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Hecke operators in Bredon (co)homology, K-(co)homology and Bianchi groups
Journal of Topology and Analysis  (IF0.457),  Pub Date : 2021-11-20, DOI: 10.1142/s1793525321500606
David Muñoz, Jorge Plazas, Mario Velásquez

In this paper, 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 $SL2(ℤ[i])$. 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.