Dynamic asymptotic dimension growth for group actions and groupoids

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 $\sigma$-compact locally compact Hausdorff étale groupoid with compact unit space having dynamic asymptotic dimension growth at most $x^{\alpha }$ $(0<\alpha <1)$ is amenable.