Relative torsion for representations in finite type Hilbert modules

For a closed manifold equipped with a Riemannian metric, a triangulation, a representation of its fundamental group on an Hilbert module of finite type (over of finite von Neumann algebra), and a Hermitian structure on the flat bundle associated to the representation, one defines a numerical invariant, the relative torsion. The relative torsion is a positive real number and unlike the analytic torsion or the Reidemeister torsion, which are defined only when the pair manifold- representation is of determinant class, is always defined. When the pair is of determinant class the relative torsionis equal to the quotient of the analytic and the Reidemeister torsion.We calculate the relative torsion.