Rokhlin dimension for actions of residually finite groups

Szabo, G., Wu, J. and Zacharias, J. (2017) Rokhlin dimension for actions of residually finite groups. Ergodic Theory and Dynamical Systems, (doi:10.1017/etds.2017.113) (Early Online Publication)

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Abstract

We introduce the concept of Rokhlin dimension for actions of residually finite groups on C*-algebras, extending previous notions of Rokhlin dimension for actions of finite groups and the integers, as introduced by Hirshberg, Winter and the third author. If the group has a box space of finite asymptotic dimension, then actions with finite Rokhlin dimension preserve the property of having finite nuclear dimension, when passing to the crossed product C*-algebra. A detailed study of the asymptotic dimension of box spaces shows that finitely generated, virtually nilpotent groups have box spaces with finite asymptotic dimension, providing a reasonably large class of examples. We then establish a relation between Rokhlin dimension of residually finite groups acting on compact metric spaces and amenability dimension of the action in the sense of Guentner, Willett and Yu. We show that for free actions of in- finite, finitely generated, nilpotent groups on finite dimensional spaces, both these dimensional values are finite. In particular, the associated transformation group C*-algebras have finite nuclear dimension. This extends an analogous result about Zm-actions by the first author. We also provide some results concerning the genericity of finite Rokhlin dimension, and permanence properties with respect to the absorption of a strongly self-absorbing C*-algebra.

Item Type:Articles
Status:Early Online Publication
Refereed:Yes
Glasgow Author(s) Enlighten ID:Zacharias, Dr Joachim
Authors: Szabo, G., Wu, J., and Zacharias, J.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Ergodic Theory and Dynamical Systems
Publisher:Cambridge University Press
ISSN:0143-3857
ISSN (Online):1469-4417
Published Online:28 November 2017
Copyright Holders:Copyright © 2017 Cambridge University Press
First Published:First published in Ergodic Theory and Dynamical Systems 2017
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher
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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
626331The Cuntz Semigroup and the Fine Stucture of Nuclear C*-AlgebrasJoachim ZachariasEngineering & Physical Sciences Research Council (EPSRC)EP/I019227/2M&S - MATHEMATICS