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

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.