Bulletin of Symbolic Logic

Hyperlinear and sofic groups: a brief guide

Vladimir G. Pestov

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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

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

First available in Project Euclid: 4 January 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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