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The paper consists of two parts. In the first one we show that a relativelyhyperbolic group $G$ splits as a star graph of groups whose central vertexgroup is finitely generated and the other vertex groups are maximal parabolicsubgroups. As a corollary we obtain that every group which admits3-discontinuous and 2-cocompact action by homeomorphisms on a compactum isfinitely generated with respect to a system of parabolic subgroups.  The second part essentially uses the methods of topological entouragesdeveloped in the first part. Using also Floyd metrics we obtain finerproperties of finitely generated relatively hyperbolic groups. We show thatthere is a system of "tight" curves satisfying the property of horosphericalquasiconvexity. We then prove that the Floyd quasigeodesics are tight and sothe parabolic subgroups of $G$ are quasiconvex with respect to the Floydmetrics. As a corollary we obtain that the preimage of a parabolic point by theFloyd map is the Floyd boundary of its stabilizer.

Non-finitely generated relatively hyperbolic groups and Floyd quasiconvexity