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Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture

  • Book
  • © 2015


  • Accessible to anyone with a graduate-level mathematical background
  • Includes many exercises to help the reader to learn quickly
  • Presents some outstanding open problems in group theory and operator algebras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2136)

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About this book

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.

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Table of contents (3 chapters)

Authors and Affiliations

  • Center for Mathematics & Computer Science, CWI, Amsterdam, The Netherlands

    Valerio Capraro

  • Department of Mathematics, California Institute of Technology, Pasadena, USA

    Martino Lupini

Bibliographic Information

  • Book Title: Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture

  • Authors: Valerio Capraro, Martino Lupini

  • Series Title: Lecture Notes in Mathematics

  • DOI:

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2015

  • Softcover ISBN: 978-3-319-19332-8Published: 20 October 2015

  • eBook ISBN: 978-3-319-19333-5Published: 12 October 2015

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: VIII, 151

  • Topics: Group Theory and Generalizations, Operator Theory

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