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Topological Rigidity of the Dynamic Asymptotic Dimension

Samantha Pilgrim

TL;DR

This paper proves that for free actions of a countable group Γ on a finite-dimensional compact metric space X, the dynamic asymptotic dimension DAD(Γ↷X) is either infinite or equal to asdim Γ. The authors develop a toolkit based on partial dynamical systems, F-chains, and (d,F,S)-covers, augmented by a finite union lemma and an inductive-dimension argument guided by a topological rigidity trick. They establish the sharp dichotomy DAD(Γ↷X) ∈ {asdim Γ, ∞} and derive corollaries identifying when DAD equals asdim Γ, notably for amenable or virtually abelian Γ, along with broader implications for boundary-type properties and Borel/arbitrary-set formulations. The results connect dynamic dimension theory to coarse geometry and C*-algebra classification, with potential impact on related problems in the Farrell–Jones program and K-theory computations.

Abstract

We show for a free action of a countable group $Γ$ on a finite-dimensional, compact metric space by homeomorphisms that the dynamic asymptotic dimension is either infinite or coincides with the asymptotic dimension of $Γ$.

Topological Rigidity of the Dynamic Asymptotic Dimension

TL;DR

This paper proves that for free actions of a countable group Γ on a finite-dimensional compact metric space X, the dynamic asymptotic dimension DAD(Γ↷X) is either infinite or equal to asdim Γ. The authors develop a toolkit based on partial dynamical systems, F-chains, and (d,F,S)-covers, augmented by a finite union lemma and an inductive-dimension argument guided by a topological rigidity trick. They establish the sharp dichotomy DAD(Γ↷X) ∈ {asdim Γ, ∞} and derive corollaries identifying when DAD equals asdim Γ, notably for amenable or virtually abelian Γ, along with broader implications for boundary-type properties and Borel/arbitrary-set formulations. The results connect dynamic dimension theory to coarse geometry and C*-algebra classification, with potential impact on related problems in the Farrell–Jones program and K-theory computations.

Abstract

We show for a free action of a countable group on a finite-dimensional, compact metric space by homeomorphisms that the dynamic asymptotic dimension is either infinite or coincides with the asymptotic dimension of .
Paper Structure (4 sections, 8 theorems, 2 equations)

This paper contains 4 sections, 8 theorems, 2 equations.

Table of Contents

  1. Abstract:
  2. Introduction
  3. Preliminaries
  4. Sharp Bounds for the DAD

Key Result

Lemma 2.6

(Finite union lemma) Let $(X, \Gamma, \{D_\gamma\}_{\gamma\in \Gamma}, \{\theta_\gamma\}_{\gamma\in \Gamma})$ be a partial dynamical system. Let $A, B\subset X$ and $F\in \mathcal{P}_{fs}(\Gamma)$. Assume that the $F^{r_A}$-components of $A$ are $F^{R_A}$-bounded, that the $F^{r_B}$-components of $B

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • ...and 11 more