Approximation of $L^2$-analytic torsion for arithmetic quotients of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$

TL;DR

This paper investigates the asymptotic behavior of regularized $L^2$-analytic torsion for arithmetic quotients of the symmetric space $ ext{SL}(n,R)/ ext{SO}(n)$, showing convergence results for principal congruence subgroups.

Contribution

It establishes the convergence of normalized analytic torsion to the $L^2$-analytic torsion for sequences of congruence subgroups, extending previous definitions to asymptotic analysis.

Findings

Logarithm of analytic torsion divided by subgroup index converges to $L^2$-analytic torsion.

Results apply to principal congruence subgroups and strongly acyclic flat bundles.

Provides new insights into the asymptotic behavior of torsion in arithmetic locally symmetric spaces.

Abstract

In [MzM] we defined a regularized analytic torsion for quotients of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ by arithmetic lattices. In this paper we study the limiting behaviour of the analytic torsion as the lattices run through sequences of congruence subgroups of a fixed arithmetic subgroup. Our main result states that for principal congruence subgroups and strongly acyclic flat bundles, the logarithm of the analytic torsion, divided by the index of the subgroup, converges to the $L^2$-analytic torsion.