Quelques contributions \`a la th\'eorie de l'action de SL(2,R) sur les espaces de modules de surfaces plates

This memoir is part of the author’s Habilitation \a Diriger des Recherches’ (HDR) dossier: it summarizes some of his researches after his PhD thesis. (The HDR diploma is general requirement for many purposes [e.g., supervising PhD students] within the French academic system.)