Relatively hyperbolic group

In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group. The motivating examples of relatively hyperbolic groups are the fundamental groups of complete noncompact hyperbolic manifolds of finite volume.
Intuitive definition.
A group "G" is relatively hyperbolic with respect to a subgroup "H" if, after contracting the Cayley graph of "G" along "H"-cosets, the resulting graph equipped with the usual graph metric becomes a δ-hyperbolic space and, moreover, it satisfies a technical condition which implies that quasi-geodesics with common endpoints travel through approximately the same collection of cosets and enter and exit these cosets in approximately the same place.
Formal definition.
Given a finitely generated group "G" with Cayley graph "Γ"("G") equipped with the path metric and a subgroup "H" of "G", one can construct the coned off Cayley graph formula_1 as follows: For each left coset "gH", add a vertex "v"("gH") to the Cayley graph "Γ"("G") and for each element "x" of "gH", add an edge "e"("x") of length 1/2 from "x" to the vertex "v"("gH"). This results in a metric space that may not be proper (i.e. closed balls need not be compact).
The definition of a relatively hyperbolic group, as formulated by Bowditch goes as follows. A group "G" is said to be hyperbolic relative to a subgroup "H" if the coned off Cayley graph formula_2 has the properties:
If only the first condition holds then the group "G" is said to be weakly relatively hyperbolic with respect to "H".
The definition of the coned off Cayley graph can be generalized to the case of a collection of subgroups and yields the corresponding notion of relative hyperbolicity. A group "G" which contains no collection of subgroups with respect to which it is relatively hyperbolic is said to be a non relatively hyperbolic group.