The 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 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.
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