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We construct a corona of a relatively hyperbolic group by blowing-up allparabolic points of its Bowditch boundary. We relate the $K$-homology of thecorona with the $K$-theory of the Roe algebra, via the coarse assembly map. Wealso establish a dual theory, that is, we relate the $K$-theory of the coronawith the $K$-theory of the reduced stable Higson corona via the coarseco-assembly map. For that purpose, we formulate generalized coarse cohomologytheories. As an application, we give an explicit computation of the $K$-theoryof the Roe-algebra and that of the reduced stable Higson corona of thefundamental groups of closed 3-dimensional manifolds and of pinched negativelycurved complete Riemannian manifolds with finite volume.

Coronae of relatively hyperbolic groups and coarse cohomologies