Fiberwise amenability of ample étale groupoids

Abstract

Let G be a locally compact σ-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 G. We define the corresponding concepts of Følner sequences and Banach densities for G, based on which, we establish a topological groupoid version of the Ornstein-Weiss quasitilling 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 (G) has the uniform property Γ and thus satisfies the Toms-Winter conjecture when G is minimal second countable (topologically) amenable and almost finite in measure. Then we prove that the topological full group [[G]] is always sofic when G is second countable minimal and admits a Følner sequence. This can be used to strengthen one of Matui's result on the commutator subgroup D[[G]] when G is almost finite. Concrete examples are provided.

Nonfree almost finite actions for locally finite‐by‐virtually Z${\mathbb {Z}}$ groups

“…See also more general versions of these theorems in [26] and [24] in the framework of étale groupoids.…”

Groupoids and Hermitian Banach *-algebras

“…For instance, they appear in the study of amenability of measured groupoids in the book of Renault and Anantharaman-Delaroche [2]; there the length function is used to show that if certain growth conditions with respect to the length function are satisfied, the measured groupoid is amenable. Ma and Wu in [30] and [29] show that length functions with the additional requirement that it is zero only on units, are in one-to-one correspondence with extended metrics on the groupoids. These metrics are then related to properties such as almost elementariness, fiberwise amenability and soficity of the topological full groups associated with groupoids.…”