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.
