A descent principle for the Dirac–dual-Dirac method

Let GG be a torsion-free discrete group with a finite-dimensional classifying space BGBG. We show that GG 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 groups is a metric, that is, coarse, invariant. We get results for groups with torsion as well.