# Hecke operators in $KK$-theory and the $K$-homology of Bianchi groups

### Bram Mesland

Universiteit Leiden, The Netherlands### Mehmet Haluk Şengün

University of Sheffield, UK

## Abstract

Let $Γ$ be a torsion-free arithmetic group acting on its associated global symmetric space $X$. Assume that $X$ is of non-compact type and let $Γ$ act on the geodesic boundary $∂X$ of $X$. Via general constructions in $KK$-theory, we endow the $K$-groups of the arithmetic manifold $X/Γ$, of the reduced group $C_{∗}$-algebra $C_{r}(Γ)$ and of the boundary crossed product algebra $C(∂X)⋊Γ$ with Hecke operators. In the case when $Γ$ is a group of real hyperbolic isometries, the $K$-theory and $K$-homology groups of these $C_{∗}$-algebras are related by a Gysin six-term exact sequence and we prove that this Gysin sequence is Hecke equivariant. Finally, when $Γ$ is a Bianchi group, we assign explicit unbounded Fredholm modules (i.e. spectral triples) to (co)homology classes, inducing Hecke-equivariant isomorphisms between the integral cohomology of $Γ$ and each of these $K$-groups. Our methods apply to case $Γ⊂PSL(Z)$ as well.

In particular we employ the unbounded Kasparov product to push the Dirac operator an embedded surface in the Borel–Serre compactification of $H/Γ$ to a spectral triple on the purely infinite geodesic boundary crossed product algebra $C(∂H)⋊Γ$.

## Cite this article

Bram Mesland, Mehmet Haluk Şengün, Hecke operators in $KK$-theory and the $K$-homology of Bianchi groups. J. Noncommut. Geom. 14 (2020), no. 1, pp. 125–189

DOI 10.4171/JNCG/361