Rokhlin dimension for actions of residually finite groups

TL;DR

This paper introduces Rokhlin dimension for residually finite group actions on C*-algebras, linking it to nuclear dimension preservation and exploring its properties for various groups and actions.

Contribution

It extends Rokhlin dimension concepts to residually finite groups, analyzes its implications for nuclear dimension, and relates it to amenability and group properties.

Findings

Finitely generated, virtually nilpotent groups have finite asymptotic dimension of box spaces.

Actions with finite Rokhlin dimension preserve finite nuclear dimension in crossed products.

Finite Rokhlin dimension is generic and stable under certain algebraic operations.

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 infinite, 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 $\mathbb{Z}^m$-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.