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Hyperlinear and Sofic Groups: A Brief Guide

Published online by Cambridge University Press:  15 January 2014

Vladimir G. Pestov
Vladimir G. Pestov*
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave. Ottawa, Ontario K1N 6N5, CanadaE-mail: vpest283@uottawa.ca
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Abstract

This is an introductory survey of the emerging theory of two new classes of (discrete, countable) groups, called hyperlinear and sofic groups. They can be characterized as subgroups of metric ultraproducts of families of, respectively, unitary groups U(n) and symmetric groups Sn, n ∈ ℕ. Hyperlinear groups come from theory of operator algebras (Connes' Embedding Problem), while sofic groups, introduced by Gromov, are motivated by a problem of symbolic dynamics (Gottschalk's Surjunctivity Conjecture). Open questions are numerous, in particular it is still unknown if every group is hyperlinear and/or sofic.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 14 , Issue 4 , December 2008 , pp. 449 - 480
Copyright
Copyright © Association for Symbolic Logic 2008

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