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Weak containment of measure-preserving group actions

Part of: Methods of category theory in functional analysis Measure-theoretic ergodic theory Abstract harmonic analysis Noncompact transformation groups Special aspects of infinite or finite groups Ergodic theory

Published online by Cambridge University Press:  17 April 2019

PETER J. BURTON  and
ALEXANDER S. KECHRIS
PETER J. BURTON
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, TX78712-1202, USA email pjburton@math.utexas.edu
ALEXANDER S. KECHRIS
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, CA91125, USA email kechris@caltech.edu
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Abstract

This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.

Keywords

measure-preserving group actionsweak containmentweak equivalence

MSC classification

Primary: 37A15: General groups of measure-preserving transformations 22F10: Measurable group actions 28D15: General groups of measure-preserving transformations
Secondary: 37A20: Orbit equivalence, cocycles, ergodic equivalence relations 20F65: Geometric group theory 43A07: Means on groups, semigroups, etc.; amenable groups 46M07: Ultraproducts
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 10 , October 2020 , pp. 2681 - 2733
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Copyright
© Cambridge University Press, 2019

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