Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups

By Vladimir G. Pestov. 2000

107 citations recorded by the source corpus.

AbstractIn 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 $$\left( \mathbb{U} \right)$$ of isometries of Urysohn’s universal complete separable metric space $$\mathbb{U}$$ , equipped with the compact-open topology, acts upon an arbitrary compact space, it has a fixed point. The same is true if $$\mathbb{U}$$ 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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dfb1eaea3a0cdc81a60839413882734a208a597d

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Citations in the corpus

Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups (2003, background): “class preserving) automorphisms of Lebesgue measure λ on I, with the weak topology (Giordano-Pestov [26]); Iso(U), the group of isometries of the Urysohn space, with the pointwise convergence topology (Pestov [ 66 ])), and Aut(h Q, <i ), the group of automorphisms of the rationals with the usual ord”
The Urysohn universal metric space is homeomorphic to a Hilbert space (2003, background): “countable base. A deep result concerning the group G = Is(U) was established by V.Pestov: the group G is extremely amenable, i.e., every compact space with a continuous action of G has a G-ﬁxed point [6, 3]. A.M.Vershik showed that the space U can be obtained as the completion of a countable metric ”
Extending partial isometries (2005, background): “(For the precise meaning of “approximate” see [9].”
On subgroups of minimal topological groups (2000, background): “). My 3 The setting considered in these papers and books is not exactly the same as in Theorem 1.8 (detailed proofs are given either for the separable case or for unbounded metrics), but, as noted in [29], the same argument works for bounded metrics verbatim. ON SUBGROUPS OF MINIMAL TOPOLOGICAL GROU”