Group topologies on automorphism groups of homogeneous structures

We classify all group topologies coarser than the topology of stabilizers of finite sets in the case of automorphism groups of countable free-homogeneous structures, Urysohn space and Urysohn sphere, among other related results.


Introduction
Minimality.A topological group (G, τ ) consists of a group (G, •) and a topology τ on G such that the map ρ : G × G → G where ρ(g, h) = gh −1 is jointly continuous.In fact (G, τ ) is minimal if and only if G does not admit a strictly coarser Hausdorff group topology than τ .Furthermore, it is also clear that every totally minimal group is minimal.
The notion of minimality for topological groups was introduced as back as 1971 as a generalization of compactness.In fact it is easy to see that any compact Hausdorff topological group is minimal.For more information about minimality, we refer the reader to the survey by Dikranjan and Megrelishvili [6].
Given a group G of permutations of some set Ω and A ⊆ Ω, let G A = {g ∈ G | ∀a ∈ A, ga = a}.Let [Ω] <ω be the set of all finite subsets of Ω.The collection {G A | A ∈ [Ω] <ω } is a base of neighbourhoods at the identity of a group topology which we call the standard topology and denote by τ st .More generally for each G-invariant X ⊆ Ω there is an associated group topology τ X st generated by {G A | A ∈ [X] <ω }.One of the earliest results on minimality due to Gaughan [9] states that (S ∞ , τ st ) is totally minimal where S ∞ denotes the group of all permutations of a countable set Ω.
Given a countable first order structure M with universe M , the automorphism group of M is a τ st -closed subgroup of S ∞ = S(M ) and vice versa: any closed subgroup of S(M ) is the automorphism group of some countable structure on M .The interplay between the dynamical properties of Aut (M) and the logical and combinatorial properties of M has been widely studied in the literature, beginning with the characterization due to Engeler, Ryll-Nardzewski, Svenonius and others of oligomorphic subgroups of S ∞ as the automorphism groups of ω-categorical countable structures.Recall that an oligomorphic group is a closed subgroup of S ∞ whose diagonal action on M n has finitely many orbits, for each n ∈ N.
In this context τ st is often referred to in the literature as the point-wise convergence topology, in implicit reference to the discrete metric on M .When discussing isometry groups it will be important for us distinguishing between τ st as above and the point-wise convergence topology relative to the metric in question which we will denote by τ m , so we will avoid this practice.
In light of the above the following is thus a natural question, already asked in [6] Problem 1.Let M be a countable ω-categorical (ω-saturated, sufficiently nice) first order structure and G = Aut (M).When is (G, τ st ) (totally) minimal?
A deep result in this direction appeared in recent work by Ben Yaacov and Tsankov [3], where the authors show that automorphism groups of countable ω-categorical, stable continuous structures are totally minimal with respect to the point-wise convergence topology.This specializes to the result that the automorphism groups of classical ω-categorical stable structures are totally minimal with respect to τ st .
Not all oligomorphic groups are minimal with respect to τ st .As pointed out in [3], an example of this is Aut (Q, <) (see Theorem 10.6 for a generalization).However even in those cases it is possible to formulate the following more general question: Problem 2. Let M be a countable ω-categorical (or sufficiently nice) first order structure and G = Aut (M).Describe the lattice of all Hausdorff group topologies on G coarser than τ st .
This work was mainly motivated by [3] and is meant as a preliminary exploration of Problems 1 and 2 in the classical setting outside the stability constraint.
In its broadest lines the strategy followed by [3] goes back to [20], where it was shown by Uspenskij that the isometry group of the Urysohn sphere is totally minimal with the point-wise convergence topology.Both proofs rely on the assumption that the group in question is Roelcke precompact and use a well behaved independece relation among (small) subsets of the structure to endow the Roelcke precompletion of the group with a topological semigroup structure.Information on the topological quotients of the original group is then recovered from the latter via the functoriality of Roelcke compactification and Ellis lemma.Recall that a topological group (G, τ ) is Roelcke precompact if for any neighbourhood W of 1 there exists a finite F ⊂ G such that W F W = G.For closed subgroups of S ∞ this is equivalent to being oligomorphic.
In contrast, our methods for obtaining (partial) minimality results are completely elementary.There are drawbacks to this lack of sophistication: for instance, we are not able to recover the result in [3] for classical structures.On the other hand we do not rely on assumptions of Roelcke pre-compactness (except for certain residual assumptions in some cases).In particular, we are able to answer in the positive the question about the minimality of the isometry group of the (unbounded) Urysohn space posed in [20] (Theorem C).It is worth emphasizing that in some cases we manage to obtain complete classifications of continuous homomorphic images of topological groups which are neither Roelcke precompact nor separable (see Theorem D).
A section by section summary with our main results can be found below.
Free amalgamation and one basedness.Section 3 provides a simple technical criterion (Lemma 3.10) of (relative) minimality for τ st from which more concrete applications are derived in Section 4.
Recall that the free amalgam of two relational structures A, B over a common substructure C is the structure resulting from taking unrelated copies of A and B and then gluing together the two copies of C without adding any extra relations.A free amalgamation class K is a collection of finite structures closed under substructures and free amalgams.Associated with any such K there is a unique Fraïssé limit: a countable structure in which every A ∈ K embeds and which is ultra-homogeneous, i.e., any finite partial isomorphism extends to an automorphism of the structure.
Theorem A. Let M be the Fraïssé limit of a free amalgamation class in a countable relational structure.Let G = Aut (M).Then any group topology τ ⊆ τ st on G is of the form τ X st , where X ⊆ M is some G-invariant set.In particular, if the action of G on M is transitive, then (G, τ st ) is totally minimal.
Simple structures (i.e.theories) occupy an important place in classification theory.We refer the reader to [18], [22] and [12] for the definition of simple theories, forking and canonical bases.We say that a simple theory T is one-based if Cb(a/A) ⊆ bdd(a) for any hyperimagianry element a and small subset A of the monster model.Theorem B. Let M be a simple, ω-saturated countable structure with elimination of hyperimaginaries, locally finite algebraic closure and weak elimination of imaginaries.Assume furthermore that T h(M) is one-based.Let G = Aut (M).Then 1.If G acts transitively on M , then (G, τ st ) is minimal.
2. If all singletons are algebraically closed, then any group topology τ on G coarser than τ st is of the form τ X st for some G-invariant X ⊆ M .By an independence relation is usually meant some ternary relation | ⌣ on (some) collection of sets of parameters of a structure such that A | ⌣C B captures the intuitive idea that B does not contain any information about A not already contained in C. The paradigmatic example is that of forking independence.The connections between the existence of an independence relations on a homogeneous structure satisfying certain axioms and the properties of the automorphism group goes back to [19] (see also [7]).Of particular relevance to us is the freedom axiom, explored in detail in [5].We explain Theorems A and B in terms of the existence of an independence relation satisfying certain sets of axioms.The roles played by stationarity and the freedom axiom in A are replaced by one basedness and the independence property respectively in B.
Generalized universal metric spaces.Urysohn universal space U R is a homogeneous space that contains all separable metric spaces due to Urysohn.It is both ω-universal, i.e. it contains any finite metric space as a subspace and ω-homogeneous, i.e. any partial isometry between finite subspaces of U extends to some global isometry.Associated with the class of metric spaces with diameter at most 1 there is an object with similar properties U [0,1] , known as the Urysohn sphere.The isometry groups Isom(U R ) and Isom(U [0,1] ) endowed with the point-wise convergence topology τ m ('m' is for 'metric') are Polish groups whose algebraic and dynamical properties have been widely studied.It is known, for instance, that any Polish group is isomorphic to a closed subgroup of Isom(U R ) and that Isom(U R ) is extremely amenable.
It is shown in [20] that any continouous quotient of (Isom(U [0,1] ), τ m ) is either trivial or a homeomorphism.Theorem C below extends this result to Isom(U R ).
We work in the framework of generalized metric spaces introduced by Conant in [5].A distance monoid is an abelian monoid endowed with a compatible linear order (see Subsection 5 for more details).Given a distance monoid R = (R, 0, ⊕, ) an R-metric space is a set X endowed with a map d : X 2 → R satisfying the obvious generalization of the axioms for a metric space.In our terminology an R-Urysohn space U will be an R-metric space satisfying the obvious generalization of ωhomogeneity and ω-universality to this setting, ignoring any separability and cardinality considerations.
) ǫ} generates a group topology on the isometry group of U.For plain metric spaces the result is the point-wise convergence topology so we keep the notation τ m in the general case.
We say that a distance monoid R as above is archimedean if for any r, s ∈ R \ {0} there exists some Theorem C. Let R = (R, 0, , ⊕) be an archimedean distance monoid, U a R-Urysohn space, G = Isom(U) and let τ 0 be either: • τ m in case for any r ∈ R \ {0} there exists s ∈ R \ {0} with s ⊕ s r; or, • τ st otherwise.Then τ 0 is the coarsest non-trivial group topology on G coarser than the stabilizer topology τ st .In particular, (G, τ 0 ) is totally minimal.
Given some S ⊆ R closed under addition and b ∈ S >0 ∪ {∞}, we let S b be the distance monoid given by the tuple {{r ∈ S | 0 r b}, 0, , + b }, where x + b y = min{x + y, b}.
Theorem D. Let S be a dense subgroup of R and b ∈ S >0 ∪ {∞}, U an S b -Urysohn space and G = Isom(U).Then there are exactly 4 group topologies on G coarser than τ st : where τ 0 + ,0 is the topology generated at the identity by the collection In Section 8 we describe a general family of group topologies on the isometry group of an R-Urysohn space U that includes all the topologies involved in the two results above.Theorems C and D, can be seen as evidence for the much more general conjecture that these are in fact all group topologies coarser than τ st on Isom(U).
Algebraic minimality: the Zariski topology.Given a group G the Zariski topology τ Z , is generated by the subbase consisting of the sets {x , where n ∈ N, g 1 , . . ., g n ∈ G, and ǫ 1 , . . ., ǫ n ∈ {−1, 1}.According to the result of Gaughan in [9] for the group S ∞ the Zariski topology τ Z and τ st coincide.In Section 9 we investigate the following general question.
Question 3.For which (sufficiently homogeneous) structures is it true that τ Z = τ st .For which of them is the Zariski topology a group topology?
First we provide a variety of Fraïssé limits for which the question above has a negative answer.In all cases this follows via Lemma 9.5 from the property that over Aut (M) of non-trivial equations in one variable have meager sets of solutions.The latter is in turn from the criterion formulated in Lemma 9.3, according to which the conclusion holds whenever any α ∈ Aut (M) \ {1} is what we call strongly unbounded (Definition 9.1).Intuitively, the latter means that points 'largely displaced' by α are in some sense dense in M .Theorem E below collects some miscellaneous results found in Corollary 9.14 (for 1.), 9.19 (for 2.), 9.21 (for 3.) and Corollary 9.25 (for 4.) below.
Theorem E. The Zarisski topology τ Z on Aut (M) is not a group topology if M = Flim (K) for a Fraïssé class K in a relational language L in each of the following cases: 1. K is a non-trivial free amalgamation class and the action of G on M is transitive; 2. M is the rational Urysohn space; 3. M is the random tournament; 4. K is of the form K 1 ⊗ K 2 (see Definition 9.23) for strong amalgamation classes K 1 and K 2 where: • K 1 is non-trivial and either it is as in 2. or the action of Aut (Flim (K 1 )) on the set Here we say that K is trivial if the equality type of a tuple from M determines its type or, equivalently, if Aut (M) is the full symmetric group.Additionally in Corollary 9.29 we can prove the following: Theorem F. Suppose M η is the Hrushovski generic structure that is obtained from a pre-dimension function with the coefficient η ∈ (0, 1].Then the Zariski topology for Aut (M η ) is not a group topology.
On the flip side there is the following positive result: Theorem G.The Zariski topology τ Z on Aut (M) is a group topology in case M is one of the following: • Some reduct of (Q, <); • A countable dense meet-tree or the lexicographically ordered dense meet-tree, in which case τ Z = τ st ; • The cyclic tournament S(2).
Topologies and partial types.Finally in Section 10, we present a natural variant of ideas of [20] and [3] in the context of automorphism groups of first order structures.Given a structure M with group of automorphisms G, we describe a semi-group of partial types R pa (M) containing G consisting of partial types and show that any idempotent in R pa (M) which is invariant under the involution and the action of G can be associated to a group topology on G coarser than τ st .
2. Review of some classical constructions of homogeneous structures 2.1.Fraïssé construction.Let us briefly review the Fraïssé construction method in a relational language.For a more detailed and general introduction see Chapter 6 in [11].
Let L be a relational signature and K be a countable class of finite L-structures closed under isomorphisms.Suppose A, B ∈ K by A ⊆ B we mean A is an L-substructure of B. We say K is a Fraïssé class if it satisfies the following properties: • (HP) It is closed under substructures; We say that a Fraïssé class K has strong amalgamation if in (AP) we might assume that h According to a theorem of Fraïssé for any Fraïssé class K there is a unique countable structure M called the Fraïssé limit of K and denoted by Flim (K), such that: • M is ultrahomogeneous, i.e. every finite partial isomorphism between substructures of M extends to an automorphism of M; • Age(M), the collection of all finite substructures of M, coincides with K. Classical examples of Fraïssé limit structures are (Q, <) and the random graph.If L is empty, then K is the class of finite sets and Flim (K) an infinite countable set.More in general, we say that K is trivial if the equality type of a finite tuple of elements from M determines its type (equivalently, if Aut (M) is the full permutation group of M ).

2.2.
Hrushovski's pre-dimension construction.Originally Hrushovski's pre-dimension construction was introduced as a means of producing countable strongly minimal structures which are not field-like or vector-space like.There are many variants of the method, but to fix notation, we consider the following basic case and later focus on a version that produces ω-categorical structures.We refer readers to [21], [1] and [7] for most of the properties that are mentioned here about Hrushovski constructions and some of their variations.Suppose s 2 and η ∈ (0, 1] .We work with the class C of finite s-uniform hypergraphs, that is, structures in a language with a single s-ary relation symbol R(x 1 , . . ., x s ) whose interpretation is invariant under permutation of coordinates and satisfies R(x 1 , . . ., x s ) → i<j (x i = x j ).
To each B ∈ C we assign the predimension where R[B] denotes the set of hyperedges on B. For A ⊆ B, we define A B iff for all A ⊆ B ′ ⊆ B we have δ(A) δ(B ′ ), and let Moreover, if N is an infinite Lstructure such that A ⊆ N , we write A N whenever A B for every finite substructure B of N that contains A. One can show C η has the -free amalgamation property (cf.Lemma 4.8 in [1]), by which we mean free amalgamation with inclusions.An analogue of Fraïssé's theorem holds in this situation: Proposition 2.2.There is a unique countable structure M η , up to isomorphism, satisfying: 1.The set of all finite substructures of M η , up to isomorphism, is precisely C η ; The structure M η , that is obtained in the above proposition, is called the Hrushovski generic structure.
2.2.1.ω-categorical case.Here we briefly discuss a variation on the Hrushovski's pre-dimension construction method as a way to generate ω-categorical structures.The original version of this is used to provide a counterexample to Lachlan's conjecture, where it is used to construct a stable ω-categorical pseudoplane (see Section 5 in [21]).Here we follow similar setting used Section 5.2. in [7].
Suppose η = m n ∈ (0, 1] where gcd(m, n) = 1.Consider the same setting of the previous subsection for L and C η .Choosing an unbounded convex increasing function f : R 0 → R 0 which is "good" enough one can consider ) has the d -free amalgamation property and d is defined as follows: A d B when δ(A ′ /A) > 0, for each A A ′ ⊆ B. In this case we have an associated countable generic structure M f η which is ω-categorical.Remark 2.3.As a good function we can take some piecewise smooth f where its right derivative f ′ satisfies f ′ (x) 1/x and is non-increasing, for x 1.The latter condition implies that f (x + y) f (x)+yf ′ (x) (for y 0).It can be shown that under these conditions, C f η has the free d -amalgamation property.
We assume that f is a good function.We will assume that f (0) = 0 and f (1) > 0, and in this case the -closure of empty set is empty.We shall also assume that f (1) = n and one can show Aut M f η acts transitively on M f η .See examples 5.11 and 5.12. in Section 5.2.[7] for details.

A relative minimality criterion for τ st
Given a topological group (G, τ ) and g ∈ G we denote by N τ (g) the filter of neighbourhoods of g in τ .Since N τ (g) = gN τ (1) = N τ (1)g for any g ∈ G, any group topology τ is uniquely determined by N τ (1).Given a filter V on G at 1 such that • U g ∈ V for every U ∈ V and g ∈ G; then there is a unique group topology τ on G such that V = N τ (1).Given a family Y of subsets of G containing 1, we say that Y generates a group topology τ at the identity if Y generates N τ (1) as a filter.
Given a set X we let [X] <ω stand for the collection of all finite subsets of X.Our setting consists of an infinite set Ω and some G S(Ω), where S(Ω) is the group of permutations of Ω.It is easy to see using the criterion above that the collection {G A | A ∈ [Ω] <ω } is a base of neighbourhoods of the identity of a unique group topology τ st , which we will refer to as the standard topology.We are mainly interested in the case in which Ω is countable, in which case S(Ω), abbreviated as S ∞ , is a Polish group.
By a closure operator on [Ω] <ω we mean a map cl : [Ω] <ω → [Ω] <ω that preserves inclusion and satisfies A ⊆ cl (A) = cl (cl (A)), for each A ∈ [Ω] <ω .There is a bijective correspondence between (G-equivariant) closure operators cl and (G-invariant) families X ⊆ [Ω] <ω closed under intersections.Each X gives a closure operator cl (−) by taking as cl (A) the smallest set in X containing A. In the opposite direction we associate cl with the class of cl-closed sets: Given tuples A, B, C of elements from Ω we write A ∼ = G B if there exists some g ∈ G such that gA = B and given an additional C we write Given A ⊂ Ω we let acl G (A) stand for the union of all elements of Ω whose orbit under G A is finite.We say acl G (−) is locally finite if acl G (A) is finite whenever A is.In that case the restriction of acl G to [Ω] <ω is a closure operator on [Ω] <ω .We write Given a family X of subsets of a set Ω, denote by (X) the collection of all tuples of elements whose coordinates enumerate some member of X.As is customary, the same letter will be used to refer to either a tuple or the corresponding set depending on the context.In particular we might use an expression such as BC to denote the union of the ranges of B and C.
Let G be the group of automorphisms of some structure M with universe M .Recall that if M is ω-saturated, then for finite A we have that acl G (A) coincides with the algebraic closure of A. If M is ω-saturated and countable, then in particular it is ω-homogeneous, i.e.
Let G be a group of permutations of a set Ω for which acl G (−) is locally finite.Suppose we are given some G-invariant X ⊆ Ω and another group topology τ * ⊂ τ X st such that for some constant K ∈ N the following property holds: (⋄) For any A, B ∈ X G and U ∈ N τ * (1) there exists Then any group topology τ ⊆ τ X st must satisfy at least one of the following two conditions: 1.Given x ∈ X there exists W ∈ N τ (1) such that gx ∈ acl G (x), for each g ∈ W ; or, 2. There exists some G-invariant X ′ X such that for all W ∈ N τ (1) there is Proof.Assume the first alternative does not hold.Then there is x 0 ∈ X such that for any W ∈ N τ (1) Our goal is to show point 2., that is, that any neighbourhood W of 1 in τ is also a neighbourhood of the identity in any topology containing τ * and τ X ′ st .Observation 3.2.For any a ∈ G • x 0 , any finite B ⊂ Ω and any W ∈ N τ (1) there exists some g ∈ W such that ga / ∈ B.
Proof.Suppose the condition above fails for some a, B, and W .By Neumann's lemma there exists some h ∈ G a such that h(B) ∩ B ⊆ acl G (a).This means that any g in W ∩ W h −1 ∈ N τ (1) must take a to a point in acl G (a), a contradiction.
The following observation follows from (⋄) by an induction argument.
Observation 3.3.There is a function µ : N → N such that given any finite collection st , there exists some finite A ⊂ X such that G A ⊆ W 0 .By local finiteness we may assume A = acl G (A).
⊆ W 0 , where µ is the function given by Observation 3.3.Let B ⊂ Ω be a finite subset such that G B ⊂ W 1 .We may assume again B ∈ X G .By Observation 3.2 for any 1 j r there exists some g j ∈ W 1 such that g j a j / ∈ B or, equivalently, By construction C ∩ A ⊆ X ′ so we are done.
Here is another instance of the same idea.
Lemma 3.4.Let G be a group of permutations of a set Ω, {X j } j∈J some collection of G-invariant subsets of Ω and Z = j∈J X j .Assume that acl G (x) = x for any x ∈ Ω and that there exists K > 0 such that for any finite A, B ⊂ Ω we have

one can show by induction:
Claim.There exists a function µ : N → N such that for any finite collection r choose some j l ∈ J such that a l / ∈ X j l and then some finite B l ⊆ X j l such that G B l ⊆ W 0 .The Claim and the choice of W 0 implies G C ⊆ W where C = A ∩ r l=1 B l .Since C ⊆ Z we are done.Lemma 3.5.Let G be the automorphism group of some structure M endowed with a G-invariant locally finite closure operator cl (−) on M and a group topology τ coarser than τ st .Assume that the action of G is transitive and there is some W ∈ N τ (1) and a ∈ M such that ga ∈ cl (a), for each g ∈ W . Then either τ is not Hausdorff or τ = τ st .
Proof.Notice that by the transitivity of the action of G on M and continuity of the inverse operation for every a ∈ M there are U a , W a ∈ N τ (1) such that f (a) ∈ cl (a) for any f ∈ W a and g −1 (a) ∈ cl (a) for any g ∈ U a .For a finite tuple A in M we write . For any V ∈ N τ (1) and any finite ∼-closed A ⊂ M consider the set Notice that this set is finite, and that given Invariance should be clear from the fact that A is ∼-closed and the definition of Y A V .Claim 3.6.Either Y A V = {id A } for some V ∈ N τ (1) and ∼-closed A or there exists f ∈ G such that for all ∼-closed A ⊂ M and all V ∈ N τ (1) we have f ↾ A ∈ Y A V . of Claim.Recall that according to the assumption the closure is locally finite.If the first alternative is not the case, then from Observation 3.3 and König's lemma follows that there is a function V for any ∼-closed A and V ∈ N τ (1).The fact that f ↾ A is a type-preserving bijection of A for any such A implies f ∈ G.
If the first possibility in Claim 3.6 holds true, then G A contains W ′ A ∩V and is thus a neighbourhood of the identity in τ , which implies that τ = τ st .We claim that if the second possibility is satisfied the resulting f ∈ G \ 1 satisfies f ∈ V ∈Nτ (1) V , so τ is not Hausdorff.Given any V ∈ N τ (1), the closure in τ st of any symmetric W ∈ N τ (1) ∩ τ st satisfying W 2 ⊂ V is itself contained in V .Hence, N τ (1) admits a basis consisting entirely of τ st -closed neighbourhoods of the identity.It is thus enough to show that f belongs to the closure of V in τ st for any V ∈ N τ (1), which is immediate from the definition of Y A V .
The following ubiquitous observation is crucial for the application of the results above.We provide a proof for the sake of completeness.Lemma 3.7.Let G be a group of permutations of a set Ω and A, B tuples of elements from Ω for which there is a chain Proof.The proof is by induction on n.In the base case n = 0 we have Definition 3.8.Suppose we are given a group G of permutations of a set Ω, and X a G-invariant family of subsets of Ω closed under intersection.We say X has the n-zigzag property (with respect to the action of G) if for every A, B ∈ (X) and any A ′ with A ∼ = G A∩B A ′ there are A 0 , . . ., A n and B 0 , . . ., B n−1 such that We will refer to the sequence A 0 , B 0 , A 1 , . . ., A n above as an (n, B)-zigzag path from A to A ′ .Observation 3.9.Given an n-zigzag path as above it is easy to show by induction that if we write Notice that for fixed A, B and n the existence of a (n, B)-zigzag path from A to A ′ depends only on the orbit of A ′ under G A .Proposition 3.10.Suppose M is a countable first order structure and G = Aut (M).Assume acl G (−) is locally finite and the corresponding X G has the n-zigzag property for some n.Then: Proof.Let us show 1. first.Let τ be a group topology on G coarser than τ st .By Lemma 3.7 it is possible to apply 3.1 with τ * = {∅, G}.If the first alternative of 3.1 holds, then by Lemma 3.5 either τ is not Hausdorff or τ = τ st .Since by assumption the only invariant subsets of M are ∅ and M , the second alternative implies that τ = {∅, G}.
Let us now show 2.. Let τ be a group topology on G coarser than τ st .By Lemma 3.4 there exists some unique minimal G-invariant set X such that τ ⊆ τ X st .Apply Lemma 3.1 with τ * = {∅, G}.The second alternative produces some G-invariant X ′ X such that τ ⊆ τ X ′ st , in contradiction with the choice of X.Since we assume acl G to be trivial, the first alternative implies τ = τ X st .

Minimality and independence
4.1.Independence.Throughout this section we work in the following setting: the associated family of closed sets.Our goal is to derive concrete applications from the results of the previous section to the case where Ω is the underlying set of a first order structure M and G = Aut (M).
Definition 4.2.We define some additional properties for a compatible pair (X, | ⌣ ): ⌣A∩B B for every A, B ∈ X.The one-basedness property admits the following generalization: ⌣C A k (notice that for k = 1 we recover the one basedness property).Lemma 4.4.Let (X, | ⌣ ) be a compatible pair that satisfies existence.Then 1.If it satisfies freedom or one-basedness, then for any A, B ∈ X there is If it satisfies transitivity, symmetry and 2m-narrowness, then for any A, B ∈ X there is Alternatively, the same conclusion follows directly from one-basedness. Let by 3.9 we conclude that A j ∩ A l = C. Arguing in a similar manner one can show that A j ∩ B l = C for any 0 j m and 0 l m − 1.This establishes that the sequence A 0 , B 0 . . .B m−1 A m satisfies the first property of the condition in the definition of 2m-narrowness, while the second follows by transitivity and construction.If we let A ′ = A m we then get A ′ | ⌣C A and A | ⌣C A ′ by symmetry, while the sequence above is an (m, B)-zigzag path from A to A ′ .
Lemma 4.5.Let (X, | ⌣ ) be a compatible pair satisfying symmetry existence and transitivity and assume that for any A, B ∈ X there exists an (m, B)-zigzag path from A to some A 1 such that If stationarity holds, then X has the 2m-zigzag property; 2. If independence holds, then X has the 4m-zigzag property.
In both cases using the assumption we start by choosing A 1 ∈ X for which there is an m-zigzag path from A to A 1 and ⌣C A ′ A by right transitivity.By weak monotonicity we get A 2 | ⌣C A ′ and by symmetry Thus, there is also an (m, B ′ )-zigzag path from A 2 to A ′ , where A ′ B ′ ∼ = G AB and combining both paths we get a (2m, B)-zigzag path from A to A ′ .
We move on to case 2..By invariance and existence there is

Transitivity and monotonicity then imply
This witnesses the existence of a (4m, B)-zigzag path from A to A ′ .Notice that symmetry is required in order to get Theorem (A).Let M be the Fraïssé limit of a free amalgamation class in a countable relational structure.Let G = Aut (M).Then any group topology τ ⊆ τ st on G is of the form τ X st , where X ⊆ M is some G-invariant set.In particular, if the action of G on M is transitive, then (G, τ st ) is totally minimal.
Proof.If we let X = [M ] <ω where M is the underlying set of M and | ⌣ = | ⌣ f r , then part 1. of Lemma 4.4 and part 1. of Lemma 4.5 apply to the pair (X, | ⌣ ).Together, they imply X has the 2-zigzag property with respect to the action of G.The result then follows from an application of Proposition 3.10.

Theorem (B).
Let M be a simple, ω-saturated countable structure with elimination of hyperimaginaries, locally finite algebraic closure and weak elimination of imaginaries.Assume furthermore that 2. If all singletons are algebraically closed, then any group topology τ on G coarser than τ st is of the form τ X st for some G-invariant X ⊆ M .Proof.As cl we take the algebraic closure acl and | ⌣ the forking independence.We claim part 1. of Lemma 4.4 and part 2. of Lemma 4.5 both apply to (X, | ⌣ ).The pair clearly satisfies invariance, weak monotonicity, transitivity and symmetry.Existence follows from the fact that M is ω-saturated, so it is left to check one-basedness and independence in sense of Definition 4.2.
Take A, B ∈ X.The fact that the theory is one-based in the sense of simplicity theory and has elimination of hyperimaginaries implies A | ⌣acl eq (A)∩acl eq (B) B. The relation A | ⌣A∩B B follows then from weak elimination of imaginaries.
Lastly, elimination of hyperimaginaries and weak elimination of imaginaries imply that the type of a tuple over a finite real closed set determines its Lascar strong type over that same set.Hence, Kim and Pillay's independence theorem [13] (see also Chapter 2.3 and Theorem 2.3.1 in [12]) translates into abstract independence (amalgamation of types) for (acl, | ⌣ ).
It is known that simple one-based ω-categorical structures have elimination of hyperimaginaries.This follows from the fact that ω-categorical theories are small and simple one-based theories admit finite coding.See section 6 and Proposition 6.1.21. in [22] for definitions and details.For stable theories the notion of being k-ample (for some k 1) generalizes the negation of one-basedness.See [8] for details.In the absence of algebraic closure being not k-ample translates into (acl, where | ⌣ f is the forking independence.From an argument similar to the one in the two theorems above we can deduce the following result: Theorem 4.6.Let M be a countable ω-saturated stable structure such that T h(M) has trivial algebraic closure, weak elimination of imaginaries and is not k-ample for some k 1.Then any group topology on G = Aut (M) coaraser than τ st is of the form τ X st for some G-invariant X ⊆ M .
Example: total minimality is not preserved under taking open finite index subgroups.
Consider the relational language L 1 = E (2) , P (1) and let K 1 be the class of all finite L 1 -structures in which E is interpreted as the edge relation of a bipartite graph with with edges only between the domain of the unary predicate P and its complement.Consider also the class K 2 in the language L 2 = {E (2) , F (2) } consisting of all finite L-structures in which F is interpreted as an equivalence relation with at most 2 classes and E as the edge relation of a bipartite graph with edges only among vertices that belong to distinct F -classes.
It is easy to check that K 1 has free amalgamation and then by Theorem A there are exactly two group topologies on G 1 strictly coarser than τ st , namely τ P (M1) st and τ ¬P (M1) ts .Notice that both are Hausdorff, since no automorphism of M 1 can fix P (M 1 ) or its complement (given any two points a, b, there exists c in P (resp ¬P ) such that tp(c, a) = tp(c, b)) so (G 1 , τ st ) is not minimal.
In this case we have an additional non-Hausdorff group topology, τ * = {∅, G 1 }.Apply Lemma 3.1 to conclude that any group topology on G 1 strictly contained in τ st is contained in τ * .
On the other hand, it follows from Theorem B that (G 2 , τ st ) is minimal.

4.2.
Simple non-modular Hrushovski structures.In Subsection 2.2 we discussed in more detail some instances of the Hrushovski construction, in particular the ω-categorical version (see section 5.2. in [7] for more).Here is a brief reminder of the setting: Choosing an unbounded convex function f which is "good" enough, one can consider C f η , a subclass of C η , with the free amalgamation property where the limit structure M f η is ω-categorical and such that Aut M f η acts transitively on M f η (underlying set of M f η ).It is shown in Lemma 5.7 in [7] that there is an independence relation defined for the class ofclosed subsets of M f η that satisfy all the properties of part 1. in Lemma 4.5.Then using Proposition 3.10 we conclude the following.

Generalized Urysohn spaces
We start by recalling some notions from [4].A distance magma R = (R, , ⊕, 0) is a set R endowed with a linear order and an operation ⊕ such that the following axioms are satisfied: When referring to a monoid R, unless anything to the contrary is said, it will be implicit in the notation that R is its underlying set, and so forth.We say that R is a distance monoid if additionally ⊕ satisfies associativity, i.e.: Given some additively closed subset S of some ordered abelian group (Λ, +, ) and b ∈ {S >0 , ∞} the structure S b = {{r ∈ S | 0 r b}, 0, , + b } is a distance monoid, where x + b y = min{b, x + y} for b ∈ S and x + ∞ y = x + y.We write S for S ∞ and Q b = S b in case S = Q 0 .We will refer to any distance monoid R of the form S b as basic.If additionally S is a subgroup of Λ with no minimal element then we will say R is standard.When talking about a standard distance monoid we may use the symbols + (as opposed to ⊕) and − to refer to the operations in the ambient group Λ without explicitly referencing Λ.Notice that in the case of basic archimedean distance monoids we can always assume Λ = R.
Given m ∈ N and r ∈ R we will write m • r for the ⊕ addition of r with itself m times.Given two elements r, s ∈ R, we write r ∼ s if there exists some positive integer n such that n • r s and n • s r.We refer to the ∼-class [r] of r as its archimedean class.A distance monoid with a single archimedean class of non-zero elements will be called archimedean.We write Fix a distance magma R := (R, , ⊕).An R-metric space (X, d) consists of a set X together with a map d : X 2 → R such that for all x, y, z ∈ X: ) is a substructure of (R 0 , ) and r ⊕ s r + s for all r, s ∈ R, then an R-metric spaces are just a particular class of metric spaces.In particular, this holds for standard archimedean distance monoids.
An isometric embedding of an R-metric spaces (X, d) into another y), for each x, y ∈ X.A surjective isometric embedding is called an isometry.Given an R-metric space (X, d), we let Isom(X, d) stand for the group of isometries from (X, d) to itself.We will use the symbol ∼ = to denote the existence of an isometry between two tuples in R-metric spaces.
In the same spirit, given finite tuples A = (a i ) k i=1 and A ′ = (a ′ i ) k i=1 inside an R-metric space we will write A ∼ =B A ′ if there is a partial isometry fixing B and sending each a i to a ′ i , that is, if for any for distinct i and j.
We will say that an R-metric space U is an R-Urysohn space if it satisfies: (U) Any finite R-metric space embeds in U; and, (H) Any isometry between finite subspaces of U extends to an isometry of U. The following strengthening of (U) is implied by the conjunction of (U) and (H) and equivalent to it under the assumption that U is countable.
(EP) For any finite R-metric space B and A ⊆ B any isometric embedding h : A → U extends to some h : B → U.This implies that the class K of all finite R-metric spaces has the joint embedding and amalgamation properties.See 2.7 in [4] for a more precise result (here we are only interested in S = R).Therefore if R is countable, then K determines a unique countable Fraïssé limit structure U R = Flim (K).This is a countable R-metric space satisfying property (EP) above and thus an R-Urysohn space (see Theorem 2.7.7 in [4]).An object satisfying the two properties above might exist even if R is not countable.The classical Urysohn space and Urysohn sphere are examples of this for R = (R 0 , 0, , +) and R = ([0, 1], 0, , + 1 ) respectively.
Given finite sets A, B of an R-metric space (X, d) we define diam (A We generalize this notation to the case in which C is not a common subsets of A and B by letting A | ⌣C B if and only if AC | ⌣C BC.

Isometry groups of archimedean Urysohn spaces
The goal of this section is to prove Theorem C of the introduction.We start with three preliminary lemmas in the following general setting: R = (R, 0, , ⊕) is a distance monoid and U an R-Urysohn space.Lemma 6.1.Suppose A and B are finite subsets of U and r ∈ R such that diam (A) r 2 • d(A, B).Then there is A ′ ⊆ U such that A ′ ∼ =B A and d(a, a ′ ) = r for all a ∈ A and a ′ ∈ A ′ .

Proof. Consider the set D = A ′
B A ′′ which is the amalgamated union of two copies A ′ , A ′′ of A over B. We define an R-valued distance function on D as follows.On A ′ B and A ′′ B the distance between two points equals the distance between the corresponding pair in U, while we set d(a ′ , a ′′ ) = r for any a ′ ∈ A ′ and a ′′ ∈ A ′′ .In order to show that the resulting function satisfies the triangle inequality it suffices to check triples {u, v, w} with u ∈ A ′ , v ∈ B and w ∈ A ′′ .We have Then the result follows from (EP).Proof.Take A ′ ∼ =B A with A ′ | ⌣B C. Construct a sequence A i , B i , for i 0 as follows.We start by taking A 0 = A ′ and B 0 = B.For any 0 i < n let C i = (CB j A j ) j i and take A, B).The result follows by an easy induction argument.
Proof.The proof for a general k follows by a simple induction argument from case k = 2, whose proof we now present.Take Given an R-metric space (X, d), a point x ∈ X and ǫ ∈ R\{0} let N x (ǫ) := {g ∈ Isom(X, d) | d(gx, x) ǫ}.The following claim is easy to check.See Lemmas 8.2 and 8.6 below for a more detailed explanation.Claim 6.4.Suppose a distance monoid R has the property that for any r ∈ R \ {0} there exists s ∈ R\{0} with s⊕s r.Then for any R-metric space (X, d) the collection {N x (ǫ) | x ∈ X, ǫ ∈ R\{0}} generates a Hausdorff group topology on G = Isom(X, d) at the identity.
We denote the topology above by τ m .For metric spaces this is just the usual point-wise convergence topology on G ⊆ X X .The following theorem generalizes Uspenski's minimality result for the isometry group of the Urysohn sphere.
Then τ 0 is the coarsest non-trivial group topology on G coarser than the stabilizer topology τ st .In particular, (G, τ 0 ) is totally minimal.
Proof.Suppose τ 0 does not satisfy the conclusion of the theorem and let τ be a group topology that is coarser than τ st but not finer τ 0 .This implies that there is Proof.Assume the conclusion fails.Take h ∈ G a such that h(B) and B are independent over a where Proof.Consider W be a neighbourhood of 1 in τ with W = W −1 such that W 6k−3 ⊂ V .Let C 0 be a finite subset of U such that G C0 ⊆ W .By Claim 6.5 there is g i ∈ W such that d(g i a i , C 0 ) > t for each 1 i k.Then d(g −1 i (C 0 ), a i ) > t for each i and by applying Lemma 6.3 with Lemma 6.7.For any V ∈ N τ (1), any finite C ⊆ U and r ∈ R\{0} there is a finite D with d(D, C) r and G D ⊆ V .
Proof.Recall that s is fixed before Lemma 6.5.We claim that there exists t 0 ∈ R such that 2t 0 s and for any r ∈ R there is m ∈ N such that mt ′ r for any t ′ > t 0 .Indeed, either there is t > 0 with 2t s, in which case we can take t 0 = t, or else 2t ′ > s for all t ′ > 0 and we can take t 0 = 0.
Fix now V ∈ N τ (1) and r ∈ R and let m be as above.Let By Lemma 6.6 applied to C and W there are A and We are now ready to finish the proof of Theorem C. Pick any neighbourhood W of 1 in τ with W = W −1 and g ∈ G\{1} where g / ∈ W 4 .Since W is a neighbourhood of identity in τ it must contain G A for some finite subset A of U. Lemma 6.7 implies the existence of some finite B ⊂ U such that G B ⊂ W and d(A g(A), B) diam (A g(A)).By Lemma 6.1 there is an isomorphic copy A ′ of A over B and s ∈ R such that d(a, a ′ ) = s for all a ′ ∈ A ′ and a ∈ A g(A).
In particular, A ∼ =A′ g(A), which implies there is B ′ such that g(A)B ′ ∼ = A ′ B ′ .By Lemma 3.7 the chain A, B, A ′ , B ′ , g(A) witnesses g ∈ (G A G B ) 2 ⊆ W 4 , contradicting the choice of W .

Group topologies on Isom(U) coarser than τ st
In the light of Theorem C one might conjecture there is a gap between τ st and the point-wise convergence topology τ m in those cases in which the latter exists.This turns out to be false.
Fix a distance monoid R, an R-metric space (X, d) and let G = Isom(X, d).For any distinct x, y ∈ X write The following is easy to check; see Lemmas 8.2 and 8.6 of the following section.
Due to the following obstruction the topology τ 0,0 + is not eliminated by an application of Lemma 3.1 to the pair (τ m , τ st ).Take for instance U = U R and two disjoint sets A, B ⊂ U R of size k 1 that lie entirely on a common line (i.e.any triangle spanned by three points in AB is degenerate) and alternate on said line.Let us say that the "leftmost" point is α ∈ A and the "rightmost" point is β ∈ B. Then for any chain , where β and β ′ are components of the same index in B and B k respectively.The main content of Theorem D is the existence of a gap between τ 0 + ,0 and τ m (Proposition 7.13), which involves a series of small technical intermediate Lemmas collected in subsections 7.1 and 7.2.In contrast, the existence of a gap between τ st and τ 0 + ,0 (Lemma 7. 19) is a direct consequence of Lemma 3.1.
The Lemmas in Subsection 7.1 highlight different aspects of the obstruction mentioned above.In particular, Lemma 7.3 can be read as saying that this is in fact the only obstruction for the assumptions of Lemma 3.1 to hold for the pair (τ m , τ st ).
Subsection 7.2 gathers Lemmas allowing one to move downwards: we are given a group topology τ and we know that there exists W ∈ N τ (1) such that all g ∈ W preserves a certain property and we want to replace it with W ′ ∈ N τ (1) such that all g ∈ W ′ preserve some different (stronger) property.
7.1.Point alignment.For future reference we state Lemma 7.3 and other lemmas in this section in greater generality than required by Theorem D. The reader might as well take R = S b where S is some dense subgroup of (R, 0, , ⊕) and b ∈ S >0 ∪ {∞}.In the definitions below R = (R, 0, , ⊕) is a distance monoid and (X, d) an R-metric space.
Given ǫ ∈ R we say that an unordered triple r 1 , r 2 , r 3 ∈ R is ǫ-flexible if r i ⊕ ǫ r j ⊕ r k where r i = max{r 1 , r 2 , r 3 } and {j, k} = {1, 2, 3} \ {i}.We say that it is strongly ǫ-flexible if moreover for any r ′ j and r ′ k such that r ′ j ⊕ ǫ r j and r ′ k ⊕ ǫ r k we have r i min{r ′ j ⊕ r k , r j ⊕ r ′ k }.A 0-flexible triple will be called simply triangular.We say that a triangular triple is (strongly) flexible if it is (strongly) ǫ-flexible for some ǫ ∈ R \ {0}.
We say that an unordered triple of points We say it is tight if it is not ǫ-flexible for any ǫ > 0• We say that an ordered triple of points (u 1 , u 2 , u 3 ) ∈ U 3 in some R-metric space is aligned if it is tight as an unordered triple and d(u 1 , u 3 ) = max {d(u i , u j ) | i, j ∈ {1, 2, 3}}.Given u, v ∈ U we let [u, v] stand for the collection of points x such that (u, x, v) is aligned.
We say that a set of three distinct points Given two finite subsets A, B ⊂ X and r ∈ R * we say that B r-cuts A if there exists a, a ′ ∈ A with d(a, a ′ ) r such that B ∩ [a, a ′ ] = ∅.We say that B cuts A if it r-cuts A for some r ∈ R. We say that B is in (ǫ-)general position relative to A if for any distinct a 1 , a 2 ∈ A and any b ∈ B the triple a 1 , a 2 , b is in (ǫ-)general position.Notice that in particular this implies that B does not cut A.
The following Lemma is the main source of motivation of the definitions above.
, where the second inequality comes from ǫ-flexibility of {a 1 , a 2 , b}.On the other hand d(a 1 , a ′ 2 ) ⊕ ǫ d(a 1 , a 2 ) so strong ǫ-flexibility of {a 1 , a 2 , b} yields: We say that R has no gaps if for any r < s there exists ǫ ∈ R \ {0} such that r ⊕ ǫ s.
Otherwise for some ǫ > 0 and i ∈ {1, 2} we have: , then for some ǫ > 0 we have: for some ǫ > 0, since R has no gaps.So in case {b 1 , b 2 } ⊆ B 2 , then the second alternative in the statement holds, while if {b 1 , b 2 } ∩ B 1 = ∅, then the first one must hold.
Corollary 7.5.Let R be a distance monoid with no gaps and U an R-Urysohn space.Let r ∈ R and A j , B j , 1 j k be finite subsets of U such that A j does not r-cut B j for any 1 j k.Then there Proof.The argument is analogous to the one in the proof of Claim 6.3.We will restrict to the case k = 2, since the general case can be deduced from it by an easy induction argument.Take B ′ 1 Both alternatives are ruled out by our assumptions on A i , B i and the fact that Lemma 7.6.Let R be a standard distance monoid and suppose we are given a triangular triple r 1 , r 2 , r 3 ∈ R \ {0} with r 1 r 2 r 3 as well as ǫ 1 , ǫ 2 , ǫ 3 , δ ∈ R such that: • 2 • ǫ i r 1 for every 1 i 3; Proof.On the one hand: where use the fact that ǫ 3 ⊕ δ ǫ 1 ⊕ ǫ 2 .On the other hand, for i ∈ {1, 2} we have: This shows that the triple is δ-flexible.Moreover, for i ∈ {1, 2} we have Lemma 7.7.Let R be a standard distance monoid with no gaps, no minimal positive element and such that for any r ∈ R there exists s ∈ R such that s ⊕ s r.Let U an R-Urysohn space and A, B ⊂ U finite sets such that B does not cut A. Then there exist A ′ ∼ =B A such that A ′ is in general position relative to A.
for all a, a ′ ∈ A and b ∈ B (we include the case a = a ′ ).Fix some symmetric injective function f : A × A → (0, ǫ) ⊂ R such that: Let us first show how to prove the Lemma using the claim above.We need to show that for any a 1 , a 2 , a 3 ∈ A, a 1 = a 2 , the triple a 1 , a 2 , ha 3 is strongly δ-flexible, where δ is the minimum between the minimum value of f and the smallest absolute difference between two values in the image of f .If a 3 = a i for some i = 1, 2, let's say If a 1 , a 2 , a 3 are all different we apply Lemma 7.6 to the three distances between a 1 , a 2 , a 3 , which we name in increasing order r 1 r 2 r 3 .Here we take ǫ i = 0 exactly for one value of i for which r i := d(a 1 , a 2 ), while for i = 1, 2 if r j = d(a i , a 3 ), we take ǫ j = f (a i , a 3 ).
The first condition in Lemma 7.6 follows immediately from the first property of f .On the other hand, our second condition on f guarantees that whichever element in {ǫ 1 , ǫ 2 } is non-zero must be also larger than ǫ 3 so the second assumption of Lemma 7.6 also holds.The third one follows from the choice of δ.
It follows from 7.6 for any a 1 , a 2 , a 3 ∈ A, a 1 = a 2 , the triple a 1 , a 2 , ha 3 is strongly flexible.This in turn implies that hA is in general position relative to A.
Let us now prove the Claim.For i = 1, 2 let A i = {a i | a ∈ A} be a copy of A and D := A 1 A 2 B. It suffices to check that the function d : D 2 → R given by: for any a 1 , a 2 ∈ A; satisfies the triangle inequality.For triples of points contained in A 1 ∪ A 2 this is part of the conclusion of Lemma 7.6.All that is left to check is the triangle inequality for triples of the form (a 1 1 , a 2 2 , b), where a 1 , a 2 ∈ A and b ∈ B. We have 2 ) by the choice of ǫ and the fact that im(f ) ⊆ (0, ǫ), while the remaining inequalities are straightforward from the inequality d(a 1 , a 2 ) d(a 1 1 , a 2 2 ).
Lemma 7.9.Let R be a standard distance monoid and U an R-Urysohn space.Assume we are given r ∈ R and finite A, B ⊂ U such that B does not r-cut A. Take A ′ such that AB ∼ = A ′ B and A ′ | ⌣B A. Then A ′ does not (r ⊕ r)-cut A.
Proof.As usual, let a ′ stand for the conjugate of any given a ∈ A in A ′ .Aiming for contradiction, suppose that there exist a 1 , a 2 , a 3 ∈ A such that d(a 1 , a 2 ) r ⊕ r and a ′ 3 ∈ [a 1 , a 2 ].This implies, in particular, that d(a 1 , a 2 ) < sup R.
Notice that by independence for all a, ã ∈ A we have d(a, ã′ ) d(a, ã), with equality only if B ∩ [a, ã] = ∅ or d(a, ã) = sup R (in the bounded case).
Since d(a 1 , a 2 ) = d(a 1 , a ′ 3 )+d(a ′ 3 , a 2 ), as d(a 1 , a 2 ) < sup R, the triangle inequality implies d(a j , a ′ 3 ) = d(a j , a 3 ) for j = 1, 2. By the previous paragraph the intersections B ∩ [a 1 , a 3 ] and B ∩ [a 2 , a 3 ] are both non-empty.However, d(a i , a 3 ) r for at least one i ∈ {1, 2}: a contradiction.7.2.Downward lemmas.Throughout this subsection R will be a standard distance monoid, U an R-Urysohn space and G = Isom(U).d(c, d) = d(a, b j ) for 1 j k.Notice also that since τ is not trivial and coarser than τ st by Theorem C it must be finer than τ m so that N x (ǫ) ∩ U ∈ N τ (1) for any x ∈ U and ǫ > 0.
For any g ∈ W ′ we have g ∈ N sp a,bj for some 1 j k.Combining this with the assumption on d(a, b l ), 1 l k, independence and the triangular inequality we get for any 1 i k: Pick some U ∈ N τ (1) such that U k ⊂ W ′ .We construct a sequence of elements g 1 , g 2 , . . ., g k ∈ U and ǫ 1 ǫ 2 . . .ǫ k ∈ R \ {0} in the inductive fashion described below.We use the notation ḡj = g j g j−1 . . .g 1 for 1 j k.
To start with, we choose . Now, suppose that for some 1 j k the elements g 1 , g 2 , . . .g j and ǫ 1 , ǫ 2 , . . .ǫ j have already been chosen.
Otherwise, the choice of W ′ together with the fact that ḡj ∈ U j ⊆ W ′ implies that d(ḡ j a, b j+1 ) = d(a, b j+1 ).Since by the second observation in the first paragraph we have U ′ := U ∩ N a (ǫ j ) ∈ N τ (1), it follows from the first observation in that same paragraph that we can choose g j+1 ∈ U ′ \ N sp ḡj a,bj+1 .Notice that in both cases we get d(ḡ j+1 a, b j+1 ) > d(a, b j+1 ).Finally, we choose ǫ j+1 ∈ (0, We claim that ḡk / ∈ N sp a,bj for any 1 j k.Since ḡk ∈ U k ⊆ W , this contradicts the initial hypothesis.For j = k this has already been shown.For j < k we have Lemma 7.12.Let τ be a group topology on G. Suppose we are given W ∈ N τ (1) and points i k and for all g ∈ W there is some i and gc i = c ′ i and consider W ′ = W g −1 ∩ W .Let h be any element in W ′ .We need to show that there exists some 1 i k ′ such that h ∈ N sp a,bi ∩ N sp a,ci .We know there exist λi , c j ) λ j + µ j while independence of {b i , c i } and {b ′ j , c ′ j } over a (and the assumption on the distances d(b l , c l )) implies ( 2) Putting this together with 1 and 2 yields: (3) Since all inequalities involved in 3 are equalities, so must be those involved in 1, so that λ i + µ i = d(b i , c i ) = λi + μi , which implies i k ′ .Together with the previous equation it also gives λi = λ i and μi = µ i .Thus any h ∈ W ′ belongs to 7.3.Proof of Theorem D. Proposition 7.13.Let R be a standard archimedean distance monoid with no least positive element.Let U be an R-Urysohn space and G = Isom(U).Then any group topology τ strictly coarser than τ 0 + ,0 is coarser than τ m .
Proof.Fix some group topology τ on G coarser than τ 0 + ,0 .Denote by ∆ the collection of all r ∈ R\{0} such that N sp u,v ∈ N τ (1) for some (equivalently, any) pair u, v ∈ U 2 with d(u, v) = r.Let Γ be the collection of all r ∈ R such that there exist a ∈ U, W ∈ N τ (1) and some finite B ⊂ U such that {ga} r-cuts B, for each g ∈ W .
The fact that ∆ is upper-closed, i.e., that s r and s ∈ ∆ implies r ∈ ∆, follows from the fact that R is closed under taking positive differences and the following observation: Observation 7.14.Let τ a group topology on G = Isom(U) where U is an R-Urysohn space.Suppose that we are given u, v, w On the other hand Γ is upper-closed by definition.Notice as well that ∆ = R\{0} implies τ = τ 0 + ,0 .Lemmas 7.11 and 7.12 come into play through the following lemma (notice s + s in the statement, rather than s ⊕ s).Proof.Take A and B as above, write A = {a j } k j=1 and pick some We may assume B ⊆ C. Assume that r / ∈ Γ.This implies that for each 1 j k there exists some h j ∈ U 0 such that {a j } does not r-cut h j C. Let C j = h j B. By Lemma 7.5 there exists g Remark 7.17.In the previous proof we are using only the fact that τ is coarser than τ st (as opposed to τ 0 + ,0 ).
Proof.Assume for the sake of contradiction that r ∈ ∆ \ Γ for some r.On the one hand, given u, v ∈ U with d(u, v) = r we have N sp u,v ∈ N τ (1), so there exists some On the other hand, by Lemma 7.16 there must be some g ∈ W such that gA does not cut {u, v}.By Lemma 7.10 this implies that G gA N sp u,v .However G gA ⊆ W 3 ⊆ N sp u,v : a contradiction.We are now ready to finish the proof of Proposition 7.13.Let S = {s ∈ R | s + s ∈ R}.Observation 7.14, Lemmas 7.18 and 7.15, together with the fact that R is archimedean imply that either Γ = ∆ = R \ {0} or Γ ∩ S = ∅.The former implies τ = τ 0 + ,0 so from now on assume Γ ∩ S = ∅.
, where in the last inequality we are using the fact that g ∈ b∈B N b (ǫ) and b ′′ ) and the result follows as well.Finally we have the case in which b 1 , b 2 , b 3 are all distinct.First of all, we have: and the same argument yields ).We are left with the situation in which the maximum distance between two points in {b One possibility is that 3 ) This suffices to settle the triangle inequality in this case, as we also have: The triangle inequality is then clear: ).This concludes the proof of the Claim and with it the proof of the Lemma.
Theorem (D).Let R be a standard archimedean distance monoid with no minimal positive element, U an R-Urysohn space and G = Isom(U).Then there are exactly 4 group topologies on G coarser than τ st τ st τ 0 + ,0 τ m {∅, G}.
Proof.The fact that {∅, G} is the only group topology strictly coarser than τ m is a particular case of Theorem C and the fact that there is no group topology τ with τ 0 + ,0 τ τ m is the content of Proposition 7. 13.
The fact that any group topology strictly coarser than τ st is coarser than τ 0 + ,0 follows from a combination Lemma 3.1 with Lemma 7.19.We are applying Lemma 3.1 with τ * = τ 0 + ,0 .The first alternative in its conclusion leads to τ = τ st , while in the second X ′ = ∅, since in this case the action is transitive.
We will explain the system behind the notation τ 0 + ,0 in the next section.As the reader might have guessed, the true identity of τ st and τ m will be τ 0,0 and τ 0 + ,0 + .

Parametrizing topologies of isometry groups of generzalized Urysohn spaces
We borrow the following construction from Conant [4].Let R be a distance monoid.By an end segment of R we mean a subset α ⊂ R with the property that t ∈ α whenever s ∈ α for some s t.Let R * be either the collection of end segments of R in case R has no maximal element or the collection of non-empty end segments in case R has a maximal element.There is a natural order * on the set R * given by α * β if and only if β ⊆ α.One can endow R * the operation ⊕ * , defined as [4][2.6.4., 2.6.5.]).This gives R * the structure of a distance monoid R * .
The natural embedding ν from R into R * sending r ∈ R to {s ∈ R | s r} respects the linear order and the operations on both sides: ν(s ⊕ t) = ν(s) ⊕ * ν(t).From now on we identify R with ν(R) and write ⊕ instead of ⊕ * .If R contains no minimal element greater than r ∈ R, then we denote the successor {s | s > r} ∈ R * as r + .Of particular interest for us will be 0 + .Notice that provided 0 + exists, the condition Let U be an R-Urysohn space and G its group of isometries.By an ideal of R we mean a non-empty closed subset of R closed under addition and such that s r ∈ R implies s ∈ R. Given an ideal µ of R let: . This is true in other situations as well, but we will skip the discussion at this point.
Let also Then S f generates the base of neighbourhoods of the identity of a group topology τ f .Moreover, µ f is an ideal and the closure of 1 in τ f coincides with the group G b µ f .Proof.To begin with observe that N −1 (u,v) (r) = N (v,u) (r), all sets N (u,v) (r) contain the identity map and the collection S f is invariant under conjugation.Take now any two points u, v ∈ U.
r. Since (s, t) ∈ T (d), there must exist some point w ∈ U such that d(w, u) = s and r) and we are done.
If r, r ′ ∈ µ f then for all q ∈ R and (s, t) ∈ T (q) we have An additional condition is needed to ensure the faithfulness of the parametrization f → τ f .Definition 8.3.We say that f : R → R * is a R-modulus of continuity if it satisfies Conditions (a) and (b) of Lemma 8.2 together with the following: Notice that this implies the inequalities in (b) and (c) are actually equalities.
Lemma 8.4.If f, g : R → R * are R-moduli of continuity, then τ f ⊆ τ g if and only if g(r) f (r) for all r ∈ R.
This is an easy consequence of Condition (c) together with the following fact: Lemma 8.5.Suppose we are given v, w ∈ U, r ∈ R and a finite collection Proof.The 'if' part is clear.For the only part, assume that neither of the two cases above holds.We want to show that (u,u ′ ,s)∈X N u,u ′ (s) N v,w (r) We may assume there is a finite set Y = {u i } q i=1 such that v = u 1 , w = u 2 (assume without loss of generality that v = w) and X contains exactly one triple (u i , u j , s i,j ) for any 1 j q and that s i,j = s j,i .We construct a new finite R-metric space as follows.As the underlying set Z we take the disjoint union of two copies Y j = {u j i } q i=1 , j = 1, 2 of Y after identifying u 1 i and u 2 i in case s i,i = 0. Consider the map d : q}, (abbreviated as sj,k ).We claim that (Z, d) is an R-metric space.Since our starting assumption translates as s1,2 > r, this will witness (u,u ′ ,s)∈X N u,u ′ (s) N v,w (r).By symmetry, all we need to check is si,j si,j ′ ⊕ sj ′ ,j as well as the symmetric inequality for all 1 i, j, j ′ q.This follows easily from the definition and the inequality d j ′ ,l ⊕ d l,j d j ′ ,j .
Given any distance monoid R, let Id(R * ) stand for the collection of all idempotents of R * .The following claim follows easily from the definitions and the fact that α ⊕ β = inf R * {s ⊕ t | s ∈ α, t ∈ β} Lemma 8.6.Let g : R → Id(R * ) satisfy the following properties: (i) g is constant on any archimedean class; (ii) g is non-increasing; (iii) If [r] < [s] and g(s) < g(r), then g(r) s; (iv) For any r ∈ R there exist s, t ∈ [r] such that s ⊕ t = r.
Then the function ĝ : R → R * given by ĝ(r) = r ⊕ g(r) is an R-modulus of continuity.
Proof.Let us check condition (c) first.If it does not hold, then there exists r ∈ R such that r ⊕ g(r) > s ⊕ t ⊕ g(t) for some (s, t) ∈ T (r).Since s ⊕ t r this implies g(t) < g(r) and thus [t] > [r], since g is non-increasing and constant on archimedean classes.Since (s, t) ∈ T (r), this implies in turn By condition (iii) it also implies g(r) t.We cannot have Notice that in general if an archimedean class [q] contains an idempotent q 0 , then q 0 = max[q] and q 0 ⊕ p = q 0 for any p q 0 .It follows that t = g(r) and both left and right-hand side of the inequality we started with are actually equal to g(r); a contradiction.Condition (b) can be proved using (iv) in a similar way.
We will refer to any map g : R → Id(R * ) as above as an R-ladder of idempotents.
Example.Given any α ∈ Id(R * ) it is easy to check that the function g with constant value α satisfies the definition above.If α = 0, then τ ĝ = τ st .If α = 0 + , then τ ĝ coincides with the generalized pointwise convergence topology τ m .The system of generating sets at the identity given this way is larger than the one in the definitions above, but it can be easily checked the extra generators are redundant.Notice that τ ĝ is Hausdorff if inf{im(g) | g ∈ G} ∈ {0, 0 + }.This is an only if in case R is countable or U the completion of a countable Urysohn space.
Proof.Consider any R-modulus of continuity f : R → R * and let g : R → R * be given by g(r) = f (r) − r.Notice that since R is closed under differences and f (r) r the right hand side is a well defined element of R * .Given any t, ′ t ∈ R with t t ′ .Property (c) implies that f (t ′ ) f (t) + (t ′ − t) from which it follows that g(t ′ ) g(t).The same property applied to (t ′ , t ′ − t) ∈ T (t) also yields g(t) g(t ′ ) + (t ′ − t).
We now claim that im(g) ⊆ Id(R * ).Property (b) yields: R * {g(s) + s + g(t) + t − r | (s, t) ∈ T (r)}.It thus suffices to show that g(s) + g(t) + ((s + t) − r) g(r) + g(r), for each (s, t) ∈ T (r).This is clear in case s, t r, since g(r) is non-increasing and s + t − r 0. If s or t are larger than r, then the result follows from inequalities g(t) + (t − r) g(r), g(s) + (s − r) g(r).Since g(r + r) r + g(r) g(r) + g(r) = g(r) we conclude that g is constant on archimedean classes.The second property of the definition follows easily from (c).
Remark 8.8.In general not all moduli of continuity need come from a ladder.Take a ∈ R + \ Q and consider R = Q 0 + Q 0 a.The sum and order inherited from R make R into a distance monoid.Let f : R → R * evaluate to r on any r / ∈ Q ∩ R and to r + on any r ∈ Q ∩ R. It is easy to check that f is a R-modulus of continuity.Question 4. Does some distance monoid R admitting moduli of continuity whose range contains non-idempotent elements?Problem 5. Classify the collection of R-moduli of continuity associated with an arbitrary R.
An alternative (and in the long run better way) of thinking about moduli of continuity is in terms of parameters of a generalized version of bi-Katetov maps as described in [20], or types of pairs of copies of U. Any function f : R → R * satisfying (c) can be associated to a G-invariant bi-Katetov map that assigns distance f (r) to any pair u ′ ,v ′′ where u ′ and v ′′ are copies of u ∈ U and v ∈ U respectively in the two copies of U. Condition (b) on the other hand states that the type is idempotent.So the following conjecture seems natural from that point of view as well.Conjecture 6.Given any distance monoid R and any R-Urysohn space U any group topology on Isom(U) strictly coarser than τ st is of the form τ f for some R-modulus of continuity f .

Zariski topology
Given a group G, an equation (inequality) over G in one variable x is an expression of the form w(x, α) = 1 (w(x, α) = 1), where w is a term over x ∪ α in the language of groups with inversion.We can think of w as an expression of the form where j l ∈ {1, . . ., r} and ǫ l ∈ {1, −1} for 0 l m − 1.This represents an element of the group G * x , where x is the cyclic free group over x.It is easy to check that if w and w ′ correspond to the same group element then the equations w = 1 and w ′ = 1 have the same set of solutions.Hence, one can always assume that the above word is reduced, i.e. α l = 1 whenever ǫ l + ǫ l+1 = 0. We say that an equation is trivial if w represents the trivial element in G * x .
A system of equations (inequalities) is just the conjunction of finitely many equations (inequalities).It can be checked that the collection of all sets of solutions of systems of inequalities over G is the base of a topology on G known as the Zariski topology, which we will denote by τ Z .
As mentioned in the introduction, according to a result of Gaughan in [9] for the group S ∞ the Zariski topology is the same as the standard topology and hence a group topology.Later, in [2] the same is shown for every subgroup of S ∞ , containing all permutations of finite support.Here, we investigate the Zariski topology for the automorphim groups of some homogeneous countable structures including the automorphism groups of Fraïssé limit structures of free amalgamation classes and rational Urysohn spaces (bounded and unbounded).Remark 9.2.Notice that being strongly unbounded is a strictly weaker notion than moving maximally in the sense of [19] (and almost moving maximally in [7]).
The following is a generalization of a classical argument for finding embeddings of free groups into automorphism groups of ω-categorical structures (see [14], Prop.4.2.3).Lemma 9.3.Suppose M is a countable ω-saturated first order structure in which acl is locally finite and put G := Aut (M).Assume α is a finite tuple of automorphisms of M where α i is either 1 or strongly unbounded and w(x, α) := α 0 x ǫ0 α 1 • • • x ǫm α m+1 a reduced word in one variable.Then the set of solutions of the equation w = 1 is meager in (G, τ st ).
Proof.As remarked above, the set of solutions of w(x, α) = 1 is closed in any group topology.We want to show it has empty interior.Aiming for contradiction suppose that is not the case.Up to performing a change of variable of the form x → xγ we can assume that there is a finite subset B such that w(G B , α) = 1.
We will construct inductively a chain of elementary maps id )); for 1 k m + 1 and c 0 = a.This finishes the proof.Indeed, given any extension We start by choosing any a ∈ α −1 m+1 (M \acl(B)) so that c m+1 = α m+1 (a) / ∈ acl(B) = acl(dom(f m+1 )).For the induction step, assume f k+1 has been successfully constructed for some 2 k m.We want to extend it to a map f k satisfying ( †).Let D k = dom(f ǫ k k+1 ) and q(x, y) = tp(c k+1 , D k ).Let p(x) := q(x, D k ) and p ′ (x) := q(x, f ǫ k k+1 (D k )).For any realization e |= p ′ (x) the map g e defined by g ǫ k e = f ǫ k k+1 ∪ {(c k+1 , e)} is elementary by construction.Our goal is thus to show that for some such e if we let f k = g e then the resulting c k ∈ M satisfies both ( †) and c k = a.Notice that by the induction hypothesis c k+1 / ∈ acl(D k ), i.e. p(x) is non-algebraic and hence so is p ′ (x).Since by assumption M is ω-saturated, p ′ (x) has infinitely many realizations in M .There are two different scenarios to consider.
If ǫ k = ǫ k−1 , then take e |= p ′ with e / ∈ {a} ∪ α −1 k (acl(D k c k+1 )).This is possible by the observation of the last paragraph and the local finiteness of acl−.Taking f k := g e we obtain: )).
Consider now the case ǫ k = −ǫ k−1 .Since w is reduced, this implies that α k = 1 and thus, by assumption, that α k is unbounded.The type p ′ (x) is non-algebraic with parameters in D ′ := f ǫ k k+1 (D k ).Unboundedness implies there exists a realization e of p ′ (x) such that α k (e) / ∈ acl(D ′ e).But , hence condition ( †) follows for c k = α k (e) as well.In the last step all we have to do is to choose e |= p ′ such that α 0 (e) = a.This is clearly possible by the fact that p ′ (x) is non-algebraic.
Remark 9.4.The actual sufficient condition given by the proof is that no two occurrences of opposite sign of x in w are separated by non-strongly unbounded element from the group.In particular, words involving only positive powers of x have always meager sets of solutions.Lemma 9.5.Suppose M is a countable first order structure such that the solution sets of all nontrivial equations of the form w(x, α) are meager in G = Aut (M) with respect to the standard topology.Then τ Z is not a group topology for G.
Proof.Indeed, fix α ∈ G and consider the equation zα −1 = 1 in G, where α ∈ G. Now, suppose we are given two systems of inequalities in one variable: where β = (β 1 , . . ., β k ) ∈ G k is the tuple of parameters appearing in the two systems, i.e., the nontrivial elements of G appearing in the corresponding normal forms.Consider the system of inequalities: where Π ′ (x, β ′ ) is the system obtained from Π(y, β) = 1 by replacing y with x −1 α (the substitution corresponds to an automorphism of G * x , so this is still a non-trivial system of equations) and β ′ the updated superset of parameters.Given a solution x 0 of Σ(x, β) = 1 and Π ′ (x, β ′ ) = 1 the pair (x 0 , x −1 0 α) belongs to the neighbourhood defined by the systems Σ = 1 and Π = 1 but their product satisfies the initial equation zα −1 = 1.Note that in a topological group any finite conjugation of group action is continuous and the pre-image of a nowhere dense set is nowhere dense.Hence the conclusion above finishes the proof.
Combining Lemma 9.3 and Lemma 9.5 one gets the following: Corollary 9.6.Suppose M is a countable homogeneous first order structure in which algebraic closure is locally finite.Assume all non-trivial automorphims of M are strongly unbounded.Then τ Z is not a group topology for Aut (M).
There is another consequence of the meagerness of solution sets of equations worth mentioning.We start with the observation that the multivariate case follows from the univariate case.Lemma 9.7.Let (G, τ ) be a non-meager Polish group.If the set of solutions of any non-trivial equality in one variable with parameters in G is meager in G then the same holds for non-trivial equalities with parameters in any number of variables.
Proof.Let w(x, α) = 1 be the equation in question, where x = (x 0 , x 1 , . . ., x k ) = (x 0 , y) by induction.For each value of y 0 := (x 0 1 , . . ., x 0 k ) consider the term in x 0 obtained by replacing each x j by the element x 0 j for j 1 in w(x, α) and then merging together all consecutive constants.If all the resulting products that lay between two consecutive occurrences of x ǫ 0 with opposite exponents are non-zero then the resulting expression is already reduced and is a non-trivial inequality in x 0 (which without loss of generality appeared in the original expression).Therefore for such y 0 comeagerly many values of x 0 satisfy the equation, by the single variable case.Now, the condition above can be expressed as a system of finitely many non-trivial inequalities in the variable y and hence holds for comeagerly many values of y by the induction hypothesis.
Using the Baire Category Theorem one can derive the following corollary: Corollary 9.8.Let (G, τ ) be a non-meager Polish group such that the set of solutions of any nontrivial equation in one variable with parameters in G is meager.Then for any countable subgroup A G there exists some free group F G over a countable base such that A, F ∼ = A * F .From Corollary 2.10 in [15] one can easily conclude the following.Fact 9.10.Suppose K is a Fraïssé class with the free amalgamation property and M = Flim (K).Assume Aut (M ) is transitive and Aut (M ) = S ∞ .Then every non-trivial automorphism of M where M = Flim (K) is strongly unbounded.
Here we introduce a more general setting for Fraïssé classes that include the setting of [15].Then Lemma 9.12 is a mild generalisation of Corollary 2.10 in [15] which we give a complete proof here.Definition 9.11.Suppose K is a Fraïssé class.We say K is non-discrete (ND) if there is m ∈ N (where min{n R : R nR ∈ L} m − 2 max{n R : R nR ∈ L}) such that for every A ∈ K with |A| m, and a 1 , a 2 ∈ A with a 1 = a 2 and A where s ′ ∼ = s.We call B a non-discrete one-point extension of A of the form a 1 ⊲ d A ′ a 2 .Occasionally, we use m-ND when we want to specify the cardinally of |A|.
Notice that in the definition above if B ⊆ Flim (K), α ∈ Aut (Flim (K)) where α ±1 (a 1 ) = a 2 and α fixes A setwise, then d ∈ Supp(α).Lemma 9.12.Suppose K is an ND Fraïssé class with the free amalgamation property.Then every non-trivial automorphism of M where M = Flim (K) is strongly unbounded.Inductively, we can build p i 's and find realisation r i 's for i ∈ N which r i |= p j and r i ∈ Supp(α) when i j.Hence we have infinitely many realisation of p 0 in Supp(α) and hence T b is infinite.
By applying Corollary 9.6, we conclude the following.
Corollary 9.14.Assume K is a non-trivial free amalgamation class.Let M = Flim (K) and G = Aut (M) and assmue G acts transitively on M .Then τ Z is not a group topology.
Proof.Since K is non-trivial there exists some m 1 such that the action G on M is (m − 1)-transitive (let us say every action is 0-transitive) but not m-transitive.Our second hypothesis implies that in fact m 2. This implies that any substructure of size at most m − 1 in K is essentially a pure set in the sense that the type of a tuple enumerating all of its members without repetitions does not depend on the order, but that this pure set of size m − 1 extends to some A ∈ K which this is not the case anymore 2 .
Consider now any A b 2 , since no relation holds for a tuple containing both b 2 and c.From Lemma 9.13 follows that K is an ND Fraïssé class.The final conclusion then follows from Corollary 9.6.
Remark 9.15.The assumption of ND in Lemma 9.12 is a necessary condition in order to conclude that every non-trivial automorphisms of a Fraïssé limits of free amalgamation classes is strongly unbounded.Notice that by the result of Gaughan in [9], for the Fraïssé class all finite (in empty signature) τ Z = τ st and hence τ Z is a group topology on S ∞ .9.1.2.Rational Urysohn spaces.Consider the distance monoids Q = (Q 0 , +, , 0) and where + b is addition truncated at b. Let U Q and U Q b be the corresponding Urysohn space respectively -see Section 5.They are precisely the classical rational Urysohn space and rational Urysohn b-spheres (or sometimes bounded rational Urysohn space).Here we prove τ Z for the automorphism groups of U Q and U Q b are not group topologies.
We briefly discuss how rational Urysohn space and rational Urysohn spheres are constructed in first order logic as Fraïssé limits.
Let L be the first-order language with a binary relation R q (x, y) for each q ∈ R. A metric space (A, d) with R-rational distances is an L-structure in the following manner: for x, y ∈ A and q ∈ R we have R q (x, y) iff d(x, y) q.Let C R be the class of all finite metric spaces with R-rational distances as L-structures.Let U R be the corresponding Fraïssé limit.On easy fact is the following: Let a 1 , a 2 ∈ A be two elements which we want to separate by a one-point extension.Let q = d(a 1 , a 2 ) and consider B = {a 1 , a 2 , b} to be an L-structure with d(a 1 , a 2 ) = q and d(a 1 , b) = q 2 and q 2 + ǫ where ǫ ∈ (0, q 2 ).It is easy to check B is a metric space with rational distance and its diameter is q hence B ∈ C R .Note that B is a one-point non-discrete extension of a 1 a 2 that separates a 1 and a 2 .Now the amalgamation of A and B over a 1 a 2 is the one-point extension of A which we are looking for.Proof.For the first part, take some R (k) ∈ L holding for some k-tuples but not for others.This implies the existence of a, b ∈ M k (with pairwise distinct coordinates) differing only in one coordinate a i = b i and such that tp(a) = tp(b).This implies ∆ k−1 ai,bi (x) is non-algebraic.By our transitivity condition, it follows that ∆ k−1 c,d (x) is non-algebraic for any distinct c, d ∈ M .The second part follows from the proof of Corollary 9.14.
We say K is dense if for any distinct a, b ∈ M = Flim (K) and any non-algebraic 1-type p over finitely many parameters of M isolated by a formula φ(x) the formula ∆ 1 a,b (x) ∧ p(x) is not algebraic.Definition 9.23.Given two Fraïssé classes K 1 and K 2 over finite relational languages L 1 and L 2 , respectively, define K 1 ⊗ K 2 to be the class of L-structures A where L = L 1 L 2 and A ↾ L1 ∈ K 1 and A ↾ L2 ∈ K 2 .Lemma 9.24.For i = 1, 2 let K i be a Fraïssé class over a finite relational language L i such that K = K 1 ⊗ K 2 is a Fraïssé class.Assume that: • K 1 is discriminating ; • K 2 is dense; and let M = Flim (K).Then any non-trivial element of G = Aut (M) is strongly unbounded.
Proof.Take α ∈ G \ {1} and some non-algebraic formula φ(z, a) in one variable over a finite tuple a of parameters which isolates a non-algebraic type.We can write φ = φ 1 ∧ φ 2 , where φ i is a quantifier free formula in the language L i .Suppose that α(c) = c ′ for distinct c, c ′ ∈ M .The fact that K 1 is discriminating implies there is some k such that the formula ∆ k,L1 c,c ′ is non-algebraic.In particular, it can be realized in M ↾ L1 by some tuple d disjoint from a. Since the L 2 formula φ 2 (x, a) is nonalgebraic, it is possible to find such d in M with the property that all of its entries satisfy φ 2 (z, a).On the other hand, α(d i ) = d i for some d i .Density of K 2 then implies φ 2 (z, a) ∧ ∆ 1,L2 d,d ′ (z) is not algebraic.Therefore, neither is φ(x) ∧ ∆ 1,L2 d,d ′ (z) (again, by quantifier elimination and the definition of K 1 ⊗ K 2 ).This implies that φ(M ) ∩ Supp(α) is non-empty.
We collect below a handful of particular cases of Lemma 9.24.Corollary 9.25.Let K 1 and K 2 be two Fraïssé classes with strong amalgamation and K = K 1 ⊗ K 2 .Assume K 1 is non-trivial and satisfies one of the following: • The action of Aut (Flim (K 1 )) on the set M 2 \ {(a, a)} a∈M is transitive; • K 1 has free amalgamation and the action of Aut (Flim (K 1 )) on Flim (K 1 ) is transitive.Assume also Flim (K 2 ) one of the following: • (Q, <); • The countable dense meet tree; • The cyclic tournament S(2).Then the solution set of any non-trivial equation with parameters in G is meager.⌣X α(Y ′ ).One can modify the definition of strongly unbounded to strongly gcl-unbounded such that for an automorphism of a structure that gcl is well-defined.Namely α is strongly gcl-unbounded if for every finite set A and b ∈ M \gcl(A) there is a realisation c ∈ M of tp(b/A) where α(c) / ∈ gcl(cA).Proposition 9.27 implies immediately the following.Fact 9.28.All non-trivial automorphisms of M η are strongly gcl-unbounded.
It has to be remarked that Proposition 9.27 is proving something stronger than just that non-trivial automorphisms are strongly gcl-unbounded.
Essentially the same arguments of the proof of Lemma 9.3 works and we only need to replace acl by gcl and apply Fact 9.28 when α i 's are non-trivial.
In order to show the starting point of the argument we provide some details and leave the rest (avoiding a repetition).We follow closely the proof of Lemma 9.3.Aiming for contradiction suppose that is not the case.Again up to performing a change of variable of the form x → xγ we can assume that there is a finite -closed subset B such that G B ⊆ w(G, α).We will construct inductively a chain of partial isomorphisms id B = f m+1 ⊆ f m ⊆ • • • ⊆ f 0 together with a ∈ M η with the property that: for 0 k m + 1   (1).Let now g ∈ W ∩ W −1 .Since g ∈ W the previous discussion implies that gc c and since g −1 ∈ W we know that g −1 c c, that is c gc so in fact gc = c.It follows that W ∩ W −1 G c and thus τ Z = τ st .9.5.a-minimality.Let G be a group.The intersection of all Hausdorff topological groups structures on G is called the Markov topology, denoted by τ M .The topology τ M is always T 1 but not necessarily a group topology.
We say that a group G is a-minimal if (G, τ M ) is a topological group.Notice that if τ Z is a group topology, then τ M = τ Z .Question 7.For which (sufficiently homogeneous) structures M is Aut (M) a-minimal?

Topologies and types
Let M be a first order structure and T = T h(M).Consider two tuples of variables x = (x m ) m∈M and y = (y m ) m∈M indexed by the elements of M .Given some finite tuple a = (a 1 , a 2 , . . ., a k ) ⊂ M we write x a in lieu of (x a1 , x a2 , . . ., x a k ).Let p M (x) = tp(M ), where variable x m is made to correspond with m ∈ M .Let R(M) stand for the collection of all T -complete types in variables x, y containing p M (x) ∪ p M (y) and let R pa (M) for the collection of partial types in variables x, y in T containing p M (x) ∪ p M (y) (i.e., of all closed subsets of R(M) in the logic topology).Here we assume types are deduction closed.Given any partial type p(x, y) we will denote the deduction closure of p(x, y) ∪ p M (x) ∪ p M (y) in T as p .The set R pa (M) can be endowed with the so-called logic topology, which we denote by τ L , generated by neighbourhoods of the form [φ] = {p ∈ R(M) | φ ∈ p}, where φ is any formula in (x, y).The result is Stone space.Given p 1 , p 2 ∈ R pa (M) we let (p 1 * p 2 )(x, y) ∈ R pa (M) denote the collection of all formulas ψ(x, y) such that there exist φ i (x, y) ∈ p i (x, y), i = 1, 2 such that φ 1 (x, z) ∧ φ 2 (z, y) ⊢ ψ(x, y).
It can be checked that * endows R pa (M) with a semigroup structure.If we let 0 = ∅ ∈ R pa then clearly p * 0 = 0 for any p ∈ R pa .We write p q for p ⊢ q.
Given p ∈ R pa , let p ∈ R pa be defined by θ(x, y) ∈ p ↔ θ(y, x) ∈ p.Every g ∈ Aut (M) is associated to some ι(g) = {x gm = y m } m∈M ∈ R pa .It can be easily checked that ι is a continuous homomorphic embedding of (G, τ st ) into (R pa (M), τ L ) whose image is contained in R(M).From now on we will write simply g instead of ι(g).Notice that p g := g −1 * p * g = {φ(x a , y b ) | φ(x g•a , y g•b ) ∈ p} for any p ∈ R pa and g ∈ G. Notice that * is a continuous map R pa (M) × R pa (M) → R pa (M) and p → p continuous with respect to τ L .For the first, notice that given p 1 , p 2 ∈ R pa (M) and a formula φ(x, y) with p 1 * p 2 ∈ N φ(x,y) , the definition of * together with compactness implies the existence of φ 1 (x, z) ∈ p 1 and φ 2 (z, y) ∈ p 2 such that T ∪ {φ 1 (x, z), φ 2 (z, y)} ⊢ φ(x, y), which implies that N φ1 * N φ2 ⊆ N φ .Definition 10.1.Suppose M is an L-structure and G = Aut (M).We say that q ∈ R pa is an invariant idempotent if the following conditions are satisfied: 1. 1 G q; 2. q = q; 3. q * q = q; and, 4. q = q g for any g ∈ G.
Notice that assumption 1. implies q = 1 * q q * q, so that item 3. could be replaced by the a priori weaker condition q * q q.

Definition 1 . 1 .
A Hausdorff topological group G is called minimal if every bijective continuous homomorphism from G to another Hausdorff topological group is a homeomorphism.The group G is totally minimal if every continuous surjective homomorphism to a Hausdorff topological group is open.
Suppose A, B and C are structures in some relational language L with A ⊆ B, C. By the freeamalgam of B and C over A, denoted by B ⊗ A C, we mean the structure with domain B A C in which a relation holds if and only if it already did in either B or C. By a free amalgamation class we mean a class K of finite structures in a relational language satisfying (HP) and such that B ⊗ A C ∈ K for any A, B, C ∈ K such that A ⊆ B, C. Note this is automatically a Fraïssé class with strong amalgamation.We write B | ⌣ f r A C if and only if the structure generated by ABC is isomorphic (with the right identifications) with the free amalgam B ⊗ A C. If B | ⌣ f r ∅ C we write B | ⌣ f r C and say B and C are free from each other.

Corollary 4 . 7 .
Let M f η be an ω-categorical Hrushovski generic structure such that G := Aut M f η acts transitively on M f η .Then (G, τ st ) is a minimal topological group.
Assume (A, d A ) and (B, d B ) are two finite R-metric spaces where C := A ∩ B = ∅ and let D be the disjoint union of A and B over C. Let d : D 2 → R be given as follows: • d restricts to d A and d B on A × A and B × B; respectively, • d(a, b) = min{d(a, c) ⊕ d(c, b) | c ∈ C} for any a ∈ A \ C and any b ∈ B \ C.It can be shown that if R is a distance monoid, then the (D, d) above is itself an R-metric space, which we will denote as A ⊗ C B.

Lemma 6 . 2 .
Suppose we are given finite A, B, C ⊆ U with A, B = ∅ and d(A, B) = 0. Then for each n ∈ N there is g ∈ G B (G A G B ) n such that d(C, g(A)) (2n + 1) • d(A, B).
We claim that d(C, A 2 ) r. Indeed, take any c ∈ C and a ∈ A 2 .There exists e ∈ A 1 B 2 such that d(c, a) = d(c, e) ⊕ d(e, a).There are two possibilities.If e ∈ B 2 , then d(c, a) d(e, a) r, by the choice of A 2 and B 2 .If e ∈ A 1 then d(c, a) d(c, e) r.

Observation 7 . 2 .
If R is a basic distance monoid, U an R-Urysohn space and u, v, w ∈ U such that u | ⌣v w and in the ambient group d(u, v) + d(v, w) < sup R ∈ R ∪ {∞} holds, then (u, v, w) is aligned.If R is standard then the opposite implication is also true: the triple of points (u, v, w) is aligned only if u | ⌣v w.

Lemma 7 . 3 .
Let R be any distance monoid, U an R-Urysohn space, G = Isom(U) and A, B finite subsets of U such that B is in ǫ-general position relative to A.Then G A G B G A ⊃ a∈A N a (ǫ).In particular, G A G B G A ∈ N τm (1)in case τ m exists (see Claim 6.4).Proof.By virtue of Lemma 3.7 the result can be rephrased as follows.Given any finite metric space (D, d) whose underlying set consists of A and an isometric copy of A ′ with the property that d(a ′ , a) ǫ for conjugate points a ∈ A, a ′ ∈ A ′ the extension of d to (D B) 2 given by d(a, b) = d(a ′ , b) = d(a, b) for a, a ′ ∈ A and b ∈ B defines an R-metric space.It suffices to check the triangular inequality for triples of points of the form (a 1 , a ′ 2 , b).On the one hand for any triples of points a 1 , a 2 , b where a 1 , a 2 ∈ A and b ∈ B we have:

Lemma 7 . 4 .
Let R be a distance monoid with no gaps and A, B, C finite subsets of some R-metric space (X, d) satisfying A | ⌣B C, where B = B 1 ∪ B 2 .If A r-cuts C, then at least one of the following holds: • B 1 r-cuts C; • A r-cuts B 2 .Proof.Pick a ∈ A and c 1 , c 2 such that d(c 1 , c 2 ) r and a ∈ [c 1 , c 2 ].For i = 1, 2 there exists b

Lemma 7 . 10 .
For any finite a, c ∈ U and finite B ⊆ U we have G B ⊆ N sp a,c if and only if B ∩[a, c] = ∅.Proof.We may assume that d(a, c) < sup R, since otherwise both the condition on the right and that on the left are trivially satisfied.For the only if direction assume B ∩ [a, c] = ∅ and choose h ∈ G B such that ha | ⌣B c.We then have d(ha, c) = min{d(a, b) ⊕ d(b, c) | b ∈ B} > d(a, c), so that h ∈ G B \ N sp a,c = ∅.The opposite direction is a direct consequence of the triangle inequality together with the existence for b ∈ B such that d(a, c) = d(a, b) ⊕ d(b, c).Lemma 7.11.Let τ be a non-trivial group topology on G strictly coarser than τ st and assume that there exists W ∈ N τ (1) and distinct points a, b 1 , b 2 , . . ., b k ∈ U such that W ⊆ 1 j k N sp a,bj and d(a, b j ) + d(a, b j ) ∈ R for all 1 j k.Then there exists 1 j 0 k such that N sp a,bj 0 ∈ N τ (1).Proof.Assume the conclusion does not hold.This means U N sp c,d for any U ∈ N τ (1) and any c, d ∈ U with a, b ′ l ), µ l := d(a, c l ) = d(a, c ′ l ) and let also λl := d(ha, b l ), μl := d(ha, c l ) and λ′ l := d(ha, b ′ l ), μ′ l := d(ha, c ′ l ) for 1 l k ′ .Our assumption on h translates into equations (1)

Lemma 7 . 15 .Lemma 7 . 16 .
If s + s + s + s ∈ R and s + s ∈ Γ then s ∈ ∆ Proof.Let a ∈ U, B a finite subset of U and W ∈ N τ (1) be such that {ga} r-cuts B for any g ∈ W .Let {b j , c j } 1 j N be an enumeration of all the pairs of points in B at distance at most s + s.If s / ∈ ∆, then ga ∈ 1 j N [b j , c j ], for each g ∈ W . Lemma 7.12 then tells us that 1 j k ′ (N sp a,bj ∩ N sp a,cj ) ∈ N τ (1) where (up to reindexing) a ∈ [b j , c j ] if and only if j k ′ .We may assume that d(a, b j ) d(a, c j ), so that d(a, b j ) s.By Lemma 7.11, there exists some 1 j 0 k ′ such that N sp a,bj 0 ∈ N τ (1) and we are done.If r / ∈ Γ, then for any U ∈ N τ (1) and finite subsets A, B ⊂ U there is h ∈ U such that hA does not r-cut B.
1 , b 2 , b 3 } is witnessed by d(b 1 , b 2 ) or d(b 2 , b 3 ).We might as well assume it is witnessed by d(b 1 , b 2 ), since the other sub-case is entirely analogous.

Definition 9 . 1 .
Let α ∈ Aut (M).We say α is strongly unbounded if for every finite subset A of M and b ∈ M \acl(A), there is a realization c |= tp(b/A) such that α(c) / ∈ acl(cA).

9. 1 .
Fraïssé constructions.Suppose M is a countable first order structure.Let G = Aut (M) and for any α ∈ G define Supp(α) := {m ∈ M | α(m) = m}.Recall the setting from Subsection 2.1; namely L is a relational signature and K is a class of finite L-structures.Suppose A, B, C are L-structures with A, B ⊆ C. Let B ′ ⊆ B\A.By ∆ C (B ′ ; A) we mean the set of all positive Boolean combinations of all φ(b, a) where b ⊆ B ′ , a ⊆ A and φ is an atomic L-formula.9.1.1.Free amalgamation classes.For a definition of free amalgamation classes see Subsection 2.1.First, we remind the reader the following fact about strong amalgamation classes (hence also about free amalgamation classes).

Fact 9 . 9 .
Suppose K is a Fraïssé class with strong amalgamation.Then acl M (A) = A for all A ⊆ M where M = Flim (K).

Lemma 9 . 13 .
Suppose α is an automorphism of a first order structure M where the algebraic closure in M is trivial.Assume for every finite subset A of M and b ∈ M \A there is a realisation c of tp(b/A) such that c ∈ Supp(α).Then α is strongly unbounded.Proof.Given a finite subset A of M and b ∈ M \A, we prove the set T b := {r ∈ M | r ∈ Supp(α), r |= tp(b/A)} is infinite.Assuming T b is infinite, because the algebraic closure is trivial, there is a realisation c ∈ T b such that α(c) / ∈ cA.This shows T b is strongly unbounded.Now we show T b is infinite.Put p 0 = tp(b/A).By the assumption there is r 0 |= p 0 where r 0 ∈ Supp(α).Now consider p 1 := tp(s 1 /r 0 A) where s 1 |= p and s 1 = r 0 A (and such s 1 exists because r 0 / ∈ A = acl(A)).Let r 1 |= p 1 where r 1 ∈ Supp(α) again using the assumption.Clearly r 1 |= p 0 .

Proposition 9 . 16 .
The class C R where R ∈ {Q, Q b | b ∈ Q >0 } has the amalgamation property.

Lemma 9 . 18 .
Non-trivial automorphisms of U Q and U Q b for b ∈ Q >0 are strongly unbounded.

9. 2 .
Products of Fraïssé classes.Given two distinct elements a, b of an ω-categorical structure M such that tp(a) = tp(b) and any k 1 denote by ∆ k a,b (x) the formula over x = (x i ) k i=1 stating that tp(x/a) = tp(x/b) and x ∩ {a, b} = ∅.Notice that for any α ∈ Aut (M) if b = α(a) then at least one component of any realisation of ∆ k a,b (x) must belong to Supp(α).We denote by ∆ k,L ′ a,b the result of calculating ∆ in M ↾ L ′ , where L ′ ⊂ L. We say that a Fraïssé class K in a relational signature L is discriminating if for each pair of distinct elements a, b ∈ M = Flim (K) there exists k 1 such that the formula ∆ k a,b (x) is non-algebraic.Observation 9.22.Let K be a non-trivial Fraïssé class in a finite relational signature and M = Flim (K).Then K is discriminating provided one of the following holds: • M has trivial algebraic closure and the action of Aut (M) on M 2 \ {(a, a)} a∈M is transitive; or, • K has free amalgamation and Aut (M) acts transitively on M .

9. 3 .
Hrushovski's pre-dimension construction.Recall the setting in subsection 2.2.Suppose s 2 and η ∈ (0, 1].Let C η := {B ∈ C | ∅ B} and M η be the countable structure that one obtains from Proposition 2.2.Suppose A is a finite subset of M η .Using the pre-dimension function δ one can define the dimension of A as d (A) := δ (cl (A)), where cl (A) is the smallest -closed finite subset of M η that contains A. Given b ∈ M η and A a finite subset of M η , we denote d (b/A) for d (bA) − d (A).From part (2) of Lemma 2.1 and part (2) Proposition 2.2 it follows that cl (A) is well-defined.Similarly to Lemma 9.3 we prove the following Lemma 9.26.Suppose α is a finite tuple of automorphisms of G = Aut (M η ).Then the set of solutions of a non-trivial equation w(x, α):= α 0 x ǫ0 α 1 • • • x ǫm α m+1 = 1 is meager in G.In order to prove Lemma 9.26 we recall some facts about non-trivial automorphisms of M η .Recall that α ∈ Aut (M η ) is gcl-bounded if there exists a finite subset B of M η such that m ∈ gcl(mB) for all m ∈ M η where gcl(X) := {m ∈ M η | d (m/X) = 0}.One can define an independence notion | ⌣ d between finite subsets of M η using the dimension function; namely A | ⌣ d B C iff d (A/B) = d (A/BC) where A, B and C are finite subsets of M η .It turns out | ⌣ d is indeed the forking-independence in M η and for simplicity we denote it by | ⌣ in M η and remove the superscript d.From Lemma 3.2.27 and Theorem 3.2.29 in [10] follows: Proposition 9.27.For every non-trivial automorphism α ∈ Aut (M ν ) and X, Y ∈ C η where X Y and Y ∩ gcl(X) = X, there is Y ′ where tp(Y ′ /X) = tp(Y /X) and Y ′ | a) / ∈ gcl(dom(f ǫ k k )); where c k = a when k = m + 1.The starting point is choosing a any element in M η \gcl(B).We have two possibilities: If α m+1 is 1, then let c m+1 = a.Suppose α m+1 is strongly gcl-unbounded.Using Fact 9.28 and Proposition 9.27 set c m+1 := α m+1 (c) where c |= tp(a/B) and α m+1 (c) / ∈ gcl(cB) and rename c to a. Assume now we have successfully constructed f k .Then the same arguments of the proof of Lemma 9.3 work only replacing acl by gcl and apply Fact 9.28 when α i 's are non-trivial.Then from Lemma 9.26 and Theorem E follows Corollary 9.29.The Zariski topology τ Z for Aut (M η ) is not a group topology.

For
the dense meet tree as I we take the collection of all sets the form I(a, b) = {a < meet(b, x) < b} where a < b and we let λ(I(a, b)) = {a, b}.Take incomparable b 1 , b 2 and let c = meet(b 1 , b 2 ).Take a < b.Then I(a, b 1 ) ∩ I(a, b 2 ) = {x ∈ M | a < x c}.Note that W := [c : I(a, b 1 ) ∩ I(a, b 2 )] ∈ N τZ There are 1 i, j k such that ga ∈ B bi (t) ∩ B hbj (t).This implies d(b i , hb j ) 2t s.Independence of B and h(B) over a implies that either d(b i , a) t or d(hb j , a) t.As ga ∈ B bi (t) ∩ B hbj (t), either of the two cases implies d(ga, a) 2t s and hence