Equivariant concentration in topological groups

Abstract

We prove that, if $G$ is a second-countable topological group with a compatible right-invariant metric $d$ and ${\left({\mu }_n\right)}_{n\in \mathbb{N}}$ 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 ${\left(spt{\mu }_n,d{\upharpoonright }_{spt{\mu }_n},{\mu }_n{\upharpoonright }_{spt{\mu }_n}\right)}_{n\in \mathbb{N}}$ concentrates to a fully supported, compact $\text{ mm}$–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.