Fiberwise amenability of ample étale groupoids

Abstract: Let $\mathcal{G}$ be a locally compact $\sigma$-compact Hausdorff ample groupoid on a compact space. In this paper, we further examine the (ubiquitous) fiberwise amenability introduced by the author and Jianchao Wu for $\mathcal{G}$. We define the corresponding concepts of F{\o}lner sequences and Banach densities for $\mathcal{G}$, based on which, we establish a topological groupoid version of the Ornstein-Weiss quasi-tilling theorem. This leads to the notion of almost finiteness in measure for ample groupoids as a weaker version of Matui's almost finiteness. As applications, we first show that $C^*_r(\mathcal{G})$ has the uniform property $\Gamma$ and thus satisfies the Toms-Winter conjecture when $\mathcal{G}$ is minimal second countable (topologically) amenable and almost finite in measure. Then we prove that the topological full group $[[\mathcal{G}]]$ is always sofic when $\mathcal{G}$ is second countable minimal and admits a F{\o}lner sequence. This can be used to strengthen one of Matui's result on the commutator subgroup $D[[\mathcal{G}]]$ when $\mathcal{G}$ is almost finite. Concrete examples are provided.