Equivariant analytic torsion for proper actions

TL;DR

This paper develops an equivariant version of Ray-Singer analytic torsion for proper group actions on manifolds, extending previous work to noncompact conjugacy classes and providing explicit computations and key properties.

Contribution

It introduces a new equivariant analytic torsion for proper group actions, generalizing earlier finite group cases and including explicit examples and fundamental properties.

Findings

Established convergence and metric independence of the torsion.

Proved vanishing for even-dimensional manifolds.

Derived a product formula and a decomposition of classical torsion.

Abstract

We construct an equivariant version of Ray-Singer analytic torsion for proper, isometric actions by locally compact groups on Riemannian manifolds, with compact quotients. We obtain results on convergence, metric independence, vanishing for even-dimensional manifolds, a product formula, and a decomposition of classical Ray-Singer analytic torsion as a sum over conjugacy classes of equivariant torsion on universal covers. We do explicit computations for the circle and the line acting on themselves, and for regular elliptic elements of $\operatorname{SO}_0(3,1)$ acting on $3$-dimensional hyperbolic space. Our constructions and results generalise several earlier constructions of equivariant analytic torsion and their properties, most of which apply to finite or compact groups, or to fundamental groups of compact manifolds acting on their universal covers. These earlier versions of equivariant analytic torsion were associated to finite or compact conjugacy classes; we allow noncompact conjugacy classes, under suitable growth conditions.