Subgroups of mapping class groups and braid groups

This thesis studies the subgroup structure of mapping class groups. We use techniques that fall into two categories: analysing the group action on a family of simplicial complexes, and investigating regular, finite-sheeted covering spaces. We use the first approach to prove that a wide class of normal subgroups of mapping class groups of punctured surfaces are geometric, that is, they have the extended mapping class group as their group of automorphisms, expanding on work of BrendleMargalit. For example, we determine that every member of the Johnson filtration is geometric. By considering punctured spheres, we also establish the automorphism groups of many normal subgroups of the braid group. The second approach is to relate subgroups of each of the mapping class groups associated to a covering space, namely, the liftable and symmetric mapping class groups. Given that the two surfaces have boundary, we consider covers in which either every mapping class lifts or every mapping class is fibre-preserving. We classify all covers that fall into one of these cases. In Chapter 1 we recall some preliminaries before stating the main results of the thesis. We then extend Brendle-Margalit's definition of complexes of regions to surfaces with punctures. Chapter 2 proves that the automorphism group of a complex of regions is the extended mapping class group, resolving in part a metaconjecture of N. V. Ivanov. In Chapter 3 we construct a complex of regions associated to a general normal subgroup of a mapping class group of a surface with punctures. We then apply the main result of the previous chapter to establish that such a normal subgroup is geometric. Finally, Chapter 4 presents joint work with Tyrone Ghaswala. We give a proof of the Birman-Hilden Theorem for surfaces with boundary and then prove the classifications of regular, finite-sheeted covering spaces of surfaces with boundary discussed above. We conclude by investigating an infinite family of branched covers of the disc. This family induces embeddings of the braid group into mapping class groups. We prove that each of these embeddings maps a standard generator of the braid group to a product of Dehn twists about curves forming a chain, providing an answer to a question of Wajnryb