Amenability and comparison for \'etale groupoids with polynomial growth

TL;DR

This paper proves that second-countable étale groupoids with polynomial growth are topologically amenable and, under certain conditions, satisfy Matui's AH conjecture, advancing understanding in groupoid theory.

Contribution

It establishes topological amenability for polynomial growth étale groupoids and links additional properties to Matui's AH conjecture under specific conditions.

Findings

Polynomial growth étale groupoids are topologically amenable.

Compact, metrizable unit spaces imply weak m-comparison.

Ample and minimal groupoids satisfy Matui's AH conjecture.

Abstract

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