We introduce a number of new tools for the study of relatively hyperbolic
groups. First, given a relatively hyperbolic group G, we construct a nice
combinatorial Gromov hyperbolic model space acted on properly by G, which
reflects the relative hyperbolicity of G in many natural ways. Second, we
construct two useful bicombings on this space. The first of these, "preferred
paths", is combinatorial in nature and allows us to define the second, a
relatively hyperbolic version of a construction of Mineyev.
As an application, we prove a group-theoretic analog of the Gromov-Thurston
2\pi Theorem in the context of relatively hyperbolic groups.Comment: 83 pages. v2: An improved version of preferred paths is given, in
which preferred triangles no longer need feet. v3: Fixed several small errors
pointed out by the referee, and repaired several broken figures. v4:
corrected definition 2.38. This is very close to the published versio
