Growth Conditions for Étale Groupoids: Amenability and Comparison

Abstract

We investigate étale Hausdorff groupoids whose source fibres have controlled growth with respect to a length function. Our first result shows that uniform subexponential growth implies topological amenability. If the unit space is compact and metrizable and such a length satisfies a suitable uniform doubling condition, then the groupoid has weak $m$-comparison for some $m$. For minimal ample groupoids, this implies comparison and almost unperforation of the type semigroup. Under the additional assumption that the unit space has no isolated points, we deduce Matui's AH-conjecture.

37 pages. Fixed gap in proof of first main result leading to a new proof. Added assumption and rewrote proof of second main result. Examples added