The Equivariant Fried Conjecture for Suspension Flow of an Equivariant Isometry

TL;DR

This paper proves an equivariant version of the Fried conjecture for suspension flows of equivariant isometries on Riemannian manifolds, extending the conjecture's validity to several important cases involving compact and non-compact groups.

Contribution

It establishes the equivariant Fried conjecture for suspension flows of equivariant isometries in multiple cases, including compact groups and specific non-compact group elements.

Findings

Proved the conjecture for compact groups.

Extended results to elements with compact centralisers.

Addressed the case of the identity element in non-compact discrete groups.

Abstract

The Fried conjecture states that the Ruelle dynamical $\zeta$-function of a flow on a compact maniofold has a well-defined value at $0$, whose absolute value equals the Ray-Singer analytic torsion invariant. The first author and Saratchandran proposed an equivariant version of the Fried conjecture for locally compact unimodular groups acting properly, isometrically, and cocompactly on Riemannian manifolds. In this paper we prove the equivariant Fried conjecture for the suspension flow of an equivariant isometry of a Riemannian manifold in several cases. These include the case where the group is compact, the case where the group element in question has compact centraliser and closed conjugacy class, and the case of the identity element of a non-compact discrete group.