A Hurewicz-type Theorem for the Dynamic Asymptotic Dimension with Applications to Coarse Geometry and Dynamics

TL;DR

This paper establishes a Hurewicz-type theorem for the dynamic asymptotic dimension, linking it to cohomology, K-theory, and asymptotic dimension of group actions and spaces, with applications to coarse geometry and C*-algebra classification.

Contribution

It introduces a new Hurewicz-type theorem for dynamic asymptotic dimension and explores its implications for group actions, box spaces, and the classification of C*-algebras.

Findings

Dynamic asymptotic dimension is subadditive over group extensions.

Upper bounds on asymptotic dimension of box spaces are established.

Many elementary amenable group actions have finite dynamic asymptotic dimension.

Abstract

We prove a Hurewicz-type theorem for the dynamic asymptotic dimension originally introduced by Guentner, Willett, and Yu. Calculations of (or simply upper bounds on) this dimension are known to have implications related to cohomology of group actions and the K-theory of their transformation group C*-algebras. Moreover, these implications are relevant to the current classification program for C*-algebras. As a corollary of our main theorem, we show the dynamic asymptotic dimension of actions by groups on profinite completions along sequential filtrations by normal subgroups is subadditive over extensions of groups, which shows that many such actions by elementary amenable groups are finite dimensional. We combine this with other novel results relating the dynamic asymptotic dimension of such an action to the asymptotic dimension of a corresponding box space. This allows us to give upper bounds on the asymptotic dimension of many box spaces (including examples from infinitely-many groups with exponential growth) using a generalization of the Hirsch length for elementary amenable groups. For some of these examples, we can also find lower bounds by utilizing the theory of ends of groups.