A permutation group G (acting on a set Ω, usually infinite) is said to be oligomorphic if G has only finitely many orbits on Ω n (the set of n-tuples of elements of Ω). Such groups have traditionally been linked with model theory and combinatorial enumeration; more recently their group-theoretic properties have been studied, and links with graded algebras, Ramsey theory, topological dynamics, and other areas have emerged. This paper is a short summary of the subject, concentrating on the enumerative and algebraic aspects but with an account of grouptheoretic properties. The first section gives an introduction to permutation groups and to some of the more specific topics we require, and the second describes the links to model theory and enumeration. We give a spread of examples, describe results on the growth rate of the counting functions, discuss a graded algebra associated with an oligomorphic group, and finally discuss group-theoretic properties such as simplicity, the small index property, and “almost-freeness”.

Abstract. 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-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-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). 1.

Let Ω be an infinite set and Sym(Ω) the full symmetric group on Ω. In his recent preprint [1] Bergman proved the following delightful result. Consider a system (Yi: i ∈ I) where |I | � |Ω | of subsets of Sym(Ω) whose union is Sym(Ω). Then there is a member Y = Yi0 of the system such

This paper contains a result on the reconstruction of certain homogeneous transitive ω-categorical structures from their automorphism group. The structures treated are relational. In the proof it is shown that their automorphism group contains a generic pair (in a slightly non-standard sense,

Research partially supported by the BSF. Publication 524 of the first author. Research partially supported by NSF Grants. 1 Typeset by AMS-TEX

ABSTRACT. A topological group G is defined to have property (OB) if any G-action by isometries on a metric space, which is separately continuous, has bounded orbits. We study this topological analogue of strong uncountable cofinality in the context of Polish groups, where we show it to have several interesting reformulations and consequences. We subsequently apply the results obtained in order to verify property (OB) for a number of groups of isometries and homeomorphism groups of compact metric spaces. We also give a proof that the isometry group of the rational Urysohn metric space of diameter 1 has strong uncountable cofinality. 1.

We show that under Martin’s Axiom, the cofinality cf(Aut(Γ)) of the automorphism group of the random graph Γ is 2 ω. 1

ABSTRACT. We show that any homomorphism from the homeomorphism group of a compact 2-manifold, with the compact-open topology, or equivalently, with the topology of uniform convergence, into a separable topological group is automatically continuous. 1.

Abstract. For finitely generated groups Γ and ultrahomogeneous countable relational structures M we study the space Rep(Γ, M) of all representations of Γ by automorphisms on M equipped with the topology it inherits seen as a closed subset of Aut(M) Γ. When Γ is the free group Fn on n generators this space is just Aut(M) n, but is in general significantly more complicated. We prove that when Γ is finitely generated abelian and M the random structure of a finite relational language or the random ultrametric space of a countable distance set there is a generic point in Rep(Γ, M), i.e., there is a comeagre set of mutually conjugate representations of Γ on M. This is analogous to results of Hrushovski, Herwig, and Herwig–Lascar for the case Γ = Fn. Contents 1. Representations of discrete groups in topological groups 1

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