Asymptotic behaviour of analytic torsion and cohomological torsion for $\mathbb{Q}$-rank $1$ arithmetic groups

TL;DR

This paper extends asymptotic analysis of analytic torsion to broader classes of reductive groups, providing new bounds on torsion growth in the cohomology of certain arithmetic groups.

Contribution

It generalizes previous results on analytic torsion asymptotics to new reductive groups and applies these to bound torsion growth in cohomology of specific arithmetic groups.

Findings

Derived new asymptotics for analytic torsion in reductive groups.

Provided bounds on torsion growth in cohomology of congruence subgroups.

Extended previous work from $ ext{SL}(n)$ to a larger family of groups.

Abstract

We extend the refined asymptotics of analytic torsion associated to congruence subgroups of $\operatorname{SL}(n)$ in previous work, to congruence subgroups in a large family of reductive groups. This is applied to give new asymptotics and bounds on the growth of torsion in the cohomology of congruence subgroups of $\operatorname{SL}(2,\mathcal{O}_F)$ for $F$ a number field, and of congruence subgroups in $\operatorname{SO}(n,1)$ with $n$ odd.