  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 obtain
similar results for groups with torsion. The framework that we develop is also
suitable for studying the Lipschitz and proper Lipschitz cohomology of Connes,
Gromov and Moscovici.
