In this paper, we study the ideal structure of reduced C * -algebras C * r (G) associated to étale groupoids G. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in C * r (G) and the open invariant subsets of the unit space G (0) of G. As a consequence, we show that if G is an inner exact, essentially principal, ample groupoid, then C * r (G) is (strongly) purely infinite if and only if every non-zero projection in C0(G (0) ) is properly infinite in C * r (G). We also establish a sufficient condition on the ample groupoid G that ensures pure infiniteness of C * r (G) in terms of paradoxicality of compact open subsets of the unit space G (0) .Finally, we introduce the type semigroup for ample groupoids and also obtain a dichotomy result: Let G be an ample groupoid with compact unit space which is minimal and topologically principal. If the type semigroup is almost unperforated, then C * r (G) is a simple C * -algebra which is either stably finite or strongly purely infinite.

We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions.

We study a Going-Down (or restriction) principle for ample groupoids and its applications. The Going-Down principle for locally compact groups was developed by Chabert, Echterhoff and Oyono-Oyono and allows to study certain functors, that arise in the context of the topological K-theory of a locally compact group, in terms of their restrictions to compact subgroups. We extend this principle to the class of ample Hausdorff groupoids using Le Gall's groupoid equivariant version of Kasparov's bivariant KK-theory. Moreover, we provide an application to the Baum-Connes conjecture for ample groupoids which are strongly amenable at infinity. This result in turn is then used to relate the Baum-Connes conjecture for an ample groupoid group bundle which is strongly amenable at infinity to the Baum-Connes conjecture for the fibres.

In this article we prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions.

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