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arXiv · 2411.19712

Dynamic asymptotic dimension growth for group actions and groupoids

Hang Wang·Yanru Wang·Jianguo Zhang·Dapeng Zhou
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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. 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 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.

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Hang Wang, Yanru Wang, Jianguo Zhang, Dapeng Zhou. 2024-11-29. Dynamic asymptotic dimension growth for group actions and groupoids. https://doi.org/10.4171/jncg%2F691

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