Moduli Spaces of Flat Tori with Prescribed Holonomy

We generalise to the genus one case several results of Thurston concerning moduli spaces of flat Euclidean structures with conical singularities on the two dimensional sphere. More precisely, we study moduli spaces of flat tori with n cone points and a prescribed holonomy ρ. In his paper `Flat Surfaces’ Veech has established that under some assumptions on the cone angles, such a moduli space F_{[ ρ ]}⊂ M_{1 , n} carries a natural geometric structure modeled on the complex hyperbolic space CH^{n - 1} which is not metrically complete. Using surgeries for flat surfaces, we prove that the metric completion F_{[ ρ ]}¯ is obtained by adjoining to F_{[ ρ ]} certain strata that are themselves moduli spaces of flat surfaces of genus 0 or 1, obtained as degenerations of the flat tori whose moduli space is F_{[ ρ ]}. We show that the CH^{n - 1}-structure of F_{[ ρ ]} extends to a complex hyperbolic cone-manifold structure of finite volume on F_{[ ρ ]}¯ and we compute the cone angles associated to the different strata of codimension 1. Finally, we address the question of whether or not the holonomy of Veech’s CH^{n - 1}-structure on F_{[ ρ ]} has a discrete image in Aut (CH^{n - 1}) = PU (1 , n- 1). We outline a general strategy to find moduli spaces F_{[ ρ ]} whose CH^{n - 1}-holonomy gives rise to lattices in PU (1 , n- 1) and eventually we give a finite list of F_{[ ρ ]} ’s whose holonomy is a complex hyperbolic arithmetic lattice.