Bulletin of Symbolic Logic

Hyperlinear and sofic groups: a brief guide

Vladimir G. Pestov

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

Permanent link to this document

Digital Object Identifier

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. http://projecteuclid.org/euclid.bsl/1231081461.

Export citation