## Geometry & Topology

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

Duc-Manh Nguyen

#### Abstract

The space $ℋhyp(4)$ is the moduli space of pairs $(M,ω)$, where $M$ is a hyperelliptic Riemann surface of genus $3$ and $ω$ is a holomorphic $1$–form having only one zero. In this paper, we first show that every surface in $ℋhyp(4)$ 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 $GL+(2,ℝ)$–orbit of the surface is dense in $ℋhyp(4)$; such surfaces are called generic. Using this criterion, we prove that there are generic surfaces in $ℋhyp(4)$ with coordinates in any quadratic field, and there are Thurston–Veech surfaces with trace field of degree three over $ℚ$ which are generic.

#### Article information

Source
Geom. Topol., Volume 15, Number 3 (2011), 1707-1747.

Dates
Revised: 12 September 2011
Accepted: 29 August 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732343

Digital Object Identifier
doi:10.2140/gt.2011.15.1707

Mathematical Reviews number (MathSciNet)
MR2851075

Zentralblatt MATH identifier
1238.57020

#### Citation

Nguyen, Duc-Manh. Parallelogram decompositions and generic surfaces in $\mathcal{H}^{\mathrm{hyp}}(4)$. Geom. Topol. 15 (2011), no. 3, 1707--1747. doi:10.2140/gt.2011.15.1707. https://projecteuclid.org/euclid.gt/1513732343

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