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 .