Rokhlin dimension for actions of residually finite groups
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Rokhlin dimension for actions of residually finite groups. / Szabó, Gábor; Wu, Jianchao; Zacharias, Joachim.
In: Ergodic Theory and Dynamical Systems, Vol. 39, No. 8, 2019, p. 2248-2304.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - Rokhlin dimension for actions of residually finite groups
AU - Szabó, Gábor
AU - Wu, Jianchao
AU - Zacharias, Joachim
PY - 2019
Y1 - 2019
N2 - We introduce the concept of Rokhlin dimension for actions of residually finite groups on-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-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-algebras have finite nuclear dimension. This extends an analogous result about-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-algebra.
AB - We introduce the concept of Rokhlin dimension for actions of residually finite groups on-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-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-algebras have finite nuclear dimension. This extends an analogous result about-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-algebra.
UR - http://www.scopus.com/inward/record.url?scp=85037987585&partnerID=8YFLogxK
U2 - 10.1017/etds.2017.113
DO - 10.1017/etds.2017.113
M3 - Journal article
AN - SCOPUS:85037987585
VL - 39
SP - 2248
EP - 2304
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
SN - 0143-3857
IS - 8
ER -
ID: 240741130