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Proceedings of the London Mathematical Society Advance Access originally published online on November 27, 2006
Proceedings of the London Mathematical Society 2007 94(2):302-350; doi:10.1112/plms/pdl007
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© 2006 London Mathematical Society

Turbulence, amalgamation, and generic automorphisms of homogeneous structures

Alexander S. Kechris1 and Christian Rosendal2,

1 386 Sloan
Department of Mathematics 253-37
California Institute of Technology
Pasadena, CA 91125
USA
kechris{at}its.caltech.edu
2 Department of Mathematics 253-37
California Institute of Technology
Pasadena, CA 91125
USA

Received 29 November 2004. Revision received 21 February 2006.

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 {omega}-stable, N0-categorical structures and of the random graph. In this connection, we also show that the infinite symmetric group S{infty} has a unique non-trivial separable group topology. For several interesting groups we also establish Serre's properties (FH) and (FA).

Current address: Department of Mathematics University of Illinois at Urbana-Champaign 273 Altgeld Hall, MC 382 1409 West Green Street Urbana, IL 61801 USA rosendal{at}math.uiuc.edu


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