The group of homeomorphisms of the Cantor set has ample generics

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 $g\cdot (h_1,h_2,...,h_m)=(g{h}_1{g}^{-1},g{h}_2{g}^{-1},...,g{h}_m{g}^{-1})$ 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.

final version, to appear in Bulletin of the London Mathematical Society

Cited by in corpus (8)

• Graph Theoretic Structure of Maps of the Cantor Space
• Connected Polish groups with ample generics
• Mapping class groups with the Rokhlin property
• On the automorphism group of the universal homogeneous meet-tree
• Compact spaces as quotients of projective Fraisse limits
• Hall universal group has ample generic automorphisms
• Arcs, hypercubes, and graphs as quotients of projective Fraïssé limits
• The generic combinatorial simplex