Hecke operators in Bredon (co)homology, K-(co)homology and Bianchi groups

2021/01/24 by Muñoz, David, Plazas, Jorge, Velásquez, Mario
#19L47 #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2101.09758

Abstract

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 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.

Related

• Hecke operators in KK-theory and the K-homology of Bianchi groups
• Arithmetic Aspects of Bianchi Groups
• Bredon Cohomology, K theory and K homology of Pullbacks of groups
• A spectral sequence in Bredon cohomology
• Unbounded symmetric operators in K-homology and the Baum-Connes Conjecture