Analytic torsion and R-torsion of Witt representations on manifolds with cusps

TL;DR

This paper proves a Cheeger-Muller theorem linking analytic torsion and R-torsion for certain unimodular representations on noncompact manifolds with cusps, including hyperbolic spaces, without assuming constant curvature.

Contribution

It extends the Cheeger-Muller theorem to noncompact manifolds with cusps under Witt conditions using renormalized traces and relates analytic torsion to intersection R-torsion.

Findings

Established Cheeger-Muller theorem for manifolds with cusps.

Connected analytic torsion to intersection R-torsion in this setting.

Applied renormalized traces to define analytic torsion.

Abstract

We establish a Cheeger-Muller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all non-compact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We use renormalized traces in the sense of Melrose to define the analytic torsion and we relate it to the intersection R-torsion of Dar of the natural compactification to a stratified space. Our proof relies on our recent work on the behavior of the Hodge Laplacian spectrum on a closed manifold undergoing degeneration to a manifold with fibered cusps.