Ample Generics Research Articles

On the automorphism groups of universal submeasures

We investigate dynamical properties of the automorphism groups of general versions of the universal submeasures defined in [3]. First, we show that, a universal submeasure D-valued exists for every countable (finite or infinite) set D of non-negative real numbers, with 0∈D. Moreover, ordered D-valued universal submeasures exist for all such D. By using the Kechris - Pestov - Todorčević theory, we prove that for all the ordered universal submeasures, automorphism groups are extremely amenable, but they do not have ample generics when D satisfies some additional conditions. Finally, we prove that the class of all finite D-valued submeasures has the Hrushovski property, and the automorphism group of the D-valued universal submeasure is amenable and has ample generics.

Automorphism groups of universal diversities

We prove that the automorphism group of the Urysohn diversity is a universal Polish group. Furthermore we show that the automorphism group of the rational Urysohn diversity has ample generics, a dense conjugacy class and that it embeds densely into the automorphism group of the (full) Urysohn diversity. It follows that this latter group also has a dense conjugacy class.

Ordered structures and large conjugacy classes

This article is a contribution to the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a Fraïssé limit) that is extremely amenable, and has ample generics. As Fraïssé limits whose automorphism groups are extremely amenable must be ordered, i.e., equipped with a linear ordering, we focus on ordered Fraïssé limits. We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager 2-dimensional diagonal conjugacy class. We formulate general conditions implying that there is no comeager conjugacy class, comeager 2-dimensional diagonal conjugacy class or non-meager 2-dimensional topological similarity class in the automorphism group of an ordered Fraïssé limit. We also provide a number of applications of these results.

Hall's universal group has ample generic automorphisms

AbstractWe show that the automorphism group of Hall's universal locally finite group has ample generics, that is, it admits comeager diagonal conjugacy classes in all dimensions. Consequently, it has the small index property, is not the union of a countable chain of non‐open subgroups, and has the automatic continuity property. Also, we discuss some algebraic and topological properties of the automorphism group of Hall's universal group. For example, we show that every generic automorphism of Hall's universal group is conjugate to all of its powers, and hence has roots of all orders.

COHERENT EXTENSION OF PARTIAL AUTOMORPHISMS, FREE AMALGAMATION AND AUTOMORPHISM GROUPS

AbstractWe give strengthened versions of the Herwig–Lascar and Hodkinson–Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraïssé classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraïssé limit of any Fraïssé class that is the class of all ${\cal F}$-free structures (in the Herwig–Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.

Automorphism groups of countable structures and groups of measurable functions

Let G be a topological group and let μ be the Lebesgue measure on the interval [0, 1]. We let L0(G) be the topological group of all μ-equivalence classes of μ-measurable functions defined on [0, 1] with values in G, taken with the pointwise multiplication and the topology of convergence in measure. We show that for a Polish group G, if L0(G) has ample generics, then G has ample generics, thus the converse to a result of Kaichouh and Le Maitre. We further study topological similarity classes and conjugacy classes for many groups Aut(M) and L0(Aut(M)), where M is a countable structure. We make a connection between the structure of groups generated by tuples, the Hrushovski property, and the structure of their topological similarity classes. In particular, we prove the trichotomy that for every tuple $$\bar f$$ of Aut(M), where M is a countable structure such that algebraic closures of finite sets are finite, either the countable group $$\langle \bar f \rangle$$ is precompact, or it is discrete, or the similarity class of $$\bar f$$ is meager, in particular the conjugacy class of $$\bar f$$ is meager. We prove an analogous trichotomy for groups L0(Aut(M)).

Extending partial isometries of generalized metric spaces

We consider generalized metric spaces taking distances in an arbitrary ordered commutative monoid, and investigate when a class $\mathcal{K}$ of finite generalized metric spaces satisfies the Hrushovski extension property: for any $A\in\mathcal{K}$ there is some $B\in\mathcal{K}$ such that $A$ is a subspace of $B$ and any partial isometry of $A$ extends to a total isometry of $B$. Our main result is the Hrushovski property for the class of finite generalized metric spaces over a semi-archimedean monoid $\mathcal{R}$. When $\mathcal{R}$ is also countable, this can be used to show that the isometry group of the Urysohn space over $\mathcal{R}$ has ample generics. Finally, we prove the Hrushovski property for classes of integer distance metric spaces omitting triangles of uniformly bounded odd perimeter. As a corollary, given odd $n\geq 3$, we obtain ample generics for the automorphism group of the universal, existentially closed graph omitting cycles of odd length bounded by $n$.

CONSEQUENCES OF THE EXISTENCE OF AMPLE GENERICS AND AUTOMORPHISM GROUPS OF HOMOGENEOUS METRIC STRUCTURES

AbstractWe define a simple criterion for a homogeneous, complete metric structure X that implies that the automorphism group Aut(X) satisfies all the main consequences of the existence of ample generics: it has the automatic continuity property, the small index property, and uncountable cofinality for nonopen subgroups. Then we verify it for the Urysohn space $$, the Lebesgue probability measure algebra MALG, and the Hilbert space $\ell _2 $, thus proving that Iso($$), Aut(MALG), $U\left( {\ell _2 } \right)$, and $O\left( {\ell _2 } \right)$ share these properties. We also formulate a condition for X which implies that every homomorphism of Aut(X) into a separable group K with a left-invariant, complete metric, is trivial, and we verify it for $$, and $\ell _2 $.

An example of a non non-archimedean Polish group with ample generics

For an analytic P P -ideal I I we denote by S I S_I the Polish group of all the permutations of N \mathbb {N} with support in I I , equipped with a topology given by the corresponding submeasure on I I . We show that if Fin ⊊ I \mbox {Fin} \subsetneq I , then S I S_I has ample generics. This implies that there exists a non non-archimedean Polish group with ample generics.

GREY SUBSETS OF POLISH SPACES

AbstractWe develop the basics of an analogue of descriptive set theory for functions on a Polish space X. We use this to define a version of the small index property in the context of Polish topometric groups, and show that Polish topometric groups with ample generics have this property. We also extend classical theorems of Effros and Hausdorff to the topometric context.

Connected Polish groups with ample generics

In this paper, we give the first examples of connected Polish groups that have ample generics, answering a question of Kechris and Rosendal. We show that any Polish group with ample generics embeds into a connected Polish group with ample generics and that full groups of type III hyperfinite ergodic equivalence relations have ample generics. We also sketch a proof of the following result: the full group of any type III ergodic equivalence relation has topological rank 2.

Generic elements in isometry groups of Polish ultrametric spaces

This paper presents a study of generic elements in full isometry groups of Polish ultrametric spaces. We obtain a complete characterization of Polish ultrametric spaces X whose isometry group Iso(X) has a neighborhood basis at the identity consisting of open subgroups with ample generics. It also gives a characterization of the existence of an open subgroup in Iso(X) with a comeager conjugacy class. We also study the transfinite sequence defined by the projection of a Polish ultrametric space X on the ultrametric space of orbits of X under the action of Iso(X).

Polish topometric groups

We define and study the notion of \emph{ample metric generics} for a Polish topological group, which is a weakening of the notion of ample generics introduced by Kechris and Rosendal in \cite{Kechris-Rosendal:Turbulence}. Our work is based on the concept of a \emph{Polish topometric group}, defined in this article. Using Kechris and Rosendal's work as a guide, we explore consequences of ample metric generics (or, more generally, ample generics for Polish topometric groups). Then we provide examples of Polish groups with ample metric generics, such as the isometry group $\Iso(\bU_1)$ of the bounded Urysohn space, the unitary group ${\mathcal U}(\ell_2)$ of a separable Hilbert space, and the automorphism group $\Aut([0,1],\lambda)$ of the Lebesgue measure algebra on $[0,1]$. We deduce from this and earlier work of Kittrell and Tsankov that this last group has the automatic continuity property, i.e., any morphism from $\Aut([0,1],\lambda)$ into a separable topological group is continuous.

Rooted trees, strong cofinality and ample generics

AbstractWe characterize those countable rooted trees with non-trivial components whose full automorphism group has uncountable strong cofinality, and those whose full automorphism group contains an open subgroup with ample generics.

The group of homeomorphisms of the Cantor set has ample generics

We show that the group of homeomorphisms of the Cantor set H(2ℕ) has ample generics, that is, for every m the diagonal conjugacy action g·(h1, h2, …, hm)=(gh1g−1, gh2g−1, …, ghmg−1) of H(2ℕ) on H(2ℕ)m has a comeager orbit. This answers a question of Kechris and Rosendal. We show that a generic tuple in H(2ℕ)m can be taken to be the limit of a certain projective Fraïssé family. We also give an example of a projective Fraïssé family, which has a simpler description than the one considered in the general case, and such that its limit is a homeomorphism of the Cantor set that has a comeager conjugacy class.

Turbulence, amalgamation, and generic automorphisms of homogeneous structures

We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has a comeager conjugacy class (answering a question of Akin, Hurley and Kennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which case the groups are said to admit ample generics). We show that Polish groups with ample generics have the small index property (generalizing results of Hodges, Hodkinson, Lascar and Shelah) and arbitrary homomorphisms from such groups into separable groups are automatically continuous. Moreover, in the case of oligomorphic permutation groups, they have uncountable cofinality and the Bergman property. These results in particular apply to automorphism groups of many ω-stable, ℵ0-categorical structures and of the random graph. In this connection, we also show that the infinite symmetric group S∞ has a unique non-trivial separable group topology. For several interesting groups we also establish Serre's properties (FH) and (FA).