Geometry & Topology

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

Duc-Manh Nguyen

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The space hyp(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 hyp(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,)–orbit of the surface is dense in hyp(4); such surfaces are called generic. Using this criterion, we prove that there are generic surfaces in hyp(4) with coordinates in any quadratic field, and there are Thurston–Veech surfaces with trace field of degree three over which are generic.

Article information

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

Received: 6 December 2010
Revised: 12 September 2011
Accepted: 29 August 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 51H25: Geometries with differentiable structure [See also 53Cxx, 53C70]
Secondary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)

translation surface unipotent flow dynamics on moduli space


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.

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