{ "id": "1107.1698", "version": "v2", "published": "2011-07-08T18:56:22.000Z", "updated": "2012-11-23T19:52:12.000Z", "title": "Generic representations of abelian groups and extreme amenability", "authors": [ "Julien Melleray", "Todor Tsankov" ], "comment": "Version 2", "journal": "Isra\\\"el Journal of Mathematics 198, 1 (2013)", "doi": "10.1007/s11856-013-0036-5", "categories": [ "math.LO", "math.DS" ], "abstract": "If $G$ is a Polish group and $\\Gamma$ is a countable group, denote by $\\Hom(\\Gamma, G)$ the space of all homomorphisms $\\Gamma \\to G$. We study properties of the group $\\cl{\\pi(\\Gamma)}$ for the generic $\\pi \\in \\Hom(\\Gamma, G)$, when $\\Gamma$ is abelian and $G$ is one of the following three groups: the unitary group of an infinite-dimensional Hilbert space, the automorphism group of a standard probability space, and the isometry group of the Urysohn metric space. Under mild assumptions on $\\Gamma$, we prove that in the first case, there is (up to isomorphism of topological groups) a unique generic $\\cl{\\pi(\\Gamma)}$; in the other two, we show that the generic $\\cl{\\pi(\\Gamma)}$ is extremely amenable. We also show that if $\\Gamma$ is torsion-free, the centralizer of the generic $\\pi$ is as small as possible, extending a result of King from ergodic theory.", "revisions": [ { "version": "v2", "updated": "2012-11-23T19:52:12.000Z" } ], "analyses": { "keywords": [ "abelian groups", "extreme amenability", "generic representations", "urysohn metric space", "infinite-dimensional hilbert space" ], "tags": [ "journal article" ], "note": { "typesetting": "TeX", "pages": 0, "language": "en", "license": "arXiv", "status": "editable", "adsabs": "2011arXiv1107.1698M" } } }