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

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 of $\Gamma$ and of the boundary crossed product algebra associated to the action of $\Gamma$ on $\partial X$, with Hecke operators. The K-theory and K-homology groups of these C*-algebras are related by a Gysin six-term exact sequence. In the case when $\Gamma$ is a group of real hyperbolic isometries, we show that this Gysin sequence is Hecke equivariant. Finally, in the case when $\Gamma$ is a subgroup of a Bianchi group, we construct explicit Hecke-equivariant maps between the integral cohomology of $\Gamma$ and each of these K-groups. Our methods apply to torsion-free finite index subgroups of $PSL(2,\mathbb{Z})$ as well. These results are achieved in the context of unbounded Fredholm modules, shedding light on noncommutative geometric aspects of the purely infinite boundary crossed product algebra.
Comment: 50 pages