PreprintCC BY 4.0arXiv
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.
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BibTeX
@misc{austad2026growth,
title = {{Growth Conditions for Étale Groupoids: Amenability and Comparison}},
author = {Are Austad and Christian Bönicke},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.16013},
eprint = {2605.16013},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.16013},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Growth Conditions for Étale Groupoids: Amenability and Comparison
AU - Are Austad
AU - Christian Bönicke
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2605.16013
UR - https://arxiv.org/abs/2605.16013
AB - 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.
ER -