The analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume

TL;DR

This paper investigates the asymptotic behavior of regularized analytic torsion for sequences of finite volume hyperbolic manifolds, showing convergence to $L^2$-torsion under specific conditions.

Contribution

It establishes the convergence of normalized analytic torsion to $L^2$-torsion for certain sequences of hyperbolic manifolds and flat bundles, extending previous understanding.

Findings

Analytic torsion divided by the covering index converges to $L^2$-torsion.

Results apply to sequences of arithmetic groups, including principal congruence and Hecke subgroups.

Provides new insights into the asymptotic properties of torsion in hyperbolic manifolds.

Abstract

In this paper we study the regularized analytic torsion of finite volume hyperbolic manifolds. We consider sequences of coverings $X_i$ of a fixed hyperbolic orbifold $X_0$. Our main result is that for certain sequences of coverings and strongly acyclic flat bundles, the analytic torsion divided by the index of the covering, converges to the $L^2$-torsion. Our results apply to certain sequences of arithmetic groups, in particular to sequences of principal congruence subgroups of $\SO^0(d,1)(\Z)$ and to sequences of principal congruence subgroups or Hecke subgroups of Bianchi groups.