#### Volume 15, issue 3 (2011)

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

### Duc-Manh Nguyen

Geometry & Topology 15 (2011) 1707–1747
##### Abstract

The space ${\mathsc{ℋ}}^{hyp}\left(4\right)$ 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 ${\mathsc{ℋ}}^{hyp}\left(4\right)$ 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 ${\mathsc{ℋ}}^{hyp}\left(4\right)$; such surfaces are called generic. Using this criterion, we prove that there are generic surfaces in ${\mathsc{ℋ}}^{hyp}\left(4\right)$ with coordinates in any quadratic field, and there are Thurston–Veech surfaces with trace field of degree three over $ℚ$ which are generic.

##### Keywords
translation surface, unipotent flow, dynamics on moduli space
Primary: 51H25
Secondary: 37B05