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

Abstract

Let $\Gamma$ be a torsion-free arithmetic group acting on its associated global symmetric space $X$. Assume that $X$ is of non-compact type and let $\Gamma$ act on the geodesic boundary $\partial X$ of $X$. Via general constructions in $KK$-theory, we endow the $K$-groups of the arithmetic manifold $X / \Gamma$, of the reduced group $C^$-algebra $C^\_r(\Gamma)$ and of the boundary crossed product algebra $C(\partial X) \rtimes\Gamma$ with Hecke operators. In the case when $\Gamma$ 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 $\Gamma$ 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 $\Gamma$ and each of these $K$-groups. Our methods apply to case $\Gamma \subset \mathbf {PSL}(\mathbf Z)$ as well. In particular we employ the unbounded Kasparov product to push the Dirac operator an embedded surface in the Borel–Serre compactification of $\mathbf H/\Gamma$ to a spectral triple on the purely infinite geodesic boundary crossed product algebra $C(\partial \mathbf H) \rtimes\Gamma$.