The purpose of this thesis is to establish various results concerning the rigidity of the mapping class group and other associated spaces. Apart from its intrinsic interest in the context of geometric group theory, these results are motivated by the comparison between mapping class groups and irreducible lattices in higher-rank Lie groups. In this setting, a natural analogue of Margulis superrigidity is whether every homomorphism between mapping class groups lifts to a map at the level of diffeomorphism groups. This questions leads the discussion for the first two chapters. We introduce in Chapter 1 a key new tool to study homomorphisms, namely a topological characterization of pairs of multitwists that satisfy the braid equation. Then, in Chapter 2 we prove that, for a range of surfaces, every homomorphism between mapping class groups is induced by a set of embeddings. As a consequence, we show that, for these surfaces, every homomorphism between mapping class groups does indeed lift to the diffeomorphism groups. A parallel rigidity result is presented in Chapter 3, where for the same range of surfaces we prove a geometric classification of holomorphic maps between their moduli spaces. In Chapter 4, we extend some of the techniques used to the setting of infinite-type surfaces, proving that epimorphisms between certain big mapping class groups are actually isomorphisms. In particular, we conclude, for a certain class of surfaces, that big mapping class groups are topologically Hopfian. The last two chapters of the thesis are devoted to simplicial actions of the mapping class group. In Chapter 5, we prove the finite rigidity of the non-separating curve complex. Finally, in Chapter 6, we introduce the matching arc complex, a variant of the arc complex that arises in the study braided Thompson groups, and we obtain conditions for this complex to be both connected and hyperbolic

Este trabajo fue realizado gracias al apoyo económico del contrato pre-doctoral PRE2020-092939 que forma parte del proyecto Severo Ochoa CEX2019000904-S-20-2. Además, parte de los contenidos de este trabajo fueron financiados por el Ministerio de Ciencia, Innovación y Universidades a través de las becas PGC2018-101179-B-I00 y CEX2023-001347-S

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de Lectura: 22-05-2025