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Weak containment of measure-preserving group actions

Burton, Peter J. and Kechris, Alexander S. (2020) Weak containment of measure-preserving group actions. Ergodic Theory and Dynamical Systems, 40 (10). pp. 2681-2733. ISSN 0143-3857.

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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.

Item Type:Article
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Additional Information:© 2019 Cambridge University Press. Received 10 August 2018 and accepted in revised form 24 February 2019. Published online by Cambridge University Press: 17 April 2019.
Subject Keywords:measure-preserving group actions, weak containment, weak equivalence
Issue or Number:10
Classification Code:2010 Mathematics Subject Classification: 37A15, 22F10, 28D15 (Primary); 37A20, 20F65, 43A07, 46M07 (Secondary)
Record Number:CaltechAUTHORS:20170712-090422831
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Official Citation:BURTON, P., & KECHRIS, A. (2020). Weak containment of measure-preserving group actions. Ergodic Theory and Dynamical Systems, 40(10), 2681-2733. doi:10.1017/etds.2019.26
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78994
Deposited By: Tony Diaz
Deposited On:12 Jul 2017 17:14
Last Modified:24 Sep 2020 21:56

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