On the Branch Loci of Moduli Spaces of Riemann Surfaces

Abstract: 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 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.Paper 1. In this paper we show that the strata corresponding to actions of order 2 and 3 belong to the same connected component for arbitrary genera. Further we show that the branch locus is connected with the exception of one isolated point for genera 5 and 6, it is connected for genus 7 and it is connected with the exception of two isolated points for genus 8.Paper 2. This paper contains a collection of results regarding components of the branch loci, some of them proved in detail in other papers. It is shown that for any integer d if p is a prime such that p > (d + 2)2, there there exist isolated strata of dimension d in the moduli space of Riemann surfaces of genus (d + 1)(p − 1)/2. It is also shown that if we consider Riemann surfaces as Klein surfaces, the branch loci are connected for every genera due to reflections.Paper 3. Here we consider surfaces of genus 4 and 5. Here we study the automorphism groups of Riemann surfaces of genus 4 and 5 up to topological equivalence and determine the complete structure of the equisymmetric stratification of the branch locus.Paper 4. In this paper we establish that the connectedness of the branch loci is a phenomenon for low genera. More precisely we prove that the only genera g where Bg is connected are g = 3, 4, 13, 17, 19, 59.