The homology of the groupoid of the self-similar dihedral group

Abstract

We compute the K-theory of the C * -algebra associated to the self-similar infinite dihedral group, and the homology of its associated étale groupoid. We see that the rational homology differs from the K-theory, strongly contradicting a conjecture posted by Matui. Moreover, we compute the abelianization of the topological full group of the groupoid associated to the self-similar infinite dihedral group.In [8] Matui conjectured that the homology groups of a minimal effective, ample groupoid G totally captures the K-theory of their associated reduced groupoid C *algebra, and called it the HK conjecture, that is,This conjecture has been proved for important classes of groupoids, like the transformation groupoids of Cantor minimal systems, products of Cuntz-Krieger groupoids andmany others (see [2,6,8,11]). More generally, it was proved in [12] that the conjecture is true whenever H n (G, Z) = 0 for n ≥ 3. However, the conjecture was shown to be false by Scarparo in [13] for a class of Odomoters of the infinite dihedral group. Then one can ask if a weakening of the conjecture could be true, the rational HK-conjecture, that isThe rational HK-conjecture it is satisfied by the transformations groupoids of Z nminimal actions on the Cantor space [8, Section 3]. Scarparo counter-example is still valid for i = 0 but not for i = 1. In here we present a complete counter-example for the rational HK-conjecture. Our example arises from a self-similar action of the infinite dihedral group over a rooted tree and it associated groupoid, that not only encodes the global action of the group but also the local self-similarities of this action [9, Section 5]. This groupoid is going to be minimal and locally contracting, and its associated (reduced) groupoid C * -algebra is going to be simple and purely infinite. This contrast with the counter-examples of Scarparo, whose associated C * -algebras were AF-algebras. Therefore, there is not obvious relationship between the example presented here and the

On groupoids and $C^*$-algebras from self-similar actions

“…The HK-conjecture was also confirmed for groupoids on one-dimensional solenoids in [20]. Recently, counterexamples to the HK-conjecture of Matui were found by Scarparo in [18] and by Ortega and Sanchez in [16].…”