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Turbulence, amalgamation, and generic automorphisms of homogeneous structures
Author(s) -
Kechris Alexander S.,
Rosendal Christian
Publication year - 2007
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdl007
Subject(s) - mathematics , conjugacy class , automorphism , separable space , homeomorphism (graph theory) , metrization theorem , totally disconnected space , countable set , group (periodic table) , group isomorphism , pure mathematics , outer automorphism group , combinatorics , automorphism group , locally compact space , mathematical analysis , chemistry , organic chemistry
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, ℵ 0 ‐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).