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      Equivariant concentration in topological groups

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          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.

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          On choosing and bounding probability metrics

          When studying convergence of measures, an important issue is the choice of probability metric. In this review, we provide a summary and some new results concerning bounds among ten important probability metrics/distances that are used by statisticians and probabilists. We focus on these metrics because they are either well-known, commonly used, or admit practical bounding techniques. We summarize these relationships in a handy reference diagram, and also give examples to show how rates of convergence can depend on the metric chosen.
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            A Topological Application of the Isoperimetric Inequality

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              On subgroups of minimal topological groups

              A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a topological group is the greatest lower bound of the left and right uniformities. A group is Roelcke-precompact if it is precompact with respect to the Roelcke uniformity. Many naturally arising non-Abelian topological groups are Roelcke-precompact and hence have a natural compactification. We use such compactifications to prove that some groups of isometries are minimal. In particular, if U_1 is the Urysohn universal metric space of diameter 1, the group Iso(U_1) of all self-isometries of U_1 is Roelcke-precompact, topologically simple and minimal. We also show that every topological group is a subgroup of a minimal topologically simple Roelcke-precompact group of the form Iso(M), where M is an appropriate non-separable version of the Urysohn space.
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                Author and article information

                Journal
                14 December 2017
                Article
                1712.05379
                aef0dba8-3d82-463f-be49-a5907c2cb9be

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                21 pages, no figures
                math.FA math.GR math.MG

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