On the analytic torsion of hyperbolic manifolds of finite volume

TL;DR

This paper investigates the analytic torsion of finite-volume hyperbolic manifolds, introducing a new method to handle weighted orbital integrals via the Selberg trace formula, advancing understanding of spectral invariants in geometric analysis.

Contribution

It presents a novel approach to analyze the regularized trace of heat operators and weighted orbital integrals on hyperbolic manifolds of finite volume.

Findings

Established asymptotic behavior of the regularized trace for small time

Developed a new method for weighted orbital integrals in the trace formula

Enhanced understanding of spectral invariants in hyperbolic geometry

Abstract

In this paper we study the analytic torsion for a complete oriented hyperbolic manifold of finite volume. This requires the definition of a regularized trace of heat operators. We use the Selberg trace formula to study the asymptotic behavior of the regularized trace for small time. The main result of the paper is a new approach to deal with the weighted orbital integrals on the geometric side of the trace formula.