Moduli spaces of flat tori with prescribed holonomy

TL;DR

This paper extends Thurston's results on moduli spaces of flat structures with conical singularities to genus one, showing their complex hyperbolic structure extends to a cone-manifold with finite volume and analyzing the holonomy's discreteness.

Contribution

It generalizes Thurston's results to genus one, describing the metric completion, cone angles, and holonomy properties of moduli spaces of flat tori with prescribed holonomy.

Findings

The metric completion adds strata of degenerations as flat surface moduli spaces.

The complex hyperbolic structure extends to a finite volume cone-manifold.

Identifies cases where holonomy is a lattice in PU(1,n-1).

Abstract

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 the moduli space of flat tori with $n$ cone points and a prescribed holonomy $\rho$. In his paper `Flat Surfaces' Veech has established that under some assumptions on the cone angles, such a moduli space ${\mathcal{F}}_{[\rho]}\subset \mathscr M_{1,n}$ carries a natural geometric structure modeled on the complex hyperbolic space ${\mathbb C}{\mathbb{H} }^{n-1}$ which is not metrically complete. Using surgeries for flat surfaces, we prove that the metric completion $\overline{{\mathcal{F}}_{[\rho]}}$ is obtained by adjoining to $ {\mathcal{F}}_{[\rho]} $ 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 $ {\mathcal{F}}_{[\rho]}$. We show that the ${\mathbb C}{\mathbb{H} }^{n-1}$-structure of $ {\mathcal{F}}_{[\rho]}$ extends to a complex hyperbolic cone-manifold structure of finite volume on $ \overline{{\mathcal{F}}_{[\rho]}}$ 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 ${\mathbb C}{\mathbb{H} }^{n-1}$-structure on $ \mathcal F_\rho$ has a discrete image in $ {\rm Aut}({\mathbb C}{\mathbb{H} }^{n-1})=\mathrm{PU}(1,n-1)$. We outline a general strategy to find moduli spaces $\mathcal F_{[\rho]}$ whose ${\mathbb C}{\mathbb{H} }^{n-1}$-holonomy gives rise to lattices in $\mathrm{PU}(1,n-1)$ and eventually we give a finite list of $\mathcal F_{[\rho]}$'s whose holonomy is a complex hyperbolic arithmetic lattice.