On the relation between homology and K-theory for étale gropoids

Abstract

In 2016 H. Matui conjectured that the K-groups of the C*-algebra associated to an effective minimal étale groupoid, with a Cantor set as unit space, could be computed as the infinite direct sum of the homology groups of given groupoid. Although a counterexample was found by E. Scarparo in 2020, the study of sufficient and/or necessary conditions for the conjecture to hold remains relevant. The main goal of this thesis is to further deepen the knowledge of this conjecture, providing some examples and counterexamples for it and, more importantly, developing new techniques for the computation of groupoids invariants. The two main classes of groupoids involved in our work are Deaconu-Renault groupoids, and self-similar groupoids