Yoshifumi Matsuda

This talk is based on a joint work in progress with Shin-ichi Oguni and Saeko Yamagata.

The notion of relatively hyperbolic groups was introduced by Gromov as a generalization of geometrically finite Kleinian groups and hyperbolic groups. When a countable group G is hyperbolic relative to a conjugacy invariant collection of infinite subgroups, this collection is called a relatively hyperbolic structure on G. For example, the collection of maximal parabolic subgroups of a geometrically finite Kleinian group G is a relatively hyperbolic structure on G.

In this talk, we propose a way to study relationship among relatively hyperbolic structures. Given a countable group G, we introduce a partial order → on the set of relatively hyperbolic structures on G. The greatest relatively hyperbolic structure with respect to → is called the universal relatively hyperbolic structure. A characterization of the universal relatively hyperbolic structure on a finitely generated group is given. This characterization enables us to recognize several groups with the universal relatively hyperbolic structure. We also discuss subgroups which are quasiconvex relative to a relatively hyperbolic structure.