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

Mean dimension, mean rank, and von Neumann–Lück rank AbstractWe introduce an invariant, called mean rank, for any module ℳ{\mathcal{M}}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 ℳ{\mathcal{M}}, the mean rank of ℳ{\mathcal{M}}, and the von Neumann–Lück rank of ℳ{\mathcal{M}}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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal für die reine und angewandte Mathematik de Gruyter

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

, Volume 2018 (739): 34 – Jun 1, 2018
34 pages

/lp/degruyter/mean-dimension-mean-rank-and-von-neumann-l-ck-rank-W2ip6OV00r
Publisher
De Gruyter
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1435-5345
eISSN
1435-5345
D.O.I.
10.1515/crelle-2015-0046
Publisher site
See Article on Publisher Site

### Abstract

AbstractWe introduce an invariant, called mean rank, for any module ℳ{\mathcal{M}}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 ℳ{\mathcal{M}}, the mean rank of ℳ{\mathcal{M}}, and the von Neumann–Lück rank of ℳ{\mathcal{M}}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.

### Journal

Journal für die reine und angewandte Mathematikde Gruyter

Published: Jun 1, 2018

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