AI 中文总结
本研究为étale群胚提出纤维型可和性及泛在纤维型可和性，构建其粗几何框架，证明极小étale群胚的Følner-悖论二分，为后续研究提供关键工具。
AI 中文摘要
我们为étale群胚引入一种新的可和性性质，称为“纤维型可和性”，以及一个更强的变体，称为“泛在纤维型可和性”。（泛在）纤维型可和性自然产生于étale群胚的粗几何视角，在变换群胚的特殊情形下，它恰好与作用群的可和性一致（而非作用的拓扑可和性），还与群胚单位空间上不变测度的存在性紧密关联。我们在本研究中系统构建的étale群胚粗几何框架，使我们能够确立（泛在）纤维型可和性的若干基础性质。作为应用，我们证明了极小étale群胚的Følner-悖论二分，这将成为后续关于étale群胚几乎初等性研究的关键工具。
英文摘要
We introduce a new amenability property for étale 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 étale 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 étale 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 Følner--paradoxical dichotomy for minimal étale groupoids, which will serve as a key tool in a sequel on almost elementariness of étale groupoids.
CommentsThe paper originally identified as arXiv:2011.01182 has been split into two parts. This submission is the first part, which focuses on the fiberwise amenability. The second part on the almost elementariness and Z-stability, has also been updated and remains available at the original arXiv identifier:2011.01182. The related work arXiv:2407.05251 has also been updated
