Mesland, B. and Sengun, M.H. (2020) Hecke operators in KK-theory and the K-homology of Bianchi groups. Journal of Noncommutative Geometry, 14 (1). pp. 125-189. ISSN 1661-6952
Abstract
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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2020 EMS Publishing House. This is an author-produced version of a paper subsequently published in Journal of Noncommutative Geometry. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | KK-theory; arithmetic groups; spectral triples; harmonic analysis |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 08 Nov 2016 15:23 |
Last Modified: | 30 Jun 2020 08:34 |
Status: | Published |
Publisher: | European Mathematical Society |
Refereed: | Yes |
Identification Number: | 10.4171/JNCG/361 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:106716 |