Coronae of relatively hyperbolic groups and coarse cohomologies

TL;DR

This paper constructs a corona for relatively hyperbolic groups by modifying their boundaries, and explores the relationship between various K-theories of these coronas and associated algebras, with applications to fundamental groups of 3-manifolds.

Contribution

It introduces a new corona construction for relatively hyperbolic groups and establishes links between its K-theory and other coarse geometric invariants.

Findings

Explicit computation of K-theory for Roe algebras of certain groups.

Establishment of duality between K-theory of the corona and Higson corona.

Development of generalized coarse cohomology theories.

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.