A descent principle for the Dirac dual Dirac method

Abstract

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.

References in corpus (3)

• Dualizing the coarse assembly map
• Generalized fixed point algebras and square-integrable group actions
• Euler characteristics and Gysin sequences for group actions on boundaries

Cited by in corpus (8)

• Equivariant representable K-theory
• Equivariant Coarse (Co-)Homology Theories
• Coronae of relatively hyperbolic groups and coarse cohomologies
• Coarse and equivariant co-assembly maps
• Slant products on the Higson-Roe exact sequence
• A categorical perspective on the Atiyah-Segal completion theorem in $\mathrm{KK}$-theory
• Proper Kasparov Cycles and the Baum-Connes Conjecture
• KK-theoretic duality for proper twisted actions