On the torsion in symmetric powers on congruence subgroups of Bianchi groups

TL;DR

This paper investigates the exponential growth of torsion in the second cohomology of congruence subgroups of Bianchi groups, extending previous results to finite-volume cases and providing bounds on growth rates.

Contribution

It extends Mueller and Marshall's results to finite-volume Bianchi groups and establishes bounds for torsion growth in cohomology with symmetric power coefficients.

Findings

Torsion grows exponentially in m^2 for fixed congruence subgroup.

Provides upper and lower bounds for the growth rate.

Establishes a limit multiplicity formula for twisted Reidemeister torsion.

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 to the m-th symmetric power of the standard representation of SL_2(C) grows exponentially in m^2. We give upper and lower bounds for the growth rate. Our result extends a result of Mueller and Marshall, who proved the corresponding statement for closed arithmetic 3-manifolds, to the finite-volume case. We also prove a limit multiplicity formula for twisted combinatorial Reidemeister torsion on higher dimensional hyperbolic manifolds.