A Descent Principle for the Dirac Dual Dirac Method

Let G be a torsion free discrete group with a finite dimensional classifying space BG. We show that G has a dual Dirac morphism if and only if a certain coarse (co)-assembly map is an isomorphism. Hence the existence of a dual Dirac morphism for such G is a metric, that is, coarse, invariant of G. We get similar results for groups with torsion as well. The framework that we develop is also suitable for studying the Lipschitz and proper Lipschitz cohomology of Connes, Gromov and Moscovici.