Parallelogram decompositions and generic surfaces in $\mathcal{H}^{\mathrm{hyp}}(4)$

Abstract

The space ${\mathcal{\mathscr{H}}}^{hyp}\left(4\right)$ is the moduli space of pairs $\left(M,\omega \right)$, where $M$ is a hyperelliptic Riemann surface of genus $3$ and $\omega$ is a holomorphic $1$–form having only one zero. In this paper, we first show that every surface in ${\mathcal{\mathscr{H}}}^{hyp}\left(4\right)$ admits a decomposition into parallelograms and simple cylinders following a unique model. We then show that if this decomposition satisfies some irrational condition, then the ${GL}^{+}\left(2,ℝ\right)$–orbit of the surface is dense in ${\mathcal{\mathscr{H}}}^{hyp}\left(4\right)$; such surfaces are called generic. Using this criterion, we prove that there are generic surfaces in ${\mathcal{\mathscr{H}}}^{hyp}\left(4\right)$ with coordinates in any quadratic field, and there are Thurston–Veech surfaces with trace field of degree three over $\mathbb{Q}$ which are generic.