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 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.

A Categorical Approach to the Baum–connes Conjecture for Étale Groupoids

“…Examples of recent research in this direction are the HK conjecture of Matui [35]a, or the relation between the homology theory of Smale spaces and the K -theory of their corresponding C * -algebras [52]. In this latter example, a special case of the methods developed here (i.e., when the groupoid is torsion-free and ample) has already been applied with great success and lead to many interesting results in topological dynamics, as is demonstrated by the papers [9,49,51,50].…”

“…In many applications, however, it is useful to have a concrete model at hand. This is certainly the case for the adjunction result in Theorem 2.3 below but has also proven to be a useful construction in [49,9].…”

On the dynamic asymptotic dimension of étale groupoids

“…It is also closely related to the diagonal dimension of sub-C * -algebras recently introduced in [14]. In [3] it was shown that the groupoid homology groups of a totally disconnected étale groupoid vanish in all degrees exceeding the dynamic asymptotic dimension of the groupoid. Estimates on the dynamic asymptotic dimension (denoted henceforth by dad(•)) are known for many concrete classes of examples (see e.g.…”

Homology and K-theory of dynamical systems IV. Further structural results on groupoid homology

“…Remark 6.11. We note that there exist maps μ i : H i (G) → K i (C * r G) for i = 0, 1 in the case of ample groupoids, and they have been studied in [BDGW23,Mat12]. For higher degrees, one can still construct analogous maps by looking at kernels of higher differentials from the associated spectral sequence from [PY22a].…”