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

Duc-Manh Nguyen

Geometry & Topology 15 (2011) 1707–1747

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.

translation surface, unipotent flow, dynamics on moduli space
Mathematical Subject Classification 2010
Primary: 51H25
Secondary: 37B05
Received: 6 December 2010
Revised: 12 September 2011
Accepted: 29 August 2011
Published: 1 October 2011
Proposed: Benson Farb
Seconded: Walter Neumann, Joan Birman
Duc-Manh Nguyen
Institut de Mathématiques de Bordeaux, Bat A33
Université Bordeaux 1
351, cours de la Libération
F-33405 Talence Cedex