Geometry & Topology

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

Duc-Manh Nguyen

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
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

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

Keywords
translation surface unipotent flow dynamics on moduli space

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


Export citation

References

  • K Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc. 17 (2004) 871–908
  • K Calta, J Smillie, Algebraically periodic translation surfaces, J. Mod. Dyn. 2 (2008) 209–248
  • A Douady, J Hubbard, On the density of Strebel differentials, Invent. Math. 30 (1975) 175–179
  • A Eskin, A Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math. 145 (2001) 59–103
  • P Hubert, E Lanneau, M Möller, Completely periodic directions and orbit closures of many pseudo-Anosov Teichm\"ller discs in $Q(1,1,1,1)$, to appear in Math. Ann.
  • P Hubert, E Lanneau, M M öller, The Arnoux–Yoccoz Teichmüller disc, Geom. Funct. Anal. 18 (2009) 1988–2016
  • P Hubert, E Lanneau, M Möller, $\mathrm{GL}_2^+(\R)$–orbit closures via topological splittings, from: “Surveys in differential geometry, Vol. XIV: Geometry of Riemann surfaces and their moduli spaces”, (L Ji, S A Wolpert, S-T Yau, editors), Surv. Differ. Geom. 14, Int. Press, Somerville, MA (2009) 145–169
  • P Hubert, T A Schmidt, An introduction to Veech surfaces, from: “Handbook of dynamical systems. Vol. 1B”, (B Hasselblatt, A Katok, editors), Elsevier, Amsterdam (2006) 501–526
  • M Kontsevich, A Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003) 631–678
  • H Masur, Interval exchange transformations and measured foliations, Ann. of Math. $(2)$ 115 (1982) 169–200
  • H Masur, S Tabachnikov, Rational billiards and flat structures, from: “Handbook of dynamical systems, Vol. 1A”, (B Hasselblatt, A Katok, editors), North-Holland, Amsterdam (2002) 1015–1089
  • C T McMullen, Dynamics of ${\rm SL}\sb 2(\mathbb R)$ over moduli space in genus two, Ann. of Math. $(2)$ 165 (2007) 397–456
  • W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. $($N.S.$)$ 19 (1988) 417–431
  • W A Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989) 553–583
  • W A Veech, Moduli spaces of quadratic differentials, J. Analyse Math. 55 (1990) 117–171
  • A Zorich, Flat surfaces, from: “Frontiers in number theory, physics, and geometry I”, (P Cartier, B Julia, P Moussa, P Vanhove, editors), Springer, Berlin (2006) 437–583