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Equivariant concentration in topological groups

Friedrich Martin Schneider

Geometry & Topology 23 (2019) 925–956

We prove that, if G is a second-countable topological group with a compatible right-invariant metric d and (μn)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 (sptμn,dsptμn,μnsptμn)n concentrates to a fully supported, compact  mm–space (X,dX,μX), 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.

topological groups, topological dynamics, measure concentration, observable distance, observable diameter, metric measure spaces
Mathematical Subject Classification 2010
Primary: 54H11, 54H20, 22A10, 53C23
Received: 18 January 2018
Revised: 2 May 2018
Accepted: 14 July 2018
Published: 8 April 2019
Proposed: Yasha Eliashberg
Seconded: Misha Gromov, Bruce Kleiner
Friedrich Martin Schneider
Institute of Algebra
TU Dresden
Departamento de Matemática
Universidade Federal de Santa Catarina
Santa Catarina