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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 320 (26 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”.
Turbulence, amalgamation, and generic automorphisms of homogeneous structures
, 2004
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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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Cited by 24 (10 self)
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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.
Generic representations of abelian groups and extreme amenability
, 2012
"... If G is a Polish group and Γ is a countable group, denote by Hom(Γ,G) the space of all homomorphisms Γ → G. We study properties of the group π(Γ) for the generic π ∈ Hom(Γ,G), when Γ is abelian and G is one of the following three groups: the unitary group of an infinitedimensional Hilbert space, th ..."
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Cited by 12 (2 self)
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If G is a Polish group and Γ is a countable group, denote by Hom(Γ,G) the space of all homomorphisms Γ → G. We study properties of the group π(Γ) for the generic π ∈ Hom(Γ,G), when Γ is abelian and G is one of the following three groups: the unitary group of an infinitedimensional Hilbert space, the automorphism group of a standard probability space, and the isometry group of the Urysohn metric space. Under mild assumptions on Γ, we prove that in the first case, there is (up to isomorphism of topological groups) a unique generic π(Γ); in the other two, we show that the generic π(Γ) is extremely amenable. We also show that if Γ is torsionfree, the centralizer of the generic π is as small as possible, extending a result of Chacon–Schwartzbauer from ergodic theory.
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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Cited by 8 (1 self)
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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.
On Bergman’s property for the automorphism group of relatively free groups
, 2004
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The Cofinality Spectrum of The Infinite Symmetric Group
 Department of Mathematics, University of Nebraska at
, 1997
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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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Cited by 6 (2 self)
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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,
Weakly almost periodic functions, model theoretic stability, and minimality of topological groups
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