## Geometry & Topology

### Equivariant concentration in topological groups

Friedrich Martin Schneider

#### Abstract

We prove that, if $G$ is a second-countable topological group with a compatible right-invariant metric $d$ and $( μ n ) n ∈ ℕ$ 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 $( spt μ n , d ↾ spt μ n , μ n ↾ spt μ n ) n ∈ ℕ$ concentrates to a fully supported, compact –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

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

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

https://projecteuclid.org/euclid.gt/1555466433

Digital Object Identifier
doi:10.2140/gt.2019.23.925

Mathematical Reviews number (MathSciNet)
MR3939055

Zentralblatt MATH identifier
07056056

#### Citation

Schneider, Friedrich Martin. Equivariant concentration in topological groups. Geom. Topol. 23 (2019), no. 2, 925--956. doi:10.2140/gt.2019.23.925. https://projecteuclid.org/euclid.gt/1555466433

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