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Oligomorphic permutation groups
 LONDON MATHEMATICAL SOCIETY STUDENT TEXTS
, 1999
"... 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 ntuples of elements of Ω). Such groups have traditionally been linked with model theory and combinatorial enumeration; more recently their grouptheoretic pro ..."
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Cited by 185 (24 self)
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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 ntuples of elements of Ω). Such groups have traditionally been linked with model theory and combinatorial enumeration; more recently their grouptheoretic 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 grouptheoretic properties such as simplicity, the small index property, and “almostfreeness”.
On Bergman’s property for the automorphism group of relatively free groups, preprint 2004
"... 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 ..."
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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
A topological version of the Bergman property
"... ABSTRACT. A topological group G is defined to have property (OB) if any Gaction 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 ..."
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ABSTRACT. A topological group G is defined to have property (OB) if any Gaction 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.
Reconstruction of homogeneous relational structures
 J. Symb. Logic
"... This paper contains a result on the reconstruction of certain homogeneous transitive ωcategorical 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 nonstandard sense, ..."
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This paper contains a result on the reconstruction of certain homogeneous transitive ωcategorical 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 nonstandard sense,
The Cofinality Spectrum of The Infinite Symmetric Group
 Journal of Symbolic Logic
, 1997
"... Research partially supported by the BSF. Publication 524 of the first author. Research partially supported by NSF Grants. 1 Typeset by AMSTEX ..."
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Cited by 4 (3 self)
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Research partially supported by the BSF. Publication 524 of the first author. Research partially supported by NSF Grants. 1 Typeset by AMSTEX
The cofinality of the random graph
 J. Symbolic Logic
"... 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 ..."
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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
AUTOMATIC CONTINUITY IN HOMEOMORPHISM GROUPS OF COMPACT 2MANIFOLDS
"... ABSTRACT. We show that any homomorphism from the homeomorphism group of a compact 2manifold, with the compactopen topology, or equivalently, with the topology of uniform convergence, into a separable topological group is automatically continuous. 1. ..."
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ABSTRACT. We show that any homomorphism from the homeomorphism group of a compact 2manifold, with the compactopen topology, or equivalently, with the topology of uniform convergence, into a separable topological group is automatically continuous. 1.
GENERIC REPRESENTATIONS OF FINITELY GENERATED GROUPS ON FRAISSE STRUCTURES
"... 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 ..."
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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
APPROXIMATION OF AUTOMORPHISMS OF THE RATIONALS AND THE RANDOM GRAPH
"... ABSTRACT. Let G be the group of orderpreserving automorphisms of the rationals Q, or the group of colourpreserving automorphisms of the Ccoloured random graph RC. We show that given any nonidentity f ∈ G, there exists g ∈ G such that every automorphism in G is the limit of a sequence of automorp ..."
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ABSTRACT. Let G be the group of orderpreserving automorphisms of the rationals Q, or the group of colourpreserving automorphisms of the Ccoloured random graph RC. We show that given any nonidentity f ∈ G, there exists g ∈ G such that every automorphism in G is the limit of a sequence of automorphisms generated by f and g. Moreover, if, in some sense, f has no finite structure, then g can be chosen with a great deal of flexibility. 1.