Asymptotics of analytic torsion for hyperbolic three--manifolds

TL;DR

This paper investigates the asymptotic behavior of analytic torsion in hyperbolic three-manifolds with cusps, showing convergence to $L^2$-torsion and Reidemeister torsion under specific conditions.

Contribution

It establishes the asymptotic relationship between regularized analytic torsion, $L^2$-torsion, and Reidemeister torsion for sequences of hyperbolic three-manifolds with cusps.

Findings

Analytic torsion approximates $L^2$-torsion under certain convergence conditions.

Asymptotic equality between analytic torsion and Reidemeister torsion for truncated manifolds.

Results apply to sequences of hyperbolic three-manifolds with cusps converging to hyperbolic space.

Abstract

We prove that for certain sequences of hyperbolic three--manifolds with cusps which converge to hyperbolic three--space in a weak ("Benjamini-Schramm") sense and certain coefficient systems the regularized analytic torsion approximates the $L^2$-torsion of the universal cover under an additional hypothesis. We also prove an asymptotic equality between the former and some Reidemeister torsion of the truncated manifolds.