Ǧan Do

Julien Bilge

Melleray
email: melleray@math.univ-lyon1.fr

ELEMENTS OF FINITE ORDER IN AUTOMORPHISM GROUPS OF HOMOGENEOUS STRUCTURES

We study properties of the automorphism groups of Fraïssé limits of classes with certain strong amalgamation properties, including classes with the free amalgamation property and classes of metric spaces. We discuss conditions on a Fraïssé class K that imply that the automorphism group G K of its limit admits generic elements of order n for all n, and show that, for many such K, any element of G K is a product of 4 conjugates of the generic element of order n for all n ≥ 2. Our constructions enable us to compute the Borel complexity of the relation of conjugacy between automorphisms of the Henson graphs, and to obtain some new results about the structure of the isometry group of the Urysohn space and the Urysohn sphere.

When Q is a metric value set, and (X, d) is a metric space whose distance function takes its values in Q, we say that (X, d) is a Q-metric space.

It is a well-known fact that one may turn finite Q-metric spaces into a Fraïssé class in a countable relational language (by adding a binary predicate for each possibel value of the distance function) and the limit of this Fraïssé class is a countable Q-metric space, which we denote by U Q and which is characterized among countable Q-metric spaces by the fact that it contains a copy of every finite Q-metric space and every isometry between finite subsets extends to an isometry of U Q . Actually, in the case Q = Q is the set of rational numbers, this space was constructed some thirty years before Fraïssé developed his famous theory [F], in a remarkable paper by Urysohn [U1].

A theorem due to Herwig and Lascar [HL], which was recently refined by Solecki [S1], led us to introduce the following definition.

INTRODUCTION

In recent years, countable homogeneous structures and their automorphism groups have attracted a considerable amount of attention, see for instance the recent survey [M1]. This domain of research features an attractive interplay between model theory and descriptive set theory. For instance, Kechris and Rosendal [KR] characterized the Fraïssé classes K such that the automorphism group G of the Fraïssé limit of K has ample generics, i.e such that there are comeager diagonal conjugacy classes in G n for all n (earlier, Ivanov [I] had obtained similar results in a slightly different setting). This notion is a good example of the aforementioned interplay: it first appeared in a paper by Hodges, Hodkinskon, Lascar and Shelah about automorphism groups of certain first-order countable structures [HHLS], then was generalized and appplied to the context of general Polish groups by Kechris and Rosendal [KR]. Another situation where Baire category and modeltheoretic techniques were used conjointly to great effect is the recent work of MacPherson and Tent, who proved the following result.

Theorem (MacPherson-Tent [MT]). Let K be a Fraïssé class in a countable relational language, and assume that K has the free amalgamation property. Let K denote the Fraïssé limit of K, and assume that Aut(K) acts transitively on K but is not the full permutation group of K. Then, for any g ∈ Aut(K) \ {1}, every other element of Aut(K) is a product of at most 32 conjugates of g.

Informally, saying that K has the free amalgamation property means that, whenever A, B, C belong to K and A is a substructure of both B and C, then the structure X with universe B ∪ C, B ∩ C = A, and relations exactly those that come from B and C (and no others -i.e, a tuple that meets both B \ A and C \ A does not satisfy any relation in the language of K) belongs to K.

In particular, whenever the assumptions of the above theorem are satisfied, the automorphism group of K is simple. Very recently, Tent and Ziegler [TZ] improved the bound above from 32 to 16 and, more interestingly, proved similar results for the group of isometries of the unbounded Urysohn space; for instance, they proved that the quotient of this group by the normal subgroup of bounded isometries is a simple group. At the moment, their techniques are not known to adapt to the bounded rational Urysohn space, and it is an open question whether the isometry group of that space is simple.

In this paper, we are particularly interested in the properties of elements of finite order in automorphism groups of Fraïssé limits of certain Fraïssé classes, including those with the free amalgamation property. In this setting, we adapt a construction originally due to Katětov [K1] and later used by Uspenskij [U2], to obtain the following basic result (below K ω denotes the set of countable structures whose age belongs to K).

Theorem. Let K be a Fraïssé class in a countable relational language, assume that K has the free amalgamation property, and denote its limit by K. Then, for any X ∈ K ω , there exists a substructure X of K isomorphic to X such that every automorphism of X extends to an automorphism of K; the extension map may be taken to be a continuous group homomorphism from Aut( X) to Aut(K). Thus, Aut(K) is universal for groups of the form Aut(X), X ∈ K ω ; this is a positive partial answer to a general problem posed by E. Jaligot in [J], which we quote now:

"Assume that C is a class of structures in a fixed language, with universal model M, and let G denote the class of groups of automorphisms of structures in C. Then is it true that the group of automorphisms Aut(M) is universal for the class of groups G? And, if the answer is positive, in what sense is it universal? "

The approach used in our construction leads to the following result.

Theorem. Let K be a Fraïssé class in a countable relational language, assume that K has the free amalgamation property, and denote its limit by K. Then for every integer n ≥ 2 and any X ∈ K ω , there exists an automorphism φ of K such that φ n = 1 and the set of fixed points of φ is isomorphic (as a substructure of K) to X.

This theorem has consequences on the Borel complexity of the conjugacy relation in the automorphism groups of the random graph and the Henson graphs.

Theorem. Let Γ be either the random graph or the universal homogeneous K m -free graph for some m ≥ 3, let G be the automorphism group of Γ and let n ≥ 2 be an integer. Then the equivalence relation induced by the conjugacy action of G on {g ∈ G : g n = 1} is universal (in the sense of Borel reducibility) for relations induced by a Borel action of S ∞ .

A consequence of the above result is that, if Γ is one of the graphs above, then conjugacy in Aut(Γ) is universal for relations induced by a Borel action of S ∞ ; in the case of the random graph, this is a recent result due to Coskey, Ellis and Schneider [CES].

The second type of Fraïssé class that we are particularly interested in is Q-metric spaces, whose definition we give now.

Definition.

A metric value set is a countable subset of R + containing 0 and which is stable under addition, or the intersection of such a subset with an interval [0, M].

Definition. Let K be a class of finite structures in a relational language L. We say that K has the isomorphic extension property (IEP) if for any A ∈ K there exists B ∈ K such that A ≤ B, and a map E from the set of partial isomorphisms of A to the set of global isomorphisms of B such that (i) For any partial isomorphism g of A, E(g) extends g.

(ii) For any A ′ ≤ A, E induces a homomorphism from Aut(A ′ ) to Aut (B).

Solecki's refinement of the Herwig-Lascar theorem presented in [S1] shows that, whenever L is a finite relational language and T is a finite set of L-structures such that the class K T of T-free structures is a Fraïssé class, K T has the isomorphic extension property. Using his technique from [S2], Solecki also showed in [S1] that, whenever Q is a metric value set, the class of Q-metric spaces has the isomorphic extension property.

Applying standard descriptive set theoretic techniques, we show that, if K is a Fraïssé class with the (IEP) and limit K, then for any k-tuple n of integers there exists a generic element in

Ω n(K) = { ḡ ∈ Aut(K) k : ∀i ∈ {1, . . . , k} g n i i = 1}
, that is, some ḡ has a comeager orbit under the diagonal conjugacy action of Aut(K) on Ω n(K). We apply this to obtain the following.

Theorem. Assume that T is a finite set of L-structures in a finite relational language L, that K is the class of finite T-free structures, and that K is a Fraïssé class with the free amalgamation property; or that K is the class of finite Q-metric sets for some metric value set Q. Denote by G the automorphism group of the Fraïssé limit of K, endowed with the permutation group topology.

Then, for any 4-tuple of integers n bigger than 2, any g ∈ G may be written in the form g = h 1 h 2 h 3 h 4 , where h i is a generic element of order n i .

Note that the results of MacPherson-Tent-Ziegler show that, in most classes with the free amalgamation property, for any element g = 1 ∈ Aut(K) any other element may be written as a product of 16 conjugates of g and g -1 ; Tent-Ziegler proved that the same thing holds in the isometry group of the rational Urysohn space whenever g is an unbounded isometry, i.e. sup d(x, g(x)) = +∞. In that respect, the results of MacPherson-Tent-Ziegler are much stronger than the above theorem; however, our proof is different and (we think) interesting in its own right, and it works in some cases where the MacPherson-Tent-Ziegler machinery is not known to work at the moment, especially classes of Q-metric spaces where Q is bounded. It can also be of interest to know that there exist generic elements of order n, a fact that does not appear in the MacPherson-Tent-Ziegler papers.

In the last section of the article, we turn to applications to the isometry group of the continuous Urysohn space U (and its bounded counterpart, the Uryohn sphere U 1 ).

Theorem. Every element of Iso(U) is a commutator. For every integer n there exists a generic element g n in {g ∈ Iso(U) : g n = 1}, and every element of Iso(U) is a product of four conjugates of g n .

This result also holds for U 1 instead of U. Our last result is the product of a failed attempt to prove that a generic element of Iso(U) embeds in a flow, i.e for a generic element g ∈ Iso(U) there exists a continuous homomorphism F : (R, +) → Iso(U) such that g = F(1). We do not answer that question, nor do we even know whether a generic element of Iso(U) admits a square root; but we do know what the situation is for generic elements of order n in Iso(U), obtaining a surprisingly strong answer in the bounded case (by d u below we mean the uniform metric on Iso(U 1 )).

Theorem. Let n be an integer. Then a generic element of order n in Iso(U) embeds in a flow. In the case of Iso(U 1 ), a generic element of order n embeds in a flow which is n-Lipschitz from (R, +) to (Iso(U 1 ), d u ).

A corollary of this, and our result that every element of Iso(U 1 ) is a product of four conjugates of the generic element of order 2, is the following result.

Corollary. The topological group (Iso(U 1 ), d u ) is path connected. Actually, for any φ ∈ Iso(U 1 ) there exists a path (φ t ) t∈ [0,1]

such that φ 0 = id, φ 1 = φ, and d u (φ t , φ s ) ≤ 8|t -s| for all t, s ∈ [0, 1].
Acknowledgements This paper is part of the first author's Ph.D thesis [B], which was supervised by E. Jaligot and the second author. Part of the second author's work on this project was done during a visit in Bogotá that was made possible by the ECOS Nord program (action ECOS Nord C10M01).; his research was also partly funded by ANR project GROUPOLOCO. Several conversations with Alexander Berenstein on the topic of this article were very helpful to us.

BACKGROUND

2.1. Fraïssé classes and limits. We refer to [H] for background on first-order logic; in this article, all languages are relational, (at most) countable and contain the equality symbol. We quickly recall the terminology of Fraïssé classes. Definition 2.1. Let L be a language, and M be a L-structure. The age of M is the collection of all finite L-structures which are isomorphic to a substructure of M. Definition 2.2. Let L be a language and K be a class of finite L-structures. One says that:

(i) K is countable if K has countably many members up to isomorphism.

(ii) K is hereditary if K is closed under embeddings, i.e if A ∈ K and B is a L-structure embedding in A then B ∈ K.

(iii) K has the joint embedding property if for any A, B ∈ K there exists C ∈ K such that both A, B embed in C. (iv) K has the amalgamation property if for any A, B, C ∈ K, and any embeddings i : A → B, j : A → C, there exists D ∈ K and embeddings β :

B → D, γ : C → D such that β • i = γ • j. A class satisying the four properties above is called a Fraïssé class.
It is clear that, whenever M is a countable L-structure, its age satisfies the first three properties above. The last one, however, is not satisfied in general.

Definition 2.3. Let L be a language, and M be a countable L-structure. We say that M is homogeneous if any isomorphism between finite substructures of M exends to an automorphism of M.

Whenever M is homogeneous, its age satisfies the amalgamation property, as was first shown by Fraïssé. The converse is true, in the following sense.

Theorem 2.4 (Fraïssé [F]). Let L be a language and K a Fraïssé class of L-structures. Then there exists a unique (up to isomorphism) L-structure K which is homogeneous and such that age(K) = K. This structure is called the Fraïssé limit of K.

Note that the Fraïssé limit M of a class K of L-structures may be characterized, up to isomorphism, by the following universal property (sometimes called Alice's Restaurant axiom):

For any finite A ⊆ M, any B ∈ K and any embedding j : A → B, there exists B ⊆ M such that A ⊆ B and an isomorphism φ : B → B such that φ • j = i, where i denotes the inclusion map from A to B.

Automorphism groups of countable structures.

Definition 2.5. Let L be a language and M be a countable L-structure. We denote its automorphism group by Aut(M), and endow it with the permutation group topology, whose basic open sets are of the form

{g ∈ Aut(M) : g(m 1 ) = n 1 , . . . , g(m k ) = n k }
where k is an integer and m i , n i are elements of M for all i ∈ {1, . . . , k} .

The topology defined above is a group topology (i.e, the group operations are continuous), and it is Polish, which means that there exists a complete separable metric inducing the permutation group topology on Aut(M).

A particularly important example is the case when L is reduced to the equality symbol, and M is a countable infinite set. Then its automorphism group is the group of all permutations of a countable infinite set; we denote this group by S ∞ . Closed subgroups of S ∞ are interesting from the decriptive set theoretic point of view for many reasons and are intimately linked with automorphism groups of countable structures because of the following folklore result.

Theorem 2.6 (see e.g. [BK]). Let L be a language and M a countable L-structure. Then Aut(M), endowed with its permutation group topology, is isomorphic (as a topological group) to a closed subgroup of S ∞ .

Conversely, for any closed subgroup G of S ∞ , there exists a language L and a homogeneous L-structure M such that G is isomorphic (as a topological group) to Aut(M) endowed with its permutation group topology.

In the remainder of this article, whenever we mention a topological property of Aut(M), it is implicitly assumed that we are talking about the permutation group topology. The fact that Aut(M) is a Polish group will be essential to us, because it enables us to use Baire-category techniques. We refer the reader to [K2], [G] and references therein for informations on Polish groups and spaces as well as the Baire-category vocabulary and techniques.

FREE AMALGATION AND CONSEQUENCES ON THE STRUCTURE OF THE AUTOMORPHISM GROUP

3.1. Free amalgams. We quickly recall the definition of a free amalgam.

Let L be a language, A a L-structure, (X j ) j∈J a family of L-structures and f j : A → X j an embedding from A to X j for all j. We define a L-structure Y, which we will call the free amalgam of the family (X j ) over the embeddings f j , or less formally the free amalgam of the X j 's over A, as follows:

• First, we consider the disjoint union Z = ⊔ j X j , and define an equivalence relation ∼ on Z by saying that f j (a) ∼ f k (a) for all j, k and there are no other nontrivial ∼-classes. Then we set Y = ⊔ j X j / ∼ and let Y be the universe of Y. • Next, we need to turn Y into a L-structure; modulo the obvious identifications, we view X j as a subset of Y, so that Y = ∪X j , X j ∩ X k = A for all j, k. Then, if n is an integer and R is a n-ary relation symbol of L, for any ȳ ∈ Y n we set

(Y |= R( ȳ)) ⇔ (∃j ∈ J ∀k ∈ {1, . . . , n} y k ∈ X j and X j |= R( ȳ)) .
Informally, the free amalgam of the X j 's over A is a L-structure Y with universe ∪X j , with X j ∩ X k = A for all j = k, such that each X j is a substructure of Y and no tuple which meets X j \ A and X k \ A for some j = k satisfies any relation in L.

Definition 3.1. Let K be a language. We say that a class K of L-structures has the free amalgamation property if whenever A, B, C ∈ K and i : A → B, j : A → C are embeddings, the free amalgam of B, C over i, j belongs to K.

Example 1. The class of all finite graphs (seen as a class of structures in the language L whose only relation symbol besides the equality is binary) has the free amalgamation property, while the class of tournaments does not (it does have the amalgamation property). If K m denotes the complete graph on m vertices, a graph which does not contain a substructure isomorphic to K m is called a K m -free graph.

The class of all K m -free graphs also has the free amalgamation property.

The other conditions defining a Fraïssé class are easy to check in all the examples above; the Fraïssé limit of the class of all finite graphs is often called the Radó graph, or the random graph, while the various Fraïssé limits of K m -free graphs (as m varies) are sometimes called the Henson graphs, as they were first introduced and studied by Henson in [H1].

3.2. Structures whose age is contained in K. In this subsection, we fix a language L and a class K of finite L-structures, and we assume that K is a Fraïssé class with the free amalgamation property. Definition 3.2. We let K ω denote the class of all countable L-structures whose age is contained in K.

Observe that, if M is a countable L-structure admitting an increasing chain of substructures M i such that each M i belongs to K ω and M = ∪M i , then M also belongs to K ω .

The following fact is clear.

Proposition 3.3. Assume that A ∈ K ω , (X i ) i∈I is a countable family of element of K ω and that f i : A → X i is an embedding for all i. Then the free amalgam of the X i 's over A belongs to K ω .

Definition 3.4. For X ∈ K ω , denote by L X the language obtained by adding to L a constant symbol c x for all x ∈ X (Here we do not respect our convention that languages are relational).

If Y = X ∪ {y} is an element of K ω , y ∈ X, seen as an L X -structure by interpreting each c x by x, the quantifier-free type of y over X is the family of all quantifier-free L X -formulas φ with one free variable z such that Y |= φ(y).

We say that a set p of quantifier-free L X -formulas with one free variable is a quantifier-free type over X if there exists Y = X ∪ {y} ∈ K ω such that p is the quantifier-free type of y. We call Y the structure associated to p; up to obvious identifications it is uniquely determined by p.

In less formal terms: if Y is a structure of the form X ∪ {y}, y ∈ X, then the quantifier-free type of y over X is the complete description of the relations between elements of X, and relations between elements of X and y. To shorten notation a bit, below we write q.f type instead of quantifier-free type. Definition 3.5. Let X ∈ K ω and p be a q.f type over X. Then, for any automorphism f of X, we define f (p) as the q.f type with one free variable z defined by:

φ(z, c x 1 , . . . , c x n ) ∈ f (p) ⇔ φ(z, c f -1 (x 1 ) , . . . , c f -1 (x n ) ) ∈ p .
Intuitively, f (p) describes an element such that, for any x 1 , . . . , x n in X, f (p) satisfies the same relations with f (x 1 ), . . . , f (x n ) as p does with x 1 , . . . , x n . Definition 3.6. Let X ∈ K ω . We say that a q.f type p over X is finitely induced if there exists a finite substructure A of X, and an element B = A ∪ {b} of K, such that the structure associated to p is isomorphic to the free amalgam of B and X over A.

Note that, if K is not reduced to a singleton, then the free amalgamation property implies that K is infinite, and for any X ∈ K ω there always exists at least one finitely-induced q.f type over X (there may exist only one, a situation that presents itself for instance when K is the class of pure sets). In the remainder of the article, we always assume that the classes we consider are infinite. Definition 3.7. Let X be an element of K ω , {p i } i∈I be an enumeration of all the finitely induced q.f types over X, and for each i let Y i denote the structure associated to p i . Let E(X) denote the free amalgam of the Y i 's over X.

Since there are countably many finitely induced q.f types over X, and K ω is stable under free amalgamation, E(X) belongs to K ω . Also, X naturally embeds in E(X), and E(X) \ X is always nonempty.

The definition of E(X) is motivated by Katětov's construction of the Urysohn space [K1]; in that context the next proposition was first pointed out by Uspenskij [U2].

Proposition 3.8. Assume X ∈ K ω . Then each automorphism of X extends uniquely to an automorphism of E(X), and the extension morphism is an embedding of topological groups from Aut(X) to Aut(E(X)).

Proof. Let φ be an automorphism of X. For each y ∈ E(X) \ X, with type denoted by p, there exists a unique z ∈ E(X) \ X such that the type of z is equal to φ(p). To extend φ to an automorphism E(φ), one has no choice but to set E(φ)(y) = z; this proves the uniqueness. The fact that this extension is indeed an automorphism of E(X) is also obvious, since by definition two different elements of E(X) \ X do not belong to any tuple satisfying a relation of L.

The uniqueness of the extension ensures that φ → E(φ) is an injective homomorphism. To show that it is continuous, pick x 1 , . . . , x n ∈ X and y 1 , . . . , y m ∈ E(X) \ X. For i ∈ {1, . . . , m}, let A i be a finite substructure of X witnessing the fact that the q.f type of y i is finitely induced, and let A = {x 1 , . . . ,

x n } ∪ ∪ i A i .
Then A is a finite substructure of X, and for any φ which coincides with the identity on A one must have φ(x i ) = x i for all i ∈ {1, . . . , n} and φ(y i ) = y i for all i ∈ {1, . . . , m}. We have just proved that, for any neighborhood V of the identity in Aut(E(X)), there exists a neighborhood U of the identity in Aut(X) such that E(U) ⊆ V, hence E is continuous. Theorem 3.9. Assume that K is a Fraïssé class with the free amalgamation property, and denote by K the Fraïssé limit of K. Then, for any X ∈ K ω there exists an embedding i : X → K such that any automorphism φ of i(X) extends to an automorphism E(φ) of K.

The extension map φ → E(φ) may be taken to be a continuous group embedding from Aut(X) to Aut(K).

In particular, this result shows that Aut(K) is universal for all permutation groups of the form Aut(X), X ∈ K ω . In the Urysohn space context, the above result was proved by Uspenskij [U2]; the proof below is essentially Uspenskij's proof, translated to our setting.

Proof. Starting from X 0 = X, we build an increasing chain of structures in K ω by setting X i+1 = E(X i ) for all i < ω, viewing X i as a substructure of X i+1 via the natural embedding from X i to E(X i ). Then X ∞ = ∪X i belongs to K ω . Using Proposition 3.8, we see that any automorphism ϕ of X i uniquely extends to an automorphism E(φ) such that E(φ)(X i ) = X i for all i, and that φ → E(φ) is a topological group embedding from Aut(X) to Aut(X ∞ ).

Fix a finite substructure A of X ∞ , and an embedding j : A → B for some

B ∈ K. Let B = j(A) ∪ {b 1 , . . . , b n }.
There must exist some i < ω such that A ⊂ X i , and an easy induction argument shows that there exists b1 , . . . , bn ∈ X i+n such that B = A ∪ { b1 , . . . , bn } is isomorphic to B via an isomorphism φ : B → B such that φ • j = i (where i stands for the inclusion map from A to B). This shows that X ∞ is isomorphic to the Fraïssé limit of K, and we are done.

Theorem 3.10. Let n ≥ 2 be an integer, assume that K is a Fraïssé class with the free amalgamation property, and let K denote the Fraïssé limit of K. Then, for any X ∈ K ω , there exists an automorphism φ of K ω such that φ n = 1 and the set of fixed points of φ, seen as a substructure of K, is isomorphic to X.

Proof. We build inductively an increasing sequence (X i , φ i ) such that each X i belongs to K ω , φ i is an automorphism of X i , and:

(i) X 0 = X, φ 0 = id X . (ii) For all i < ω, X i ⊆ E(X i ) ⊆ X i+1 and φ i+1 exends φ i . (iii) For all i < ω, one has φ n i = 1. (iv) For all i < ω, the set of fixed points of φ i is equal to X 0 . In point (ii), the inclusion X i ⊆ E(X i ) is to be understood as the natural embedding of X i in E(X i ). Then, set X ∞ = ∪X i , φ ∞ = ∪φ i .
As in the proof of Theorem 3.9, (ii) ensures that X ∞ is the Fraïssé limit of K; points (iii) and (iv) ensure that φ n ∞ = 1 and that the set of fixed points of φ ∞ is equal to X 0 , hence, as as substructure of X ∞ , is isomorphic to X.

Thus, we only need to explain how to carry out the construction. The first step is imposed, so we set X 0 = X, φ 0 = id X . Assume that (X i , φ i ) has been constructed. We let X i+1 be the free amalgam of n copies of

E(X i ) over X i . To define φ i+1 , let Y 0 . . . , Y n-1 be n copies of E(X i ) such that X i+1 = ∪Y k , Y j ∩ Y k = X i for all j = k ∈ {0, . . . , n -1},
and pick y ∈ Y \ X i . There exists a unique j such that y ∈ Y j ; let p denote the q.f type of y over X i . There exists a unique element z of Y j+1 (addition here being modulo n) such that the q.f type of z is equal to φ(p); we set φ i+1 (y) = z.

It is clear that φ i+1 is then an automorphism of X i+1 , whose set of fixed points is the same as the set of fixed points of φ i , hence is equal to X 0 . To check that φ n i+1 = 1, it suffices to notice that, for any y ∈ Y j \ X i , with q.f type denoted by p, φ n i+1 (y) is by definition the unique element of Y j+n = Y j whose type is equal to φ n i (p) = p. Hence φ n i+1 (y) = y, and the proof is complete. We note in passing that the result above also holds for the rational Urysohn space, with a similar proof. The second author proved in [M2] a slightly weaker result, namely, that any countable rational metric space is isometric to the set of fixed points of some isometry of the rational Urysohn space. The construction provided there enabled one to obtain a similar result for the Urysohn space (which is the completion of the rational one), while the construction presented here is not enough for that purpose.

3.

3. An application to the complexity of some conjugation problems. In this subsection, we apply Theorem 3.10 to show that, if K is a Fraïssé class with the free amalgamation property, then the isomorphism relation on K ω is always reducible to the conjugacy relation in Aut(K). This enables one to compute the exact complexity of the conjugacy relation in the automorphism group of the Henson graphs and the random graph.

Since this section is only tangentially related to our interests in this paper, we will not give any background on the theory of Borel reducibility of definable equivalence relations. For information on this rich theory we refer to [BK], [G], [H2] and [K3]. Let us simply set our notation: we fix a language L = (R i , n i ) i∈I . For any i ∈ I, we let X i = 2 N n i , endowed with the product topology, and set

X L = ∏ i∈I X i .
Endowed with the product topology, this is a compact topological space (homeomorphic to the Cantor space); for each i we denote by π i : X L → X i the coordinate projection.

To each element X of X L we may associate an infinite, countable L-structure X by setting, for all n ∈ N n i :

X |= R i ( n) ⇔ π i (X)( n) = 1 .
Conversely, any L-structure with universe N defines an element of X L ; thus we may see X L as the space of all infinite countable L-structures. We say that X, Y ∈ X L are isomorphic if X, Ỹ are isomorphic as L-structures. The permutation action of S ∞ on N naturally extends to a continuous action of S ∞ on X L , defined by setting, for all

X ∈ X L , σ ∈ S ∞ , i ∈ I and n ∈ N n i , π i (σ • X)( n) = π i (X)(σ -1 ( n)) .
Then, X and Y are isomorphic if, and only if, there exists σ ∈ S ∞ such that σ • X = Y; thus, thinking of X L as the space of all infinite countable L-structures, the relation of isomorphism of infinite countable L-structures is given by a continuous action of S ∞ on the compact space X L , called the logic action of

S ∞ on X L . Now, if K is a Fraïssé class of L-structures, define X K = {X ∈ X L : X ∈ K ω }. Then X K is a Borel subset of X L (it is even G δ if L is finite)
, and X K is invariant under the logic action of S ∞ . We call the relation induced by the logic action the isomorphism relation on infinite elements of K ω .

Theorem 3.11. Let n ≥ 2 be an integer, L be a countable relational language, and K be a Fraïssé class of L-structures with the free amalgamation property. Denote by K the Fraïssé limit of K. Then, for any integer n, the isomorphism relation on infinite elements of K ω (as encoded above) is Borel reducible to the conjugacy relation in {g ∈ Aut(K) : g n = 1}.

Proof. It is possible to define a Borel map X → φ(X) from X K to Aut(K) such that φ(X) is conjugate to the automorphism of K built in the proof of Theorem 3.10. We do not give the tedious technical details of this construction; they are presented in full in the first author's doctoral dissertation [B]). Let us now see why this proves our result.

First, letting ( Xi , φ i (X)) and ( Ỹi , φ i (Y)) denote the sequences constructed by applying the construction of 3.10 to X, Ỹ, we see that if X, Ỹ are isomorphic we may inductively build a sequence of isomorphisms g n : Xi → Ỹi such that g i φ i (X)g -1 i = φ i (Y). This yields an isomorphism g : X∞ → Ỹ∞ such that gφ(X)g -1 = φ(Y); identifying X∞ , Ỹ∞ with K we see that φ(X), φ(Y) are conjugate. Conversely, if φ(Y) = gφ(X)g -1 for some g ∈ Aut(K), then g must map the set of fixed points of φ(X) onto the set of fixed points of φ(Y). Hence the sets of fixed points of φ(X), φ(Y) are isomorphic substructures of M, that is, X and Y are isomorphic.

The fact that the conjugacy relation in the automorphism group of the random graph is S ∞ -universal, which follows from point (i) below, was first proved by Coskey, Ellis and Schneider [CES] Corollary 3.12. Let n ≥ 2 be an integer. Then:

(i) The conjugacy relation on {g ∈ Aut(R) : g n = 1}, where R denotes the random graph, is S ∞ -universal. (ii) Let m ≥ 3 be an integer and G m denote the Fraïssé limit of the class of K mfree graphs. Then the conjugacy relation on {g ∈ Aut(G m ) :

g n = 1} is S ∞ - universal.
Proof. Theorem 3.11 shows that the relation of isomorphism of graphs Borel reduces to the conjugacy relation on {g ∈ Aut(R) : g n = 1} for all n ≥ 2, so the first result is an immediate corollary of Theorem 3.11 and the fact that isomorphism of countable graphs is S ∞ -universal.

For the second one, we claim that, for any m, the isomorphism relation for graphs Borel reduces to the isomorphism relation for K m -free graphs, from which the result follows. One way to build a reduction is as follows: let G denote a countable graph; for each edge of G, add four new vertices, remove the edge, and add new edges as in the picture below.

x y x y u 1 u 2 v 2 v 1
Let G denote the graph produced by applying this transformation to G; for example, the following picture shows what G is if G is a triangle.

Then G is K m -freee for all m ≥ 3, and the map G → G can be coded by a Borel map (in the space of codes of graphs), thus we only need to show that, for two countable graphs G, H, one has

(G ∼ = H) ⇔ ( G ∼ = H) .
The implication from left to right is obvious. To prove the converse, assume that φ : G → H is an isomorphism. The v i 's constructed above are the only elements of G with exactly one neighbour, and the u i 's are the only elements of G with neighbour a v i . The same thing happens in H, so φ must map the u i 's, v i 's that were added to G to the u i 's, v i 's that were added to H, hence φ maps the set V (G) of vertices of G to the set V(H) of vertices of H. Finally, two elements of V (G) have an edge between them in G if and only if they have a common u i as a neighbour in G, and this is preserved by φ, hence φ induces an isomorphism from G to H, and we are done.

Before moving on, let us note that we do not know any example of a Fraïssé class K with the free amalgamation property and such that the isomorphism relation for infinite elements of K ω is neither smooth not S ∞ -universal.

GENERIC ELEMENTS OF FINITE ORDER

4.1. Coherent extensions. In this subsection, we fix a finite language L. We discuss an improvement by Solecki of a theorem of Herwig-Lascar. Definition 4.1 (see [HL], p. 1994). Let A, B be two L-structures. A map f : A → B is a weak homomorphism if for any integer n, any n-ary relational symbol R of L and any a 1 , . . . , a n ∈ A one has

A |= R(a 1 , . . . , a n ) ⇒ B |= R( f (a 1 ), . . . , f (a n )) .
If A is a L-structure and T is a set of L-structures, say that A is T-free if there is no weak homomorphism from a structure in T to A (this coincides with the usual terminology in the case of graphs). Definition 4.2 (Solecki [S1]). Let A be a set, and p 1 , p 2 , p 3 three partial bijections of A.

Say that (p 1 , p 2 , p 3 ) is coherent if dom(p 1 ) = rng(p 2 ), dom(p 2 ) = dom(p 3 ), rng(p 1 ) = rng(p 3 ) and p 3 = p 1 • p 2 .
If A, B are sets and φ is a function from the sets of partial bijections of A to the set of partial bijections of B,

say that φ is coherent if (φ(p 1 ), φ(p 2 ), φ(p 3 )) is coherent whenever (p 1 , p 2 , p 3 ) is coherent.
The following is Solecki's aforementioned refinement of the Herwig-Lascar theorem.

Theorem 4.3 (Solecki [S1]). Let T be a finite family of structures, A be a finite T-free structure, and P be a set of partial isomorphisms of A. If there exists a T-free structure M containing A such that each element of P extends to an automorphism of M, then there exists a finite T-free structure B containing A and E : P → Aut (B) such that (i) E(p) is an extension of p for all p ∈ P.

(ii) E is coherent.

Using the notations of the above theorem, note that if A ′ is a substructure of A, and P contains Aut(A ′ ), then the coherence of the map E implies that E : Aut(A ′ ) → Aut(B) is a homomorphism (which is necessarily injective).

The assumption of Theorem 4.3 is satisfied if T is a finite set of L-structures such that the set of all T-free structures is a Fraïssé class. Solecki also used this result to show a similar theorem for classes of metric spaces, proving the following: Theorem 4.4 (Solecki [S1]). Let A be a finite metric space. There exists a finite metric space B such that A ⊆ B as metric spaces, each partial isometry p of A extends to an isometry E(p) of B and the function E is coherent. Moreover, the distances between points in B belong to the additive semi-group generated by the distances between points in A.

Metric spaces as relational structures.

To prove Theorem 4.4, one of the steps is to view finite metric spaces as relational structures. We quickly explain how one may do this; first, to stay within the realm of countable structures, one has to impose a condition on the set of possible values for the metric, and we introduce some ad hoc terminology. Definition 4.5. Let Q be a countable subset of [0, +∞) containing 0. We say that Q is a metric value set if one of the following conditions is satisfied:

(1) Q is a subsemigroup of (R; +).

(2) Q is the intersection of a subsemigroup of (R, +) and a bounded interval, and

M Q = sup(Q) ∈ Q. Definition 4.6. If Q is a metric value set, a Q-metric space is a metric space (X, d) such that d(x, x ′ ) ∈ Q for all (x, x ′ ) ∈ X 2 .
Q-metric spaces may be turned into relational structures in a countable language L Q , containing a binary predicate R q for all q ∈ Q: simply put R q (x, y) if and only if d(x, y) = q. Thus one may think of Q-metric spaces as relational structures in a countable language; it is well-known that, if Q is a metric value set, the class of finite Q-metric spaces is a Fraïssé class, with limit denoted by U Q ; the countable metric space U Q is characterized up to isometry, within the class of countable Q-metric spaces, by the fact that it contains a copy of any finite Q-metric space and any isometry between finite subsets extends to an isometry of the whole space.

We now discuss the amalgamation procedure we use for Q-metric spaces: let (A, d A ), (B, d B ), (C, d C ) be three finite Q-metric spaces, i : A → B, j : A → C two isometric embeddings and assume that A is nonempty. We let X denote the disjoint union of B and C and define a pseudometric ρ on X as follows:

(

i) If Q is unbounded, ρ(b, c) = min{d B (b, i(a)) + d C (j(a), c) : a ∈ A}. (ii) If Q is bounded and M Q = sup(Q), ρ(b, c) = min{M Q , min{d B (b, i(a)) + d C (j(a), c) : a ∈ A}} .
In less formal terms, we identified the two copies of A in B, C and for b ∈ B \ A, c ∈ C \ C, set d D (b, c) to be equal to length of the shortest path between b and c going through A (cut off at M Q if Q is bounded). Note that we do not really need sup(Q) to belong to Q to make amalgamation work, but this assumption simplifies slightly the exposition and is more than enough for the applications to the Urysohn space that we have in mind.

Since we allow metric spaces to be empty, we should also explain how we amalgamate over the empty set. Assume that Q is a metric value set and B, C are two finite Q-metric spaces. We extend the metric on B, C to a metric on B ⊔ C by setting d(b, c) = max(diam (B), diam(C)) for all b ∈ B, c ∈ C.

Here, it may be worth pointing out a somewhat counterintuitive phenomenon: the Fraïssé limit of the class of Q-metric spaces, when seen as a structure in a relational language as above, is not ℵ 0 -categorical as soon as Q is infinite: there are models of its theory which are not metric spaces, as there is no first-order way to ensure that for any x, y there exists q such that M |= R q (x, y).

The isomorphic extension property.

Definition 4.7. Let K be a class of finite structures in a relational language L. We say that K has the isomorphic extension property (IEP) if for any A ∈ K there exists B ∈ K such that A ≤ B, and a map E from the set of partial isomorphisms of A to the set of global automorphisms of B such that (i) For any partial isomorphism g of A, E(g) extends g.

(ii) For any A ′ ≤ A, E induces a homomorphism from Aut(A ′ ) to Aut (B). If there exists E as above such that only condition (i) is satisfied, then we say that K has the extension property.

As mentioned above, whenever L is a finite relational language, and T is a finite set of L-structures such that the class K of all finite T-free structures is a Fraïssé class, it follows from the Herwig-Lascar-Solecki theorem that K has the isomorphic extension property. Similarly, it follows from Theorem 4.4 that, whenever Q is a metric value set, the class M Q of all finite Q-metric spaces has the isomorphic extension property. Definition 4.8. Let L be a relational language and K a class of L-structures. We say that K has the isomorphic amalgamation property (IAP) if, for any A, B, C ∈ K and any embeddings i : A → B, j : A → C, there exists D ∈ K and embeddings α :

B → D, β : C → D such that (i) α • i = β • j.
(ii) Whenever φ ∈ Aut(B) fixes i(A), ψ ∈ Aut(C) fixes j(A), and i -1 φi = j -1 ψj, there exist a partial isomorphism σ of D fixing α (B), β(C) respectively and such that α -1 σα = φ, β -1 σβ = ψ.

Note that any Fraïssé class with the free amalgamation property satisfies the isomorphic amalgamation property, as does the class of Q-metric spaces for any metric value set Q. Also, if K has both the (IAP) and the extension property (which will be the case in all our examples), then σ can be taken to be an automorphism of D.

Notation. Let K be a Fraïssé class with limit K, and denote by G the automorphism group of K. For any integer k and any n = (n 1 , . . . , n k ) ∈ N k , we set

Ω n(G) = { ḡ ∈ G k : ∀i ∈ {1, . . . , k} g n i i = 1} .
We endow it with the topology induced by the topology of G k , which turns it into a Polish space on which G acts by diagonal conjugacy: g • (g 1 , . . . , g k ) = (gg 1 g -1 , . . . , gg n g -1 )

We say that ḡ ∈ Ω n(G) is generic if its diagonal conjugacy class is comeager (equivalently, dense G δ ).

Similarly, if A is a structure, we write Ω n( A) to denote the set of n-tuples (g 1 , . . . , g n ) of automorphisms of A such that g n i i = 1 for all i. Proposition 4.9. Let K be a Fraïssé class with the isomorphic exension property, and let G denote the automorphism group of its limit K. Then for any integer k, and any k-tuple n of non-negative integers, there exists a generic element in Ω n (G). The orbit of any x ∈ K under ḡ is finite.

Proof. First, let us show that for a comeager set of ḡ ∈ Ω n(G), the orbit of any x ∈ K under ḡ is finite. Using the Baire category theorem, and the fact that for a fixed x the set G); without loss of generality, we may assume that there exist finite substructures B 1 , . . . , B k ⊂ K and automorphisms g 1 , . . . , g k such that g

A x = { ḡ ∈ Ω n(G) : ḡ • x is finite} is open, it is enough to show that for fixed x ∈ K the set A x is dense. To see this, fix a nonempty open subset V of Ω n(
n i i = 1, g i (B i ) = B i and ∀ h ∈ Ω n(G) (∀i h i|B i = g i |B i ) ⇒ h ∈ V . Set B = {x} ∪ ∪ k i=1 B i .
Applying the isomorphic extension property to B, we may find a finite C ⊆ K containing B and automorphisms h 1 , . . . , h n of C such that h i coincides with g i on B i , and h n i i = 1. Using the usual back-and-forth construction and the isomorphic exension property, we may extend each h i to an automorphism of K such that h

n i i = 1. By construction, h = (h 1 , . . . , h k ) ∈ V and h • x ⊆ C is finite.
Now, we show that, for a generic ḡ ∈ Ω n(G), the following holds:

For any finite ḡ -invariant B ⊂ K, any finite C ∈ K such that B ≤ C and any h ∈ Ω n(C) fixing B setwise and coinciding with ḡ on B , there exists C ⊆ K such that ( C, ḡ| C) ∼ = (C, h|C ).

To see this, let Σ denote the set of ḡ ∈ Ω n(G) satisfying the above condition. It is clear that Σ is G δ ; to see that it is dense, fix B ≤ C and p ∈ Ω n (B). Pick some nonempty open set O in Ω n (G). We may assume without loss of generality that there exists a finite D ⊂ K containing B and a tuple of partial isomorphisms q ∈ Ω n(D) such that O consists of all tuples extending q. We need to find ḡ extending q and such that either some g i does not coincide with p i on B, or there exists C ⊆ K such that ( C, ḡ| C) ∼ = (C, h|C ). If some q i does not coincide with p i on E = B ∩ D, we have nothing to prove. In the other case, E must be fixed by q 1 , . . . , q n , hence by p 1 , . . . , p n . Then, we may use (IAP) and (IEP) to amalgamate D and C over E, obtaining a substructure F ∈ K and automorphisms g 1 , . . . , g n of F with g n i i = 1 for all i, g i coinciding with q i on D and with p i on B. We may assume that D ⊆ F ⊆ K. Then we may extend each f i to an element of Aut(K) still denoted by g i and such that g n i i = 1. This is the tuple we were looking for. Thus, the set of elements ḡ satisfying the following two conditions is dense G δ in Ω n(G):

(i) The orbit of any x ∈ K under ḡ is finite.

(ii) For any finite ḡ -invariant B ⊆ K, any finite C ∈ K such that B ≤ C and any h ∈ Ω n(C) fixing B setwise and coinciding with ḡ on B , there exists C ⊆ K such that ( C, ḡ| C) ∼ = (C, h|C ).

Since any two tuples satisfying both conditions above are conjugate, we are done.

Note that essentially the same proof as above would enable one to show that if K is a Fraïssé class with (IAP) and (EP), then the automorphism group G of its limit has ample generics, i.e. there is a comeager diagonal conjugacy class in G n for each integer n. This would also follow easily from the results of [I] or [KR]. Below we will only use that, under the above conditions on K, G has a dense conjugacy class.

Remark 1. In the case K is the class of Q-metric spaces for some metric value set Q, the conclusion of Proposition 4.9 follows from [START_REF]Finitely approximable groups and actions part II: Generic representations[END_REF]Theorem 5], since a tuple of elements of finite order n 1 , . . . , n k is the same thing as a homomorphism from Z n 1 * . . . * Z n k to the isometry group of U Q , and Z n 1 * . . . * Z n k has property (RZ). Rosendal's proof also uses a version of the Herwig-Lascar-Solecki theorem.

From now on, we assume that, whenever K is a Fraïssé class with the (IAP), A, B, C ∈ K and i : A → C, j : B → C are embeddings, the isomorphism type of the triple (D, α, β) produced by the (IAP) only depends on the isomorphism type of (A, B, C, i, j). For a class with the free amalgamation property, we choose D to be the free amalgam of B and C over A; for a class of metric spaces, we choose for D the metric amalgam as presented in Section 4.2. Slightly abusing notation, we will often call D the I-amalgam of B and C over A. Definition 4.10. Let K be a Fraïssé class with the (IAP), and n an integer. We say that n allows extensions if whenever B ∈ K, φ a partial isomorphism of B and A ≤ B are such that φ(A) = A, φ n |A = 1, and C is the I-amalgam of dom(φ) and φ(dom(φ)) over A, there exists D ∈ K containing C and φ D an automorphism of D extending φ and such that φ n D = 1. This is a technical asumption on n that is very often, but not always, satisfied, as witnessed by the folowing lemma.

Lemma 4.11. If K is a Fraïssé class with the free amalgamation property, or the class of finite Q-metric spaces for some metric value set Q, then any integer n ≥ 2 allows extensions. If K is the class of all finite tournaments, then 3 allows extensions but 2 does not.

Proof. We begin with the case when K has the free amalgamation property or is the class of finite Q-metric spaces. Pick A, B, φ as in the definition of numbers allowing exensions. We may assume that B is the I-amalgam of dom(φ) and φ(dom(φ)) over A.

Enumerate B as {b 1 , . . . , b m } with A = {b 1 , . . . , b p }. Fix j ∈ {0, . . . , n -1}. We define a structure B (j) with universe a m-element set {b

(j) 1 , . . . , b (j)
m } as follows: if R is a q-ary relational symbol of the language of K, and (b

(j) k 1 , . . . , b (j) k q ) is a q-tuple of elements of B (j) , we first define (x 1 , . . . , x q ) ∈ B q by setting ∀i ≤ q x i = φ -j (b k i ) if k i ≤ p b k i else .
Then we set x q ) . Then the map φ (j) : B → B (j) defined by

B (j) |= R(b (j) k 1 , . . . , b (j) k q ) ⇔ B |= R(x 1 , . . . ,
φ (j) (b i ) = b (j) k if i ≤ p and φ j (b i ) = b k b (j) i
else is an isomorphism from B onto B (j) , so each B (j) belongs to K. Each φ (j) naturally induces an embedding i j of A into B (j) . Using these identifications of A with a substructure of B (j) , we may form the I-amalgam D of the B (j) 's over A. It is straightforward to check that φ extends to an isomorphism of D of order n.

For tournaments, we should first explain our amalgamation procedure: if B, C are tournaments with a common subtournament A, we amalgamate them by saying that any element of B \ A loses to any element of C \ A. This amalgamation procedure satisfies the (IAP) and, using the same reasoning as above, one sees that 3 allows extensions. The fact that 2 does not allow extensions is obvious, as the automorphism group of any tournament cannot contain an element of order 2.

We are almost ready to state, and prove, the main result of this section. Before that, we need to recall a definition and a well-known lemma. Definition 4.12. Let X, Y be Polish spaces, f : X → Y a continuous map and x ∈ X. Say that x is locally dense for f if, whenever U is a neighborhood of x, f (U) is a neghborhood of f (x). Lemma 4.13 ("Dougherty's lemma", see e.g [K4]). Assume that X, Y are Polish spaces, f : X → Y is continuous and the set of points which are locally dense for f is dense. Then f (X) is not meager.

Proof. Assume f (X) is meager, and let F n be a countable family of subsets of Y with empty interior such that f (X) ⊆ ∪ n F n . Then X = ∪ f -1 (F n ), so some f -1 (F n ) must have nonempty interior, hence contain a point of local density for f . Then f ( f -1 (F n )) ⊆ F n has nonempty interior, a contradiction. Proposition 4.14. Let K be a Fraïssé class with the (IAP) and the (IEP), and n, m be two integers allowing extensions. Define π : Ω (n,m) (G) → G by π(σ, τ) = στ. Then any generic element of Ω (n,m) (G) is locally dense for π.

Proof. Let (σ, τ) be a generic element of Ω (n,m) (G), and U be a neighborhood of (σ, τ). There exists a finite A ⊆ K which is σ, τ -invariant and such that

∀(φ, ψ) ∈ Ω (n,m) (G) (φ |A = σ |A , ψ |A = τ |A ) ⇒ (φ, ψ) ∈ U . We claim that π(U) ⊇ O = {g ∈ G : g |A = στ |A }
. Proving this will yield the desired result. To see that it is true, let V be a nonempty open subset of O; we may assume without loss of generality that there exists a finite B ⊆ K containing A, and an automorphism g of K such that g

(B) = B and ∀h ∈ G h |B = g |B ⇒ h ∈ V .
Let B = {b 1 , . . . , b P } be enumerated in such a way that A = {b 1 , . . . , b p } (with p < P otherwise there is nothing to prove). As in the proof of Lemma 4.11, we pick an abstract M-point set C = {c 1 , . . . , c P } and turn it into a L-structure as follows. Given R a q-ary relational symbol of L, and (c k 1 , . . . , c k q ) a q-tuple of elements of C, we first define (x 1 , . . . , x q ) ∈ B by setting

∀i ≤ q x i = τ -1 (b k i ) if k i ≤ p b k i else .
Then we set

C |= R(c k 1 , . . . , c k q ) ⇔ B |= R(x 1 , . . . , x q ) . By construction, τ induces an isomorphism τ : B → C, defined by τ(b i ) = c j if i ≤ p and τ(b i ) = b j c i else .
We may also use σ to define σ : C → B by setting

σ(c i ) = σ(b i ) if i ≤ p h(b i ) else
It is straightforward to check from the definition (and the fact that στ = g on A) that σ is an isomorphism from C to B, and σ τ = g.

We may form the I-amalgam D of B, C over A, using the map τ|A for our embedding from A to C. Then, (D, τ) fits our setup for numbers allowing extensions, so by assumption on m we can find E containing D such that τ extends to an isomorphism of Ω m (E), still denoted by τ. Applying the same reasoning to (D, σ-1 ) and n, we find F containing D and such that σ extends to an isomorphism in Ω n (F).

Amalgamating E, F over D, and applying the (IEP) one last time, we finally find H ∈ K containing D and ( σ, τ) extending the original ( σ, τ) and such that σn = 1, τm = 1. We may assume that H ≤ K and extend σ, τ to automorphisms of K; by construction ( σ, τ) ∈ Ω n,m (G). The construction ensures that ( σ, τ) ∈ U and σ τ|B = g |B , showing that σ τ ∈ V, and we are done.

Theorem 4.15. Let K be a Fraïssé class with the (IAP) and the (IEP), and g i be a generic element of order i. Then, for any quadruple ī of integers allowing extensions, and any g ∈ G, there exist h 1 , . . . , h 4 such that each h j is conjugate to g i j and g = h 1 . . . h 4 .

When i is a single integer allowing extensions, this shows that every element of G is a product of four conjugates of g i .

Proof. Applying Proposition 4.14 (whose notations we reuse here), we know that π(Ω (n 1 ,n 2 ) (G)) is not meager. Since this set is analytic and conjugacy-invariant, and there is a dense conjugacy class in G, the 0 -1-topological law implies that π(Ω (n 1 ,n 2 ) (G)) is comeager, and an easy Baire category argument using the fact that points of local density are dense shows that for any comeager subset A of Ω (n 1 ,n 2 ) (G) π(A) is comeager. The same argument works for (n 3 , n 4 ). Now, let B (n 1 ,n 2 ) be the set of elements which are a product of a conjugate of g 1 and a conjugate of g 2 , and B (n 3 ,n 4 ) be the set of elements which are a product of a conjugate of g 3 and a conjugate of g 4 . Since each of those sets is comeager, we may apply Pettis' lemma ([K2, 8.9]) and obtain that

G = B (n 1 ,n 2 ) • B (n 3 ,n 4 ) .
Similar arguments would show, for instance, that in the automorphism group of the random tournament, there exists an element g of order 3 such that every other element is a conjugate of four conjugates of g.

APPLICATIONS TO THE ISOMETRY GROUP OF THE URYSOHN SPACE.

We turn to applying our results to the isometry group of the Urysohn space U.

Notation. We say that a metric value set Q is divisible if q n ∈ Q for all q ∈ Q and all n ∈ N * . We recall that metric value sets are by definition countable; also, whenever Q is divisible, it must be dense in [0, sup(Q)].

We denote by U the (unbounded) Urysohn space and by U R the Urysohn space of diameter R.

Note that if Q is divisible and unbounded, then the completion of

U Q is isomet- ric to U, while if Q is divisible and bounded with sup(Q) = R then the completion of U Q is isometric to U R .
Lemma 5.1. Let g 1 , . . . , g n be a finite family of isometries of U. Then there exists a divisible countable metric value set Q and an isometric copy X of U Q which is dense in U and such that g i (X) = X for all i ∈ {1, . . . , n}.

The same lemma, with obvious modifications, would also be true for U 1 instead of U; this lemma was proved, earlier and independently, by Tent and Ziegler [TZ]. Their proof used a model-theoretic argument based on the L öwenheim-Skolem theorem, which is essentially the same idea as the proof below.

Proof. We proceed by induction: we build an increasing sequence of countable dense subspaces (Y i ) of U and divisible metric value sets Q i such that:

(1) For all i, Y i is a Q i -metric space and Y i is ḡ -invariant.

(2) For any finite subset A of Y i , and any one-point Q i -metric extension A ∪ {z} of A, there exists z ∈ Y i+1 such that d( z, a) = d(z, a) for all a ∈ A.

To begin the construction, let X 0 be any countable dense subspace of U, Y 0 be the countable set ḡ • X 0 , and Q 0 the smallest divisible metric value set containing the values of the distance on Y 0 (this is indeed a countable set); Now, assume that (Y i , Q i ) has been built. Then, there are only countably many one-point Q i -extensions of finite subsets of Y i , and each of them is realized in U, so we may add a countable set {x j } to Y i so that all one-point

Q i -metric extensions of Y i are realized in X i+1 = Y i ∪ {x j }. Let Y i+1 = ḡ • X i+1
, and Q i+1 be the smallest divisible metric value set containing all the values of the distance on Y i+1 .

At the end of this construction, Q = Q i is a countable, divisible metric value set, and Y i is a Q-metric space which is ḡ -invariant, isometric to U Q and dense in U.

Theorem 5.2. Every element of Iso(U) is a commutator and a product of at most four elements of order n for all n ≥ 2. The same result is true for Iso(U 1 ).

Proof. Pick g ∈ Iso(U), and apply lemma 5.1 to find a dense, countable metric value set Q and a dense copy X of U Q such that g(X) = X. Since Iso(U Q ) has a comeager conjugacy class, every element of Iso(X) is a commutator; thus there exist a, b ∈ Iso(X) such that g |X = aba -1 b -1 . These elements a, b uniquely extend to isometries of U (still denoted by a, b) and we obtain g = aba -1 b -1 .

The proof that every element of Iso(U) is a product of at most four elements of order n for all n ≥ 2 follows similarly from Lemma 5.1 and Theorem 4.15.

It is clear that exactly the same argument works for U 1 .

Below, we explain how to obtain a stronger result: for any n there exists an element g of order n in Iso(U) such that every element of Iso(U) is a product of at most four conjugates of g (and the same fact holds for Iso(U 1 ); to avoid unecessary repetitions, we focus on the case Iso(U) below). For this, we first need to show that, as in the discrete case, there exists a generic element of order n in Iso(U). This might be a bit surprising, since Kechris proved that, as opposed to the discrete case, conjugacy classes are meager in Iso(U); this fact was published in Glasner and King's paper [GK].

Notation.

In what follows we let G denote Iso(U) and, for any integer n, set

Ω n (G) = {g ∈ G : g n = 1} .
Definition 5.3. Let n ≥ 2 and σ ∈ Ω n (G). Say that σ has the n-approximate extension property if for any ǫ > 0, any finite σ-invariant subset A = {a 1 , . . . , a m } of U and any (B, d B , τ) such that B = {a 1 , . . . , a m , b, τ(b), . . . , τ n-1 (b)} is a metric space containing A, τ coincides with σ on A, and τ n = 1, there exists b ∈ U such that :

(i) ∀i ∈ {1, . . . , m} ∀j ∈ {0, . . . , n -1} |d B (a i , τ j (b)) -d(a i , σ j (b))| ≤ ǫ. (ii) ∀i, j ∈ {0, . . . , n -1} |d(σ i ( b), σ j ( b)) -d(τ i (b), τ j (b))| ≤ ǫ.
If the condition above is satisfied for ǫ = 0, we say that σ has the n-extension property.

Note that, in the definition of the n-approximate extension property, we allow the possibility that τ q (b) = b for some strict divisor q of n -in that case ({τ(b), . . . , τ n-1 (b)}, d B ) is a pseudometric space rather than a metric space, which is why we will have to manipulate pseudometrics below.

We turn to proving that the n-approximate extension property and the n-extension property are equivalent; as a first step for that proof, we need the following lemma. In the case A below is empty, this lemma is due to Uspenskij [U3, Proposition 7.1]; the proof we give is essentially the same as Uspenskij's. The lemma below was also known for a long time to Henson, who never published it, and the second author heard it from him.

Lemma 5.4. Let A = {a 1 , . . . , a n } be an enumerated finite metric space. Let B = A ∪ {b 1 . . . , b p } and C = A ∪ {c 1 , . . . , c p } be two enumerated finite pseudometric spaces containing A, and ǫ > 0 be such that

(i) ∀i, j ∈ {1, . . . , p} d B (b i , b j ) -d C (c i , c j ) ≤ 2ǫ. (ii) ∀i ∈ {1, . . . , n} ∀j ∈ {1, . . . , p} d B (a i , b j ) -d C (a i , c j ) ≤ ǫ.
Then there exists a pseudometric ρ on X = B ∪ C such that:

(1) ρ extends d B , d C .

(2) ∀i ∈ {1, . . . , p} ρ(b i , c i ) ≤ ǫ.

(3) If f : A → A is an isometry, and π is a permutation of {1, . . . , p} such that setting f (b i ) = b π(i) extends f to an isometry of (B, d B ), and setting f

(c i ) = c π(i) extends f to an isometry of C, then setting f (b i ) = b π(i) and f (c i ) = c π(i) extends f to an isometry of (X, ρ). This pseudometric is such that ρ(b, c) > 0 for all b ∈ B \ A, c ∈ C \ A.
Proof. We use the same idea as in [U3]. First, define a partial function ω on X by the following conditions:

•

∀b, b ′ ∈ B ω(b, b ′ ) = d B (b, b ′ ). • ∀c, c ′ ∈ C ω(c, c ′ ) = d B (c, c ′ ). • ∀i ∈ {1, . . . , p} ω(b i , c i ) = ǫ.
Note that, if a, a ′ belong to A then the first two conditions both give ω(a, a ′ ) = d(a, a ′ ). If x, y ∈ X, we say that a finite sequence x 0 , . . . , x k of elements of X is a path from x to y if x 0 = x, y = x k , and ω(x i , x i+1 ) is defined for all i ∈ {0, . . . , k -1}. Then we set

ρ(x, y) = inf{ k-1 ∑ i=0 ω(x i , x i+1 ) : (x 0 , . . . , x k ) is a path from x to y}
It is clear that ρ is a pseudometric, and conditions (2) and (3) follow immediately from the definition. The only fact that remains to be checked is (1); this is straightforward but a bit tedious, reducing to cases.

Lemma 5.5. Let n ≥ 2 and σ ∈ Ω n (G). Then σ has the n-extension property if, and only if, it has the n-approximate extension property.

Proof. Fix A = {a 1 , . . . , a m }, (B, d B , τ) as in the definition of the n-approximate extension property. Using the completeness of U, it is enough to build a sequence (c q ) of elements of U such that:

(1) ∀q d(c q , c q+1 ) ≤ 2 1-q .

(2) ∀q ∀i ∈ {1, . . . , m} ∀j ∈ {0, . . . , n -1} d(σ j (c q ), a i )

-d B (τ j (b), a i ) ≤ 2 -q . (3) ∀q ∀i, j ∈ {0, . . . , n -1} d(σ i (c q ), σ j (c q )) -d B (τ i (b), τ j (b)) ≤ 2 -q .
Use the approximate extension property of σ to define c 0 , then assume that c q has been defined. Let C q = A ∪ {c q , . . . , σ n-1 (c q )}. Using Lemma 5.4 applied to the extensions B, C q of A, one may find a metric ρ on X = B ∪ C q extending the original metrics, such that ρ(b, c q ) ≤ 2 -q and the map f : X → X defined by

f (x) =      σ(x) if x ∈ A τ j+1 (b) if x = τ j (b) σ j+1 (c q ) if x = σ j (c q )
is an isometry of X. We may see (X, f ) as an abstract extension of C q , and apply the approximate extension property of σ with ǫ = 2 -q-1 , to find c q+1 . All the desired properties are straightforward to check from the definitions and the triangle inequality.

There remains one last piece of bookkeeping ahead of us.

Lemma 5.6. Let Q be a divisible metric value set, and see U Q as a dense subset of U. Let n ≥ 2 be an integer and σ be a generic element of Ω n (Iso(U Q )). Then the extension of σ to an element of Ω n (G) has the n-almost extension property (hence also the n-extension property).

Proof. We still denote by σ the extension of σ to U. To see that it has the napproximate extension property, fix a finite, σ-invariant set A = {a 1 , . . . , a m }, a finite enumerated pseudometric space B = ({a 1 , . . . , a m } ∪ {b, . . . , τ n-1 (b)}, d B ) containing A, an isometry τ of B that extends σ |A and such that τ n = 1, and ǫ > 0. We may assume that ǫ < diam (B). To simplify notation, we write b i = τ i (b) for i = 0, . . . , n -1. Our proof is in three steps.

First step: Let ρ denote the metric on B defined by B). It is easy to check that ρ is a metric on B, and that τ is an isometry of (B, ρ)

• ∀a, a ′ ∈ A ρ(a, a ′ ) = d(a, a ′ ). • ∀b ∈ B \ A ∀a ∈ A ρ(a, b) = diam(B). • ∀b = b ′ ∈ B ρ(b, b ′ ) = diam(
. Define d on B by setting, for all b, b ′ ∈ B, d(b, b ′ ) = (1 - ǫ diam(B) )d B (b, b ′ ) + ǫ diam(B) ρ(b, b ′ ) .
Then d is a metric on B, coinciding with d on A, and we have:

∀i, j ∈ {0, . . . , n -1} | d(b i , b j ) -d B (b i , b j )| ≤ 2ǫ . (5.1) ∀i ∈ {1, . . . , m} ∀j ∈ {1, . . . , p} | d(a i , b j ) -d B (a i , b j )| ≤ 2ǫ . (5.2)
What we have gained with introducing this new metric is that now there exists some δ > 0 such that, for any triple {x, y, z} of distinct elements of B not contained in A, one has

(5.3) d(x, y) + d(y, z) ≥ d(x, z) + 3δ .
We may, and do, assume that δ < ǫ and fix such a δ for the remainder of the proof. Now we want to approximate our original metric space by a Q-metric space, so as to use genericity of σ in Iso(U Q ) in order to prove that σ has the n-almost extension property.

Second step:

Pick a ′ 1 , . . . , a ′ m ∈ U Q such that d(a i , a ′ i ) ≤ δ. Define A ′ = {σ k (a ′ i ) : k ∈ {0, . . . , n -1}, i ∈ {1, . . . , m}} We may extend d to a metric on X = B ∪ A ′ ,

still denoted by d, and extending d on

A ∪ A ′ by setting, for all b ∈ B, a ′ ∈ A ′ : d(b, a ′ ) = min{ d(b, a) + d(a, a ′ ) : a ∈ A} .
For any triple {x, y, z} of distinct elements of X not contained in A ∪ A ′ , (5.3) ensures that we still have

(5.4) d(x, y) + d(y, z) ≥ d(x, z) + 3δ .
The metric d may not have be Q-valued. We modify it as follows: let r 1 , . . . , r j denote the values of d which do not belong to Q. For any i ∈ {1, . . . , j}, find

q i ∈ Q such that |q i -r i | ≤ δ. Then define a new metric D on Y = {b 0 , . . . , b n-1 } ∪ A ′ by setting D(x, x ′ ) = q i whenever d(x, x ′ ) = r i , D(x, x ′ ) = d(x, x ′ ) otherwise.
The values taken by D belong to Q, and we claim that it is indeed a metric. To see this, pick a triple {x, y, z} of distinct elements of Y. If {x, y, z} ⊆ A ′ then D coincides with d on {x, y, z} so the triangle inequality holds. Otherwise, we have:

D(x, y) + D(y, z) ≥ d(x, y) + d(y, z) -2δ ≥ d(x, z) + 3δ -2δ (by (5.4)) ≥ d(x, z) + δ ≥ D(x, z) .
Also, for any i, j ∈ {0, . . . , n -1} we have

|D(b i , b j ) -d B (b i , b j )| ≤ δ + | d(b i , b j ) -d B (b i , b j )| ≤ δ + 2ǫ ≤ 3ǫ . Define a map φ on Y by setting φ(b i ) = τ(b i ) for i ∈ {0 . . . , n -1}, φ(a ′ ) = σ(a ′ ) for a ′ ∈ A ′ .
We have φ n = 1; of course, φ does not need to be an isometry, and this is what we have to take care of in the last step of the proof. Letting c i = σ i (c), the only thing that remains to be checked in order to prove that σ has the n-almost extension property is whether |d(c i , a j )d B (b i , a j )| is small for all i ∈ {0, . . . , n -1}, j ∈ {1, . . . , m}. We have, since d(c i , a ′ j ) = D(b i , a ′ j ), that

Third step: For x, y ∈ Y, we set D(x, y) = 1 n n-1 ∑ k=0 D(φ k (x), φ k (y)) . Then D is a metric on Y, taking its values in Q because Q is divisible,
| d(τ k (b i ), σ k (a ′ j ) -d(τ k (b i ), σ k (a j )| ≤ d(σ k (a ′ j )
, σ k (a j )) = d(a j , a ′ j ) ≤ δ . Using these inequalities, and the fact that τ is a d-isometry exending σ, (5.1) and (5.5) give, for all i ∈ {0, . . . , n -1} and all j ∈ {1, . . . , m},

|d(c i , a ′ j ) -d B (b i , a j )| ≤ 1 n n-1 ∑ k=0 (2δ + | d(b i , a j ) -d(b i , a j )|) ≤ 2ǫ + 2δ ≤ 4ǫ .
Finally, since d(a ′ j , a j ) ≤ δ ≤ ǫ, we obtain |d(c i , a j )d B (b i , a j )| ≤ 5ǫ and, since ǫ was arbitrary, this concludes the proof.

Essentially the same reasoning as above enables one to check that the n-approximate extension property is equivalent to the following condition:

For any ǫ > 0, any finite σ-invariant subset A = {a 1 , . . . , a m } of U Q and any (B, Thus, the set of σ ∈ Ω n (G) having the n-approximate extension property is G δ . It is easy to check from Lemmas 5.1 and 5.6 that this set is dense in Ω n (G). Since any two elements of Ω n (G) with the n-approximate extension property are conjugate (because they actually have the extension property), we have finally obtained the following result.

Theorem 5.7. For any integer n there exists an element g n whose conjugacy class is comeager in {g ∈ Iso(U) : g n = 1}. Any g ∈ Iso(U) is a product of four conjugates of g n .

Proof. We have already explained why the first sentence is true. The second one follows as in the proof of Theorem 5.2 from the fact that whenever Q is a countable, divisible metric value set and U Q is densely embedded in U, the extension to U of a generic element of Iso(U Q ) is a generic element of Ω n (Iso(U)) (because it has the n-extension property).

We conclude the paper by discussing an open problem and a partial answer: it is known that there are elements of Iso(U) without roots of order n for any n ≥ 2; however it is unknown whether a generic element must have roots of any order (or even, square roots); note that by the 0 -1 topological law ([K2, 8.46]), the existence of dense conjugacy classes implies that the set of elements admitting a n-th root is either meager or comeager for all n.

It is also unknown whether a stronger condition holds, namely whether a generic element g may be embedded in a flow, i.e. whether there exists a continuous homomorphism from (R, +) to Iso(U) such that g = F(1). We can answer this question in the particular case of generic elements of order n, and obtain a stronger result than expected in the bounded case.

Definition 5.8. Let X be a bounded metric space. We let d u denote the uniform metric on Iso(X), defined by d u (g, h) = sup{d(g(x), h(x)) : x ∈ X} .

The uniform metric d u is bi-invariant, complete, and is not separable in general.

Lemma 5.9. Let n ≥ 2 be an integer, X a compact metric space of diameter at most 1, and Y a dense countable Q-metric subspace of X. Assume that F = (F t ) is a flow of isometries of X such that F 1 = σ fixes Y, σ n = 1, and d u (F t , F s ) ≤ n|t -s| for all t, s ∈ R.

Let Z = Y ∪ A be a Q-metric space containing Y with A finite, and τ an isometry of Z extending σ |Y such that τ n = 1. Then there exists a compact metric space X containing X, of diameter at most 1, a dense countable Q-metric subspace Ỹ of X, and a flow G = (G t ) of isometries of X, such that Proof. By induction, we may assume that A = {a, τ(a), . . . , τ n-1 (a)} for some a ∈ A. For t, s ∈ R we set δ(t, s) = n|t -s|; for t ∈ R and x ∈ X we denote F t (x) = t • x. Our assumptions imply that we have, for all t ∈ R:

(5.6)

δ(t, 0) ≥ sup x∈X d(t • x, x)
We define a map ω on (R × X) 2 by the following conditions:

• For all t, s ∈ R, ω(t, s) = min{d(a, τ i (a)) + δ(t + i, s)} • For all t ∈ R, x ∈ X, ω(t, x) = d(a, (-t).x).

• For all x, y ∈ X, ω(x, y) = d(x, y).

We let ρ be the pseudometric on R × X associated to ω, i.e.

∀a, b ∈ R × X ρ(a, b) = inf{ n ∑ i=0 ω(x i , x i+1 ) : x 0 = a, x n+1 = n}
Then one may check the following facts (the verifications, if done in the same order as below, are completely straightforward so we omit them):

(i) ω |R 2 is an invariant pseudometric. (ii) For any triple x, y, z with at least two of its elements in X one has ω(x, y) + ω(y, z) ≥ ω(x, z). (iii) For any s, t ∈ R, for any x ∈ X, ω(t, s) + ω(s, x) ≥ ω(t, x). (iv) On X, ρ and d coincide. (v) For all i ∈ Z, ρ(0, i) = ω(0, i) = d(a, τ i (a)).

  and φ is an isometry of (Y, D), extendingσ |A ′ . Also, D(b i , b j ) = D(b i , b j ) for all i, j ∈ {0, . . . , n -1}. Using the fact that σ is a generic element of Ω n (U Q ), we can find c ∈ U Q such that (A ′ ∪ {c, σ(c), . . . , σ n-1 (c)}, σ) ∼ = (A ′ ∪ {b, τ(b), . . . , τ n-1 (b)}, φ, D)We already know that ∀i, j ∈ {0, . . . , n -1} |d(σ i (c),σ j (c))d B (τ i (b), τ j (b))| = |D(b i , b j )d B (b i , b j )| ≤ 3ǫ .

  i , a ′ j )d B (b i , a j )| ≤ k (b i ), σ k (a j )d B (b i , a j )| . By definition of D, for all k ∈ {0, . . . , n -1} |D(τ k (b i ), σ k (a ′ j )) -d(τ k (b i ), σ k (a ′ j ))| ≤ δ . Since d was a metric on B ∪ A ∪ A ′ coinciding with d on A ∪ A ′ ,we have, for all k ∈ {0, . . . , n -1},

  d B , τ) such that B = {a 1 , . . . , a m , b, τ(b), . . . , τ n-1 (b)} is a Q-metric space containing A, τ coincides with σ on A, and τ n = 1, there exists b ∈ U such that :(i) ∀i ∈ {1, . . . , m} ∀j ∈ {0, . . . , n -1} |d B (a i , τ j (b))d(a i , σ j (b))| < ǫ. (ii) ∀i, j ∈ {0, . . . , n -1} |d(σ i ( b), σ j ( b))d(τ i (b), τ j (b))| < ǫ.

  (i) G |X = F. (ii) G 1|A = τ and G 1 ( Ỹ) = Ỹ. (iii) ∀t, s ∈ R d u (G t , G s ) ≤ n|t -s|.

 (vi)The infimum in the definition of ρ is actually a minimum, so ρ(a, b) ∈ Q for all a, b ∈ Q ∪ Y. (vii) For all t, s, ρ(t, s) ≤ n|t -s|.

Once all these things are checked, we are essentially done: let X be the metric space obtained by identifying points a, b such that ρ(a, b) = 0. Then X is naturally isometric to X ∪ (R/nZ, ρ), with ρ coinciding with d on X; R acts isometricaly on X, extending the action given by F, the action on R/nZ being the quotient of the translation of (R, +) on itself. We may identify Y ∪ A with Y ∪ {0, . . . , n -1}, and under this identification we have j • a = τ j (a) for all j ∈ Z. Define Ỹ as Y ∪ Q/nZ, which is a dense Q-metric subspace of Ỹ on which ρ takes only rational values. Replacing ρ by min(ρ, 1), the proof is complete.

Theorem 5.10. Let n be an integer. Then a generic element of Ω n (Iso(U)) embeds in a flow. A generic element of Ω n (Iso(U 1 )) embeds in a flow which is n-Lipschitz from (R, +) to (Iso(U 1 ), d u ).

Before the proof, we need to introduce some notations.

Notation. Let Y be a countable Q-metric space of diameter at most 1, n be an integer and σ an element of Iso(Y) such that σ n = 1. For any finite, σ-invariant subset A ⊆ Y, any finite metric space B containing A and any isometry τ of B coinciding with σ on A and such that τ n = 1, one may form the metric amalgam (for spaces of diameter at most 1) Z of Y and B over A and then extend τ to an isometry of Y.

If (Z, τ) has been obtained by the procedure above, we say that (Z, τ) is a finite extension of (Y, σ) attached to (A, B, τ |B ); we denote by E(Y, σ) the countable set of all (isomorphism types of ) (Z, τ) which can be obtained by this procedure.

Proof of Theorem 5.10. We do not give the proof for Iso(U), which is based on an easy modification of Lemma 5.9 and the construction below. Fix a bijection f : N × N → N such that f (p, q) > p for all (p, q). Then, using lemma 5.9, we may build a sequence of compact metric spaces (X m ), with dense Q-metric subspaces Y m , and a sequence of isometric flows F m such that:

the flow on X ∞ produced by the above construction, and σ ∞ = F ∞ (1). Then the construction (the last condition in particular) implies that Y ∞ is isometric to U Q 1 and σ ∞ is a generic element of Iso(U Q 1 ). The completion of X ∞ contains Y ∞ as a dense subspace, so it is isometric to the completion of U Q 1 , i.e. to U 1 . Thus σ ∞ extends to a generic element of Iso(U 1 ) and the flow F ∞ extends to a flow such that

Proof. Let σ be a generic element in Ω 2 (Iso(U 1 )), and g ∈ G. We know that there is a 2-Lipschitz flow F : (R, +) → (Iso(U 1 ), d u ) such that σ = F(1), and elements

and φ is continuous from [0, 1] to (Iso(U 1 ), d u ) (actually, d u (φ(t), φ(s)) ≤ 8|t -s| for all t, s).

To close the paper, we note that the analogue of Corollary 5.11 for (Iso(U), d u ) is false, since for a generic g ∈ Iso(U) one has d u (g, 1) = +∞, hence g is not in the path-connected component of 1. Similarly, the generic element g n of order n in Iso(U) is such that d u (g n , 1) = +∞, so one cannot improve the statement of Theorem 5.10 to obtain d u -continuity in the case of Iso(U). Also, Corollary 5.11 would follow from simplicity of Iso(U 1 ), since the path-connected component of 1 is a nontrivial normal subgroup; and simplicity of Iso(U 1 ) would itself follow from simplicity of Iso(U Q ), where Q = Q ∩ [0, 1]. Unfortunately, deciding whether this group is simple appears to be out of reach at the moment; in particular, the MacPherson-Tent-Ziegler techniques seem not to work in this case, or at least not to adapt in a straightforward way. Still, their work seems to represent the best hope to solve this problem positively in the near future (provided, of course, if admits a positive answer!).