Dynamical comparison for local homeomorphisms
Abstract
We prove dynamical comparison for Deaconu--Renault groupoids associated to minimal surjective non-injective local homeomorphisms of compact metrizable spaces with finite Lebesgue covering dimension. As a corollary, the associated C*-algebras are UCT Kirchberg algebras, recovering results by Carlsen--Thomsen via dynamical methods. In the zero-dimensional case, our result also verifies Matui's AH-conjecture for these groupoids using a recent breakthrough of Xin Li. Our proof combines techniques from both the purely infinite and stably finite regimes: We construct partial actions of non-abelian free groups as suitable ``large subgroupoids'' and establish comparison properties for these using the paradoxical towers technique developed by Gardella--Geffen--Kranz--Naryshkin. The boundary of the subgroupoid is controlled by a groupoid version of the topological small boundary property which we deduce from finite covering dimension of the unit space. As a byproduct, we prove the classical small boundary property for minimal actions of countable discrete groups on finite-dimensional compact metrizable spaces without any freeness assumption.
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BibTeX
@misc{geffen2026dynamical,
title = {{Dynamical comparison for local homeomorphisms}},
author = {Shirly Geffen and Shanshan Hua and Julian Kranz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.13000},
eprint = {2608.13000},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.13000},
}
RIS
TY - UNPB
TI - Dynamical comparison for local homeomorphisms
AU - Shirly Geffen
AU - Shanshan Hua
AU - Julian Kranz
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2608.13000
UR - https://arxiv.org/abs/2608.13000
AB - We prove dynamical comparison for Deaconu--Renault groupoids associated to minimal surjective non-injective local homeomorphisms of compact metrizable spaces with finite Lebesgue covering dimension. As a corollary, the associated C^*-algebras are UCT Kirchberg algebras, recovering results by Carlsen--Thomsen via dynamical methods. In the zero-dimensional case, our result also verifies Matui's AH-conjecture for these groupoids using a recent breakthrough of Xin Li. Our proof combines techniques from both the purely infinite and stably finite regimes: We construct partial actions of non-abelian free groups as suitable ``large subgroupoids'' and establish comparison properties for these using the paradoxical towers technique developed by Gardella--Geffen--Kranz--Naryshkin. The boundary of the subgroupoid is controlled by a groupoid version of the topological small boundary property which we deduce from finite covering dimension of the unit space. As a byproduct, we prove the classical small boundary property for minimal actions of countable discrete groups on finite-dimensional compact metrizable spaces without any freeness assumption.
ER -