Theta Functions and the Structure of Torelli Groups in Low Genus

Abstract

The Torelli group Tg of a closed orientable surface Sg of genus g >1 is the groupof isotopy classes of orientation-preserving diffeomorphisms of Sg which act triviallyon its first integral homology. The hyperelliptic Torelli group TDg is the subgroupof Tg whose elements commute with a fixed hyperelliptic involution. The finitenessproperties of Tg and TDg are not well-understood when g > 2. In particular, it is notknown if T3 is finitely presented or if TD3 is finitely generated. In this thesis, we begina study of the finiteness properties of genus 3 Torelli groups using techniques fromcomplex analytic geometry. The Torelli space T3 is the moduli space of non-singulargenus 3 curves equipped with a symplectic basis for the first integral homology and isa model of the classifying space of T. Each component of the hyperelliptic locus T hyp 3in T3 is a model of the classifying space for TD3. We will investigate the topologyof the zero loci of certain theta functions and thetanulls and explain how these arerelated to the topology of T3 and T3 hyp. We show that the zero locus in h 2 x C2 of any genus 2 theta function is isomorphic to the universal cover of the universal framed genus 2 curve of compact type and that it is homotopy equivalent to an infinite bouquet of 2-spheres. We also derive a necessary and sufficient condition for the zero locus of any genus 3 even thetanull to be homotopy equivalent to a bouquet of 2-spheres and 3-spheres.