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Rokhlin dimension for actions of residually finite groups

Published online by Cambridge University Press:  28 November 2017

GÁBOR SZABÓ ,
JIANCHAO WU  and
JOACHIM ZACHARIAS
GÁBOR SZABÓ
Affiliation:
Department of Mathematical Sciences, Universitetsparken 5, 2100 København Ø, Denmark email gabor.szabo@math.ku.dk
JIANCHAO WU
Affiliation:
Department of Mathematics, Pennsylvania State University, 109 McAllister Building, University Park, PA 16802, USA email jianchao.wu@psu.edu
JOACHIM ZACHARIAS
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow Q12 8QW, UK email joachim.zacharias@glasgow.ac.uk
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Abstract

We introduce the concept of Rokhlin dimension for actions of residually finite groups on $\text{C}^{\ast }$-algebras, extending previous such notions for actions of finite groups and the integers by Hirshberg, Winter and the third author. We are able to extend most of their results to a much larger class of groups: those admitting box spaces of finite asymptotic dimension. This latter condition is a refinement of finite asymptotic dimension and has not previously been considered. In a detailed study we show that finitely generated, virtually nilpotent groups have box spaces with finite asymptotic dimension, providing a large class of examples. We show that actions with finite Rokhlin dimension by groups with finite-dimensional box spaces preserve the property of having finite nuclear dimension when passing to the crossed product $\text{C}^{\ast }$-algebra. 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 infinite, finitely generated, nilpotent groups on finite-dimensional spaces, both these dimensional values are finite. In particular, the associated transformation group$\text{C}^{\ast }$-algebras have finite nuclear dimension. This extends an analogous result about $\mathbb{Z}^{m}$-actions by the first author to a significantly larger class of groups, showing that a large class of crossed products by actions of such groups fall under the remit of the Elliott classification programme. We also provide results concerning the genericity of finite Rokhlin dimension, and permanence properties with respect to the absorption of a strongly self-absorbing $\text{C}^{\ast }$-algebra.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 8 , August 2019 , pp. 2248 - 2304
Copyright
© Cambridge University Press, 2017 

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