Ramsey-Milman phenomenon,Urysohn metric spaces,and extremely amenable groups |
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Authors: | Vladimir Pestov |
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Institution: | (1) School of Mathematical and Computing Sciences, Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand |
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Abstract: | In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed
points, and Ramsey-type theorems for metric spaces. We prove that whenever the group Iso
of isometries of Urysohn’s universal complete separable metric space
, equipped with the compact-open topology, acts upon an arbitrary compact space, it has a fixed point. The same is true if
is replaced with any generalized Urysohn metric spaceU that is sufficiently homogeneous. Modulo a recent theorem by Uspenskij that every topological group embeds into a topological
group of the form Iso(U), our result implies that every topological group embeds into an extremely amenable group (one admitting an invariant multiplicative
mean on bounded right uniformly continuous functions). By way of the proof, we show that every topological group is approximated
by finite groups in a certain weak sense. Our technique also results in a new proof of the extreme amenability (fixed point
on compacta property) for infinite orthogonal groups. Going in the opposite direction, we deduce some Ramsey-type theorems
for metric subspaces of Hilbert spaces and for spherical metric spaces from existing results on extreme amenability of infinite
unitary groups and groups of isometries of Hilbert spaces. |
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Keywords: | |
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