Hyperlinear and sofic groups: a brief guide

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dc.creatorPestov, Vladimir G.
dc.date2008-04-24
dc.date2008-08-04
dc.date.accessioned2026-07-07T12:47:24Z
dc.date.available2026-07-07T12:47:24Z
dc.descriptionRelatively recently, two new classes of (discrete, countable) groups have been isolated: hyperlinear groups and sofic groups. They come from different corners of mathematics (operator algebras and symbolic dynamics, respectively), and were introduced independently from each other, but are closely related nevertheless. Hyperlinear groups have their origin in Connes' Embedding Conjecture about von Neumann factors of type $II_1$, while sofic groups, introduced by Gromov, are motivated by Gottschalk Surjunctivity Conjecture (can a shift $A^G$ contain a proper isomorphic copy of itself, where $A$ is a finite discrete space and $G$ is a group?). Groups from both classes can be characterized as subgroups of metric ultraproducts of families of certain metric groups (formed in the same way as ultraproducts of Banach spaces): unitary groups of finite rank lead to hyperlinear groups, symmetric groups of finite rank - to sofic groups. We offer an introductory guide to some of the main concepts, results, and sources of the theory, following Connes, Gromov, Benjamin Weiss, Kirchberg, Ozawa, Radulescu, Elek and Szabó, and others, and discuss open questions which are for the time being perhaps more numerous than the results.
dc.description28 pages, 2 figures, latex 2e. This version incorporates minor corrections made in the Bulletin of Symbolic Logic galley proofs
dc.identifierhttps://arxiv.org/abs/0804.3968
dc.identifierhttp://arxiv.org/abs/0804.3968
dc.identifierThe Bulletin of Symbolic Logic 14 (2008), pp. 449-480.
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/221707
dc.subjectGroup Theory
dc.subject03C20; 20F69; 37B10; 46L10
dc.titleHyperlinear and sofic groups: a brief guide
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