Dynamic Asymptotic Dimension: relation to dynamics, topology, coarse geometry, and $C^{\ast }$-algebras

Abstract

We introduce dynamic asymptotic dimension, a notion of dimension for actions of discrete groups on locally compact spaces, and more generally for locally compact étale groupoids. We study our notion for minimal actions of the integer group, its relation with conditions used by Bartels, Lück, and Reich in the context of controlled topology, and its connections with Gromov's theory of asymptotic dimension. We also show that dynamic asymptotic dimension gives bounds on the nuclear dimension of Winter and Zacharias for C*-algebras associated to dynamical systems. Dynamic asymptotic dimension also has implications for K-theory and manifold topology: these will be drawn out in subsequent work.

References in corpus (3)

• The variability of Sun-like stars: reproducing observed photometric trends
• The nuclear dimension of C*-algebras
• The nuclear dimension of C*-algebras associated to homeomorphisms

Cited by in corpus (6)

• Ideal structure and pure infiniteness of ample groupoid $C^{\ast }$-algebras
• Quantitative $K$-theory for Banach algebras
• Going-Down functors and the Künneth formula for crossed products by étale groupoids
• On equivariant asymptotic dimension
• Scale Structures and C*-algebras
• Coarse geometry of Hecke pairs and the Baum-Connes conjecture