In this paper we obtain an isoperimetric characterization of relatively hyperbolicity of a groups with respect to a collection of subgroups. This allows us to apply classical combinatorial methods related to van Kampen diagrams to obtain relative analogues of some well-known algebraic and geometric properties of ordinary hyperbolic groups. We also introduce and study the notion of a relatively quasi-convex subgroup of a relatively hyperbolic group and solve some natural algorithmic problems.

Table of Contents

• Introduction
• Relative isoperimetric inequalities
• Geometry of finitely generated relatively hyperbolic groups
• Algebraic properties
• Algorithmic problems
• Open questions
• Appendix. Equivalent definitions of relative hyperbolicity
• Bibliography