Rigidity of Mapping Class Group Mod Powers of Dehn Twists

In this thesis, which is the extended version of the paper "Rigidity of Mapping Class Groups mod powers of Twists" with Alessandro Sisto (https://arxiv.org/abs/2212.11014), we study quotients of mapping class groups of punctured spheres by suitable large powers of Dehn twists. In particular, we show that these groups coincide with the simplicial automorphisms of the corresponding quotients of curve graphs, thus establishing an analogue of Ivanov's theorem. Then we use this fact to show that these quotient groups are quasi-isometrically rigid and that their automorphism groups coincide with the groups themselves