Dynamic asymptotic dimension and Matui's HK conjecture

TL;DR

This paper establishes a link between the homology of principal ample groupoids and their K-theory, proving vanishing results in higher dimensions and verifying Matui's HK conjecture for certain cases.

Contribution

It introduces a new model of groupoid homology using Tor groups and proves the HK conjecture for groupoids with dynamic asymptotic dimension at most two.

Findings

Homology groups vanish above the dynamic asymptotic dimension.

K-theory can be computed from groupoid homology for finite dynamic asymptotic dimension.

Explicit maps from homology to K-theory are constructed and analyzed.

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 $C^*$-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 $C^*$-algebras in degrees zero and one, and investigate their properties.