aa r X i v : . [ m a t h . L O ] A p r Automorphism groups of universal diversities
Andreas Hallbäck27. april 2020
Abstract
We prove that the automorphism group of the Urysohn diversity is auniversal Polish group. Furthermore we show that the automorphismgroup of the rational Urysohn diversity has ample generics, a denseconjugacy class and that it embeds densely into the automorphismgroup of the (full) Urysohn diversity. It follows that this latter groupalso has a dense conjugacy class.
Diversities were introduced by Bryant and Tupper in [6] and further de-veloped in [7] in order to generalise applications of metric space theory tocombinatorial optimisation and graph theory to the hypergraph setting. Theidea is very simple: Instead of only assigning real numbers to pairs of ele-ments, a diversity assigns a real number to every finite subset of the space.This turns out to generalise metric spaces quite nicely, and in [6, 7] theauthors prove diversity versions of a number of results concerning or usingmetric spaces. The term diversity comes from a special example of a diver-sity that appears in phylogenetics and ecological diversities demonstratingthe broad variety of applications of diversities and of mathematics in generalof course. The precise definition of a diversity is as follows:
Definition 1.1. A diversity is a set X equipped with a map δ , the diver-sity map , defined on the finite subsets of X to R such that for all finite A, B, C ⊆ X we have(D1) δ ( A ) ≥ and δ ( A ) = 0 if and only if | A | ≤ .(D2) If B = ∅ then δ ( A ∪ B ) + δ ( B ∪ C ) ≥ δ ( A ∪ C ) . As an abuse of language we will follow [4, 5] and from time to time referto the diversity map as a diversity as well. Hopefully this confusion of nameswill not cause confusion for the reader.The following observation is useful and is easy to verify.1 emma 1.2. (D2) holds for δ if and only if the following two conditionshold:(D2’) Monotonicity , i.e. if A ⊆ B then δ ( A ) ≤ δ ( B ) .(D2”) Connected sublinearity , i.e. if A ∩ B = ∅ then δ ( A ∪ B ) ≤ δ ( A ) + δ ( B ) . Another general observation to make is that any diversity is automaticallya metric space since the map d ( a, b ) = δ ( { a, b } ) defines a metric. We refer tothis metric as the induced metric . A diversity is complete , respectively separable , if the induced metric is complete, respectively separable. Abijective map f : X → Y between two diversities that preserves the valuesof the diversity map will be called an isoversity . If Y = X we will call f an autoversity or simply an automorphism of X . The group of all autoversitiesof a diversity X is denoted by Aut( X ) .An important observation to make is that for each n ∈ N the diversitymap induces a uniformly continuous map δ n on X n given by δ n ( x , . . . , x n − ) = δ ( { x , . . . , x n } ) . We will use this fact several times, so we include it here for the convenienceof the reader. The proof can be found both in [4] and in [5, Lemma 21].
Lemma 1.3 ([5, Lemma 21]) . Let ( X, δ ) be a diversity and for n ∈ N let δ n denote the map on X n that δ induces. Then δ n is -Lipschitz in eachargument. It follows that for all ¯ x, ¯ y ∈ X n we have | δ n (¯ x ) − δ n (¯ y ) | ≤ X δ ( x i , y i ) . In particular δ n is uniformly continuous. In fact, to ease our notation, we will hardly discern between the maps δ n and the diversity map itself. Hence we will from time to time write δ (¯ a ) foran ordered tuple ¯ a = ( a , . . . , a n ) instead of writing δ ( { a , . . . , a n } ) . It willbe clear from the context what is meant, so this causes no confusion.The interest in diversities from the viewpoint of Polish group theorybegan with the paper [4]. There the authors construct a diversity analogue ofthe Urysohn metric space by adapting Katětov’s construction to the diversitysetting. They call the resulting space the Urysohn diversity , denoted by U ,and show, among other things, that the metric space it induces is the Urysohnmetric space. The existence of such a universal object among diversities givesrise to a plethora of questions concerning its automorphism group, Aut( U ) ,since this group is virtually unstudied. The first main result of this paper isthe following (cf. Theorem 2.11 below): Theorem 1.
Aut( U ) is a universal Polish group. Theorem 2.
Aut( U ) has a dense conjugacy class. Theorem 3.
The automorphism group of the the rational Urysohn diversityhas ample generics.
Here a rational diversity simply means that the diversity map only takesrational values. We will denote the rational Urysohn diversity by U Q . Ofcourse the first thing we need to do, is to show that U Q actually exists. Wedo this by showing that the class of all finite rational diversities is a so-called Fraïssé class . It follows that this class has a
Fraïssé limit and this limit isthe rational Urysohn diversity. Moreover we show that the completion of U Q is the Urysohn diversity, thus providing a new proof of the existence of U . For the convenience of the reader we have included a short introductionto Fraïssé theory in Section 3 where we also define a useful amalgamation ofdiversities that generalises the free amalgamation of metric spaces.Once the existence of U Q is established, it is easy to show that Aut( U Q ) has a dense conjugacy class by applying a theorem of Kechris and Rosendalfrom [18]. Furthermore, we will show that Aut( U Q ) embeds densely into Aut( U ) (cf. Theorem 4.8 below) from which Theorem 2 follows immediately.Afterwards we show that Aut( U Q ) has ample generics . Ample generics isa property with many strong implications such as the automatic continuityproperty , the small index property and the fact that the group cannot be theunion of countably many non-open subgroups. All of these notions will beexplained below. Theorem 3 follows from an extension theorem for diversitiesinspired by a result of Solecki in [26] and another theorem of Kechris andRosendal from [18]. Aut( U ) The first thing we need is to introduce some terminology and definitions from[4]. In that paper, the authors construct the Urysohn diversity by adaptingKatětov’s construction of the ditto metric space to the diversity setting. AsUspenskij did in [30], we shall use this construction to prove that
Aut( U ) is auniversal Polish group. The first thing we need to introduce is the diversityanalogue of Katětov functions. These are the so-called admissible maps.First we let [ X ] <ω denote the finite subsets of X . Definition 2.1.
Let ( X, δ X ) be a diversity. A map f : [ X ] <ω → R is ad-missible if the following holds: i ) f ( ∅ ) = 0 , ( ii ) f ( A ) ≥ δ X ( A ) for every A , ( iii ) f ( A ∪ C ) + δ X ( B ∪ C ) ≥ f ( A ∪ B ) for all A, B, C with C = ∅ , ( iv ) f ( A ) + f ( B ) ≥ f ( A ∪ B ) .The set of all admissible maps on ( X, δ X ) is denoted E ( X ) . The reason why these maps are called admissible is because they definediversity extensions as the lemma below tells us.
Lemma 2.2 ([4, Lemma 2]) . Let ( X, δ ) be a diversity and let f : [ X ] <ω → R .Then f ∈ E ( X ) if and only if for some y the map b δ : [ X ∪ { y } ] <ω → R givenby b δ ( A ) = δ ( A ) , b δ ( A ∪ { y } ) = f ( A ) for A ⊆ X finite, defines a diversity map on X ∪ { y } . Similarly to the metric setting, we can define a diversity map on the setof admissible maps. This is defined as follows.
Definition 2.3 ([4, Page 5]) . Let ( X, δ ) be a diversity. On [ E ( X )] <ω wedefine a map b δ by b δ ( { f , . . . , f n } ) = max j ≤ k sup n f j (cid:0) [ i = j A i (cid:1) − X i = j f i ( A i ) (cid:12)(cid:12)(cid:12) A i ⊆ X finite o whenever n ≥ and b δ ( f ) = b δ ( ∅ ) = 0 . Observe that b δ ( f , f ) = sup B finite | f ( B ) − f ( B ) | . Moreover, as the notation suggests, b δ is a diversity map on E ( X ) . Theorem 2.4 ([4, Theorem 3]) . Let ( X, δ ) be a diversity. Then ( E ( X ) , b δ ) is a diversity and ( X, δ ) embeds into ( E ( X ) , b δ ) via the map x κ x where κ x ( A ) = δ ( A ∪ { x } ) . Unfortunately, just like in the metric setting, E ( X ) need not be sepa-rable even if X is. Therefore we need to restrict ourselves to a subspace of E ( X ) to maintain separability. This is the subspace of the finitely supportedadmissible maps. These are defined as follows:
Definition 2.5.
Let ( X, δ ) be a diversity and let S ⊆ X be any subset. If f ∈ E ( S ) then we define the extension of f to X by f XS ( A ) = inf n f ( B ) + X b ∈ B δ ( A b ∪ { b } ) (cid:12)(cid:12)(cid:12) B ⊆ S finite , [ b ∈ B A b = A o here A ⊆ X is finite. We say that S is the support of f XS .The set of all finitely supported admissible maps on X is the set ofall those h ∈ E ( X ) such that for some finite S ⊆ X and some f ∈ E ( S ) wehave h = f XS . This set will be denoted by E ( X, ω ) . Of course one needs to check that the extension map f XS is in fact admis-sible. We refer the reader to [4, Lemma 6] for the details. It is also easy tocheck that κ x is supported on { x } for any x ∈ X and hence that X embedsinto E ( X, ω ) . Therefore ( E ( X, ω ) , b δ ) is a diversity extension of X . Moreover E ( X, ω ) is separable. Theorem 2.6 ([4, Theorem 9]) . Let ( X, δ ) be a separable diversity. Then ( E ( X, ω ) , b δ ) is a separable diversity as well. One can then iterate this construction and obtain a
Katětov tower on agiven diversity X consisting of a sequence ( X n , δ n ) where X i embeds into X i +1 . The union X ω of all of these diversities turns out to have an analogueof the extension property for metric spaces that characterises the Urysohndiversity. This extension property is defined as follows: Definition 2.7.
A diversity ( X, δ X ) has the approximate extension prop-erty if for any finite subset F ⊆ X , any admissible map f defined on F andany ε > , there is x ∈ X such that | f ( A ) − δ X ( A ∪ { x } ) | ≤ ε for A ⊆ F .If the above holds for ε = 0 , ( X, δ X ) has the extension property . Just like in the metric setting, it turns out that complete diversitieswith the approximate extension property actually has the extension property.Moreover, the completion of any separable diversity with the approximateextension property has the approximate extension property. Therefore wehave:
Proposition 2.8 ([4, Lemmas 16 and 17]) . Suppose ( X, δ X ) is a separablediversity with the approximate extension property. Then its completion hasthe extension property. Furthermore, as mentioned, this property characterises the Urysohn di-versity, meaning that any two
Polish diversities, i.e. with complete and sepa-rable induced metrics, that have the extension property are isomorphic. Thisis one of the main results of [4].
Theorem 2.9 ([4, Theorems 14 and 22]) . Any two Polish diversities bothhaving the extension property are isomorphic. In particular any Polish di-versity with the extension property is isomorphic to the Urysohn diversity.
With these preliminaries we move on to show that
Aut( U ) is a univer-sal Polish group. The strategy to show this is the following: Any Polishgroup G can be embedded into the automorphism group of a separable di-versity ( X, δ X ) . Denote the diversity Katětov tower on X by X ω . Then5 ut( X ) embeds into Aut( X ω ) , which in turn embeds into Aut( U ) becausethe completion of X ω is isomorphic to U . Moreover, these embeddings areall continuous with continuous inverses. Below we elaborate each of thesesteps. First, a lemma: Lemma 2.10.
Let ( X, δ X ) be a separable diversity and let X := E ( X, ω ) denote the diversity of admissible maps on X with finite support. Then Aut( X ) embeds as a topological group into Aut( X ) .Proof. Let
Φ : Aut( X ) → Aut( X ) be the map defined by Φ( g )( f XS ) = f ′ Xg ( S ) where f ′ ( g ( A )) = f ( A ) for A ⊆ S . It is straightforward to check that Φ( g ) is a bijection of X extending g . Moreover we note that Φ( g )( f XS )( A ) = f XS ( g − A ) (1)for any finite A ⊆ X . Using this, it is straightforward to verify that Φ( g ) isan automorphism of X and that Φ is injective. Furthermore, continuity of Φ follows either from Pettis’ theorem (cf. [22]) or simply by a direct argumentusing (1). Finally, continuity of the inverse of Φ can be seen as follows:Suppose Φ( g n ) → Φ( g ) and let x ∈ X be given. We must show that g n ( x ) → g ( x ) . For this, let κ x ∈ X denote the image of x under theembedding of X into X . Then Φ( g n )( κ x ) → Φ( g )( κ x ) which means that sup B finite | Φ( g n )( κ x )( B ) − Φ( g )( κ x )( B ) | → . In particular this is true for B = { gx } . Therefore we have | Φ( g n )( κ x )( { gx } ) − Φ( g )( κ x )( { gx } ) | = | δ ( { g − n gx, x } ) − δ ( { g − gx, x } ) | = δ ( { gx, g n x } ) → where we have used that Φ( h )( κ x ) = κ hx for any h ∈ Aut( X ) , which easilyfollows from (1) above. We conclude that g n → g in Aut( X ) .With this lemma we can now show that Aut( U ) is a universal Polishgroup. Theorem 2.11.
Aut( U ) is a universal Polish group.Proof. First, any Polish group G can be embedded into the isometry group ofits left completion ( G L , d L ) equipped with a left-invariant metric d L (cf. [23]for details on completions of Polish groups). We turn G L into a diversity byusing the diameter diversity, denoted here by δ L , associated to d L , i.e. δ L ( A ) is simply the diameter of A . Then Aut( G L , δ L ) is still just Iso( G L , d L ) so G embeds into this group.Given any separable diversity X we let X denote E ( X, ω ) and for any n ∈ N we let X n denote E ( X n − , ω ) . By X ω we denote the union S X n . In6he lemma above we saw that Aut( X i ) embeds into Aut( X i +1 ) for every i .Hence we obtain a chain of embeddings Aut( X ) ֒ → Aut( X ) ֒ → Aut( X ) ֒ → . . . . Moreover, it is easy to see that the resulting map
Aut( X ) → Aut( X ω ) isan embedding as well. Finally, by [4, Theorem 19] the completion of X ω isisomorphic to U . Therefore it follows from uniform continuity of δ (cf. Lemma1.3) that Aut( X ω ) embeds into Aut( U ) .In conclusion, we have seen that given any Polish group G , we can embed G into Aut( G L , δ L ) , which in turn may be embedded into Aut( U ) using theconstruction above. Hence Aut( U ) is a universal Polish group, which waswhat we wanted. In this section we briefly recall the Fraïssé theory that we will need to con-struct the rational Urysohn diversity as the Fraïssé limit of the class of allfinite rational diversities. We will also define a useful free amalgamation ofdiversities, that generalises the usual free amalgamation of metric spaces.First let us fix some notation. Given two structures A and B in somesignature L , we denote by A (cid:22) B that A embeds into B , i.e. that there is aninjective map f : A → B that preserves the structure on A . Fraïssé classesfor relational signatures are then defined as follows: Definition 3.1.
Let L be a countable relational signature for a first-orderlanguage and let K be a class of finite L -structures. Then K is a Fraïsséclass if it has the following properties: ( i ) (HP) K is hereditary , i.e. if B ∈ K and A (cid:22) B then A ∈ K . ( ii ) (JEP) K has the joint embedding property , i.e. if A, B ∈ K thenthere is some C ∈ K such that A, B (cid:22) C . ( iii ) (AP) K has the amalgamation property , i.e. if A, B, C ∈ K and f : A → B and g : A → C are embeddings, then there is D ∈ K andembeddings h B : B → D and h C : C → D such that h B ◦ f = h C ◦ g .In diagram form: ∀ BA (cid:9) ∃ D ∀ C ∀ f ∀ g ∃ h B ∃ h C We call such a structure D an amalgam of B and C over A . iv ) K contains countably many structures (up to isomorphism), and con-tains structures of arbitrarily large (finite) cardinality. The main reason for studying Fraïssé classes is that any Fraïssé class K has a so-called Fraïssé limit K , which is universal and ultrahomogeneous .Universality in this case means that the class of all finite structures thatembeds into K equals K . This class is the so-called age of K and is denotedAge ( K ) . Ultrahomogeneity is defined as follows: Definition 3.2.
A structure A is ultrahomogeneous if any isomorphismbetween finite substructures of A extends to an automorphism of A . Fraïssé’s theorem then reads:
Theorem 3.3 (Fraïssé, [11, 10], cf. also [13, Theorem 7.1.2]) . Let L be acountable relational signature and let K be a Fraïssé class of L -structures.Then there exists a unique (up to isomorphism) countable structure K satis-fying: ( i ) K is ultrahomogeneous. ( ii ) Age ( K ) = K . The structure K in the theorem above is the Fraïssé limit of K . Us-ing this theorem we will show that there is a universal ultrahomogeneouscountable rational diversity. First we need a couple of definitions and anamalgamation lemma to make it simpler for us to verify the AP for the classof finite rational diversities. Definition 3.4.
Let Y be a set and let X ⊆ Y . A connected cover of X is a collection { E i } of subsets of Y such that X ⊆ S E i and such thatthe intersection graph G defined on { E i } by E i G E j ⇐⇒ E i ∩ E j = ∅ isconnected. Remark 3.5. If ( Y, δ ) is a diversity and X ⊆ Y is finite, then for any finiteconnected cover { E i } of X with each E i finite, we have that δ ( X ) ≤ P δ ( E i ) .This inequality is the main reason why we are interested in connected covers.With this terminology established we can define a free amalgamation oftwo diversities sharing a common sub-diversity. This is a diversity versionof the free amalgamation of metric spaces. Definition 3.6.
Let ( A, δ A ) , ( B, δ B ) and ( C, δ C ) be non-empty finite di-versities such that A = B ∩ C and such that A is a subdiversity of B and C . The free amalgam of B and C over A is the diversity ( D, δ D ) where D = B ∪ C and where δ D ( X ) is given by the minimum over all sums P i δ ( E i ) for { E i : i ≤ n } a connected cover of X such that for each i either E i ⊆ B or E i ⊆ C . emark 3.7. If X has elements from both B and C , the definition of δ D ( X ) above requires the connected cover to include elements from A . Hence, if werestrict δ D to pairs we obtain the usual free amalgamation of metric spaces,i.e. δ D ( b, c ) = min a ∈ A { δ B ( b, a ) + δ D ( a, c ) } for b ∈ B and c ∈ C .Of course it is not necessarily evident that δ D above defines a diversitymap and that both ( B, δ B ) and ( C, δ C ) embeds into ( D, δ D ) . We proceed toverify this. Lemma 3.8. δ D defined above is a diversity map on B ∪ C extending both δ B and δ C . It follows that ( D, δ D ) is an amalgam of B and C over A .Proof. First we show δ D agrees with δ B and δ C on B and C , respectively.Suppose therefore X ⊆ B (the other case is similar). Then { X } is aconnected cover of X so δ D ( X ) ≤ δ B ( X ) . To show equality, let { E i } be a connected cover of X . Then we can assume E i ⊆ B as well. Bymonotonicity of δ B we have δ B ( X ) ≤ δ B ( S E i ) . By connectivity we have δ B ( S E i ) ≤ P δ B ( E i ) . We conclude that δ B ( X ) ≤ δ D ( X ) as well, so in fact δ D ( X ) = δ B ( X ) . In particular δ D ( X ) = 0 if | X | ≤ .Next we show monotonicity. Let therefore X ⊆ Y be given. Then anyconnected cover of Y whose elements are contained in either B or C mustalso cover X . Hence δ D ( X ) ≤ δ D ( Y ) .Lastly we show connected sublinearity. Suppose therefore that X ∩ Y = ∅ .Let { E i } and { F j } be connected covers realising δ D ( X ) and δ D ( Y ) , respec-tively. Then, since X and Y intersect, we have that { E i , F j } is a connectedcover of X ∪ Y whose elements are either contained in B or C . Hence wemust have δ D ( X ∪ Y ) ≤ X δ ( E i ) + X δ ( F j ) = δ D ( X ) + δ D ( Y ) . It follows that δ D is a diversity map.Observe that if the diversities A , B and C above are all rational, thenthe amalgam D will also be a rational diversity. It follows that the class offinite rational diversities, denoted D , has the AP and hence that this classis a Fraïssé class. Proposition 3.9. D is a Fraïssé class with limit U Q . Moreover, the com-pletion of U Q is (isomorphic to) the Urysohn diversity.Proof. We first note that clearly there are rational diversities of arbitrarilylarge finite cardinality. Moreover, up to isomorphism, there are only count-ably many possible finite rational diversities. Hence D has property ( iv ) of9efinition 3.1 above. We verify that D has the three other properties: HP,JEP and AP.HP is clearly satisfied and JEP is also easily seen to hold: If A, B ∈ D then we find some rational N > δ A ( A ) , δ B ( B ) and define δ on the disjointunion A ⊔ B to be δ A on A , δ B on B , and if X ⊆ A ⊔ B contains elementsfrom both A and B , then δ ( X ) = N . It is easy to check that this defines adiversity map. Thus A, B (cid:22) A ⊔ B and of course ( A ⊔ B, δ ) ∈ D .Finally, AP follows from Lemma 3.8 above. To see this, suppose we aregiven A, B, C ∈ D with A (cid:22) B, C via embeddings f B and f C . Then we let D = B ∪ A C be the union of B and C where we identify f B ( A ) with f C ( A ) while leaving B \ f B ( A ) and C \ f C ( A ) disjoint. Identifying A with its imageinside D we now have that A = B ∩ C . Therefore Definition 3.6 applies, andwe obtain an amalgam ( D, δ D ) of B and C over A .We conclude that D is a Fraïssé class and hence that it has a Fraïssélimit: U Q .The "moreover" part follows since U Q has the approximate extensionproperty: If F ⊆ U Q is finite, f ∈ E ( F ) is admissible and ε > , we canfind an admissible map f ′ with rational values such that | f ′ ( A ) − f ( A ) | < ε .Then f ′ defines a rational diversity on F ∪ { z } for some new element z . Byuniversality and ultrahomogeneity of U Q we find x ∈ U Q such that for all A ⊆ F we have | δ ( A ∪ { x } ) − f ( A ) | = | f ′ ( A ) − f ( A ) | < ε . It now follows fromProposition 2.8 above that the completion of U Q has the extension property.Moreover, from Theorem 2.9 it follows that this completion is isomorphic to U as claimed. With the existence of U Q established, we set out to show that Aut( U Q ) and Aut( U ) have a dense conjugacy class. First recall that the conjugacy action of a group on itself is given by g · h := ghg − . Having a dense conjugacyclass is then defined as follows. Definition 4.1.
A Polish group G is said to have a dense conjugacy class if there is some element of G whose orbit under the conjugacy action of G on itself is dense. In [18] Kechris and Rosendal characterise when the automorphism groupof a Fraïssé limit of a class K has a dense conjugacy class. They do this interms of the JEP not for K itself, but for the class of all K -systems . Below, A Ă ∼ B denotes that A is a substructure of B , i.e. that A ⊆ B and that theinclusion is an embedding of A into B . Definition 4.2.
Let K be a Fraïssé class. A K -system consists of a struc-ture A in K together with a substructure A Ă ∼ A and a partial automorphism : A → A . Such a system is denoted A = ( A, ( f, A )) . The class of all K -systems is denoted K p .An embedding of a K -system A = ( A, ( f, A )) into another K -system B = ( B, ( g, B )) is a map Φ : A → B that embeds A into B , A into B and f ( A ) into g ( B ) such that Φ ◦ f ⊆ g ◦ Φ . In diagram form: A f ( A ) (cid:9) B g ( B ) f ΦΦ g Kechris and Rosendal then obtain the following characterisation of havinga dense conjugacy class.
Theorem 4.3 ([18, Theorem 2.1]) . Let K be a Fraïssé class with limit K .Then the following are equivalent: ( i ) There is a dense conjugacy class in
Aut( K ) . ( ii ) K p has the JEP. As an immediate corollary to this, we obtain that
Aut( U Q ) has a denseconjugacy class. Corollary 4.4. D p has the JEP. Hence Aut( U Q ) has a dense conjugacyclass.Proof. Let A = ( A, ( f, A )) and B = ( B, ( g, B )) be D -systems. Then let C = ( C, ( h, C )) be the system where C = A ⊔ B , C = A ⊔ B and h = f ∪ g and where the diversity map δ C is defined to be δ A on A , δ B on B and onsubsets with elements from both A and B , δ C is constant, equal to somerational N > δ A ( A ) , δ B ( B ) . It is easy to check that C is in K p and that both A and B embeds into C .We now wish to show the same thing for the automorphism group ofthe full Urysohn diversity. In order to do that, we will show that Aut( U Q ) embeds densely into Aut( U ) . This will follow from a homogeneity-like prop-erty that the rational and complete Urysohn diversities and metric spacesall share. In short, the property says that if two finite subspaces are close tobeing isomorphic, then we can find an isomorphic copy of one space close tothe other space. In [31] the author refers to this property for metric spacesas pair propinquity . To emphasise that we are working with diversities wewill call this property diversity propinquity . It is defined as follows: Definition 4.5.
Let ( X, δ X ) be a diversity and let ¯ a = ( a i ) i ∈ I and ¯ b = ( b i ) i ∈ I be two tuples of elements of X . For ε > we say that ¯ a and ¯ b are ε -isomorphic if we have | δ X (¯ a J ) − δ X (¯ b J ) | < ε or all J ⊆ I where ¯ b J := ( b j ) j ∈ J . Definition 4.6.
Let ( X, δ X ) be a diversity. We say that ( X, δ X ) has di-versity propinquity if for all ε > there is an ε ′ > such that for all ε ′ -isomorphic tuples ¯ a and ¯ b in X there is some ¯ a ′ isomorphic to ¯ a andpointwise within ε of ¯ b , i.e. max i δ X ( a ′ i , b i ) < ε . We now have the following lemma, the proof of which is modelled on theproof of the corresponding fact for the Urysohn metric space in [24, Lemma6.5].
Lemma 4.7.
The Urysohn diversity and the rational Urysohn diversity bothhave diversity propinquity. Moreover the ε ′ of the definition may simply bechosen to be the given ε .Proof. The proof for the two diversities is the same. In the rational case allone needs to check is that the diversity maps defined below are rational, butsince we are dealing with finite sets this is easily verified.Let n ∈ N and let ε > . The first thing we need, is to introduce somenotation for dealing with the various diversities one may assign to an n -tuple. Thus let D ¯ x be the set of all diversity assignments to the n -tuple ¯ x = ( x , . . . , x n − ) . That is, if we denote { x i : i ∈ I } by ¯ x I , then D ¯ x isthe set of those maps on the power set of ¯ x , ¯ r : P (¯ x ) → R (or into Q for therational case), such that( i ) ¯ r ( ∅ ) = 0 and ¯ r (¯ x I ) = 0 if and only if | I | ≤ ,( ii ) For all I , I and all I = ∅ we have ¯ r (¯ x I ∪ ¯ x I ) ≤ ¯ r (¯ x I ∪ ¯ x I )+¯ r (¯ x I ∪ ¯ x I ) .Of course any ¯ r ∈ D ¯ x corresponds to an element of R n that we will alsodenote by ¯ r . Thus we will use the notation ¯ r ( I ) for ¯ r (¯ x I ) which will beconvenient below.Let now d ∞ denote the maximum metric on D ¯ x , i.e. d ∞ (¯ r, ¯ r ′ ) = sup I ⊆ n {| ¯ r ( I ) − ¯ r ′ ( I ) |} . Next we define another metric on D ¯ x that measures how close together wecan embed two diversities with n elements into a third diversity. To definethis metric, let ¯ y be another n -tuple of elements disjoint from ¯ x . Then define d to be the metric given by d (¯ r , ¯ r ) = inf ¯ r { max i ≤ n { ¯ r ( x i , y i ) } : ¯ r ∈ D ¯ x ∪ ¯ y , ¯ r ↾ ¯ x = ¯ r , ¯ r ↾ ¯ y = ¯ r } where ¯ r , ¯ r ∈ D ¯ x are two different diversity assignments. If ¯ r = ¯ r we set d (¯ r , ¯ r ) = 0 . Of course here ¯ r ↾ ¯ y = ¯ r means that the diversity assignmenton ¯ y given by ¯ r (i.e. ¯ y I ¯ r (¯ x I ) ) is equal to ¯ r ↾ ¯ y . That d is in fact ametric follows from Lemma 1.3. We now claim that d (¯ r , ¯ r ) ≤ d ∞ (¯ r , ¯ r ) .12oreover we claim that this will imply the lemma, but let’s do one thing ata time.Let therefore ¯ r , ¯ r ∈ D ¯ x be two different diversity assignments and set c := d ∞ (¯ r , ¯ r ) . We need to define some ¯ r ∈ D ¯ x ∪ ¯ y such that ¯ r ↾ ¯ x = ¯ r , ¯ r ↾ ¯ y = ¯ r and such that max ¯ r ( x i , y i ) ≤ c . In order to define such an ¯ r ,we need to introduce some notation. Given a subset s = { y i , . . . , y i k } ⊆ ¯ y ,we denote the corresponding set { x i , . . . , x i k } ⊆ ¯ x by s ′ . A collection ofsubsets { E i } of ¯ x or ¯ y is said to be connected if the intersection graph on { E i } forms a connected graph. Let now ¯ r be the diversity assignment wherefor each s ⊆ ¯ x ∪ ¯ y , ¯ r ( s ) is defined to be the minimum over sums of the form P i ¯ r ( E i ) + P j ¯ r ( F ′ j ) + c/ where • E i ⊆ ¯ x , • F j ⊆ ¯ y , • { E i , F ′ j } is connected, • s ∩ ¯ x ⊆ S E i , • s ∩ ¯ y ⊆ S F j .Let us argue why ¯ r is a diversity assignment. If s ⊆ s then any collectionsatisfying the properties of the minimum above for s will also satisfy theproperties for s . Hence ¯ r ( s ) ≤ ¯ r ( s ) . If s ∩ s = ∅ we let { E i } , { F j } realise ¯ r ( s ) and { E l } , { F k } realise ¯ r ( s ) . Then it is easy to check that { E i , E l } , { F j , F k } satisfiy the properties of the minimum for s ∪ s . There-fore ¯ r ( s ∪ s ) ≤ ¯ r ( s ) + ¯ r ( s ) as required. We conclude that ¯ r is in facta diversity assignment. Moreover, we see that sup i ¯ r ( x i , y i ) = c/ since thesingletons { x i } and { y i } satisfy the properties of the minimum. This showsthat d (¯ r , ¯ r ) ≤ c/ < d ∞ ( r , r ) as we claimed.It now follows that both U and U Q have diversity propinquity. Sincethe argument for both diversities is the same, we only provide it for U .Let n ∈ N and ε > be given. Then we claim that ε works as the ε ′ of Definition 4.6. To see this, let ¯ a and ¯ b be n -tuples of elements of U andsuppose sup I ⊆ n | δ (¯ a I ) − δ (¯ b I ) | < ε . Let ¯ r ¯ a and ¯ r ¯ b be the diversity assignmentscorresponding to ¯ a and ¯ b . Then d ∞ (¯ r ¯ a , ¯ r ¯ b ) < ε and so d (¯ r ¯ a , ¯ r ¯ b ) < ε as well.Therefore we find a diversity assignment ¯ r on ¯ a ∪ ¯ b such that restricted to ¯ a we get ¯ r ¯ a and restricted to ¯ b we get ¯ r ¯ b and such that sup i ¯ r ( a i , b i ) < ε .By universality of U we find ¯ a ′ , ¯ b ′ ∈ U n isomorphic as diversities to ¯ a and ¯ b , respectively, such that sup δ ( a ′ i , b ′ i ) < ε . By ultrahomogeneity we find anautomorphism g of U such that δ ( a i , g · b i ) < ε which was what we wanted.We can now show that Aut( U Q ) embeds densely into Aut( U ) . Theorem 4.8.
Aut( U Q ) continuously embeds into Aut( U ) as a dense sub-group.Proof. Recall that U Q is dense in U by Proposition 3.9. Furthermore, sincethe diversity map defines uniformly continuous maps on finite powers of U g ∈ Aut( U Q ) uniquely extendsto an autoversity of U . Thus Aut( U Q ) embeds into Aut( U ) . Moreover, thisembedding must be continuous by Pettis’ theorem (cf. [22]).We move on to show that Aut( U Q ) is dense in Aut( U ) . Recall that thetopology on Aut( U ) is the pointwise convergence topology generated at theidentity by sets of the form U ¯ a,r := { g ∈ Aut( U ) : δ ( g (¯ a ) , ¯ a ) < r } for a tuple ¯ a = ( a , . . . , a n ) of elements of U and some r > . In each ofthese sets we must find an autoversity extending a rational autoversity. Lettherefore U ¯ a,r be given and let g ∈ U ¯ a,r . Set ε := r − max i δ ( g ( a i ) , a i ) > andfind a tuple ¯ x of n elements of U Q with max i δ ( a i , x i ) < ε/ . Let moreover ¯ y be an n -tuple of elements of U Q such that max i δ ( y i , g ( x i )) < ε/ (4 n ) . Notethat g ( x i ) is not necessarily in U Q - hence this approximation. By Lemma 1.3it follows that (¯ y, δ ) is ε/ -isomorphic to (¯ x, δ ) and therefore, by propinquityand ultrahomogeneity of U Q , we find an autoversity g of U Q such that max i δ ( g ( x i ) , y i ) < ε/ . We claim that the extension of g to U is in U ¯ a,r .Let therefore ˜ g denote this extension. We have δ ( a i , ˜ g ( a i )) ≤ δ ( a i , g ( a i )) + δ ( g ( a i ) , g ( x i )) + δ ( g ( x i ) , y i ) + δ ( y i , ˜ g ( x i ))+ δ (˜ g ( x i ) , ˜ g ( a i )) < r − ε + ε/ ε/ (4 n ) + ε/ ε/ ≤ r. We conclude that ˜ g ∈ U ¯ a,r and hence that Aut( U Q ) is a dense subgroup of Aut( U ) .As an immediate corollary we obtain that Aut( U ) has a dense conjugacyclass. Corollary 4.9.
Aut( U ) has a dense conjugacy class.Proof. This follows easily since
Aut( U Q ) has a dense conjugacy class and isdensely embedded into Aut( U ) . We move on to our next endeavour: Ample generics of
Aut( U Q ) . Let usbegin by defining this notion. Definition 5.1.
A Polish group G has ample generics if for each n ∈ N there is a comeagre orbit for the diagonal conjugacy action of G on G n definedby g · ( g , . . . , g n ) = ( gg g − , . . . , gg n g − ) . Examples.
The following groups have ample generics. • The automorphism group of the random graph, [15], cf. also [14]. • The free group on countably many generators, [8]. • The group of measure preserving homeomorphisms of the Cantor space,[18]. • The automorphism group of N <ω seen as the infinitely splitting regularrooted tree, [18]. • The isometry group of the rational Urysohn metric space, [26].In [18], where these examples are taken from, Kechris and Rosendal show,as mentioned, a number of powerful consequences of ample generics. We havecollected the most important ones in the theorem below.
Theorem 5.2.
Let G be a Polish group with ample generics. Then G hasthe following properties:(1) Automatic continuity property , i.e. any homomorphism from G to a separable group H is continuous.(2) Small index property , i.e. any subgroup of G of index < ℵ is open.(3) G cannot be the union of countably many non-open subgroups. Another important result from [18] is a characterisation of when theautomorphism group of a Fraïssé limit has ample generics in terms of the JEPand a weak form of the AP. This weaker form of amalgamation is, naturallyenough, called the weak amalgamation property (or
WAP for short) and isdefined as follows:
Definition 5.3.
Let K be a class of finite structures. Then K has the weakamalgamation property (WAP) if for any A ∈ K there is A ∈ K and anembedding f : A → A such that whenever g B : A → B and g C : A → C areembeddings into B, C ∈ K , there is D ∈ K and embeddings h B : B → D and h C : C → D such that h B ◦ g B ◦ f = h C ◦ g C ◦ f . In diagram form: ∀ B ∃ A (cid:9) ∃ D ∀ CA ∃ f ∀ g B ∀ g C ∃ h B ∃ h C However, it is not the Fraïssé class itself that must have the WAP and theJEP in order for the automorphism group to have ample generics, but theclass of so-called n -systems for n ∈ N . This is the class of finite structures A , together with n substructures of A and n partial automorphisms of A defined on these substructures. The exact definition is as follows:15 efinition 5.4. Let K be a Fraïssé class and let n ≥ be given. An n -system in K consists of a structure A in K together with n substructures A , . . . , A n Ă ∼ A and n partial automorphisms f : A → A, . . . , f n : A n → A .We denote such a system by A = ( A, ( f i , A i ) i ≤ n ) . The class of all n -systemsin K is denoted K np .An embedding of an n -system A = ( A, ( f i , A i )) into another n -system B = ( B, ( g i , B i )) is a map Φ : A → B that embeds A into B , A i into B i and f i ( A i ) into g i ( B i ) and such that Φ ◦ f i ⊆ g i ◦ Φ for each i ≤ n . In diagramform, for each i ≤ n : A i f i ( A i ) (cid:9) B i g i ( B i ) f i ΦΦ g i Note that since we have defined embeddings between n -systems, we cantalk about the WAP and the JEP for the class K np . In [18] Kechris andRosendal show that these two properties for K np actually characterise amplegenerics. Theorem 5.5 ([18, Theorem 6.2]) . Let K be a Fraïssé class and let K denoteits limit. Then the following are equivalent: ( i ) Aut( K ) has ample generics. ( ii ) For all n ≥ , K pn has the JEP and the WAP. Using this theorem, we can show that
Aut( U Q ) has ample generics. Theproof uses the following extension result inspired by Solecki’s [26, Theorem2.1]. Theorem 5.6.
Let ( A, δ A ) be a finite diversity. Then there is a finite diver-sity ( B, δ B ) containing A as a subdiversity and such that any partial isover-sity of A extends to a full autoversity of B .Proof. We can without loss of generality assume that | A | ≥ . Let D be theset D := { ( δ A ( X ) , | X | ) : X ⊆ A } \ { (0 , , (0 , } . That is, D is all pairs of the non-zero values of δ A together with the size ofthe set the value comes from. For each ( r, n ) ∈ D we let R ( r,n ) be an n -aryrelation symbol and let L be the (finite) relational language consisting ofthese symbols. We call a tuple of elements of D , α = (( r , n ) , . . . , ( r k , n k )) ,a configuration if we have that k X i =1 r i < r and k X i =1 ( n i − ≥ n . Given a configuration α = (( r i , n i )) let Y , Y , . . . , Y k be sets such that16 i ) Y ⊆ S ki =1 Y i ,( ii ) | Y i | = n i ,( iii ) The intersection graph on { Y , . . . , Y k } is connected.We call such a family of sets { Y i : 0 ≤ i ≤ k } an α -family . Note that since α is a configuration it is always possible to find at least one α -family. Moreover,we note that there are only finitely many α -families.Given a configuration α = (( r i , n i ) : 0 ≤ i ≤ k ) and an α -family β = { Y i } we define an L -structure M α,β with universe S Y i by declaring that the onlyrelations satisfied by M α,β are the following: M α,β (cid:15) R ( r i ,n i ) ( σ ( Y i )) for any permutation σ of the elements of Y i (considered here as an orderedtuple and not just a set). The permutations merely ensure that the relationsare symmetric and do not really serve any other purpose. Let T denote thefamily of all M α,β for all configurations α and all α -families β . Note that T is finite.Any diversity ( X, δ X ) is naturally also an L -structure by letting X (cid:15) R ( r,n ) ( Y ) ⇐⇒ δ X ( Y ) = r and | Y | = n for any finite subset/tuple Y of elements of X , meaning that we are con-sidering Y as a subset on the right-hand side and as an ordered tuple onthe left-hand side. Note, however, that the order we choose on Y is notimportant. Observe that any partial autoversity of X is also a partial auto-morphism of X as an L -structure.Given a configuration α = (( r i , n i ) : 0 ≤ i ≤ k ) and an α -family β = { Y , . . . , Y k } we now claim that there are no weak homomorphisms h : M α,β → X , i.e. there is no map h such that if we have M α,β (cid:15) R ( r,n ) ( Y ) then X (cid:15) R ( r,n ) ( h ( Y )) . To see this, suppose h was such a map. Then since β is an α -family we would have that h ( Y ) ⊆ S ki =1 h ( Y i ) . By the monotonicityof the diversity map this would imply δ X ( h ( Y )) ≤ δ X ( S ki =1 h ( Y i )) . Sincethe intersection graph on { Y i : 1 ≤ i ≤ k } is connected, it would follow thatthe intersection graph on the images { h ( Y i ) : 1 ≤ i ≤ k } was connectedtoo. Therefore we could find Y i such that the intersection graph on thefamily { h ( Y i ) : i = i } remained connected (this is always possible for finiteconnected graphs). Hence, by the triangle inequality for the diversity map,we would have that δ X ( k [ i =1 h ( Y i )) ≤ δ X ( h ( Y i )) + δ X ( [ i = i h ( Y i )) . By induction this would imply that δ X ( k [ i =1 h ( Y i )) ≤ k X i =1 δ X ( h ( Y i )) . h was a weak homomorphism and β is an α -family, we wouldhave that X (cid:15) R ( r i ,n i ) ( h ( Y i )) and hence δ X ( h ( Y i )) = r i for each i . Thus wewould have that r = δ X ( h ( Y )) ≤ k X i =1 δ X ( h ( Y i )) = k X i =1 r i < r , which is, of course, a contradiction.An L -structure with this property, i.e. such that there are no weak ho-momorphisms from any M α,β into it, is said to be T -free .Next, let ( U , δ U ) denote the Urysohn diversity. By universality we canembed ( A, δ A ) into ( U , δ U ) and by ultrahomogeneity we can extend eachpartial isoversity of A to an autoversity of U . Note that since D includes allvalues of δ A , any partial L -automorphism of A is a partial isoversity of A (and vice versa of course). Hence we can extend any partial L -automorphismof A to a full L -automorphism of U viewed as an L -structure. By Herwigand Lascar’s [12, Theorem 3.2] we can find a finite T -free L -structure C containing A as a substructure such that each partial L -automorphism of A extends to a full one of C . Given a partial automorphism of A , g , wewill denote its extension to C by ˜ g . By convention, we will assume that theempty map is extended to the identity map.A sequence of subsets e , . . . , e k ⊆ C is called a connection if the inter-section graph on { e i } is connected and if there are ( r , n ) , . . . , ( r k , n k ) ∈ D such that for each i ( i ) | e i | = n i ,( ii ) C (cid:15) R ( r i ,n i ) ( σ ( e i )) for any permutation σ of e i considered as an orderedtuple.Given c, c ′ ∈ C we say that they are connected if there is a connection e , . . . , e k such that c ∈ e and c ′ ∈ e k . Let B ⊆ C be those b ∈ C thatare connected to some a ∈ A . Note that any b ∈ B is connected to all a ∈ A since if b is connected to a ′ ∈ A via the connection e , . . . , e k , then { a, a ′ } , e , . . . , e k is a connection between a and b . Moreover, clearly A ⊆ B since given a ∈ A we pick a ′ ∈ A \ { a } (remember that we have assumed | A | ≥ ) and see that { a, a ′ } is a connection between a and a ′ .Given a partial automorphism of A , g , we claim that ˜ g ( B ) = B . To showthis, it is enough to show that ˜ g ( B ) ⊆ B since we are dealing with finite sets.If g is the empty map, then we extend it to the identity and there is nothingto show. If not, pick a in the domain of g and let b ∈ B . Then, as notedabove, we can find a connection between a and b . Let e , . . . , e k denote sucha connection. Since ˜ g is an automorphism it follows that ˜ g ( e ) , . . . , ˜ g ( e k ) is a connection between ˜ g ( a ) = g ( a ) ∈ A and ˜ g ( b ) , because clearly theintersection graph on { ˜ g ( e i ) : 1 ≤ i ≤ k } is connected and ˜ g preserves therelations. We conclude that ˜ g ( b ) ∈ B as we claimed.18efine now a diversity δ B on B by letting δ B ( X ) be if | X | ≤ andotherwise letting it be the minimum over all sums P ki =1 r i where for someconnection e , . . . , e k with e i ⊆ B we have C (cid:15) R ( r i , | e i | ) ( σ ( e i )) for any per-mutation of e i considered as a tuple and where X ⊆ S e i . Note that since X ⊆ B each element of X is connected to the same element of a ∈ A . Hencethe collection of all these connections, one for each x ∈ X , forms a connec-tion containing X . Therefore this minimum is not taken over the empty setand hence δ B is well-defined.We must argue why δ B is a diversity map, i.e. we must show that foreach X, Y, Z ⊆ B with Z = ∅ we have δ B ( X ∪ Y ) ≤ δ B ( X ∪ Z ) + δ B ( Z ∪ Y ) . Let { e i } and { r i } realise δ B ( X ∪ Z ) and let { f j } and { s j } realise δ B ( Z ∪ Y ) .Then since Z ⊆ X ∪ Z ⊆ S e i and Z ⊆ Z ∪ Y ⊆ S f j it follows that theintersection graph on { e i } ∪ { f j } is connected. Hence { e i } ∪ { f j } forms aconnection. Moreover, this connection covers X ∪ Y . Therefore we havethat δ B ( X ∪ Y ) ≤ P r i + P s i = δ B ( X ∪ Z ) + δ B ( Z ∪ Y ) as we wanted.Moreover, if g is a partial isoversity of A it follows that the extension ˜ g andits inverse ˜ g − maps connections to connections. Therefore we must havethat ˜ g : B → B is an autoversity with respect to δ B .Finally, we must show that δ B extends δ A . First of all it is clear thatwe must have δ B ( X ) ≤ δ A ( X ) for all X ⊆ A since { X } is itself a con-nection covering X as C (cid:15) R ( δ A ( X ) , | X | ) ( X ) . Suppose next that we have δ B ( X ) < δ A ( X ) . Then let e , . . . , e k be a connection with correspondingvalues r , . . . , r k witnessing this, i.e. P r i < δ A ( X ) . It follows that ( δ A ( X ) , | X | ) , ( r , | e | ) , . . . , ( r k , | e k | ) is a configuration because X ⊆ S e i so | X | ≤ | e | + | e \ e | + . . . + | e k \ ( k − [ i =1 e i ) |≤ X ( | e i | − , where the second inequality follows since the first sum counts each elementof S e i exactly once and the second sum counts each element at least once,given that the intersection graph on { e i } is connected. If we denote thisconfiguration by α then { e i , X : 1 ≤ i ≤ k } is an α -family, β . Therefore M α,β is in T and the identity map on M α,β is a weak homomorphism into C . This contradicts that C is T -free. We conclude that δ B ( X ) = δ A ( X ) .All in all we have extended each partial isoversity of A to an autoversityof ( B, δ B ) , and this diversity contains ( A, δ A ) as a subdiversity. This waswhat we wanted. 19e are now ready to prove that Aut( U Q ) has ample generics. Theorem 5.7.
Aut( U Q ) has ample generics.Proof. We show that for each n ∈ N , the class D np of n -systems in D has theWAP. Since it clearly has the JEP, it follows from Kechris and Rosendal’sTheorem 5.5 above that Aut( U Q ) has ample generics.Let therefore A = ( A, ( f i , A i )) be an n -system in D np . By the extensiontheorem above we find a rational diversity B containing A where the partialisoversities of A extend to autoversities of B . Let ˜ f i denote the extensionof f i to B and let B denote the resulting n -system in D np . Suppose nowthat we are given n -systems C = ( C , ( g i , C i )) and C = ( C , ( g i , C i )) andembeddings Φ j : B → C j , j = 1 , . We need to construct an amalgam of C and C over B . To do that we apply the extension theorem to both C and C and get e C and e C where the partial isoversities g i and g i extend to fullautoversities ˜ g i and ˜ g i of e C and e C , respectively. Denote the resulting n -systems by e C and e C . As usual we can assume that B = e C ∩ e C . Thereforewe can construct the free amalgam D of e C and e C over B . Moreover, we candefine an n -system using D by letting h i be ˜ g i ∪ ˜ g i , which is an autoversity of D . Denote the resulting n -system by D . In diagram form for the n -systems: C B (cid:9) C e C e C DA and in diagram form for j = 1 , and each i : BBA i f i ( A i ) (cid:9)(cid:9) C ij g ij ( C ij ) e C j e C j (cid:9) (cid:9) DD ˜ f i g ij ˜ g ij h i f i It is easy to check that D is an amalgam of C and C over B . Therefore,we conclude that D np has the WAP and hence that Aut( U Q ) has amplegenerics. References [1] Howard Becker and Alexander S. Kechris.
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