, called peripheral subgroups, in a way that enables "hyperbolic reduction" of problems for
Further generalizations such as acylindrical hyperbolicity are also explored by current research.
The original insight by Gromov, motivated by examples from Riemannian geometry and later elaborated by Bowditch, is to say that
s fix points at infinity and that the action becomes cocompact after truncating horoballs around them.
[3] The second kind of definition, first due to Farb, roughly says that after contracting the left-cosets of the
[4] The resulting notion, known today as weak hyperbolicity, turns out to require extra assumptions on the behavior of quasi-geodesics in order to match the Gromov-Bowditch one.
to act on a hyperbolic graph with certain additional properties, including that the conjugates of the
[6] Druțu and Sapir gave a characterization in terms of asymptotic cones being tree-graded metric spaces, a relative version of real trees.
This allows for a notion of relative hyperbolicity that makes sense for more general metric spaces than Cayley graphs, and which is invariant by quasi-isometry.
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
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.