Coronae of relatively hyperbolic groups and coarse cohomologies

Abstract

We construct a corona of a relatively hyperbolic group by blowing-up all parabolic points of its Bowditch boundary. We relate the $K$-homology of the corona with the $K$-theory of the Roe algebra, via the coarse assembly map. We also establish a dual theory, that is, we relate the $K$-theory of the corona with the $K$-theory of the reduced stable Higson corona via the coarse co-assembly map. For that purpose, we formulate generalized coarse cohomology theories. As an application, we give an explicit computation of the $K$-theory of the Roe-algebra and that of the reduced stable Higson corona of the fundamental groups of closed 3-dimensional manifolds and of pinched negatively curved complete Riemannian manifolds with finite volume.

48 pages

References in corpus (2)

• Coronae of product spaces and the Coarse Baum-Connes conjecture
• The coarse Baum-Connes conjecture for relatively hyperbolic groups

Cited by in corpus (3)

• A coarse Cartan-Hadamard theorem with application to the coarse Baum-Connes conjecture
• The coarse Baum-Connes conjecture for Busemann non-positively curved spaces
• Blown-up boundaries associated with ample cones of K3 surfaces