Asymptotics of analytic torsion for congruence quotients of $\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n)$

TL;DR

This paper establishes a precise asymptotic growth rate for the analytic torsion of congruence quotients of $ ext{SL}(n, ext{R})/ ext{SO}(n)$, with implications for arithmetic group cohomology.

Contribution

It provides a sharpened asymptotic formula for analytic torsion growth, utilizing bounds on heat kernel traces and orbital integrals, and introduces the concept of $ ext{lambda}$-strongly acyclic representations.

Findings

Derived explicit asymptotics for analytic torsion growth.

Proved the existence of $ ext{lambda}$-strongly acyclic representations for all $ ext{lambda}>0$.

Potential applications to torsion in arithmetic group cohomology.

Abstract

In this paper we prove a sharpened asymptotic for the growth of analytic torsion of congruence quotients of $\SL(n,\R)/\SO(n)$ in terms of the volume. The result is based on bounds on the trace of the heat kernel, allowing control of the large time behaviour of certain orbital integrals, as well as a careful analysis of error terms. The result requires the existence of $\lambda$-strongly acyclic representations, which we define and show exists in plenitude for any $\lambda>0$. The motivation is possible applications to torsion in the cohomology of arithmetic groups.