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

Samantha Pilgrim

TL;DR

This work develops a Hurewicz-type framework for the dynamic asymptotic dimension ($DAD$) of group actions, establishing a subadditivity principle that mirrors classical dimension theory in topology. It proves a key inequality $DAD_{ ext{free}}(Γ\curvearrowright X) \le DAD(f) + DAD_{ ext{free}}(Λ\curvearrowright Y)$ for compatible maps and uses a central 'trick' to control open/closed covers, enabling product and extension results. The authors connect $DAD$ to box-space asymptotics via odometer actions, show subadditivity over extensions, and apply these ideas to compute box-space dimensions for Baumslag–Solitar groups, including families BS$(1,n)$ with exponential growth. These results have implications for coarse geometry and dynamics, including links to the $K$-theory of transformation-group $C^*$-algebras and the classification program. Overall, the paper extends dynamic dimension theory to include extension theorems and concrete computations, enriching the toolkit for studying actions, box spaces, and their coarse-geometric properties.

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.

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

TL;DR

This work develops a Hurewicz-type framework for the dynamic asymptotic dimension () of group actions, establishing a subadditivity principle that mirrors classical dimension theory in topology. It proves a key inequality for compatible maps and uses a central 'trick' to control open/closed covers, enabling product and extension results. The authors connect to box-space asymptotics via odometer actions, show subadditivity over extensions, and apply these ideas to compute box-space dimensions for Baumslag–Solitar groups, including families BS with exponential growth. These results have implications for coarse geometry and dynamics, including links to the -theory of transformation-group -algebras and the classification program. Overall, the paper extends dynamic dimension theory to include extension theorems and concrete computations, enriching the toolkit for studying actions, box spaces, and their coarse-geometric properties.

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.
Paper Structure (8 sections, 5 theorems, 4 equations)

This paper contains 8 sections, 5 theorems, 4 equations.

Table of Contents

  1. Abstract:
  2. Introduction
  3. (Dynamic) Asymptotic Dimension
  4. Odometers and Box Spaces
  5. Product Theorem
  6. Hurewicz-type Theorem
  7. Applications
  8. Baumslag-Solitar groups

Key Result

Theorem A

Suppose $\Gamma\curvearrowright X$ and $\Lambda\curvearrowright Y$ are actions on compact metric spaces by homeomorphisms, and that $f: X\to Y$ and $\alpha: \Gamma\times X\to \Lambda$ are continuous functions such that $f(\gamma\cdot x) = \alpha(\gamma, x)\cdot f(x)$ for all $x\in X$ and $\gamma\in

Theorems & Definitions (34)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Example
  • proof
  • proof
  • proof
  • proof
  • ...and 24 more