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. 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 and dynamic asymptotic dimension growth at most x^{\alpha}\,(0<\alpha<1) is amenable. As an application, we show that the Baum–Connes conjecture with coefficients holds for such groupoids.