orem 1.4 of [6]). Moreover, the weaker notion of coherence is sufficient for construct-ing ultraextensive metric spaces. Theorem 2.17.
Let ∆ be a countable distance value set. Then U ∆ is ultraextensive.In particular, G ∆ contains a countable dense locally finite subgroup.Proof. Recall that Age( U ∆ ) is the class of all finite ∆-metric spaces. Since U ∆ isultrahomogeneous, Solecki’s Theorem 2.11 gives (ii) of Definition 2.16. Similarly,(iii) of Definition 2.16 follows from Solecki’s Theorem 2.11 and Lemma 2.15. (cid:3) We remark that Theorem 2.17 can be proved without using Solecki’s constructionof strongly coherent S-extensions. For instance, condition (ii) of Definition 2.16 for
UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 9 ∆-metric spaces follows implicitly from all proofs of the existence of S-extensions,including Solecki’s original proof in [27]. Condition (iii) of Definition 2.16 for ∆-metric spaces follows implicitly from the proof of Theorem 4.5 of [6].3.
Hall’s Group and Vershik’s Conjecture
Hall’s universal countable locally finite group.
Recall that a locallyfinite group is a group in which every finitely generated subgroup is finite. Thefollowing theorem is due to P. Hall [10].
Theorem 3.1 (Hall [10]) . There exists a countable locally finite group H that isdetermined up to isomorphism by the following properties:( A ) any finite group can be embedded in H , and( B ) any two isomorphic finite subgroups of H are conjugate by an element of H . It follows easily from the characterizing properties of H that every countablelocally finite group is a subgroup of H . Thus H is called Hall’s universal countablelocally finite group . For simplicity, we refer to it as
Hall’s group .Hall [10] also established the following strengthening of condition ( B ) above. Proposition 3.2 (Hall [10]) . For every triple ( G , G , Ψ), where G , G are finitesubgroups of H and Ψ : G → G is a group isomorphism, there exists h ∈ H suchthat for every g ∈ G we have Ψ( g ) = hgh − .In particular, we see that H is ultrahomogeneous and universal for finite groups:as such, it the Fra¨ıss´e limit of the class of finite groups. Thus it can also becharacterized as follows. Proposition 3.3.
Let H be a countable locally finite group with the followingproperty:( E ) for every triple ( G , G , Ψ ), where G ≤ G are finite groups and Ψ : G → H is a group embedding, there exists a group embedding Ψ : G → H such that Ψ ↾ G = Ψ .Then H is isomorphic to H .Hall [10] also proved that the commutator group of H is H , and therefore H hasa trivial abelianization. Consider the collection of all groups of the form H ⊕ A ,where A is an abelian p -group. The abelianization of H ⊕ A is isomorphic to A .This implies that H ⊕ A ∼ = H ⊕ A ′ iff A ∼ = A ′ . Thus there are continuum manynon-isomorphic countable locally finite groups which are universal for all countablelocally finite groups.3.2. A proof of Vershik’s conjecture.
Vershik’s conjecture [32] states that theisometry group of the universal Urysohn space and the automorphism group of thecountable random graph each contain a dense subgroup that is isomorphic to H .We will show that Hall’s group H actually arises in some sense as a canonical densesubgroup of Iso( U ∆ ).We will be using the following lemma due to Rosendal (Lemma 16 of [24]). Lemma 3.4 (Rosendal [24] ) . Let ∆ be a countable distance value set. Let Γ bea group, Λ ≤ Γ a subgroup. Assume that X ⊆ Y are ∆-metric spaces, and thatΛ y Y , Γ y X are compatible isometric actions. Then there exists a ∆-metricspace Z containing Y , and an isometric action Γ y Z compatible with the Λ-actionon Y .Moreover, if Γ and Y are both finite then one can find a finite Z as above. Proof.
The proof goes exactly like that of Lemma 16 of [24]. We only note that thespace Z defined in the proof in [24] is a ∆-metric space. (cid:3) Definition 3.5.
A countable group Γ has the
RZ-property (standing for Ribes–Zalesski˘ı) if any finite product Γ · · · Γ n of finitely generated subgroups of Γ is closedin the profinite topology.It was proved by Ribes–Zalesski˘ı [22] that countable free groups have the RZ-property. Moreover, they essentially showed in [22] that, if Λ ≤ Γ has finite indexthen Λ has the RZ-property iff Γ has the RZ-property. This gives the following factwe will need in our proof.
Proposition 3.6.
Let Γ , Γ be two finite groups, and Λ be a common subgroupof Γ , Γ . Then Γ ∗ Λ Γ , the amalgamated free product of Γ and Γ over Λ, hasthe RZ-property. Proof.
It is known that the amalgamated free product of finite groups is virtuallyfree, i.e., it contains a free group as a subgroup of finite index (cf., e.g., Serre [25],Corollary to Proposition 11 on p. 120). By the above results of Ribes–Zalesski˘ı[22], virtually free groups have the RZ-property. (cid:3)
Although not needed in our proof here, we note that Coulbois [4] showed that theRZ-property is preserved under taking free products. The RZ-property will playan important role in our proof because of the following theorem, due to Rosendal.
Theorem 3.7 (Rosendal [23]) . Let ∆ be a countable distance value set. Let Γ bea countable group with the RZ-property. Assume that π : Γ y X is an isometricaction of Γ on a ∆ -metric space X . Then, for any finite A ⊆ X and F ⊆ Γ thereexists a finite ∆ -metric space Y containing A , and an isometric action π ′ : Γ y Y such that for all γ ∈ F and all a ∈ A one has π ′ ( γ ) a = π ( γ ) a . Actually, Rosendal’s theorem is an equivalence (the RZ-property is equivalentto the so-called finite approximability of actions on metric spaces; cf. [23]) but weonly need the implication mentioned above.
Definition 3.8.
Let K ∆ be the class of all structures ( X, G ) such that • X is a finite ∆-metric space. • G is a finite group. • G acts isometrically on X .Note that we accept the case where X is empty, considering that any group actsisometrically on the empty set. Also, G is allowed to be equal to { } . UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 11
Theorem 3.9. K ∆ is a Fra¨ıss´e class.Proof. The hereditary property is obvious. The joint embedding property is alsoeasily witnessed by the product action. We only need to prove the amalgamationproperty. Assume that X is a subspace contained in two finite ∆-metric spaces Y , Y , and that Λ is a subgroup of two finite groups Γ , Γ . Assume further thatΓ y Y , Γ y Y isometrically, in such a way that X is Λ-invariant for bothactions, and the two Λ-actions coincide on X . Let Γ = Γ ∗ Λ Γ .We first define a ∆-metric space Z amalgamating Y and Y over X , in sucha way that Z = Y ∪ Y , Y ∩ Y = X and the action of Λ on Y induced by theactions of Γ , Γ on Y , Y is by isometries. To ensure Z is a ∆-metric space, weonly need to define, for y ∈ Y \ X and y ∈ Y \ X , d Z ( y , y ) = min (cid:26) inf x ∈ X { d Y ( y , x ) + d Y ( x, y ) } , sup(∆) (cid:27) . Then we define by induction an increasing sequence { Z n } n ≥ of ∆-metric spaces,as well as Γ actions on each Z n − and Γ actions on each Z n , so that all actionsare compatible with each other and with the original actions of Γ on Y and Γ on Y . To define Z , apply Lemma 3.4 to the Γ action on Y and the Λ action on Z .To define Z , apply Lemma 3.4 to the Γ action on Y and the induced Λ actionon Z . In general, obtain Z n +1 by applying Lemma 3.4 to the Γ action on Z n − and the induced Λ action on Z n , and obtain Z n +2 by applying Lemma 3.4 to theΓ action on Z n and the induced Λ action on Z n +1 . Let Z ∞ = S n ≥ Z n . Thenour construction gives an action of Γ on Z ∞ .Now, using the fact that Γ has the RZ-property from Proposition 3.6, we applyRosendal’s theorem 3.7 to the action of Γ on Z ∞ , with A = Z and F = Γ ∪ Γ ,to find a finite ∆-metric space Y containing Z , and an isometric action Γ on Y that extends the original actions of Γ , Γ on Y , Y respectively. Then, let G bethe subgroup of Iso( Y ) generated by F = Γ ∪ Γ . The actions of elements of F extends to an action of G on Y , which gives an amalgam ( Y, G ) ∈ K ∆ of ( Y , Γ )and ( Y , Γ ) over ( X, Λ). (cid:3)
Denote the Fra¨ıss´e limit of K ∆ by ( X ∆ , H ∆ ), where X ∆ is a ∆-metric space and H ∆ is a locally finite group acting isometrically on X ∆ . The following lemmas willgive us the main result of this section. Lemma 3.10. X ∆ is isometric to U ∆ . Proof.
We verify the Urysohn property from Proposition 2.7 for X ∆ . Let A be afinite subset of X ∆ , and f : A → ∆ satisfying | f ( a ) − f ( b ) | ≤ d ( a, b ) ≤ f ( a ) + f ( b ) for all a, b ∈ A . Then, viewing A ∪ { f } as a metric space, ( A, { } ) ∈ K ∆ is a substructure embedded in ( A ∪ { f } , { } ) ∈ K ∆ . By the universality andultrahomogeneity of a Fra¨ıss´e limit we may find x ∈ X ∆ such that d ( x, a ) = f ( a )for all a ∈ A . (cid:3) Lemma 3.11. H ∆ acts faithfully on X ∆ , i.e., if g ∈ H ∆ \ { } there is x ∈ X ∆ suchthat g · x = x . Proof.
To see that H ∆ acts faithfully, let g ∈ H ∆ \ { } and let Λ be the finitesubgroup of H ∆ generated by g . Let c ∈ ∆ and let X = (Λ , d ) be a ∆-metric spacewith d ( a, b ) = c for any distinct a, b ∈ Λ. Then the left multiplication of Λ is afaithful action of Λ on X by isometries. Thus ( X, Λ) ∈ K ∆ . By the universality of X ∆ , X can be realized as a subset of X ∆ . By the ultrahomogeneity of ( X ∆ , H ∆ ),the action of Λ on X ∆ extends the Λ action on X . Since the action of Λ on X isfaithful, it follows that the action of Λ on X ∆ is faithful. (cid:3) Lemma 3.12. H ∆ is isomorphic to H . Proof.
We verify the property ( E ) from Proposition 3.3 for H ∆ . Let Λ be a finitesubgroup of H ∆ , and i : Λ → Γ be a group embedding of Λ into a finite group Γ.Then i induces an embedding from the structure ( ∅ , Λ) ∈ K ∆ into ( ∅ , Γ) ∈ K ∆ . Bythe universality and ultrahomogeneity of ( X ∆ , H ∆ ) we see that there is a groupembedding j : Γ → H ∆ such that j ◦ i ( g ) = g for all g ∈ Λ. (cid:3) Lemma 3.13. H ∆ is dense in Iso( X ∆ ). Proof.
Let g ∈ Iso( X ∆ ), and let A be a finite subset of X ∆ . We need to find anelement of H ∆ coinciding with g on A . Let Γ be the subgroup of Iso( X ∆ ) generatedby g . Then Γ acts on X ∆ by isometries. We claim that there is a finite subset B of X ∆ containing A and an isometry h of B which coincides with g on A . Indeed,if Γ is finite, then we can let B = Γ · A and h = g . If Γ is infinite, then it is a freegroup (isomorphic to Z ), and it has the RZ-property. We can then apply Rosendal’stheorem 3.7 to find a finite B containing A and an isometry h of B which coincideswith g on A .Letting H denote the finite group generated by h , we see that ( B, H ) ∈ K ∆ . Wemay realize the embedding from ( B, { } ) into ( B, H ) inside ( X ∆ , H ∆ ), which givesus a finite subgroup of H ∆ isomorphic to H and acting like H on B . In particularthere exists an element of H ∆ coinciding with g on A . (cid:3) We have thus proved the following result.
Theorem 3.14.
Let ∆ be any countable distance value set. Then Iso( U ∆ ) containsa dense subgroup that is isomorphic to H . Theorem 3.14 immediately gives (2), (3), (5) of Theorem 1.1.
Corollary 3.15.
Iso( QU ), Aut( R ), and S ∞ all contain H as a dense subgroup.The other part of Vershik’s conjecture, Theorem 1.1 (1), is a corollary of Theo-rem 1.1 (2) from a standard argument. Theorem 3.16.
Iso( U ) contains H as a dense subgroup.Proof. Since QU is a countable dense subset of U , the map h h sending h ∈ Iso( QU ) to its completion h ∈ Iso( U ) is a well-defined group embedding. SinceIso( QU ) has the permutation group topology and Iso( U ) has the pointwise conver-gence topology, this map is continuous. If H is a dense subgroup of Iso( QU ) then H = { h : h ∈ H } is a dense subgroup of Iso( U ) isomorphic to H . (cid:3) UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 13
To conclude this section, we note that the idea of considering finite metric spaces,with finite groups acting on them, as forming a Fra¨ıss´e class has already beenconsidered in Doucha’s paper [5] (though his formalism is different from ours).In particular, Theorem 0.2 in [5] is closely related to the results in this section,though the proof is different. One may think of Theorem 0.2 of [5] as a precursorto Theorem 3.9. We also note that our approach gives an answer to Question 3.7from [5]: it follows from our results that the group H appearing in [5] is equal toIso( U ). 4. Automorphism Groups of Relational Structures
In this section we show some analogous results to the main results of the preced-ing section for ultraextensive relational structures. As a corollary, we will obtainTheorem 1.1 (6). We first recall the concept of the HL-property defined in [7] anddevelop some results necessary for our proof.4.1.
The HL-property of a group.Definition 4.1 (Herwig–Lascar [11]) . Let G be a group and let H , . . . , H n ≤ G .A left system of equations on H , . . . , H n is a finite set of equations with variables x , . . . , x m and constants g , . . . , g l , where each equation is of the form x i H j = g k H j or x i H j = x r g k H j for 1 ≤ i, r ≤ m , 1 ≤ k ≤ l and 1 ≤ j ≤ n . Definition 4.2.
Let G be a group. We say that G has the HL-property (standingfor Herwig–Lascar) if for every finitely generated H , . . . , H n ≤ G and left systemΣ of equations on H , . . . , H n , if Σ does not have a solution, then there existnormal subgroups of finite index N , . . . , N n E G such that, letting Σ( ~N ) be the leftsystem of equations obtained from Σ by replacing all occurrences of H , . . . , H n by N H , . . . , N n H n respectively, Σ( ~N ) does not have a solution either.Herwig–Lascar [11] proved that the HL-property implies the RZ-property forgroups. They also essentially showed in [11] that finitely generated free groupshave the HL-property. As a strengthening of Coulbois’s result on the preservationof the RZ-property under taking free products [4], it was shown in [7] that theHL-property is also preserved under taking free products. Lemma 4.3.
Let G be a group and H ≤ G be a subgroup of finite index. Then G has the HL-property iff H has the HL-property. Proof.
Recall the fact that if H ≤ G is a subgroup of finite index, then G has anormal subgroup N of finite index such that N ≤ H . Thus to prove the lemma,we may assume without loss of generality that H is normal in G .First assume G has the HL-property. Let Σ be a left system on finitely generated H , . . . , H n ≤ H . We claim that if Σ has a solution in G then it also has a solutionin H . Let V be the set of all variables x such that x appears in an equation inΣ of the form xH j = gH j . Since the constants in Σ are in H , any solution for an x ∈ V must be in H . Now let V be the set of all variables which appear inan equation in Σ of the form x i H j = x r gH j , where at least one of x i and x r isin V . We see that any solution for an x ∈ V must also be in H . Repeat thisconstruction and define V , V , etc. Since Σ is finite, we obtain a maximal set ofvariables V = V ∪ V ∪ V ∪ . . . so that any solution for an x ∈ V must be in H . LetΣ ′ ⊆ Σ be the subsystem of all equations that contain (only) variables in V . Thenthe subsystem Σ \ Σ ′ contains only equations of the form x i H j = x r gH j whereboth x i , x r V . Now if Σ has a solution in G , say x = γ , . . . , x m = γ m , then γ i , . . . , γ i k are in H where V = { x i , . . . , x i k } and all the other variables are fromthe same coset of H . Let gH be this coset. Then x i j = γ i j for j = 1 , , . . . , k and x i = g − γ i for i / ∈ { i , i , . . . , i k } is a solution of Σ that consists of only elements in H . We have thus shown the claim. Now assume Σ does not have a solution in H ,then by the claim it does not have a solution in G . Since G has the HL-property,there are N , . . . , N n E G of finite index such that Σ( ~N ) on N H , . . . , N n H n doesnot have a solution in G . Let K j = N j ∩ H for 1 ≤ j ≤ n . Then K j E H is offinite index, and Σ( ~K ) on K H , . . . , K n H n does not have a solution in H , sinceany solution of Σ( ~K ) is also a solution of Σ( ~N ).For the converse, assume H has the HL-property and H E G is of finite index.Let Σ be a left system on finitely generated H , . . . , H n ≤ G . Let L j = H j ∩ H for 1 ≤ j ≤ n . Then for each j , L j ≤ H is finitely generated and has finite indexin H j . For each j , let h j, L j , . . . , h j,S j L j enumerate all the left cosets of L j in H j .Then each equation of the form xH j = ygH j (here y could be 1 or a variable) isequivalent to xL j = ygh j,s L j for some 1 ≤ s ≤ S j . Now consider the collection S of all left systems Σ ′ where each Σ ′ is obtained from Σ by replacing each equationin Σ of the form xH j = ygH j by an equation of the form xL j = ygh j,s L j for some1 ≤ s ≤ S j . There are only finitely many left systems in S , and Σ has a solutionin G iff a Σ ′ ∈ S has a solution in G . To verify the HL-property for G , supposeΣ does not have a solution in G . Then none of Σ ′ ∈ S has a solution. Assumingthat the HL-property holds for L , . . . , L n for G , then for each Σ ′ ∈ S , there existnormal subgroups N Σ ′ , . . . , N Σ ′ n E G of finite index such that Σ ′ ( ~N Σ ′ ) does not havea solution. Let N j = T Σ ′ ∈S N Σ ′ j . Then N j E G is still of finite index, and for eachΣ ′ ∈ S , Σ ′ ( ~N ) still does not have a solution, since a solution for Σ ′ ( ~N ) would bea solution for Σ ′ ( ~N Σ ′ ). This implies that Σ( ~N ) does not have a solution, since asolution for it would be a solution of Σ ′ ( ~N ) for some Σ ′ ∈ S . To finish the proof,it suffices to check that the HL-property holds for L , . . . , L n for G .The above argument shows that it suffices to prove the HL-property for H , . . . , H n in G when H , . . . , H n ≤ H E G , which we demonstrate below. Let Hg , . . . , Hg T be the right cosets of H in G . Then G = Hg ∪ · · · ∪ Hg T . First, suppose Σ has asolution x = γ , . . . , x m = γ m in G . Then there are 1 ≤ t , . . . , t m ≤ T such that γ i ∈ Hg t i for all 1 ≤ i ≤ m . If x l H j = gH j is an equation in Σ, then the solution x l = γ l ∈ Hg t l witnesses that y l g t l H j g − t l = gg − t l g t l H j g − t l UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 15 has a solution y l = λ l = γ l g − t l ∈ H . If we let H lj = g t l H j g − t l , then H lj ≤ H since H is normal, and the above equation becomes y l H lj = ( gg − t l ) H lj . Similarly,if x l H j = x k gH j is an equation in Σ, then the solution x l = γ l , x k = γ k witnessesthat y l H lj = y k ( g t k gg − t l ) H lj has a solution y l = λ l = γ l g − t l , y k = λ k = γ k g − t k in H . Now for each ~t = ( t , . . . , t m )where 1 ≤ t , . . . , t m ≤ T , we obtain a left system Σ ~t from Σ by replacing eachequation in Σ by an equation of the above form. Note that all the constantsappeared in Σ ~t are elements of H . Let S be the collection of all such Σ ~t . By ourconstruction, Σ has a solution in G iff a Σ ~t ∈ S has a solution in H . To verify theHL-property for H , . . . , H n in G , suppose Σ does not have a solution in G . Thennone of Σ ~t ∈ S has a solution in H . By the HL-property of H , for each Σ ~t ∈ S ,there exist normal subgroups N Σ ~t , . . . , N Σ ~t n ≤ H of finite index such that Σ ~t ( ~N Σ t )does not have a solution in H . Let N j = T Σ ~t ∈S N Σ ~t j . Then N j E H is still of finiteindex, and for each Σ ~t ∈ S , Σ ~t ( ~N ) still does not have a solution in H . Now each N j is of finite index in G since H is of finite index in G . Let M j E G be of finiteindex such that M j ≤ N j . It follows from our construction of S that Σ( ~M ) doesnot have a solution in G , since any solution of Σ( ~M ) would give rise to a solutionfor some Σ ~t ( ~N ) where Σ ~t ∈ S . (cid:3) Similar to Proposition 3.6 we obtain the following proposition from the abovelemma and the Herwig–Lascar theorem on the HL-property of finitely generatedfree groups.
Proposition 4.4.
Let Γ , Γ be two finite groups, and Λ be a common subgroupof Γ , Γ . Then Γ ∗ Λ Γ has the HL-property.4.2. The Fra¨ıss´e class of actions by automorphisms.
Let L be a finite rela-tional language. If C and D are L -structures, a homomorphism from C to D is amap π : C → D such that for every n -ary relation R ∈ L and every a , . . . , a n ∈ C , R C ( a , . . . , a n ) ⇒ R D ( π ( a ) , . . . , π ( a n )) . If T is a set of L -structures and D is an L -structure, then D is T -free if there is no C ∈ T and homomorphism π : C → D .An L -structure C is called a Gaifman clique if for every a, b ∈ C there is a relationsymbol R ∈ L with arity m ≥ c , . . . , c m ∈ C with a, b ∈ { c , . . . , c m } and R C ( c , . . . , c m ). It is clear that if C is a Gaifman clique and D is a homomorphicimage of C (i.e., there is a surjective homomorphism π : C → D ), then D is also aGaifman clique. Moreover, if C is a finite Gaifman clique, then it has only finitelymany homomorphic images, up to isomorphism. Definition 4.5.
Let T be a finite set of finite L -structures each of which is aGaifman clique. Let K be the class of all pairs ( M, G ) such that • M is a finite T -free L -structure, • G is a finite group, and • G acts on X by isomorphisms. Suppose T is a finite set of finite L -structures each of which is a Gaifman clique.Let ˜ T be the set of all homomorphic images of structures in T . Then ˜ T is also afinite set of L -structures each of which is a Gaifman clique. The class of all finite T -free L -structures coincides with the class of all finite L -structures that do not allowan isomorphic embedding from any ˜ T ∈ ˜ T , and by Lemma 4.5 of Siniora–Solecki[29], this is a Fra¨ıss´e class closed under taking free amalgams.The following characterization of the HL-property was proved in [7] as an analogof Rosendal’s theorem 3.7. Theorem 4.6.
Let G be a group. Then the following are equivalent: (i) G has the HL-property; (ii) Let L be a finite relational language with unary relation symbols S , . . . , S n ∈L . Let T be a finite set of finite L -structures. Let D be a T -free L -structuresuch that { S D , . . . , S Dn } is a partition of the domain of D . Let C be a finitesubstructure of D . Let F be a finite subset of G . Suppose that π : G y D is a faithful action by isomorphisms and that π is transitive on each S Di for i = 1 , . . . , n . Then there exists a finite T -free L -structure D ′ extending C ,and an action π ′ : G y D ′ by isomorphisms such that for all γ ∈ F and a ∈ C one has π ′ ( γ ) a = π ( γ ) a . (iii) Clause (ii) with the additional assumption that every structure T ∈ T is aGaifman clique. In the following we also prove an analog of Rosendal’s lemma 3.4.
Lemma 4.7.
Let L be a finite relational language. Let Γ be a group, Λ ≤ Γ asubgroup. Assume that M ⊆ N are L -structures, and that Λ y N , Γ y M arecompatible actions by isomorphisms. Then there exists an L -structure P extending N , and an action Γ y P by isomorphisms that is compatible with the Λ-action on N .Moreover, if Γ and N are both finite then one can find a finite P as above. Proof.
Define an equivalence relation ∼ on N × Γ by ( a , g ) ∼ ( a , g ) iff( g − g ∈ Λ and g − g · a = a ) or ( a , a ∈ M and g · a = g · a ) . To see that ∼ is an equivalence relation, we only need to note that if a , a ∈ M , g · a = g · a , g − g ∈ Λ, and g − g · a = a , then a ∈ M and g · a = g · a ,and thus ( a , g ) ∼ ( a , g ).Let P = N × Γ / ∼ . Let [ a, g ] denote the equivalence class [( a, g )] ∼ . For R ∈ L an n -ary relation symbol, define R P ([ a , g ] , . . . , [ a n , g n ]) iff there are b , . . . , b n ∈ N and g ∈ Γ such that [ a , g ] = [ b , g ] , . . . , [ a n , g n ] = [ b n , g ], and R N ( b , . . . , b n ).To see that this is well-defined, suppose [ a , g ] = [ b , g ] = [ c , h ] , . . . , [ a n , g n ] =[ b n , g ] = [ c n , h ]. We need to show that R N ( b , . . . , b n ) iff R N ( c , . . . , c n ). In allcases we have h − g · b = c , . . . , h − g · b n = c n . Since both Γ y M and Λ y N are by isomorphisms, we have R N ( b , . . . , b n ) iff R N ( c , . . . , c n ).Now it is easy to see that a [ a,
1] is an isomorphic embedding of N into P .Define Γ y P by letting g · [ a, h ] = [ a, gh ]. If g ∈ Λ and a ∈ N , g · [ a,
1] = [ a, g ] =[ g · a, y N . UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 17
It is also obvious that if N and Γ are finite then P is finite. (cid:3) Now the proofs of Theorem 3.9 and the lemmas following it can be repeated toestablish the following theorem.
Theorem 4.8. K is a Fra¨ıss´e class. Furthermore, if ( N ∞ , H ∞ ) is the Fra¨ıss´e limitof K , then H ∞ ∼ = H , H ∞ acts faithfully on N ∞ , and H ∞ is dense in Aut( N ∞ ) . Corollary 4.9.
For all n ≥
3, the automorphism group of the Henson graph H n contains H as a dense subgroup.5. Many Dense Locally Finite Subgroups
Omnigenous locally finite groups.
We define a concept of omnigenousgroups and show that all countable omnigenous locally finite groups are embeddableas a dense subgroup of Iso( U ∆ ). We will need the following extension lemma. Lemma 5.1.
Let ∆ be any countable distance value set. Let X be a finite ∆-metricspace. Let Λ ≤ Γ be finite groups and π : Λ → Iso( X ) be an isomorphic embedding.Then there is a finite ∆-metric space Y extending X and an isomorphic embedding π ′ : Γ → Iso( Y ) such that for any γ ∈ Λ and x ∈ X , π ′ ( γ )( x ) = π ( γ )( x ). Proof.
Define a pseudometric δ on X × Γ by δ (( x , g ) , ( x , g )) = (cid:26) d X ( π ( g − g )( x ) , x ) , if g − g ∈ Λ,diam( X ) , otherwise.Define ( x , g ) ∼ ( x , g ) iff δ (( x , g ) , ( x , g )) = 0. Then ∼ is an equivalence rela-tion on X × Γ. Let Y = X × Γ / ∼ . Then δ gives rise to a metric d Y ([ x , g ] , [ x , g ]) = δ (( x , g ) , ( x , g )). Y is obviously a finite ∆-metric space.It is easy to see that the map x [ x,
1] is an isometric embedding from X into Y .For any γ ∈ Γ, x ∈ X and g ∈ Γ, let π ′ ( γ )([ x, g ]) = [ x, γg ]. Then π ′ : Γ → Iso( Y )is an isomorphic embedding. We check that for any γ ∈ Λ and x ∈ X , π ′ ( γ )( x ) = π ′ ( γ )([ x, x, γ ] = [ π ( γ )( x ) ,
1] = π ( γ )( x ) . (cid:3) Definition 5.2.
Let G be a group. We say that G is omnigenous if for every finitesubgroup G ≤ G , finite groups Γ ≤ Γ and group isomorphism Ψ : G ∼ = Γ ,there is a finite subgroup G ≤ G with G ≤ G and an onto homomorphismΨ : G → Γ such that Ψ ↾ G = Ψ .If we strengthen the requirement on Ψ to be an isomorphism, then this becomesthe property ( E ) from Proposition 3.3. Thus H is omnigenous. Theorem 5.3.
Let H be a countable omnigenous locally finite group. Then for anycountable distance value set ∆ , Iso( U ∆ ) contains H as a dense subgroup.Proof. Let q , q , . . . be an enumeration of all partial isometries of U ∆ . Fix also anenumeration of all elements of H . We will define by induction infinite sequences offollowing objects: • finite subsets D n of U ∆ , for n ≥ • elements h , . . . , h n ∈ H , and H n = h h , . . . , h n i ≤ H , for n ≥ • group embeddings π n : H n → Iso( D n ), for n ≥ n ≥ q n ⊆ π n +1 ( h n +2 ); in particular dom( q n ) ∪ rng( q n ) ⊆ D n +1 ;(ii) for each n ≥ D n ⊆ D n +1 ;(iii) for each n ≥ g ∈ H n , and x ∈ D n , π n +1 ( g )( x ) = π n ( g )( x );(iv) for each n ≥ h n +1 is the least element of H \ { h , . . . , h n } in the fixedenumeration of elements of H .Granting such sequences, it follows from (i) that S ∞ n =1 D n = U ∆ . From (ii) and(iii), it follows that for any g ∈ H n , we have π n ( g ) ⊆ π n +1 ( g ) ⊆ · · · ⊆ π n + m ( g ) ⊆ · · · and the limit lim m →∞ π n + m ( g ) exists and is a full isometry of U ∆ extending π n ( g ).Let Γ n = π n ( H n ) and let i n : Γ n → Γ n +1 be the isomorphic embedding with i n ( π n ( g )) = π n +1 ( g ) for all g ∈ H n . We have a direct systemΓ i −→ Γ i −→ · · · giving a direct limit Γ that is a dense locally finite subgroup of Iso( U ∆ ). We mayregard the group embeddings as inclusions, and the direct limit of the system asan increasing union Γ = S ∞ n =1 Γ n . Moreover, we have S ∞ n =1 H n ∼ = Γ. By (iv), S ∞ n =1 H n = H , and thus H ∼ = Γ.Assume that D n , h , . . . , h n , and π n have been defined. We proceed to define D n +1 , h n +1 , h n +2 , and π n +1 : H n +1 = h H n , h n +1 , h n +2 i → Iso( D n +1 ).First, let h n +1 be the least element of H \{ h , . . . , h n } in the fixed enumerationof elements of H . We define a ∆-metric space X extending D n , and an isomorphicembedding σ n : h H n , h n +1 i → Iso( X ) such that for all g ∈ H n and x ∈ D n ,we have σ n ( g )( x ) = π n ( g )( x ). If n = 0, let a be the order of h and c ∈ ∆ bearbitrary, and define X to be a set with a many elements with d X ( x, y ) = c iff x = y ∈ X . Define σ : h h i → Iso( X ) by letting σ ( h ) be a cyclic permutationon X . If n ≥
1, we check if π n : H n → Iso( D n ) can be extended to an isomorphicembedding σ n : h H n , h n +1 i → Iso( D n ). If so, then we let X = D n and σ n be suchan extension. Assume that π n cannot be extended to an isomorphic embeddingfrom h H n , h n +1 i to Iso( D n ). We apply Lemma 5.1 to obtain a finite ∆-metricspace X extending D n and an isomorphic embedding σ n : h H n , h n +1 i → Iso( X )such that for any g ∈ H n and x ∈ D n , σ n ( g )( x ) = π n ( g )( x ).Using the universality and ultrahomogeneity of U ∆ , we may assume the above X are defined as a subset of U ∆ . Next we further extend X to define D n +1 . ApplyTheorem 2.11 to obtain a strongly coherent S -extension ( Y, φ ) of X ∪ dom( q n ) ∪ rng( q n ). Now φ ( q n ) is an element of Iso( Y ) extending q n . By Lemma 2.12 φ givesrise to an isomorphic embedding from Iso( X ) to Iso( Y ), which we still denote by φ : Iso( X ) → Iso( Y ).Let G = h H n , h n +1 i ≤ H . Let Ψ = φ ◦ σ n be the isomorphic embeddingfrom G into Iso( Y ). Let Λ = Ψ ( G ) and Λ = h Λ , φ ( q n ) i ≤ Iso( Y ). Since H is UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 19 omnigenous, there is a finite G ≤ H and an onto homomorphism Ψ : G → Λ such that Ψ ↾ H n = Ψ = φ ◦ σ n . Let h n +2 ∈ Ψ − ( { φ ( q n ) } ). Thus Ψ ( h n +2 ) = φ ( q n ). By redefining, we can assume G = h G , h n +2 i .Let b = diam( Y ) and let G be given the discrete metric with constant value b .Then G becomes a ∆-metric space. We define a finite ∆-metric space Z = Y ∪ G to be the disjoint union of Y and G , with d Z ( y, g ) = b for all y ∈ Y and g ∈ G .Appealing again to the universality and ultrahomogeneity of U ∆ , we may assumethat all of these extensions took place inside U ∆ . We let D n +1 = Z ⊆ U ∆ .We have H n +1 = h H n , h n +1 , h n +2 i = h G , h n +2 i = G . Define π n +1 : H n +1 → Iso( D n +1 ) = Iso( Z ) by letting π n +1 ( g ) ↾ Y = Ψ ( g ) and π n +1 ( g )( h ) = gh for all h ∈ G . Then for any g ∈ H n and x ∈ D n , π n +1 ( g )( x ) = Ψ ( g )( x ) = σ n ( g )( x ) = π n ( g )( x ). To complete the proof, we need to verify that π n +1 thusdefined is a group isomorphism. For this, we show that for all g , . . . , g k ∈ { h , . . . , h n , h n +1 , h n +2 } and ǫ , . . . , ǫ k ∈ { +1 , − } , we have g ǫ · · · g ǫ k k = 1 ⇐⇒ π n +1 ( g ) ǫ · · · π n +1 ( g k ) ǫ k = 1 . First, suppose g ǫ · · · g ǫ k k = 1. Observe that, as an element of Iso( Z ), the action of π n +1 ( g ) ǫ · · · π n +1 ( g k ) ǫ k on the Y part of Z is given by Ψ ( g ) ǫ · · · Ψ ( g k ) ǫ k . SinceΨ is a homomorphism, we have Ψ ( g ) ǫ · · · Ψ ( g k ) ǫ k = 1. On the other hand, onthe G part of Z , the action of π n +1 ( g ) ǫ · · · π n +1 ( g k ) ǫ k is the same as the leftmultiplication by g ǫ · · · g ǫ k k = 1. Since both these actions are identity, we have π n +1 ( g ) ǫ · · · π n +1 ( g k ) ǫ k = 1. Conversely, if π n +1 ( g ) ǫ · · · π n +1 ( g k ) ǫ k = 1, then itsaction on the G part is the left multiplication by g ǫ · · · g ǫ k k ; thus g ǫ · · · g ǫ k k = 1. (cid:3) A family of omnigenous locally finite groups.
In this subsection we con-struct an uncountable family of pairwise non-isomorphic, omnigenous, universalcountable locally finite groups.Let P be a set of prime numbers. If P = ∅ , enumerate its elements as p < p
0, there are infinitely many elements of order p n in H . Fix a subset S = { s , s , s , . . . } ⊆ H where each s n is of order p n , suchthat H \ S still generates a universal countable locally finite group. This is easy toarrange since H ⊕ H is embedded as a subgroup of H and we may choose S as asubset of the first copy of H .Let X be the disjoint union of H with a copy of Z p n = Z /p n Z for each n ≥ H as a subset of X as Y , and for each n ≥ Z p n as a subset of X as Z n . Thus X = Y ∪ S n ≥ Z n .For T ⊆ H , we say that T is of type P if T = { t , t , t , . . . } where each t n is oforder p n ∈ P . For any T ⊆ H of type P , we define a map λ T : H → Sym( X ) asfollows. For all g ∈ H , λ T ( g ) acts on Y = H as the left multiplication by g . If g T ,then λ T ( g ) acts on each Z n as identity. If g ∈ T and the order of g is p n , then λ T ( g ) acts on Z n = Z p n as +1, and acts on other Z m , m = n , as identity. Let G T be the subgroup of Sym( X ) generated by the set λ T ( H ). Note that for any subset A ⊆ H \ T , λ T gives an isomorphism between h A i ≤ H and h λ T ( A ) i ≤ Sym( X ).For each t ∈ T , λ T ( t ) has the same order as t . Also note that for any B ⊆ H , the map λ T ( b ) b induces a homomorphismfrom h λ T ( B ) i onto h B i . To see this, we need to show that for all g , . . . , g l ∈ B and ǫ , . . . , ǫ l ∈ { +1 , − } , if λ T ( g ) ǫ · · · λ T ( g l ) ǫ l = 1, then g ǫ · · · g ǫ l l = 1. However,this is obvious by observing the action of λ T ( g ) ǫ · · · λ T ( g l ) ǫ l on Y = H .We will construct a omnigenous, universal countable locally finite group H P asa direct limit of a direct system H e −→ H e −→ H e −→ · · · · · · where e k : H k − → H k is an isomorphic embedding for all k ≥
1. In fact, each H k will be of the form G T k for some T k ⊆ H of type P .We define the H k , e k by induction on k . For k = 0, let T = S and H = G T .Since H \ S generates a universal countable locally finite group, H is universal forall countable locally finite groups. Since H will be embedded as a subgroup of H P , H P is thus universal as well. In the rest of our construction we focus on theomnigenous property of H P .In general, suppose H k = G T k has been defined. Let i k +1 : H k → H be anisomorphic embedding. Let T k +1 = i k +1 ( λ T k ( T k )). Then T k +1 is of type P . Let H k +1 = G T k +1 . Define a map f : λ T k ( H ) → H k +1 by f ( γ ) = λ T k +1 ( i k +1 ( γ )) for all γ ∈ λ T k ( H ). We claim that f extends uniquely to an isomorphic embedding from H k into H k +1 . For this, we show that for all γ , . . . , γ l ∈ λ T k ( H ) and ǫ , . . . , ǫ l ∈ { +1 , − } , we have γ ǫ · · · γ ǫ l l = 1 ⇐⇒ f ( γ ) ǫ · · · f ( γ l ) ǫ l = 1 . Suppose γ i = λ T k ( g i ) for g i ∈ H for all 1 ≤ i ≤ l . First, assume γ ǫ · · · γ ǫ l l = 1.Then, by observing the action of this element on the Y part, we get that g ǫ · · · g ǫ l l =1. By observing the action of this element on the Z n parts, we conclude that, if t ∈ T k of order p n appears as g i for 1 ≤ i ≤ l , then N t = P g i = t ǫ i is a multiple of p n .It follows that i k +1 ( γ ) ǫ · · · i k +1 ( γ k ) ǫ k = 1, and consequently f ( γ ) ǫ · · · f ( γ k ) ǫ k acts on the Y part as identity. Moreover, for any t ∈ T k of order p n , letting t ′ = f ( λ T k ( t )) ∈ T k +1 , then N t ′ = X f ( γ i )= t ′ ǫ i = X g i = t ǫ i is a multiple of p n . Thus f ( γ ) ǫ · · · f ( γ l ) ǫ l acts on the Z n parts also as iden-tity. Therefore f ( γ ) ǫ · · · f ( γ l ) ǫ l = 1. Conversely, suppose f ( γ ) ǫ · · · f ( γ l ) ǫ l =1. Then by observing the action of this element on the Y part, we get that i k +1 ( γ ) ǫ · · · i k +1 ( γ l ) ǫ l = 1. Thus γ ǫ · · · γ ǫ l l = 1. We have thus established theclaim. From the claim, let e k +1 : H k → H k +1 be the unique isomorphic embeddingextending f .This finishes our definition of the direct system. As usual, we view all e k asinclusions, and H P as an increasing union of H k .We verify that H P is omnigenous. For this, let G ≤ H P be a finite subgroup.Let k be sufficiently large that G ≤ H k . Let Γ ≤ Γ be finite and Ψ : G ∼ = Γ .Now consider i k +1 ( G ) ≤ H and note that i k +1 ◦ Ψ − is an isomorphic embedding UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 21 from Γ into H with image i k +1 ( G ). By property ( E ) from Proposition 3.3 for H ,there is an isomorphic embedding j : Γ → H extending i k +1 ◦ Ψ − . Let G bethe group generated by λ T k +1 ( j (Γ )). As noted before, there is a homomorphism ψ from G onto j (Γ ) such that ψ ↾ G is an isomorphism. Let Ψ : G → Γ be j − ◦ ψ . It is straightforward to check that Ψ ↾ e k +1 ( G ) = Ψ ◦ e − k +1 . This showsthat H P is omnigenous.The next lemma characterizes the isomorphism type of H P in terms of the set P . Lemma 5.4. P is exactly the set of all primes p such that there are order- p elements α, β ∈ H P that are not conjugate in H P . Proof.
First let p n ∈ P . Let a ∈ H be an element of order p n such that a = s n ∈ S = T . We claim that α = λ S ( a ) ∈ H and β = λ S ( s n ) ∈ H are not conjugatein H P . Toward a contradiction, assume α, β are conjugate in H P . Then there is k ≥ H k . By the construction of H k , we have α = λ T k ( g ) for some g ∈ H \ T k and β = λ T k ( h ) for some h ∈ T k of order p n . Let g , . . . , g l ∈ H and ǫ , . . . , ǫ l ∈ { +1 , − } such that λ T k ( g ) ǫ · · · λ T k ( g l ) ǫ l λ T k ( g ) λ T k ( g l ) − ǫ l · · · λ T k ( g ) − ǫ = λ T k ( h ) . The action of the element on the left hand side on Z n is identity, while the actionof λ T k ( h ) on Z n is +1, a contradiction.On the other hand, suppose α, β ∈ H P both have order p P . Then there is k ≥ α = λ T k ( g ) and β = λ T k ( h ) for g, h ∈ H \ T k . By Proposition 3.2 g, h are conjugate in H , i.e., there is g ∈ H such that g gg − = h . Then weclaim λ T k ( g ) λ T k ( g ) λ T k ( g ) − = λ T k ( h ). This is because, the action of the elementon the left hand side on Y is by left multiplication of g gg − , while the action of λ T k ( h ) on Y is by left multiplication of h , which are the same; on the other hand,the action of the element on the left hand side on all Z n is identity regardless ofwhether g ∈ T k , which is the same as the action of λ T k ( h ) on Z n . Thus the claimholds true, and α and β are conjugate in H k . (cid:3) By Lemma 5.4, if P = P ′ are distinct sets of primes, then H P and H P ′ are notisomorphic. Since all H P are omnigenous, by Theorem 5.3 we can embed H P intoIso( U ∆ ) as a dense subgroup. We have thus proved the following theorem. Theorem 5.5.
There are continuum many pairwise nonisomorphic countable uni-versal locally finite groups each of which can be embedded into
Iso( U ∆ ) as a densesubgroup. When P = ∅ , it is easy to see that H P has the property ( E ) from Proposition 3.3,and hence is isomorphic to H . Thus we obtain another proof of Theorem 3.14.5.3. Ultrametric spaces.
In this subsection we deal with ultrametric Urysohnspaces and their isometry groups. We first recall some basic facts about ultrametricUrysohn spaces (cf., e.g., [9] and [19]).Recall that an ultrametric d on a space X is a metric such that d ( x, y ) ≤ max { d ( x, z ) , d ( y, z ) } for all x, y, z ∈ X . In any separable ultrametric space, the ultrametric can takeonly countably many values. Consequently, there is no separable ultrametric spacethat is universal for all separable ultrametric spaces.Given any countable set R of positive real numbers, an R -ultrametric space isan ultrametric space in which the ultrametric takes positive values only in R . Theclass of all finite R -ultrametric spaces is a Fra¨ıss´e class, and we let K uR denote itsFra¨ıss´e limit. K uR is a universal countable, ultrahomogeneous R -ultrametric space,and we call it the universal countable R -ultrametric Urysohn space . We endowIso( K uR ) with the permutation group topology.Consider the completion of K uR under the pointwise convergence topology, whichwe denote as U uR and call the R -ultrametric Urysohn space . U uR is a Polish R -ultrametric space which is universal for all Polish R -ultrametric spaces and is itselfultrahomogeneous. We endow Iso( U uR ) with the pointwise convergence topology.By the standard argument in the proof of Theorem 3.16, any dense subgroupof Iso( K uR ) gives rise to an isomorphic dense subgroup of Iso( U uR ). We prove thefollowing theorem. Theorem 5.6.
For any non-empty countable set R of positive real numbers, thefollowing hold:(i) Iso( K uR ) and Iso( U uR ) contain H as a dense subgroup.(ii) Every countable omnigenous locally finite group can be embedded into Iso( K uR ) or Iso( U uR ) as a dense subgroup.(iii) There are continuum many non-isomorphic universal countable locally finitegroups that can be embedded into each of Iso( K uR ) and Iso( U uR ) as a densesubgroup. Our plan is to repeat the proof in the preceding subsections for Iso( K uR ). Wefirst need a lemma for R -ultrametric spaces that is analogous to Lemma 5.1. Itturns out that the proof is verbatim the same as that of Lemma 5.1, only notingthat the pseudometric defined there is indeed a pseudo-ultrametric. We state thelemma below without proof. Lemma 5.7.
Let R be any nonempty countable set of positive numbers. Let X bea finite R -ultrametric space. Let λ ≤ Γ be finite groups and π : Λ → Iso( X ) be anisomorphic embedding. Then there is a finite R -ultrametric space Y extending X and an isomorphic embedding π ′ : Γ → Iso( Y ) such that for any γ ∈ Λ and x ∈ X , π ′ ( γ )( x ) = π ( γ )( x ).The next thing we need is a result analogous to Solecki’s Theorem 2.11 for R -ultrametric spaces. This is an easy consequence of the techniques used to prove themetric case (cf., e.g. [29]), but we give a self-contained proof here. Lemma 5.8.
Let R be any nonempty countable set of positive real numbers. Let X be a finite R -ultrametric space. Then X has a strongly coherent S-extension( Y, φ ) where Y is a finite R -ultrametric space. Proof.
Let D ( X ) = { d X ( x, y ) : x = y ∈ X } . We prove this by induction on | D ( X ) | . UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 23
First consider the case | D ( X ) | = 1. In this case Iso( X ) = Sym( X ). Fix a linearorder < on X . Given any partial isometry (permutation) p of X , define φ ( p ) to bethe extension of p by the (unique) < -order-preserving bijection between X \ dom( p )and X \ rng( p ). Then ( X, φ ) is easily seen to be a strongly coherent S-extension of X .Suppose | D ( X ) | > r be the least element of D ( X ). For each x ∈ X let B r ( x ) = { y ∈ X : d X ( x, y ) ≤ r } = { x } ∪ { y ∈ X : d X ( x, y ) = r } . Define X = { B r ( x ) : x ∈ X } and d on X by d ( B r ( x ) , B r ( x )) = (cid:26) d X ( x , x ) , if d X ( x , x ) > r ,0 , otherwise.Then | D ( X ) | = | D ( X ) |−
1. By the inductive hypothesis applied to R = D ( X ) \{ r } , X has a finite R -ultrametric strongly coherent S-extension ( Y , φ ), where D ( Y ) ⊆ R . Let N = max {| B r ( x ) | : x ∈ X } and fix an x ∈ X such that | B r ( x ) | = N .Fix a linear order < x on each B r ( x ); however, make < x depend only on B r ( x ) butnot on x . Let X = B r ( x ) and d = d X on X ⊆ X . For each of B r ( x ), let e x : B r ( x ) → X be the order-preserving embedding so that e x ( B r ( x )) is an initialsegment in X . Every B r ( x ) is identified as a subset of X via e x . We will view e x as an inclusion. For each partial isometry p of B r ( x ) (viewed also as a partialisometry of X ), let φ x ( p ) be the extension of p by the order-preserving bijectionbetween X \ dom( p ) and X \ rng( p ).Let Y = Y × X and define d Y (( u , u ) , ( v , v )) = max { d ( u , v ) , d ( u , v ) } .Every x ∈ X is identified with ( B r ( x ) , e x ( x )) ∈ Y . If p is a partial isometry of X , then p induces a partial isometry of X , which we denote by p . For every x ∈ dom( p ) ⊆ X , p induces a partial isometry between B r ( x ) and B r ( p ( x )), whichis identified as a partial isometry of X via e x and e p ( x ) , which we denote by p x .Note that p x depends on B r ( x ) but not on x . Define φ ( p ) ∈ Iso( Y ) by φ ( p )( u, v ) = (cid:26) ( φ ( p )( u ) , φ x ( p x )( u, v )) , if x ∈ dom( p ) and d Y ( x, ( u, v )) ≤ r ,( φ ( p )( u ) , v ) , if there is no such x .Then it is straightforward to check that ( Y, φ ) is a strongly coherent S-extension of X . (cid:3) The rest of the proof in the preceding section works verbatim. In particular, weobtain the space X using Lemma 5.7 and Y using Lemma 5.8, and then the space Z constructed is an ultrametric space.6. Properties of Dense Locally Finite Subgroups
In this section we study some properties of all dense locally finite subgroups ofIso( U ∆ ) from the point of view of model theory and combinatorial group theory. Discerning types and discerning structures.
We first define some con-cepts and fix some notation. Throughout this section we let L be a countablerelational language (with equality). Given an L -structure M and A ⊆ M , we de-note by L A the language L expanded by a constant symbol for each element of A . Definition 6.1.
Fix an L -structure M and A ⊆ M . A 1 -type over A is a set p of L A -formulas, with (at most) one free variable x , for which there exists m ∈ M suchthat p = { ϕ ( x ) : M | = ϕ ( m ) } . Such m is called a realization of p in M . We say that p is nontrivial if p has arealization not belonging to A . We denote by S ( A ) the set of all 1-types over A .Our terminology differs slightly from the usual definition in that these types areusually referred to as types realized in M . In our context there will always be anunderlying structure M , and any 1-type over A is realized in M . Note that a 1-typeover A is nontrivial iff all realizations of p do not belong to A , because if a ∈ A and b ∈ M \ A , then they cannot have the same 1-type over A (one satisfies x = a andthe other one does not). Definition 6.2.
Let M be a countable L -structure. Given A ⊆ M and p a 1-typeover A , we denote by [ p ] the set of all realizations of p in M , i.e., [ p ] = { m ∈ M : ∀ ϕ ∈ p M | = ϕ ( m ) } .We say that p is algebraic over A if [ p ] is finite.As an example, consider the structure U ∆ , where ∆ is a countable distance valueset. As illustrated in Subsection 2.2, U ∆ can be viewed as a relational structurein a language L with a binary relation symbol for each value of ∆. If A ⊂ U ∆ isfinite, the 1-type over A of x ∈ U ∆ is entirely determined by the distance function a d ( a, x ) (by ultrahomogeneity). Thus we may identify 1-types over A with Katˇetov maps over A , i.e. maps f : A → ∆ ∪ { } such that ∀ a, b ∈ A | f ( a ) − f ( b ) | ≤ d ( a, b ) ≤ f ( a ) + f ( b ) . We denote by E ( A ) the set of all Katˇetov maps over A . Note that if f ∈ E ( A ) issuch that f ( a ) = 0 for some a , then f = d ( a, · ). Below we will sometimes identify A with the subset of E ( A ) made up of trivial 1-types. Definition 6.3.
Let M be a countable L -structure, A a finite substructure and p a nontrivial 1-type over A . We say that p is discerning if for every non-identity g ∈ Aut( M ) there exists x ∈ [ p ] such that g ( x ) = x .Note that, if A is finite and p is both discerning and algebraic over A , then { g ∈ Aut( M ) : ∀ x ∈ [ p ] g ( x ) = x } = { } , hence Aut( M ) is discrete. Equivalently,if Aut( M ) is non-discrete, A is finite and p ∈ S ( A ) is discerning, then p cannot bealgebraic, thus has infinitely many realizations. Definition 6.4.
Let M be a countable L -structure. We say that M is discerning if for any finite A ⊆ M , every nontrivial 1-type over A is discerning. UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 25
Note that a countable infinite set, viewed as a first-order structure with onlythe equality symbol in its language, is not discerning, because of the existence ofelements of Aut( M ) with finite support. Definition 6.5.
Let M be a countable ultrahomogeneous L -structure. We say that M has rich types if it satisfies the following conditions: • For any finite A ⊆ M , and any g ∈ Aut( M ) \ { } , there exists x A suchthat g ( x ) = x ; • For any finite A ⊆ M , any x = y ∈ M with x A , and any nontrivial p ∈ S ( A ), there exist z ∈ M which is a realization of p and an L -formula ϕ such that M | = ϕ ( x, z ) and M | = ¬ ϕ ( y, z ).Observe that the first condition above forbids the existence of nontrivial elementsof Aut( M ) with finite support. Proposition 6.6.
Let M be a countable ultrahomogeneous L -structure. If M hasrich types, then M is discerning. Proof.
Let A ⊆ M be finite, p a nontrivial type over A , and g ∈ Aut( M ) \ { } .By assumption, g cannot fix every element in the complement of A . So we pick x ∈ M \ A such that g ( x ) = x . Apply the fact that M has rich types to p , x ,and y = g ( x ). We thus obtain a realization z of p and an L -formula ϕ such that M | = ϕ ( x, z ) and M | = ¬ ϕ ( y, z ). Since g ∈ Aut( M ), we have M | = ϕ ( y, g ( z )), andthus g ( z ) = z . This z witnesses that p is discerning. (cid:3) The random graph R has rich types. In particular, for ∆ = { , } , U ∆ has richtypes. In the above we already saw that U { } does not have rich types because it isnot discerning. In general, U ∆ can also fail to have rich types for the reason that itdoes not satisfy the second condition of the definition: it is possible that the typeof z over A forces that ( x, z ) and ( y, z ) satisfy the same L -formulas. Nevertheless,this happens for fairly few types, so that one can still prove the following. Proposition 6.7.
Let ∆ be a countable distance value set with | ∆ | ≥
2. Then U ∆ is discerning. Proof.
We first note that when | ∆ | ≥ U ∆ does satisfy the first condition of thedefinition of rich types. In fact, let A ⊆ U ∆ be finite and g ∈ Iso( U ∆ ) \ { } . If g fixes all elements of A , then there is x ∈ U ∆ \ A with g ( x ) = x , since g is notidentity. Otherwise, let a ∈ A be such that g ( a ) = a . If g ( a ) A , we can let x = g ( a ), and then g ( x ) = x since d ( a, x ) = d ( g ( a ) , g ( x )) = d ( x, g ( x )). Suppose g ( a ) ∈ A and let r = d ( a, g ( a )). Since | ∆ | ≥
2, there is an s ∈ ∆ \ { r } . Supposefirst s < r . Then let k be the positive integer with ks < r ≤ ( k + 1) s . Define aKat˘etov map f over A by letting f ( c ) = min { d ( a, c ) + ks, d ( g ( a ) , c ) + r, sup(∆) } for all c ∈ A .In particular f ( a ) = ks = r = f ( g ( a )). Let x ∈ U ∆ be such that f ( c ) = d ( x, c )for all c ∈ A . Then x A since d ( x, c ) = f ( c ) > c ∈ A . Then d ( g ( x ) , g ( a )) = d ( x, a ) = f ( a ) = ks = r = f ( g ( a )) = d ( x, g ( a )). Thus g ( x ) = x . Assume next r < s . Let k be the positive integer with kr < s ≤ ( k + 1) r . Define aKat˘etov map f over A by letting f ( c ) = min { d ( a, c ) + kr, d ( g ( a ) , c ) + s, sup(∆) } for all c ∈ A .Then by a similar argument we can find x A with g ( x ) = x .To prove U ∆ is discerning, fix a finite set A ⊆ U ∆ and nontrivial f ∈ E ( A ).Then f ( c ) > c ∈ A . If A = ∅ the statement is obvious. So we assume A = ∅ . Fix a non-identity g ∈ Iso( U ∆ ) and x A ∪ g − ( A ) ∪ g − ( A ) such that x = g ( x ). Then x, g ( x ) A ∪ g − ( A ).Let B = A ∪ g − ( A ) ∪ { x, g ( x ) } . Let R = max { f ( c ) , d ( a, b ) : c ∈ A, a = b ∈ B } and r = min { f ( c ) , d ( a, b ) : c ∈ A, a = b ∈ B } > R > r . Find t ∈ ∆ such that R ≤ max { r, R − r } ≤ t < R .If r ≥ R we can just let t = r . Otherwise, let k be the positive integer with kr < R ≤ ( k + 1) r and let t = kr . Define a Kat˘etov map F over B by letting F ( b ) = (cid:26) t, if b = g ( x ), R, otherwise.Let y ∈ U ∆ be such that d ( y, b ) = F ( b ) for all b ∈ B . Since d ( y, x ) = R and d ( y, g ( x )) = t , we get that g ( y ) = y . Moreover, for any c ∈ A , since d ( y, g − ( c )) = R , we have d ( g ( y ) , c ) = R . Finally, define a Kat˘etov map ˆ f over A ∪ { y, g ( y ) } that is an extension of f by letting ˆ f ( y ) = R and ˆ f ( g ( y )) = t . Let z ∈ U ∆ be arealization of ˆ f . Then z is a realization of f , and d ( z, y ) = R = t = d ( z, g ( y )).Thus g ( z ) = z . (cid:3) Mixed identities and MIF properties.
In this subsection we considermixed identities in a group and show that all dense subgroups of Iso( U ∆ ) do notsatisfy any nontrivial mixed identities, i.e., they are MIF (mixed identity free). Forlocally finite groups, we also define the more general notion of ∞ -MIF and showthat it coincides with omnigenity.We refer the reader to Hull–Osin [15] (particularly Section 5 there) for an accountof the study of mixed identities in group theory and for many existing results aboutthe notion of MIF groups. The argument used to prove Theorem 6.10 below is anadaptation to our context of arguments of Theorem 5.9 of [15] about highly transitivegroups (dense subgroups of the permutation group S ∞ of an infinite countable set). Definition 6.8.
Let G be a group. Let F n be the free group generated by variables x , . . . , x n . A nontrivial mixed identity in G is a word w ( x , . . . , x n ) ∈ G ∗ F n \ G such that w ( g , . . . , g n ) = 1 for all g , . . . , g n ∈ G . If there is no nontrivial mixedidentity in G , we say that G is mixed identity free ( MIF ).As noted in [15] (Remark 5.1), a group G is MIF iff there is no nontrivial mixedidentity w ( x ) ∈ G ∗ h x i \ G , as there is an isomorphism from G ∗ F n into G ∗ h x i that sends x , . . . , x n to xgx, . . . , x n gx n for a g ∈ G \ { } . In the sequel we onlyconsider mixed identities with one free variable x . Such identities are of the form g x m g x m · · · g k x m k for g , . . . , g k ∈ G , g , . . . , g k ∈ G \ { } , and m , . . . , m k ∈ Z \ { } . We denote such identities by w ( x ; g , . . . , g k ). UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 27
Proposition 6.9.
Let G be a topological group. If G is MIF then any densesubgroup of G is MIF. Proof.
Let Γ be a dense subgroup of G , and assume that Γ satisfies a nontrivialmixed identify w ( γ ; γ , . . . , γ k ) = 1 for some fixed γ , . . . , γ k ∈ Γ and all γ ∈ Γ.Since { g ∈ G : w ( g ; γ , . . . , γ k ) = 1 } is closed, and contains Γ, we conclude that wehave w ( g ; γ , . . . , γ k ) = 1 for all g ∈ G . (cid:3) Theorem 6.10.
Let L be a countable relational language and M be a countable ul-trahomogeneous L -structure. Suppose M is discerning and Aut( M ) is non-discrete.Then for every w ( x ) ∈ Aut( M ) ∗ h x i \ Aut( M ) the set { g ∈ G : w ( g ) = 1 } is dense.In particular, Aut( M ) is MIF, and so is any dense subgroup of Aut( M ) .Proof. Let g , ..., g k ∈ Aut( M ) with g , ..., g k = 1, and m , ..., m k ∈ Z \ { } .We have to prove that the set consisting of all elements g ∈ Aut( M ) such that g g m · · · g k g m k = 1 is dense. We will use the following lemmas. Lemma 6.11.
Let
A, B be finite subsets of M and t be a nontrivial 1-type over A . Then there is b B which is a realization of t . Proof.
Since Aut( M ) is non-discrete, A is finite, and t ∈ S ( A ) is discerning, t hasinfinitely many realizations. Since B is finite, there is a realization of t outside of B . (cid:3) Lemma 6.12.
Let p be any partial automorphism of M and let a dom( p ). Thenthere is a partial automorphism q extending p such that q ( a ) dom( q ). Proof.
Since M is ultrahomogeneous, p can be extended to a ψ ∈ Aut( M ). Let t be the 1-type of ψ ( a ) over rng( p ). Since a dom( p ), ψ ( a ) rng( p ), and therefore t is nontrivial. By Lemma 6.11, t has a realization b dom( p ) ∪ { a } . Since b ∈ [ t ],the map p ∪ { ( a, b ) } is a partial automorphism. Let q = p ∪ { ( a, b ) } , then q extends p . Moreover, q ( a ) = b dom( p ) ∪ { a } = dom( q ). (cid:3) Lemma 6.13.
Let g ∈ Aut( M ), p be a partial automorphism of M , and let a dom( p ). Then there is a partial automorphism q extending p such that q ( a ) , gq ( a ) dom( q ) and gq ( a ) = q ( a ). Proof.
Again, since M is ultrahomogeneous, p can be extended to a ψ ∈ Aut( M ).Let t be the 1-type of ψ ( a ) over rng( p ). Since a dom( p ), ψ ( a ) rng( p ), andtherefore t is nontrivial. By Lemma 6.11, t has a realization b dom( p ) ∪ { a } ∪ g − (dom( p )) ∪{ g − ( a ) } . Let s be the 1-type of b over dom( p ) ∪{ a } ∪ g − (dom( p )) ∪{ g − ( a ) } . Then s is nontrivial, hence is discerning. Let c be a realization of s with g ( c ) = c . Then g ( c ) dom( p ) ∪ { a } , because otherwise c ∈ g − (dom( p ) ∪ { a } )and would satisfy the negation of some formula in s . Now t ⊆ s , so we have that c is a realization of t . As before, it follows that the map p ∪ { ( a, c ) } is a partialautomorphism. Let q = p ∪ { ( a, c ) } , then q extends p . Moreover, q ( a ) = c, gq ( a ) = g ( c ) dom( p ) ∪ { a } = dom( q ). (cid:3) Let p be any partial automorphism of M and fix a dom( p ) ∪ rng( p ). Byapplying Lemma 6.12 repeatedly, to either p if m k > p − if m k <
0, we obtain a partial automorphism q extending p or p − accordingly, such that a, q ( a ) , q ( a ) , . . . , q | m k |− ( a )are pairwise distinct. Then, by applying Lemma 6.13 once to g k , q and q | m k |− ( a ),we obtain a partial automorphism r extending q , such that a, r ( a ) , r ( a ) , . . . , r | m k |− ( a ) , r | m k | ( a ) , g k r | m k | ( a )are pairwise distinct. Let p k = r if m k > p k = r − if m k <
0. Then we get apartial automorphism p k extending p such that a, g k p m k k ( a ) are distinct.Repeating the argument k − p ⊆ p k ⊆ p k − ⊆ · · · ⊆ p such that a, g k p m k k ( a ) , g k − p m k − k − g k p m k k − ( a ) , . . . , g p k · · · g k p m k ( a )are all pairwise distinct. In the very last step of the construction, we applyLemma 6.13 as above if g = 1 and apply Lemma 6.12 if g = 1. In particular, weobtain that for any g ∈ Aut( M ) extending p , we have that g g m · · · g k g m k ( a ) = a .Since p was arbitrary, we conclude that the set of g such that g g m · · · g k g m k = 1is dense. (cid:3) The following corollary follows immediately from Proposition 6.7, Proposition 6.9,and Theorem 6.10.
Corollary 6.14.
For any countable distance value set ∆ of cardinality ≥
2, Iso( U ∆ )is MIF. Moreover, any dense subgroup of Iso( U ∆ ) is MIF. Example 6.15. (1) H is MIF.(2) Let Alt( N ) be the group of all finitely supported even permutations on N .Hull–Osin showed that Alt( N ) is not MIF (Theorem 5.9 of [15]). SinceAlt( N ) is dense in S ∞ , S ∞ is not MIF.(3) Let P be a set of primes and T ⊆ H be a subset of type P . The group G T constructed in Subsection 5.2 is MIF. To see this, assume γ x m · · · γ k x m k isa mixed identity in G T . By observing its action on Y = H , there are g ∈ H and g . . . , g k ∈ H \ { } such that for any g ∈ H , g g m · · · g k g m k = 1. Thiscontradicts the fact that H is MIF.(4) In Subsection 3.1 we noted that the family of groups H ⊕ A , where A isan abelian p -group, consists of continuum many non-isomorphic countableuniversal locally finite groups. It is easy to see that H ⊕ A , when A isnontrivial, is not MIF: a nontrivial mixed identity in H ⊕ A is w ( x ) = xgx − g − , where g ∈ A \{ } . Thus none of the groups H ⊕ A are embeddableas a dense subgroup of Iso( QU ) when A is nontrivial.We mention another corollary of our results. Corollary 6.16.
The group Iso( U ) is MIF. Consequently, any dense subgroup ofIso( U ) is MIF. UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 29
Proof.
Consider a nontrivial word w ( g ; g , . . . , g k ) with g , . . . , g k ∈ Iso( U ). Us-ing Lemma 5.1 of [2], we obtain a distance value set ∆ and a dense subspace X of U which is isometric to U ∆ and such that g i ( X ) = X for all i ∈ { , . . . , k } .Since Iso( U ∆ ) is MIF, we can find g ∈ Iso( X ) such that w ( g ; g | X , . . . , g k | X ) = 1.Extending g to an isometry of U , we have w ( g ; g , . . . , g k ) = 1. (cid:3) In the rest of this subsection we consider some notions stronger than MIF.For notational simplicity we will write a word w ( x , . . . , x n ) ∈ G ∗ F n \ G as w ( x , . . . , x n ; g , . . . , g k ) if the constants occurring in the normal form of w ( x , . . . , x n )are among g , . . . , g k . Definition 6.17.
Let G be a group and k ≥ G is k -MIF if for any w ( x , . . . , x n ) , . . . , w k ( x , . . . , x n ) ∈ G ∗ F n \ G there are h , . . . , h n ∈ G such that w ( h , . . . , h n ) , . . . , w k ( h , . . . , h n ) = 1.If G is a locally finite group, we say that G is ∞ -MIF if for any g , . . . , g k ∈ G and any infinite sequence w ( x , . . . , x n ; g , . . . , g k ) , w ( x , . . . , x n ; g , . . . , g k ) , . . . ofelements of G ∗ F n \ G , whenever there is a finite group Γ which is an overgroup of h g , . . . , g k i in which there are γ , . . . , γ n ∈ Γ such that w i ( γ , . . . , γ n ; g , . . . , g k ) = 1for all i ≥
1, there are h , . . . , h n ∈ G such that w i ( h , . . . , h n ; g , . . . , g k ) = 1 forall i ≥ k ≥
1, the definition of k -MIF is equivalent to the version where we consider words with only one variable. Proposition 6.18.
Let G be any group. Then the following hold:(i) G is MIF iff G is k -MIF for some k ≥ G is k -MIF for all k ≥ G is a locally finite group and G is ∞ -MIF, then G is MIF.(iii) If G is a locally finite group, then G is ∞ -MIF iff G is omnigenous. Proof.
MIF is exactly 1-MIF, and it is clear that k -MIF implies 1-MIF. The factthat 1-MIF implies k -MIF is a direct consequence of Proposition 5.3 of [15]. We givea full proof here for the convenience of the reader. We first show that 1-MIF implies2-MIF. Let w ( x , . . . , x n ) , w ( x , . . . , x n ) ∈ G ∗ F n \ G . Let w ( x , . . . , x n , x ) ∈ G ∗ F n ∗ h x i be the word [ w , x − w x ] = w − x − w − xw x − w x . Then w is nontrivial.Since G is 1-MIF, there are h , . . . , h n , h ∈ G such that w ( h , . . . , h n , h ) = 1. Itfollows that w ( h , . . . , h n ) , w ( h , . . . , h n ) = 1. Thus G is 2-MIF. In general, let k ≥ w , . . . , w k ∈ G ∗ F n \ G . Then similarly define a nontrivial w ∈ G ∗ F n ∗h x i as w = [ w , x − w x, x − w x , · · · , x − k +1 w k x k − ]where the commutator is inductively defined by[ u , u ] = u − u − u u and [ u , . . . , u m ] = [[ u , . . . , u m − ] , u m ] . Reasoning as before, we find h , ..., h n , h ∈ G satisfying w ( h , ..., h n , h ) = 1, whichimplies that all the elements w ( h , ..., h n ),..., w k ( h , ..., h n ) are non trivial. Thisfinishes the proof that 1-MIF implies k -MIF for every k ≥
2, so (i) is established.To prove (ii), let G be an ∞ -MIF locally finite group. In view of (i), we onlyneed to show that G is MIF. Let w ( x ; g , ..., g k ) be a nontrivial word in G ∗ h x i . Since h g , ..., g k i is finite and h x i ∼ = Z is residually finite, the group h g , ..., g k i ∗ h x i is residually finite. We can thus find a finite group Γ and a homomorphism ρ : h g , ..., g k i ∗ h x i → Γ such that ρ ( w ) = 1 and ρ ( g ) = 1 for every g ∈ h g , ..., g k i \ { } .Since ρ ↾ h g , . . . , g n i is an isomorphism, we can then view Γ as an overgroup of h g , ..., g k i , in which w ( ρ ( x ); g , . . . , g k ) = 1. Applying the ∞ -MIF property, wefind h ∈ G such that w ( h ; g , ..., g k ) = 1 as required.To prove (iii), suppose G is a locally finite group. First, assume G is omnigenous.Let g , . . . , g k ∈ G and w , w , · · · ∈ G ∗ F n \ G be given. Let Γ be a finite overgroupof h g , . . . , g k i and γ , . . . , γ n ∈ Γ are such that w i ( γ , . . . , γ n ; g , . . . , g k ) = 1 for all i ≥
1. Since G is omnigenous, there are h , . . . , h n ∈ G such that the map given by g j g j for j = 1 , . . . , k and h l γ l for l = 1 , . . . , n generates a homomorphismfrom h h , . . . , h n , g , . . . , g k i onto h γ , . . . , γ n , g , . . . , g k i ≤ Γ. Thus, for any i ≥ w i ( h , . . . , h n ; g , . . . , g k ) = 1, since otherwise by applying the homomorphism wewould get w i ( γ , . . . , γ n ; g , . . . , g k ) = 1.Conversely, assume G is ∞ -MIF. Suppose G = { } ∪ { g , . . . , g k } is a fi-nite subgroup of G and Γ is a finite overgroup of G with additional generators γ , . . . , γ n G , i.e., Γ = h g , . . . , g k , γ , . . . , γ n i . Let w , w , . . . enumerate allwords w ∈ G ∗ F n \ G such that w ( γ , . . . , γ n ; g , . . . , g k ) = 1. Since G is ∞ -MIF,there are h , . . . , h n ∈ G such that for all i ≥ w i ( h , . . . , h n ; g , . . . , g k ) = 1.Then the map given by g j g j for j = 1 , . . . , k and h l γ l for l = 1 , . . . , n gener-ates a homomorphism from h g , . . . , g n , h , . . . , h n i onto Γ. To see this, note that if w ( x , . . . , x n ; g , . . . , g k ) ∈ G ∗ F n \ G is such that w ( h , . . . , h n ; g , . . . , g k ) = 1,then w ( γ , . . . , γ n ; g , . . . , g k ) = 1 by our construction. (cid:3) We have thus established the following implications for countable locally finitegroups G : G is omnigenous m G is ∞ -MIF ⇓ G is embeddable as a dense subgroup of Iso( U ∆ ) for | ∆ | ≥ ⇓ G is k -MIF for all k ≥ m G is MIFIt follows from our results that H is ∞ -MIF, along with all groups H P constructedin Subsection 5.2. We now show that being ∞ -MIF is not equivalent to being MIF.To this end, we will use the following recent result of Jacobson, which relies on aconstruction due to Ol’shanskii, Osin and Sapir [20]. Theorem 6.19 ([16]) . There is a MIF locally finite p -group. Jacobson’s examples are explicit, but let us observe that the above result can beshown directly as follows.
UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 31
Proof.
We construct an increasing sequence of finite p -groups { G i } ∞ i =1 such that G = S ∞ i =1 G i is MIF. Let G be any finite p -group and assume that we haveconstructed the sequence { G i } ni =1 of finite p -groups. For every 1 ≤ i ≤ n , let W i = { w i , w i , . . . } be the set of all nontrivial mixed identities of G i of the form w ( g ; g , . . . , g j ) where g , g , . . . , g j ∈ G i . Furthermore, let σ : N × N → N be abijection such that for every s, t ∈ N we have max { s, t } ≤ σ ( s, t ). We find G n +1 such that:(1) G n +1 is a finite p -group,(2) G n ≤ G n +1 , and(3) w σ − ( n ) is not a nontrivial mixed identity of G n +1 .Assume w σ − ( n ) = g g n · · · g k g n k where g , . . . , g k ∈ G n (this is possible since σ − ( n ) = ( s, t ) where s, t ≤ n ). If w σ − ( n ) is not a nontrivial mixed identity of G n , then we can take G n +1 = G n . Otherwise, w σ − ( n ) ∈ W n . Let m ∈ N besuch that n , n , . . . , n k < p m . Then w σ − ( n ) is not a mixed identity of the freeproduct H = G n ∗ Z /p m Z . Since both G n and Z /p m Z are p -groups, by Higman[12] H is a residually p -finite group. Let w ′ σ − ( n ) = [ w σ − ( n ) , h , . . . , h l ] where G n = { , h , h , . . . , h l } . Since H is residually p -finite there exists N E H suchthat [ H : N ] is equal to a power of p and w ′ σ − ( n ) / ∈ N . Now G n +1 = H/N is asdesired. By construction, it is clear that G = S ∞ i =1 G i is MIF. (cid:3) Proposition 6.20.
There is a MIF locally finite group which is not ∞ -MIF. Proof.
By Proposition 6.18 it suffices to construct a MIF locally finite group whichis not omnigenous. Consider a MIF p -group G as constructed above, let G be anyfinite subgroup of G . Then consider some q ≥ p , and embed G into G × Z /q Z . Since G is a p -group, it is clear that no subgroup of G surjects onto G × Z /q Z , so G fails to be omnigenous as wanted. (cid:3) Note that a p -group can not be dense in Iso( U ∆ ) since for every q ∈ N (inparticular for q ∈ N where ( q, p ) = 1) there exists g ∈ Iso( U ∆ ) such that g has anorbit of length q . Therefore, by Proposition 6.20 there are MIF groups that are notdense in Iso( U ∆ ).Let us end this section with another class of potentially relevant examples in theabove implications, namely some MIF highly transitive groups coming from Cantordynamics. Recall that, if ϕ is a minimal homeomorphism of a Cantor space X , itstopological full group [[ ϕ ]] is the group of all homeomorphisms g of X such thatthere is a clopen partition A , . . . , A n of X and integers i , . . . , n with the propertythat g ( x ) = ϕ i k ( x ) for all x ∈ A k . These groups are amenable (by a celebratedresult of Juschenko-Monod [17]), and provide many examples of highly transitivecountable amenable groups (their action on any ϕ -orbit is highly transitive andfaithful). Now if one fixes some x ∈ X and consider the subgroup Γ of all elementsof [[ ϕ ]] which map the positive semi-orbit of x on itself, then Γ is locally finiteand also highly transitive. A basic example is the group of dyadic permutations,obtained for ϕ being the odometer. All these groups are MIF; we would guessthey cannot be embedded as dense subgroups of any Iso( U ∆ ) besides S ∞ . If this were true, then in particular the stabilizers of positive semi-orbits would never be ∞ -MIF. 7. A Characterization of Isomorphism
In this section we turn to a different problem, namely to study how the isomor-phism type of Iso( U ∆ ) is dependent on ∆.7.1. Topological simplicity of
Iso( U ∆ ) . The main theorem of this subsectionis a result about topological simplicity for pointwise stabilizers in Iso( U ∆ ), whichwe will use later. Tent–Ziegler [30], [31] have studied simplicity for many relatedautomorphism groups, and the result here also follows from their techniques. Wegive a direct proof using techniques developed in the previous section. Lemma 7.1.
Let ∆ be any countable distance set. Let A ⊆ U ∆ be a finite set, p a nontrivial 1-type over A , x, y ∈ [ p ], and s ≤ d ( x, A ) an element of ∆. Thenthere exist an integer m and elements x = x, x . . . , x m − , x m = y ∈ [ p ] such that d ( x i − , x i ) = s for all i = 1 , . . . , m . Proof.
Since x and y have the same 1-type over A , we have d ( x, y ) ≤ d ( x, A ). If s ≥ d ( x, A ), then consider the Kat˘etov map f : A ∪ { x, y } → ∆ defined by f ( a ) = (cid:26) d ( x, a ) , if a ∈ A , s, if a ∈ { x, y } .Let x ∈ U ∆ be such that f ( x ) = d ( x , a ) for all a ∈ A ∪ { x, y } . Then x ∈ [ p ]and d ( x, x ) = d ( x , y ) = s . If s < d ( x, A ), then let m be the least integer so that ms ≥ d ( x, A ). Define a metric space A ∪ { x = x , x , . . . , x m − , x m = y } as anextension of A ∪ { x, y } by letting, for all 1 ≤ i < m , d ( x i , a ) = d ( x, a ) for all a ∈ A (i.e. x i ∈ [ p ]), and for all 1 ≤ i ≤ m and j < i , d ( x j , x i ) = ( i − j ) s if i − j < m . Bythe Urysohn property of U ∆ , we can find such x , . . . , x m − ∈ U ∆ . (cid:3) For a finite subset A ⊆ U ∆ , denote by G A the pointwise stabilizer of A , i.e., G A = { g ∈ Iso( U ∆ ) : ∀ a ∈ A g ( a ) = a } . Lemma 7.2.
Let ∆ be any countable distance value set with | ∆ | ≥
2. Let A ⊆ U ∆ be a finite set, p a nontrivial 1-type over A , and distinct x, y ∈ [ p ]. Let g be a nontrivial element of G A . Then there exist h , . . . , h n ∈ G A such that h n gh − n · · · h gh − ( x ) = y . Proof.
Since U ∆ is discerning by Proposition 6.7, there is some z ∈ [ p ] such that d ( g ( z ) , z ) = s >
0. Since U ∆ is homogeneous, there is h ∈ Iso( U ∆ ) with h ( z ) = x .Then d ( h gh − ( x ) , x ) = d ( g ( z ) , z ) = s . Note that s ≤ d ( x, A ) since g fixes A pointwise, so by Lemma 7.1 we may find x = x , x , . . . , x n − , x n = y ∈ [ p ] suchthat d ( x i − , x i ) = s for all i = 1 , . . . , n . By the ultrahomogeneity of U ∆ we can find g , . . . , g n ∈ G A such that g i ( x ) = x i − and g i ( h gh − ( x )) = x i . Letting h i = g i h ,we thus have h i gh − i ( x i − ) = x i for all i = 1 , . . . , n and we are done. (cid:3) Theorem 7.3.
Let ∆ be any countable distance value set. Let A ⊂ U ∆ be a finiteset. Then the pointwise stabilizer G A is topologically simple. UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 33
Proof.
When | ∆ | = 1, Iso( U ∆ ) is isomorphic to S ∞ . It is well-known that S ∞ istopologically simple, and in S ∞ pointwise stabilizers of finite tuples are isomorphicto S ∞ . In the rest of this proof, we assume | ∆ | ≥
2. Let g be a nontrivial elementof G A . We show that the normal subgroup of G A generated by g is dense in G A .Let N be the set of all products of conjugates of g by elements of G A , plus theidentity. It suffices to show that N is dense in G A .For this we show that for any finite A ⊆ U ∆ and any partial isometry ψ suchthat A ⊆ dom( ψ ) and ψ ( a ) = a for all a ∈ A , there is an h ∈ N extending ψ .Without loss of generality assume ψ is not the identity. We prove this by inductionon | dom( ψ ) \ A | . Suppose dom( ψ ) = A ∪ { x , . . . x n } , and let y i = ψ ( x i ) for i = 1 , . . . , n . For n = 1, this is exactly the content of Lemma 7.2. Assume that n > n −
1. Using our inductive hypothesis,we find f ∈ N such that f ( x ) = y , . . . , f ( x n − ) = y n − . Let ϕ = f ψ − . Thendom( ϕ ) = A ∪ { y , . . . , y n − , y n } , ϕ ( a ) = a for all a ∈ A , ϕ ( y i ) = y i for i =1 , . . . , n − ϕ ( y n ) = f ( x n ). Applying Lemma 7.2 to A ′ = A ∪{ y , . . . , y n − } and ϕ , we get f ′ ∈ N such that f ′ ( a ) = a for all a ∈ A , f ′ ( y i ) = y i for i = 1 , . . . , n − f ′ ( y n ) = f ( x n ). Let h = f ′− f . Then h ∈ N , h ( a ) = a for all a ∈ A , h ( x i ) = y i for all i = 1 , . . . , n . (cid:3) Open subgroups and reconstruction.Definition 7.4.
Let ∆ be a distance value set. A ∆ -triangle is a triple ( d , d , d )of elements of ∆ which satisfies the triangle inequalities, i.e. for all distinct i, j, k ∈{ , , } , | d j − d k | ≤ d i ≤ d j + d k . Definition 7.5.
Two distance value sets ∆ , Λ are equivalent if there exists a bijec-tion θ : ∆ → Λ such that, for any ( d , d , d ) ∈ ∆ , we have( d , d , d ) is a ∆-triangle ⇔ ( θ ( d ) , θ ( d ) , θ ( d )) is a Λ-triangle . The following is the main theorem of this subsection, whose remainder is devotedto its proof.
Theorem 7.6.
Let ∆ , Λ be two countable distance value sets. The following areequivalent:(1) ∆ and Λ are equivalent.(2) Iso( U ∆ ) and Iso( U Λ ) are isomorphic (as abstract topological groups).(3) There exists a (continuous) group homomorphism ϕ : Iso( U ∆ ) → Iso( U Λ ) with dense image. The implication (1) ⇒ (2) is easy: starting from U ∆ , and a map θ witnessingthat ∆ and Λ are equivalent, define a new distance d θ on U ∆ by setting d θ ( x, y ) = θ ( d ( x, y )). Then ( U ∆ , d θ ) is isometric to U Λ , and its isometry group has not changedunder this operation.The implication (2) ⇒ (3) follows from the automatic continuity property forIso( U ∆ ), which in turn follows from the fact that it has ample generics (cf. Corollary4.1 of [27]). Next we prove (3) ⇒ (2). We will need to use stabilizers of Iso( U ∆ ) for elements x ∈ U ∆ , which we will denote as G ∆ x for notational simplicity. More generally, forany tuple ¯ x ∈ U n ∆ , we also denote by G ∆¯ x the pointwise stabilizer of Iso( U ∆ ) for ¯ x .We will be using the following theorem which is essentially due to Slutsky (cf.Theorems 4.12, 4.16 and Corollary 4.17 of [26]). Theorem 7.7 (Slutsky [26]) . Let ∆ be any countable distance value set. If A, B are finite subsets of U ∆ , then h G A , G B i is dense in G A ∩ B . Lemma 7.8.
Let V be a proper open subgroup of Iso( U ∆ ). Then there exists anelement x ∈ U ∆ with a finite V -orbit. Proof.
Assume that V is open and has only infinite orbits. We claim that V isdense, so that V = Iso( U S ), since any open subgroup is also closed. Let ¯ a ∈ U n ∆ besuch that G ∆¯ a ⊆ V . By Neumann’s lemma (cf. Corollary 4.2.2 of [13]), there exists v ∈ V such that g (¯ a ) ∩ ¯ a = ∅ . By Theorem 7.7, h G ∆¯ a , G ∆ g (¯ a ) i is a dense subgroup ofIso( U ∆ ). This group is contained in V , and we are done. (cid:3) Lemma 7.9.
Assume that τ is a Hausdorff, separable group topology on Iso( U ∆ )which admits a proper open subgroup. Then τ coincides with the usual Polishgroup topology of Iso( U ∆ ). Proof.
Consider the identity embedding from Iso( U ∆ ) into (Iso( U ∆ ) , τ ). By auto-matic continuity, this embedding is continuous. Let V be a proper τ -open subgroup.Then V as the pullback is open for the usual Polish group topology. By the pre-vious lemma, there is an a ∈ U ∆ with a finite V -orbit A . Using universality andultrahomogeneity of U ∆ , we may find a g ∈ Iso( U ∆ ) such that g ( A ) ∩ A = { a } is asingleton. Then V ∩ gV g − is τ -open, and contained in G ∆ a . Thus G ∆ a is τ -open.Since Iso( U ∆ ) acts transitively on U ∆ , any point stabilizer is thus τ -open. Sincethe point stabilizers form a nbhd subbasis for 1 in the usual Polish group topology,we have that τ is finer than the usual topology, hence they are the same. (cid:3) Proposition 7.10.
Assume that ϕ : Iso( U ∆ ) → S ∞ is a nontrivial group homo-morphism. Then ϕ is a topological group isomorphism onto its image. Proof.
We know that ϕ is continuous by automatic continuity. By Theorem 7.3,Iso( U ∆ ) is topologically simple, so ϕ is also injective. Let τ be the topology onIso( U ∆ ) pulled back via ϕ . Then τ is Hausdorff, separable and admits a properopen subgroup by the nature of the topology of S ∞ . By the previous lemma, τ coincides with the usual Polish topology on Iso( U ∆ ). This means that ϕ is atopological group isomorphism onto its image. (cid:3) Now the implication (3) ⇒ (2) follows immediately from the previous proposi-tion. In fact, let ϕ : Iso( U ∆ ) → Iso( U Λ ) be a nontrivial group homomorphism witha dense image. Since Iso( U Λ ) is a closed subgroup of S ∞ , ϕ satisfies the assumptionof the previous proposition. It follows that ϕ is a topological group isomorphismonto its image, which is closed in Iso( U Λ ). Thus ϕ is onto. Lemma 7.11.
Let V be a proper open subgroup of Iso( U ∆ ). The following areequivalent: UTOMORPHISM GROUPS OF ULTRAEXTENSIVE SPACES 35 (i) There exists a (unique) x ∈ U ∆ such that V = G ∆ x .(ii) V is topologically simple, and is maximal among proper closed subgroupsof Iso( U ∆ ). Proof.
First suppose V = G ∆ x for some x ∈ U ∆ . Then by Theorem 7.3, V istopologically simple. Assume H is a proper closed overgroup of G ∆ x . Let g ∈ H \ G ∆ x .Then g ( x ) = x and G ∆ g ( x ) = gG ∆ x g − ≤ H . By Theorem 7.7, h G ∆ x , G ∆ g ( x ) i , which isa subgroup of H , is dense in Iso( U ∆ ). Since H is closed, H = Iso( U ∆ ).For the converse, assume that V is an open subgroup which is maximal amongproper closed subgroups of G . By Lemma 7.8, there is a ∈ U ∆ with a finite V -orbit, which we denote as A . Now { g ∈ Iso( U ∆ ) : g ( A ) = A } is a closed subgroupof Iso( U ∆ ) containing V . By maximality we know that V = { g ∈ G S : g ( A ) = A } .We claim that A = { a } , and so V = G ∆ a . Otherwise, A = { a } , then G ∆ A ≤ V is aproper normal open subgroup of V , thus V is not topologically simple.For uniqueness, if V = G ∆ x = G ∆ y for x = y , then by Theorem 7.7 h G ∆ x , G ∆ y i isdense in Iso( U ∆ ), which, together with the openness (and therefore closedness) of V , would imply that V is not proper. (cid:3) Note that in the characterization above, we may replace the condition “ V istopologically simple” by “ V has a comeagre conjugacy class” or “ V has a denseconjugacy class”, since these two properties are true for stabilizers of singletons(which have ample generics) while the other maximal open subgroups admit aproper clopen normal subgroup (hence do not have a dense conjugacy class).We can now prove the last remaining implication (2) ⇒ (1) of Theorem 7.6.Assume that ϕ : Iso( U ∆ ) → Iso( U Λ ) is an isomorphism. By the previous lemma,we know that for any x ∈ U ∆ there exists a unique y ∈ U Λ such that ϕ ( G ∆ x ) = G Λ y ,and vice versa. We write y = ψ ( x ); since ϕ is an isomorphism, ψ is a bijection from U ∆ to U Λ .Note that, for any ( x, y ) , ( x ′ , y ′ ) ∈ U ∆ the groups G ∆ { x,y } = G ∆ x ∩ G ∆ y and G ∆ { x ′ ,y ′ } = G ∆ x ′ ∩ G ∆ y ′ are conjugate iff d ( x, y ) = d ( x ′ , y ′ ). Since ϕ is a group isomor-phism which maps point stabilizers to point stabilizers, this implies that ∀ x, y ∈ U ∆ d ( x, y ) = d ( x ′ , y ′ ) ⇔ d ( ψ ( x ) , ψ ( y )) = d ( ψ ( x ′ ) , ψ ( y ′ )) . Given d ∈ ∆, we can then define θ ( d ) ∈ Λ by finding x, y ∈ U ∆ such that d ( x, y ) = d , and then setting θ ( d ) = d ( ψ ( x ) , ψ ( y )). Then θ is a bijection from∆ to Λ. It remains to prove that θ is an equivalence between S and T . Pick( d , d , d ) ∈ ∆ . Then ( d , d , d ) is a ∆-triangle iff there exists x, y, z ∈ U ∆ suchthat d ( x, y ) = d , d ( y, z ) = d and d ( z, x ) = d . Then ( ψ ( x ) , ψ ( y ) , ψ ( z )) ∈ U ,so that ( θ ( d ) , θ ( d ) , θ ( d )) = ( d ( ψ ( x ) , ψ ( y )) , d ( ψ ( y ) , ψ ( z )) , d ( ψ ( z ) , ψ ( x )) is a Λ-triangle. By symmetry, ( d , d , d ) is a ∆-triangle if ( θ ( d ) , θ ( d ) , θ ( d )) is a Λ-triangle.We have thus completed the proof of Theorem 7.6. Open Problems
The first general problem is to characterize all dense countable locally finitesubgroups of Iso( U ∆ ).When | ∆ | = 1, Iso( U ∆ ) is just S ∞ , so we are asking which locally finite groupsare highly transitive. Note that the Hull-Osin dichotomy states that such groupsare either MIF or contain a normal subgroup isomorphic to Alt( N ).For | ∆ | ≥
2, the likely answer to the characterization problem for locally finitegroups is a condition strictly in between ∞ -MIF and MIF, but we do not knowthat the following question has a negative answer. Question 1.
For | ∆ | ≥
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Department of Mathematics, University of North Texas, 1155 Union Circle
E-mail address : [email protected] Department of Mathematics, University of North Texas, 1155 Union Circle
E-mail address : [email protected] Universit´e Paris Diderot, Sorbonne Universit´e, CNRS, Institut de Math´ematiques deJussieu-Paris Rive Gauche, IMJ-PRG, F-75013, Paris, France.
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