The primary goal of the project is to obtain an understanding of geometric and dynamical properties of moduli spaces and mapping class groups. For a mapping class group of a surface of finite type, we are interested in subgroups, in particular in the trace fields of Veech groups beyond the case of genus 2. Convex cocompact surface subgroups are word hyperbolic surface-by-surface groups, and we aim at clarifying whether or not such groups exist.Fine asymptotics of the distribution of periodic orbits for the Teichmüller flow on strata of quadratic or abelian differentials can be related to dynamical zeta functions. A Borel conjugacy of the Teichmüller flow on the moduli space of quadratic differentials into the Weil-Petersson flow will be used to analyze dynamical properties of the Weil-Petersson flow.The handlebody is a finitely presented subgroup of the mapping class group which however is not quasi-isometrically embedded. A new geometric model for the group will be used towards obtaining a comprehensive understanding of the geometry of this group, in particular with respect to calculating the Dehn function and quasi-isometric rigidigy.A similar geometric model for the outer automorphism group of the free group may yield hyperbolicity of the electrified sphere graph on which this group acts by simplicial automorphisms..