Dynamic asymptotic dimension and Matui's HK conjecture
Abstract
We prove that the homology groups of a principal ample groupoid vanish in dimensions greater than the dynamic asymptotic dimension of the groupoid (as a side-effect of our methods, we also give a new model of groupoid homology in terms of the Tor groups of homological algebra, which might be of independent interest). As a consequence, the K-theory of the -algebras associated with groupoids of finite dynamic asymptotic dimension can be computed from the homology of the underlying groupoid. In particular, principal ample groupoids with dynamic asymptotic dimension at most two and finitely generated second homology satisfy Matui's HK-conjecture. We also construct explicit maps from the groupoid homology groups to the K-theory groups of their -algebras in degrees zero and one, and investigate their properties.
Cite
@article{arxiv.2104.05885,
title = {Dynamic asymptotic dimension and Matui's HK conjecture},
author = {Christian Bönicke and Clément Dell'Aiera and James Gabe and Rufus Willett},
journal= {arXiv preprint arXiv:2104.05885},
year = {2023}
}
Comments
Version 3 adds a new description of the comparison map in dimension one based on the ABC spectral sequence, and shows that it is the same as the description based on mapping cones. This is the final version, to appear in Proceedings of the LMS