2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65131The aim of this paper is to introduce a group containing the mapping class groups of all genus zero surfaces. Roughly speaking, such a group is intended to be a discrete analogue of the diffeomorphism group of the circle. One defines indeed a {\it universal mapping class group of genus zero}, denoted $\B$. The latter is a nontrivial extension of the Thompson group $V$ (acting on the Cantor set) by an inductive limit of pure mapping class groups of all genus zero surfaces. We prove that $\B$ is a finitely presented group, and give an explicit presentation of it.G.A.F.A., to appear, 46 p. The paper has been split, this version is the revision of the first partGeometric TopologyGroup Theory57 N 05, 20 F 38text