On the Branch Loci of Moduli Spaces of Riemann Surfaces

The spaces of conformally equivalent Riemann surfaces, Mg where g ≥ 1, are not manifolds. However the spaces of the weaker Teichmüller equivalence, Tg are known to be manifolds. The Teichmüller space Tg is the universal covering of Mg and Mg is the quotient space by the action of the modular group. This gives Mg an orbifold structure with a branch locus Bg . The branch loci Bg can be identified with Riemann surfaces admitting non-trivial automorphisms for surfaces of genus g ≥ 3. In this thesis we consider the topological structure of Bg . We study the connectedness of the branch loci in general by considering families of isolated strata and we establish that connectedness is a phenomenon for low genera. Further, we give the orbifold structure of the branch locus of surfaces of genus 4 and genus 5 in particular, by studying the equisymmetric stratification of the branch locus.