Dynamic asymptotic dimension growth for group actions and groupoids

TL;DR

We address how large-scale dimension growth can be quantified beyond finite asymptotic dimension by introducing dynamic asymptotic dimension growth for actions of discrete groups and for étale groupoids. Using BLR-type conditions and coarse-geometry techniques, the authors establish three equivalent notions of asymptotic dimension growth and prove their compatibility with group actions, coarse spaces, and groupoids, including an exact equivalence between $ad_X$ for a space $X$ and $dad_{G(X)}$ for its coarse groupoid $G(X)$. They show subexponential growth in these dynamical settings implies amenability (and, in the coarse case, property A), and provide a broad framework that connects dimension-growth behavior to amenability without freeness assumptions. The results yield concrete examples, such as actions on the Cantor set for groups with subexponential growth, and extend existing GWY-type theorems to slower growth regimes, highlighting the interplay between dynamical complexity and large-scale amenability. Overall, the paper offers a unified approach to dynamic asymptotic dimension growth across spaces, actions, and groupoids with significant implications for coarse Baum–Connes-type results and operator-algebraic amenability criteria.

Abstract

We introduce the notion of dynamic asymptotic dimension growth for actions of discrete groups on compact spaces, and more generally for locally compact étale groupoids. Using the work of Bartels, Lück, and Reich, we bridge asymptotic dimension growth for countable discrete groups with our notion for their group actions, thereby providing numerous concrete examples. Moreover, we demonstrate that the asymptotic dimension growth for a discrete metric space of bounded geometry is equivalent to the dynamic asymptotic dimension growth for its associated coarse groupoid. Consequently, we deduce that the coarse groupoid with subexponential dynamic asymptotic dimension growth is amenable. More generally, we show that every $σ$-compact locally compact Hausdorff étale groupoid with compact unit space having dynamic asymptotic dimension growth at most $x^α$ $(0<α<1)$ is amenable.

Theorems & Definitions (83)

• Theorem 1.1: see Theorem \ref{['th 4.16']}
• Corollary 1.2: see Corollary \ref{['cor 4.19']}
• Theorem 1.3: see Theorem \ref{['th 4.22']}
• Theorem 1.4
• Definition 2.1: Dra06
• Remark 2.2
• Proposition 2.3: Bel05Dra06
• Lemma 2.4: ANWZ18
• Proposition 2.5
• proof