Bulletin of Symbolic Logic

Hyperlinear and sofic groups: a brief guide

Vladimir G. Pestov

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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 S$_n$, 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.

Article information

Source
Bull. Symbolic Logic Volume 14, Issue 4 (2008), 449-480.

Dates
First available in Project Euclid: 4 January 2009

Permanent link to this document
http://projecteuclid.org/euclid.bsl/1231081461

Digital Object Identifier
doi:10.2178/bsl/1231081461

Mathematical Reviews number (MathSciNet)
MR2460675

Zentralblatt MATH identifier
05495887

Subjects
Primary: 03C20: Ultraproducts and related constructions 20F69: Asymptotic properties of groups 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx] 46L10: General theory of von Neumann algebras

Citation

Pestov, Vladimir G. Hyperlinear and sofic groups: a brief guide. Bulletin of Symbolic Logic 14 (2008), no. 4, 449--480. doi:10.2178/bsl/1231081461. http://projecteuclid.org/euclid.bsl/1231081461.


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