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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”.
Hyperlinear and sofic groups: a brief guide
 Bull. Symbolic Logic
"... Relatively recently, two new classes of (discrete, countable) groups have been isolated: hyperlinear groups and sofic groups. They come from different corners of mathematics (operator algebras and symbolic dynamics, respectively), and were introduced independently from each other, but are closely re ..."
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Cited by 67 (1 self)
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Relatively recently, two new classes of (discrete, countable) groups have been isolated: hyperlinear groups and sofic groups. They come from different corners of mathematics (operator algebras and symbolic dynamics, respectively), and were introduced independently from each other, but are closely related nevertheless.
Automatic Continuity of Homomorphisms and Fixed Point on Metric Compacta
, 2005
"... We prove that arbitrary homomorphisms from one of the groups Homeo(2 N), Homeo(2 N) N, Aut(Q, <), Homeo(R), or Homeo(S 1) into a separable group are automatically continuous. This has consequences for the representations of these groups as discrete groups. For example, it follows, in combination ..."
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Cited by 24 (1 self)
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We prove that arbitrary homomorphisms from one of the groups Homeo(2 N), Homeo(2 N) N, Aut(Q, <), Homeo(R), or Homeo(S 1) into a separable group are automatically continuous. This has consequences for the representations of these groups as discrete groups. For example, it follows, in combination with a result on V.G. Pestov, that any action of the discrete group Homeo+(R) by homeomorphisms on a compact metric space has a fixed point. 1
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.
Strongly bounded groups and infinite powers of finite groups
 Comm. Algebra
"... Abstract. We define a group as strongly bounded if every isometric action on a metric space has bounded orbits. This latter property is equivalent to the socalled uncountable strong cofinality, recently introduced by Bergman. Our main result is that G I is strongly bounded when G is a finite, perfe ..."
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Cited by 19 (4 self)
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Abstract. We define a group as strongly bounded if every isometric action on a metric space has bounded orbits. This latter property is equivalent to the socalled uncountable strong cofinality, recently introduced by Bergman. Our main result is that G I is strongly bounded when G is a finite, perfect group and I is any set. This strengthens a result of Koppelberg and Tits. We also prove that ω1existentially closed groups are strongly bounded. 1.
Generically there is but one self homeomorphism of the Cantor set
 arXiv:math.DS/0603538 v1. DAVID KERR AND HANFENG LI
"... Abstract. We describe a selfhomeomorphism R of the Cantor set X and then show that its conjugacy class in the Polish group H(X) of all homeomorphisms of X forms a dense Gδ subset of H(X). We also provide an example of a locally compact, second countable topological group which has a dense conjugacy ..."
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Cited by 16 (1 self)
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Abstract. We describe a selfhomeomorphism R of the Cantor set X and then show that its conjugacy class in the Polish group H(X) of all homeomorphisms of X forms a dense Gδ subset of H(X). We also provide an example of a locally compact, second countable topological group which has a dense conjugacy class.
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.
Random Orderings and Unique Ergodicity of Automorphism Groups
, 2012
"... We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, ..."
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Cited by 10 (1 self)
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We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first examples of uniquely ergodic groups, other than compact groups and extremely amenable groups, after Glasner and Weiss’s example of the group of all permutations of the integers. We also contrast these results to those for certain special classes of graphs and metric spaces in which such random orderings can be found that are not uniform.
Distortion in transformation groups
, 2010
"... We exhibit rigid rotations of spheres as distortion elements in groups of diffeomorphisms, thereby answering a question of J Franks and M Handel. We also show that every homeomorphism of a sphere is, in a suitable sense, as distorted as possible in the group Homeo(S n), thought of as a discrete grou ..."
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Cited by 9 (1 self)
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We exhibit rigid rotations of spheres as distortion elements in groups of diffeomorphisms, thereby answering a question of J Franks and M Handel. We also show that every homeomorphism of a sphere is, in a suitable sense, as distorted as possible in the group Homeo(S n), thought of as a discrete group. An appendix by Y de Cornulier shows that Homeo(S n) has the strong boundedness property, recently introduced by G Bergman. This means that every action of the discrete group Homeo(S n) on a metric space by isometries has bounded orbits.