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The group of homeomorphisms of the Cantor set has ample generics

By Aleksandra Kwiatkowska

Abstract

We show that the group of homeomorphisms of the Cantor set $H(K)$ has ample generics, that is, for every $m$ the diagonal conjugacy action $g\cdot(h_1,h_2,..., h_m)=(gh_1g^{-1},gh_2g^{-1},..., gh_mg^{-1})$ of $H(K)$ on $H(K)^m$ has a comeager orbit. This answers a question of Kechris and Rosendal. We show that the generic tuple in $H(K)^m$ can be taken to be the limit of a certain projective Fraisse family. We also present a proof of the existence of the generic homeomorphism of the Cantor set in the context of the projective Fraisse theory.Comment: final version, to appear in Bulletin of the London Mathematical Societ

Topics: Mathematics - Dynamical Systems, Mathematics - Group Theory, Mathematics - Logic
Year: 2012
DOI identifier: 10.1112/blms/bds039
OAI identifier: oai:arXiv.org:1104.3340
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