A survey of measured group theory (Q2908716)

The present survey studies infinite countable groups using measure-theoretic tools as well as ergodic theory of group actions. Emphasis is put on the impact of the group structure on these actions. The origins of measured group theory go back to the paper [Ann. Math. (2) 112, 511--529 (1980; Zbl 0468.22011)] by \textit{R. J. Zimmer}. That paper established a deep connection between questions on orbit equivalence in ergodic theory and Margulis' superrigidity theorem for lattices in semisimple groups. The aim of the present survey is to give an overview of the current state of the subject restricted to the interaction of infinite groups with ergodic theory, omitting the connections to the theory of von Neumann algebras and descriptive set theory.NEWLINENEWLINE This survey is organized as follows: after the first introductory section, the second section focuses on a general preliminary discussion emphasizing the relations among measure equivalence, quasi-isometry, and orbit equivalence in ergodic theory. The third section deals with the notion of measure equivalence between countable groups. In the fourth section, equivalence relations with orbit relations as a primary example are studied. It is noteworthy that in Sections 3 and 4, the author considers separately the invariants of the studied objects (groups and relations) and rigidity results. The main techniques used in all these theories are described in the fifth section, where some proofs of the aforementioned results are also given.NEWLINENEWLINEFor the entire collection see [Zbl 1225.00042].