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Hecke operators for non-congruence subgroups of Bianchi groups


Author: Saeid Hamzeh Zarghani
Journal: Proc. Amer. Math. Soc. 139 (2011), 3853-3865
MSC (2010): Primary 11F03, 11F25, 11F75, 20G30; Secondary 19B37
DOI: https://doi.org/10.1090/S0002-9939-2011-10807-4
Published electronically: March 25, 2011
MathSciNet review: 2823032
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Abstract: We prove that the action of the Hecke operators on the cohomology of a finite index non-congruence subgroup $ \Gamma$ of a Bianchi group is essentially the same as the action of Hecke operators on the cohomology groups of $ \hat{\Gamma}$, the congruence closure of $ \Gamma$. This is a generalization of Atkin's conjecture, first confirmed in a special case by Serre in $ 1987$ and proved in general by Berger in $ 1994$.


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Additional Information

Saeid Hamzeh Zarghani
Affiliation: Department of Mathematics, Heinrich-Heine University Düsseldorf, Düsseldorf, Germany
Email: zarghani@math.uni-duesseldorf.de, zarghani.s@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2011-10807-4
Received by editor(s): May 13, 2010
Received by editor(s) in revised form: September 21, 2010
Published electronically: March 25, 2011
Additional Notes: The author was supported in part by Graduirtenkolleg Homotopie und Kohomologie (GRK1150)
Dedicated: Before the first draft of this work was completed, Fritz Grunewald tragically passed away. Indeed, without his guidance and support this work would never have been done. I dedicate this paper to his memory with admiration, gratitude and love.
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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