A. Yaman

We characterise relatively hyperbolic groups as geometrically finite convergence groups. More precisely, we show the following. Suppose $M$ is a non-empty perfect compact metrisable space, and suppose that a group, $\Gamma$, acts as a convergence group on $M$ such that $M$ consists only conical limit points and bounded parabolic points. Suppose also that the stabiliser of each bounded parabolic point is finitely generated. Then $\Gamma$ is relatively hyperbolic, and $M$ is equivariantly homeomorphic to the boundary of $\Gamma$. We also give an other aspect of this characterisation by showing under the above assumptions that $\Gamma$ acts also as a cusp uniform group on the space of triples of $M$.