EQUIVARIANT ANALYTIC TORSION AND AN EQUIVARIANT RUELLE DYNAMICAL ZETA FUNCTION

Analytic torsion was introduced by Ray and Singer as a way to realise Reidemeister-Franz torsion analytically. (The equality was independently proved by Cheeger and Müller.) The Ruelle dynamical zeta function is a topological way to count closed curves of flows on compact manifolds. The Fried conjecture states that, for a suitable class of flows, the Ruelle dynamical zeta function has a well-defined value at zero, and that the absolute value of this value equals analytic torsion. With Hemanth Saratchandran, we define equivariant versions of analytic torsion and of the Ruelle dynamical zeta function, for proper actions by locally compact groups, with compact quotients. These have some natural fundamental properties, generalising properties of their non-equivariant counterparts. The resulting equivariant version of Fried’s conjecture does not hold in general, but it does hold in some classes of examples. This motivates the search for general conditions under which the equivariant Fried conjecture is true.