Geometry & Topology

Equivariant concentration in topological groups

Friedrich Martin Schneider

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We prove that, if G is a second-countable topological group with a compatible right-invariant metric d and ( μ n ) 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 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.

Article information

Geom. Topol., Volume 23, Number 2 (2019), 925-956.

Received: 18 January 2018
Revised: 2 May 2018
Accepted: 14 July 2018
First available in Project Euclid: 17 April 2019

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Zentralblatt MATH identifier

Primary: 54H11: Topological groups [See also 22A05] 54H20: Topological dynamics [See also 28Dxx, 37Bxx] 22A10: Analysis on general topological groups 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

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


Schneider, Friedrich Martin. Equivariant concentration in topological groups. Geom. Topol. 23 (2019), no. 2, 925--956. doi:10.2140/gt.2019.23.925.

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