#### Volume 23, issue 2 (2019)

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Equivariant concentration in topological groups

### Friedrich Martin Schneider

Geometry & Topology 23 (2019) 925–956
##### Abstract

We prove that, if $G$ is a second-countable topological group with a compatible right-invariant metric $d$ and ${\left({\mu }_{n}\right)}_{n\in ℕ}$ 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 ${\left(spt{\mu }_{n},d\phantom{\rule{0.3em}{0ex}}{↾}_{spt{\mu }_{n}},{\mu }_{n}\phantom{\rule{0.3em}{0ex}}{↾}_{spt{\mu }_{n}}\right)}_{n\in ℕ}$ concentrates to a fully supported, compact –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.

##### Keywords
topological groups, topological dynamics, measure concentration, observable distance, observable diameter, metric measure spaces
##### Mathematical Subject Classification 2010
Primary: 54H11, 54H20, 22A10, 53C23