Turbulence, amalgamation, and generic automorphisms of homogeneous structures

By Alexander S. Kechris, Christian Rosendal. 2004

338 citations recorded by the source corpus.

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 and 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 and 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).

Subfields: Genetics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/01f4786318c51840c203ece7ca27c0f2777e842e

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Citations in the corpus

Graph theoretic structure of maps of the Cantor space (2012, methodology): “ion In recent years H({0,1}N), the group of homeomorphisms of the Cantor space, has enjoyed attention from mathematicians working in diverse ﬁelds such as dynamical systems [4], [16] and model theory [18]. From the dynamics point of view, {0,1}N is the symbol space on two variables and a fundamental”
Big Mapping Class Groups: An Overview (2020, background): “For examples, none of which are trivial, the homeomorphism group of the Cantor set [73] as well as the homeomorphism group of any closed manifold [82,107] has the automatic continuity property.”
Extending partial isometries (2005, background): “Remark 1.1: Kechris and Rosendal [ 5, Theorem 1.2 ] and, independently, Glasner and Pestov proved a precursor to Corollaries 4.1 and 4.4.”
Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups (2013, methodology): “rom the introduction. Recall that a Polish group G has ample generics if the conjugation action G yGn has a comeager orbit for every n. (Note that the deﬁnitions of ample generics given in [HHLS] and [KR] are somewhat different; we use the one of [KR].) Lemma 4.11. Let G be the automorphism group of”
Generically there is but one self homeomorphism of the Cantor set (2006, background): “preserve a special kind of a probability measure µ on X has the property that its action on itself by conjugation admits a dense Gδ conjugacy class. Recently this was shown by Kechris and Rosendal in [KR] to be the case for many other closed subgroups of G = H(X), including G itself. However the aut”
Automatic continuity of homomorphisms and fixed points on metric compacta (2006, result): “For example, it follows, in combination with a result on V.G. Pestov, that any action of the discrete groupHomeo+(R) by homeomorphisms on a compact metric space has a fixed point.”
Independence in topological and C*-dynamics (2006, background): “Kechris and Rosendal showed by modeltheoretic means that the Polish group of homeomorphisms of the Cantor set has a dense Gδ conjugacy class [47], and a description of the generic homeomorphism has been given by Akin, Glasner, and Weiss in [1].”
A topological version of the Bergman property (2005, neutral): “L O ] 2 8 Se p”