%0 Generic %1 schneider2017equivariant %A Schneider, Friedrich Martin %D 2017 %K concentration geometry observable-diameter topological-groups %T Equivariant concentration in topological groups %U http://arxiv.org/abs/1712.05379 %X We prove that, if $G$ is a second-countable topological group with a compatible right-invariant metric $d$ and $(\mu_n)_n N$ is a sequence of compactly supported Borel probability measures on $G$ converging to invariance with respect to the mass transportation distance over $d$ such that $łeft(spt \, \mu_n, d\!\!\upharpoonright_spt \, \mu_n, \mu_n\!\!\upharpoonright_spt \, \mu_n\right)_n N$ concentrates to a fully supported, compact $mm$-space $łeft(X,d_X,\mu_X\right)$, 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.

@misc{schneider2017equivariant, 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 $d$ 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.}, added-at = {2017-12-15T10:36:20.000+0100}, author = {Schneider, Friedrich Martin}, biburl = {https://www.bibsonomy.org/bibtex/29d0ca6c249e72204fd678de555a45420/tomhanika}, description = {Equivariant concentration in topological groups}, interhash = {bace5cb2a9c0dca0ee9997dbfc7c2a8c}, intrahash = {9d0ca6c249e72204fd678de555a45420}, keywords = {concentration geometry observable-diameter topological-groups}, note = {cite arxiv:1712.05379Comment: 21 pages, no figures}, timestamp = {2018-09-16T15:38:24.000+0200}, title = {Equivariant concentration in topological groups}, url = {http://arxiv.org/abs/1712.05379}, year = 2017 }