Topology Proceedings 33 (2009), pp. 1-12

On extremely amenable groups of homeomorphisms

Vladimir Uspenskij

A topological group G is extremely amenable if every compact G-space has a G-fixed point. Let X be compact and G ⊂ Homeo(X). We prove that the following are equivalent: (1) G is extremely amenable; (2) every minimal closed G-invariant subset of Exp R is a singleton, where R is the closure of the set of all graphs of g ∈ G in the space Exp (X²) (Exp stands for the space of closed subsets); (3) for each n = 1, 2, ... there is a closed G-invariant subset Y_n of (Exp X)ⁿ such that ∪_n=1^∞ Y_n contains arbitrarily fine covers of X and for every n ≥ 1 every minimal closed G-invariant subset of \Exp Y_n is a singleton. This yields an alternative proof of Pestov's theorem that the group of all order-preserving self-homeomorphisms of the Cantor middle-third set (or of the interval [0, 1]) is extremely amenable.

Keywords: greatest ambit, minimal flow, Vietoris topology, exponent

Mathematics Subject Classification: 54H11 (22A05 22A15 22F05 37B05 54H15 54H20 54H25 57S05)

Document formats: PDF file 158.6 Kb

Topology Proceedings, Volume 33 (2009)
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