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