Hyperbolic Group - Generalizations

Generalizations

An important generalization of hyperbolic groups in geometric group theory is the notion of a relatively hyperbolic group. Motivating examples for this generalization are given by the fundamental groups of non-compact hyperbolic manifolds of finite volume, in particular, the fundamental groups of hyperbolic knots, which are not hyperbolic in the sense of Gromov.

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 is a δ-hyperbolic space and, moreover, it satisfies an additional 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.