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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 312 (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
 Israel J. Math
"... Generic representations of abelian groups and extreme amenability ..."
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Cited by 15 (4 self)
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Generic representations of abelian groups and extreme amenability
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 9 (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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Weakly almost periodic functions, model theoretic stability, and minimality of topological groups
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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,
The Cofinality Spectrum of The Infinite Symmetric Group
 Department of Mathematics, University of Nebraska at
, 1997
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