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Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture (Paperback, 1st ed. 2015)
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Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture (Paperback, 1st ed. 2015)
Series: Lecture Notes in Mathematics, 2136
Expected to ship within 12 - 17 working days
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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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