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

TL;DR

This paper develops a framework for studying Hecke operators on Bredon (co)homology of arithmetic groups, especially Bianchi groups, enabling explicit computations in K-theory via the Baum-Connes conjecture.

Contribution

It introduces a novel approach linking Hecke operators with Bredon (co)homology and K-theory, providing explicit calculations for Bianchi groups using spectral sequences.

Findings

Explicit Hecke operator computations for SL_2(Z[i])

Demonstration of the method's effectiveness via the Baum-Connes conjecture

Use of Atiyah-Segal spectral sequence in Bredon homology context

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 $SL_2(\mathbb{Z}[i])$. 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.