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Metric mean dimension for algebraic actions of Sofic groups


Author: Ben Hayes
Journal: Trans. Amer. Math. Soc. 369 (2017), 6853-6897
MSC (2010): Primary 37A35, 37A55, 37B40; Secondary 22D25
DOI: https://doi.org/10.1090/tran/6834
Published electronically: March 30, 2017
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Abstract: We prove that if $ \Gamma $ is a sofic group and $ A$ is a finitely generated $ \mathbb{Z}(\Gamma )$-module, then the metric mean dimension of $ \Gamma \curvearrowright \widehat {A},$ in the sense of Hanfeng Li, is equal to the von Neumann-Lück rank of $ A.$ This partially extends the results of Hanfeng Li and Bingbing Liang from the case of amenable groups to the case of sofic groups. Additionally we show that the mean dimension of $ \Gamma \curvearrowright \widehat {A}$ is the von Neumann-Lück rank of $ A$ if $ A$ is finitely presented and $ \Gamma $ is residually finite. It turns out that our approach naturally leads to a notion of $ p$-metric mean dimension, which is in between mean dimension and the usual metric mean dimension. This can be seen as an obstruction to the equality of mean dimension and metric mean dimension. While we cannot decide if mean dimension is the same as metric mean dimension for algebraic actions, we show in the metric case that for all $ p$ the $ p$-metric mean dimension coincides with the von Neumann-Lück rank of the dual module.


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Additional Information

Ben Hayes
Affiliation: Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, California 90095-155
Address at time of publication: Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville, Tennessee 37240
Email: benjamin.r.hayes@vanderbilt.edu

DOI: https://doi.org/10.1090/tran/6834
Keywords: Sofic groups, metric mean dimension, von Neumann rank
Received by editor(s): February 11, 2015
Received by editor(s) in revised form: June 21, 2015, and September 28, 2015
Published electronically: March 30, 2017
Additional Notes: The author is grateful for support from NSF Grants DMS-1161411 and DMS-0900776
Article copyright: © Copyright 2017 American Mathematical Society