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.