Abstract
The Minimal Model Program (MMP) is one of the greatest theories in Algebraic Geometry developped to classify algebraic varieties. For some families of algebraic varieties, the MMP has been studied in depth. In particular, for toric and horospherical varieties, it comes down to a quite easy study of families of polytopes, called moment polytopes, and it could be adapted to weaker hypothesis of singularities. The goal of this thesis is to show that this reduction can be extended to spherical compactifications of a group. First of all we describe these varieties and classify all moment polytopes of such compactifications. Then we prove that the MMP applied on this spherical compactifications reduces to a study of a families of this moment polytopes. Finaly we give a computer program, coded in SageMath, which gives all polytopes appearing in the MMP of a simple group's spherical compactification.