Rokhlin dimension for actions of residually finite groups

Abstract

We introduce the concept of Rokhlin dimension for actions of residually finite groups on C * -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 been considered previously. 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 C * -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 C * -algebras have finite nuclear dimension. This extends an analogous result about 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 C * -algebra.

Rokhlin Dimension for Flows

“…We can establish a connection between the nuclear dimension A α G and that of A α| H H , where H is a closed, cocompact subgroup. If one has sufficient information about the restricted action α| H : H → Aut(A) or its crossed product, the method discussed in this section can have some advantages that global Rokhlin dimension, in the sense of the previous section or [HWZ15,SWZ15], does not have in the non-compact case. For example, obtaining bounds concerning decomposition rank of crossed products by non-compact groups appears to be difficult, and would require different techniques and more severe constraints on the action than just finite Rokhlin dimension.…”

“…[HWZ15,SWZ15,HP15]), finite Rokhlin dimension does not appear sufficient for the purpose of proving that D-absorption passes to the crossed product. Therefore, we consider a stronger variant of finite Rokhlin dimension, namely with commuting towers.…”

“…In [HWZ15] Rokhlin dimension was defined for actions of the integers and of finite groups; in [Sza15a] the notion was generalized to Z d -actions, and in [SWZ15] to residually finite groups. All of these work nicely and yield very convincing results.…”

Warped cones, (non‐)rigidity, and piecewise properties

“…In general, equivariant asymptotic dimension is computed relative to a family of subgroups of , and above (and in the sequel) we consider simply the family , the singleton of the trivial subgroup . We refer the interested reader to for details, see also . We remark that when consists only of the trivial subgroup as above, equivariant asymptotic dimension is also called amenability dimension by some authors .…”

“…We refer the interested reader to for details, see also . We remark that when consists only of the trivial subgroup as above, equivariant asymptotic dimension is also called amenability dimension by some authors . Equivariant asymptotic dimension was introduced as a technical condition (without an explicit definition, hence the variety of names in the literature) in the proof of the Farrell–Jones conjecture for hyperbolic groups by Bartels, Lück, and Reich .…”