Amenability and comparison for étale groupoids with polynomial growth

Abstract

We show that any second-countable locally compact Hausdorff étale groupoid with polynomial growth is topologically amenable. If moreover the groupoid is compactly generated with compact and metrizable unit space, it has weak $m$-comparison. Thus if the groupoid is also ample and minimal, it satisfies Matui's AH-conjecture.

24 pages. Weakened assumptions on the length functions considered. Added more references, introduction reworked