  In this article we provide a framework for the study of Hecke operators
acting on the Bredon (co)homology of an arithmetic discrete group. Our main
interest lies in the study of Hecke operators for Bianchi groups. Using the
Baum-Connes conjecture, we can transfer computations in Bredon homology to
obtain a Hecke action on the $K$-theory of the reduced $C^{*}$-algebra of the
group. We show the power of this method giving explicit computations for the
group $SL_2(\mathbb{Z}[i])$. In order to carry out these computations we use an
Atiyah-Segal type spectral sequence together with the Bredon homology of the
classifying space for proper actions.

K-Theory and Homology