Equivariant concentration in topological groups

TL;DR

This paper proves a new connection between measure concentration, topological groups, and Samuel compactification, confirming a conjecture by Pestov and extending Gromov and Milman's results on extreme amenability.

Contribution

It establishes a link between measure concentration in topological groups and their Samuel compactification, confirming Pestov's conjecture and generalizing classical results.

Findings

Homeomorphism of the limit space to a G-invariant subspace of the Samuel compactification.

Confirmation of Pestov's conjecture on measure concentration and extreme amenability.

Connection between orbit diameter in flows and Gromov's observable diameters.

Abstract

We prove that, if $G$ is a second-countable topological group with a compatible right-invariant metric $d$ and $(\mu_{n})_{n \in \mathbb{N}}$ 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 $\left(\mathrm{spt} \, \mu_{n}, d\!\!\upharpoonright_{\mathrm{spt} \, \mu_{n}}, \mu_{n}\!\!\upharpoonright_{\mathrm{spt} \, \mu_{n}}\right)_{n \in \mathbb{N}}$ concentrates to a fully supported, compact $mm$-space $\left(X,d_{X},\mu_{X}\right)$, 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.