Fiberwise amenability of étale groupoids

What the math gives to ML

The paper provides a constructive Følner–paradoxical dichotomy for étale groupoids: fiberwise amenability supplies finite sets whose compact-neighborhood growth is arbitrarily close to one, while non-amenability supplies many disjoint translates and therefore unavoidable expansion. This transfers naturally to graph neural networks by treating message-passing neighborhoods as the compact set K and node subsets as finite Følner candidates. A model can estimate local expansion online and switch between ordinary propagation, pooling, and skip-local processing when receptive fields become expansively large. The key falsifiable prediction is that neighborhood-growth ratios should sharply separate amenable-like graph families such as cycles and grids from non-amenable-like families such as regular trees and expanders.