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arXiv · 2608.09796

Fiberwise amenability of \'{e}tale groupoids

Xin Ma·Jianchao Wu
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Abstract

We introduce a new amenability property for \'{e}tale groupoids, termed \textit{fiberwise amenability}, along with a stronger variant termed \emph{ubiquitous fiberwise amenability}. (Ubiquitous) fiberwise amenability emerges naturally from a coarse-geometric perspective on \'{e}tale groupoids and, in the special case of transformation groupoids, it coincides precisely with the amenability of the acting group (rather than topological amenability of the action). It is also tightly linked to the existence of invariant measures on the unit space of the groupoid. The coarse-geometric framework for \'{e}tale groupoids that we develop systematically in this work allows us to establish several foundational properties of (ubiquitous) fiberwise amenability. As an application, we prove a F\o lner--paradoxical dichotomy for minimal \'{e}tale groupoids, which will serve as a key tool in a sequel on almost elementariness of \'{e}tale groupoids.

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Xin Ma, Jianchao Wu. 2026-08-10. Fiberwise amenability of \'{e}tale groupoids. https://arxiv.org/abs/2608.09796

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