The torsion in symmetric powers on congruence subgroups of Bianchi groups

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 m th symmetric power of the standard representation of SL 2 ⁡ ( C ) \operatorname {SL}_2(\mathbb {C}) grows exponentially in m 2 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 3 -manifolds, to the finite-volume case. We also prove a limit multiplicity formula for combinatorial Reidemeister torsions on higher-dimensional hyperbolic manifolds.