Non-finitely generated relatively hyperbolic groups and Floyd quasiconvexity

Abstract

We regard a relatively hyperbolic group as a group acting non-trivially by homeomorphisms on a compactum $T$ discontinuously on the set of distinct triples and cocompactly on the set of distinct pairs of points of $T$.

In the first part of the paper we prove that such a group $G$ admits a graph of groups decomposition given by a star graph whose central vertex group is finitely generated relatively hyperbolic with respect to the edge groups, and the other vertex groups are stabilizers of non-equivalent parabolic points. It follows from this result that every relatively hyperbolic group is relatively finitely generated with respect to the parabolic subgroups. Another corollary is that the definition of the relative hyperbolicity which we are using is equivalent to those of Bowditch and Osin (taken with respect to finitely many peripheral subgroups) and they are all equivalent to the existence of the above star graph of groups decomposition.

The second part of the paper uses the method of the first part. Considering the induced action of $G$ on the space of distinct pairs of $T$ we construct a connected graph on which $G$ acts properly and cofinitely on edges. Equipping the graph with Floyd metrics we prove that the quasigeodesics in this metric are close somewhere to the geodesics in the word metric. This allows us to prove that the parabolic subgroups of $G$ are quasiconvex with respect to the Floyd metrics. As a corollary we prove that the preimage of a parabolic point by the Floyd map is the Floyd boundary of its stabilizer.