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 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 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. Contents 1. Introduction Outline of the paper 2. Models for groupoid homology 2.1. The category of G-modules and the coinvariant functor 2.2. Semi-simplicial G-spaces and homology 2.3. Projective resolutions and Tor 3. Colourings and homology 3.1. Colourings and nerves 3.2. Homology vanishing 3.3. Maps between nerves and G 3.4. Anti-Čech homology 3.5. Dynamic asymptotic dimension 4. The HK conjecture 4.1. The one-dimensional comparison map 4.2. Dynamic asymptotic dimension one 4.3. The spectral sequence of Proietti and Yamashita 5. Examples and applications 5.1. Free actions on totally disconnected spaces 5.2. Smale spaces with totally disconnected (un)stable sets 5.3. Bounded geometry metric spaces 5.4. Examples with topological property (T) References

Categorical approach to the Baum-Connes conjecture for étale groupoids

“…Examples of recent research in this direction are the HK conjecture of Matui [5], or the relation between the homology theory of Smale spaces and the K-theory of their corresponding C * -algebras [8]. 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 [9,10,11].…”

“…) is a dual Dirac morphism for G. The final assertion then follows from the commutative diagram (11).…”

A counterexample to the HK-conjecture that is principal

“…The first counterexample is due to Scarparo [19] and a stronger counterexample (it does not even satisfy the rational version of the conjecture) can be found in [14]. On the other hand, there have been a number of positive results, starting with Matui's original work [12], also see [1,8,13,15,22]. In particular, there has been quite a bit of success verifying the conjecture for particular classes of principal groupoids, see in particular [1, Corollary C] and [15,Remark 3.5].…”

“…One approach to this conjecture would be to study the range of the K-theory of groupoids satisfying the HK-conjecture (e.g., by satisfying the hypotheses of [1,Corollary C] or ideally generalizations of it). As stated Conjecture 0.3 would not be useful for computations.…”

Long exact sequences of homology groups of étale groupoids

“…Later, in [11,12,13], when the underlying space is totally disconnected, the homology groups were computed for some specific examples, and the connection with topological full groups as well as K-theory of C * -algebras was discussed. See [5,15,28,29,16,2] for further developments.…”