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Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture

Capraro, Valerio and Lupini, Martino (2015) Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture. Lecture Notes in Mathematics. No.2136. Springer , Cham, Switzerland. ISBN 978-3-319-19332-8. http://resolver.caltech.edu/CaltechAUTHORS:20180806-085228595

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Abstract

This monograph presents some cornerstone results in the study of sofic and hyperlinear groups and the closely related Connes' embedding conjecture. These notions, as well as the proofs of many results, are presented in the framework of model theory for metric structures. This point of view, rarely explicitly adopted in the literature, clarifies the ideas therein, and provides additional tools to attack open problems. Sofic and hyperlinear groups are countable discrete groups that can be suitably approximated by finite symmetric groups and groups of unitary matrices. These deep and fruitful notions, introduced by Gromov and Radulescu, respectively, in the late 1990s, stimulated an impressive amount of research in the last 15 years, touching several seemingly distant areas of mathematics including geometric group theory, operator algebras, dynamical systems, graph theory, and quantum information theory. Several long-standing conjectures, still open for arbitrary groups, are now settled for sofic or hyperlinear groups. The presentation is self-contained and accessible to anyone with a graduate-level mathematical background. In particular, no specific knowledge of logic or model theory is required. The monograph also contains many exercises, to help familiarize the reader with the topics present.


Item Type:Book
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/978-3-319-19333-5DOIArticle
ORCID:
AuthorORCID
Lupini, Martino0000-0003-1588-7057
Additional Information:© 2015 Springer International Publishing.
Subject Keywords:Connes' embedding conjecture; Hyperlinear groups; Model theory for metric structures; Sofic groups
Classification Code:MSC: 20F65, 20F69, 03C20, 03C98, 46L10, 46M07
Record Number:CaltechAUTHORS:20180806-085228595
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20180806-085228595
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:88585
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:06 Aug 2018 16:49
Last Modified:06 Aug 2018 16:49

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