Equivariant concentration in topological groups

Schneider, Friedrich Martin

doi:10.2140/gt.2019.23.925

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Equivariant concentration in topological groups

Friedrich Martin Schneider

Geometry & Topology 23 (2019) 925–956

DOI: 10.2140/gt.2019.23.925

Abstract

We prove that, if $G$ is a second-countable topological group with a compatible right-invariant metric $d$ and ${(μ_{n})}_{n \in ℕ}$ is a sequence of compactly supported Borel probability measures on $G$ converging to invariance with respect to the mass transportation distance over $d$ and such that ${(spt μ_{n}, d ↾_{spt μ_{n}}, μ_{n} ↾_{spt μ_{n}})}_{n \in ℕ}$ concentrates to a fully supported, compact $mm$ –space $(X, d_{X}, μ_{X})$ , then $X$ is homeomorphic to a $G$ –invariant subspace of the Samuel compactification of $G$ . In particular, this confirms a conjecture by Pestov and generalizes a well-known result by Gromov and Milman on the extreme amenability of topological groups. Furthermore, we exhibit a connection between the average orbit diameter of a metrizable flow of an arbitrary amenable topological group and the limit of Gromov’s observable diameters along any net of Borel probability measures UEB–converging to invariance over the group.

Keywords

topological groups, topological dynamics, measure concentration, observable distance, observable diameter, metric measure spaces

Mathematical Subject Classification 2010

Primary: 54H11, 54H20, 22A10, 53C23

References

Publication

Received: 18 January 2018

Revised: 2 May 2018

Accepted: 14 July 2018

Published: 8 April 2019

Proposed: Yasha Eliashberg

Seconded: Misha Gromov, Bruce Kleiner

Authors

Friedrich Martin Schneider
Institute of Algebra TU Dresden Dresden Germany	Departamento de Matemática Universidade Federal de Santa Catarina Trindade Florianópolis Santa Catarina Brazil

Volume 23, issue 2 (2019)