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

We study the connection between the Baum-Connes conjecture for an ample groupoid G with coefficient A and the Künneth formula for the K-theory of tensor products by the crossed product A⋊rG. To do so we develop the machinery of Going-Down functors for ample groupoids.As an application we prove that both the uniform Roe algebra of a coarse space which uniformly embeds into a Hilbert space and the maximal Roe algebra of a space admitting a fibred coarse embedding into a Hilbert space satisfy the Künneth formula. We also provide a stability result for the Künneth formula using controlled K-theory, and apply it to give an example of a space that does not admit a coarse embedding into a Hilbert space, but whose uniform Roe algebra satisfies the Künneth formula. As a by-product of our methods, we also prove a permanence property for the Baum-Connes conjecture with respect to equivariant inductive limits of the coefficient algebra.2010 Mathematics Subject Classification. 46L80, 22A22, 19K35.

We introduce a notion of topological property (T) for étale groupoids. This simultaneously generalizes Kazhdan's property (T) for groups and geometric property (T) for coarse spaces. One main goal is to use this property (T) to prove the existence of so-called Kazhdan projections in both maximal and reduced groupoid C ˚-algebras, and explore applications of this to exactness, K-exactness, and the Baum-Connes conjecture. We also study various examples, and discuss the relationship with other notions of property (T) for groupoids and with a-T-menability.

We introduce a notion of topological Property (T) for étale groupoids. This simultaneously generalizes Kazhdan's Property (T) for groups and geometric Property (T) for coarse spaces. One main goal is to use this Property (T) to prove the existence of so-called Kazhdan projections in both maximal and reduced groupoid C * -algebras, and explore applications of this to exactness, K-exactness, and the Baum-Connes conjecture. We also study various examples, and discuss the relationship with other notions of Property (T) for groupoids and with a-T-menability.Résumé. -Nous définissons une notion de propriété (T) pour les groupoïdes étales. Elle généralise à la fois la propriété (T) de Kazhdan pour les groupes, et la propriété (T) géométrique pour les espaces grossiers. Notre but principal est l'application de cette propriété (T) à l'existence de projecteurs de type Kazhdan dans les C * -algèbres réduites et maximales des groupoïdes, dont nous explorons les conséquences sur l'exactitude, l'exactitude en K-théorie, et sur la validité de la conjecture de Baum-Connes. Nous étudions aussi divers exemples, et comparons cette notion à d'autres versions de la propriété (T) ainsi qu'à la a-T-moyennabilité.

We develop a generalization of quantitative K-theory, which we call controlled K-theory. It is powerful enough to study the K-theory of crossed product of C * -algebras by action ofétale groupoids and discrete quantum groups. In this article, we will use it to study groupoids crossed products. We define controlled assembly maps, which factorize the Baum-Connes assembly maps, and define the controlled Baum-Connes conjecture. We relate the controlled conjecture for groupoids to the classical conjecture, and to the coarse Baum-Connes conjecture. This allows to give applications to Coarse Geometry. In particular, we can prove that the maximal version of the controlled coarse Baum-Connes conjecture is satisfied for a coarse space which admits a fibred coarse embedding, which is a stronger version of a result of M. Finn-Sell.