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Transactions of the American Mathematical Society

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The torsion in symmetric powers on congruence subgroups of Bianchi groups


Authors: Jonathan Pfaff and Jean Raimbault
Journal: Trans. Amer. Math. Soc. 373 (2020), 109-148
MSC (2010): Primary 11F75
DOI: https://doi.org/10.1090/tran/7875
Published electronically: July 1, 2019
MathSciNet review: 4042870
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Abstract: In this paper we prove that for a fixed neat principal congruence subgroup of a Bianchi group the order of the torsion part of its second cohomology group with coefficients in an integral lattice associated with the $ m$th symmetric power of the standard representation of $ \operatorname {SL}_2(\mathbb{C})$ grows exponentially in $ m^2$. We give upper and lower bounds for the growth rate. Our result extends a result of W. Müller and S. Marshall, who proved the corresponding statement for closed arithmetic $ 3$-manifolds, to the finite-volume case. We also prove a limit multiplicity formula for combinatorial Reidemeister torsions on higher-dimensional hyperbolic manifolds.


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Additional Information

Jonathan Pfaff
Affiliation: Mathematisches Institut, Universität Bonn, Bonn, Germany
Email: jnthnpfff@gmail.com

Jean Raimbault
Affiliation: Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France
Email: Jean.Raimbault@math.univ-toulouse.fr

DOI: https://doi.org/10.1090/tran/7875
Received by editor(s): January 22, 2019
Received by editor(s) in revised form: April 15, 2019
Published electronically: July 1, 2019
Additional Notes: The first author was financially supported by DFG grant PF 826/1-1. He gratefully acknowledges the hospitality of Stanford University in 2014 and 2015.
Article copyright: © Copyright 2019 American Mathematical Society