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

  • Bram Mesland

    Universiteit Leiden, The Netherlands
  • Mehmet Haluk Şengün

    University of Sheffield, UK
Hecke operators in $KK$-theory and the $K$-homology of Bianchi groups cover

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Let be a torsion-free arithmetic group acting on its associated global symmetric space . Assume that is of non-compact type and let act on the geodesic boundary of . Via general constructions in -theory, we endow the -groups of the arithmetic manifold , of the reduced group -algebra and of the boundary crossed product algebra with Hecke operators. In the case when is a group of real hyperbolic isometries, the -theory and -homology groups of these -algebras are related by a Gysin six-term exact sequence and we prove that this Gysin sequence is Hecke equivariant. Finally, when 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 and each of these -groups. Our methods apply to case as well.

In particular we employ the unbounded Kasparov product to push the Dirac operator an embedded surface in the Borel–Serre compactification of to a spectral triple on the purely infinite geodesic boundary crossed product algebra .

Cite this article

Bram Mesland, Mehmet Haluk Şengün, Hecke operators in -theory and the -homology of Bianchi groups. J. Noncommut. Geom. 14 (2020), no. 1, pp. 125–189

DOI 10.4171/JNCG/361