Free products of abelian groups in mapping class groups

The notions of convex cocompactness and geometric finiteness originally come from the study of Kleinian groups. The analogous notion of convex cocompactness for mapping class groups is due to Farb-Mosher. In recent work, Dowdall-Durham-Leininger-Sisto have proposed a definition of “parabolic” geometric finiteness for mapping class groups. In this thesis, we construct a new family of examples of parabolically geometrically finite mapping class subgroups and prove that they are undistorted in Mod(S). We also prove that a subfamily of our examples are also geometrically finite in the sense of Durham-Hagen-Sisto