Parallelogram decompositions and generic surfaces in H^{hyp}(4)

TL;DR

This paper investigates the structure of translation surfaces in the stratum H^{hyp}(4), demonstrating a unique parallelogram decomposition, establishing a criterion for orbit density, and identifying conditions for generic surfaces with specific algebraic properties.

Contribution

It introduces a unique parallelogram decomposition for surfaces in H^{hyp}(4) and provides a criterion for orbit density, linking geometric decompositions to dynamical properties.

Findings

Every surface in H^{hyp}(4) admits a unique parallelogram decomposition.

A specific condition on the decomposition guarantees dense SL(2,R) orbits.

Existence of generic surfaces with coordinates in any quadratic field and from Thurston-Veech constructions.

Abstract

In this paper we are interested in the stratum H^{hyp}(4) of translation surfaces, which consists of pairs (M,\omega), where M is a hyper-elliptic Riemann surface of genus 3, and \omega is a holopmorphic 1-form on M having only one zero. We first show that every surface in this stratum can be decomposed into parallelograms following a unique model. We then single out a condition on this decomposition, and show that if this condition is satisfied then the SL(2,R) orbit of the surface is dense in the stratum. Using this criterion, we show that there are generic surfaces in this stratum with coordinates in any quadratic field, and that surfaces arising from the Thurston-Veech construction with cubic trace field can be generic.