The torsion in symmetric powers on congruence subgroups of Bianchi groups
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- by Jonathan Pfaff and Jean Raimbault PDF
- Trans. Amer. Math. Soc. 373 (2020), 109-148 Request permission
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.References
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Additional Information
- Jonathan Pfaff
- Affiliation: Mathematisches Institut, Universität Bonn, Bonn, Germany
- MR Author ID: 936834
- 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
- MR Author ID: 951452
- Email: Jean.Raimbault@math.univ-toulouse.fr
- 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.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 109-148
- MSC (2010): Primary 11F75
- DOI: https://doi.org/10.1090/tran/7875
- MathSciNet review: 4042870