Dynamic asymptotic dimension growth for group actions and groupoids

TL;DR

This paper introduces the concept of dynamic asymptotic dimension growth for group actions and groupoids, linking it to asymptotic dimension growth in groups and coarse spaces, and explores its implications for amenability.

Contribution

It defines dynamic asymptotic dimension growth for group actions and groupoids, establishing connections with asymptotic dimension and amenability, and provides numerous concrete examples.

Findings

Dynamic asymptotic dimension growth is equivalent for a space and its coarse groupoid.

Subexponential growth in this dimension implies amenability.

Groups with controlled dimension growth are shown to be amenable.

Abstract

We introduce the notion of dynamic asymptotic dimension growth for actions of discrete groups on compact spaces, and more generally for locally compact \'etale groupoids. Using the work of Bartels, L\"uck, 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 $\sigma$-compact locally compact Hausdorff \'etale groupoid with compact unit space having dynamic asymptotic dimension growth at most $x^{\alpha}$ $(0<\alpha<1)$ is amenable.