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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

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Volume 2018, Issue 739


Mean dimension, mean rank, and von Neumann–Lück rank

Hanfeng Li / Bingbing Liang
Published Online: 2015-09-12 | DOI: https://doi.org/10.1515/crelle-2015-0046


We introduce an invariant, called mean rank, for any module of the integral group ring of a discrete amenable group Γ, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced Γ-action on the Pontryagin dual of , the mean rank of , and the von Neumann–Lück rank of all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin–Schnirelmann theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups, algebraic actions with zero mean dimension are inverse limits of finite entropy actions.


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About the article

Received: 2014-07-30

Revised: 2015-04-28

Published Online: 2015-09-12

Published in Print: 2018-06-01

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1001625

Award identifier / Grant number: DMS-1266237

H. Li was partially supported by NSF grants DMS-1001625 and DMS-1266237.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018, Issue 739, Pages 207–240, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2015-0046.

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