My research is a blend of algebraic topology and representation theory with some applications towards number theory. I focus on asymptotic behavior of braid groups and certain congruence subgroups of braid groups. I am particularly interested in homological and representation stability. When we look at important subgroups of the braid group like the pure braid group or the level k braid group, their homology grows infinitely large as the number of strands increases. The goal of representation stability is to use techniques from representation theory to argue the homology still grows in a very controlled way. My thesis investigates the homology of the level 4 braid group by combining representation theory with recent stability results for Hurwitz spaces.