Geometric and Functional Analysis

, Volume 29, Issue 6, pp 1617–1637 | Cite as

Rank One Orbit Closures in \(\varvec{\mathcal {H}}^{\varvec{\lowercase {hyp}}}(\varvec{\lowercase {g}}-1,\varvec{\lowercase {g}}-1)\)

  • Paul ApisaEmail author


All \(\mathrm {GL}(2, \mathbb {R})\) orbits in hyperelliptic components of strata of abelian differentials in genus greater than two are closed, dense, or contained in a locus of branched covers.



The author thanks Alex Eskin and Alex Wright for useful conversations and encouragement. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1144082 and upon work supported by the National Science Foundation under Award No. 1803625. The author gratefully acknowledges their support.


  1. AEM17.
    A. Avila, A. Eskin, and M. Möller. Symplectic and isometric \({\rm SL}(2,\mathbb{R})\)-invariant subbundles of the Hodge bundle. J. Reine Angew. Math., 732 (2017), 1–20MathSciNetCrossRefGoogle Scholar
  2. Api18.
    P. Apisa. \({\rm GL}_2\mathbb{R}\) orbit closures in hyperelliptic components of strata. Duke Math. J., (4)167 (2018), 679–742MathSciNetCrossRefGoogle Scholar
  3. EFW18.
    A. Eskin, S. Filip, and A. Wright. The algebraic hull of the Kontsevich-Zorich cocycle. Ann. of Math. (2), (1)188 (2018), 281–313MathSciNetCrossRefGoogle Scholar
  4. EM18.
    A. Eskin and M. Mirzakhani. Invariant and stationary measures for the \({\rm SL}(2,\mathbb{R})\) action on moduli space. Publ. Math. Inst. Hautes Études Sci., 127 (2018), 95–324MathSciNetCrossRefGoogle Scholar
  5. EMM15.
    A. Eskin, M. Mirzakhani, and A. Mohammadi. Isolation, equidistribution, and orbit closures for the \({\rm SL}(2,\mathbb{R})\) action on moduli space. Ann. of Math. (2), (2)182 (2015), 673–721MathSciNetCrossRefGoogle Scholar
  6. KZ03.
    M. Kontsevich and A. Zorich. Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Invent. Math., (3)153 (2003), 631–678MathSciNetCrossRefGoogle Scholar
  7. Lin15.
    K.A. Lindsey. Counting invariant components of hyperelliptic translation surfaces. Israel J. Math., (1)210 (2015), 125–146MathSciNetCrossRefGoogle Scholar
  8. McM03.
    C.T. McMullen. Billiards and Teichmüller curves on Hilbert modular surfaces. J. Amer. Math. Soc., (4)16 (2003), 857–885 (electronic)MathSciNetCrossRefGoogle Scholar
  9. McM05.
    C.T. McMullen. Teichmüller curves in genus two: discriminant and spin. Math. Ann., (1)333 (2005), 87–130MathSciNetCrossRefGoogle Scholar
  10. McM06.
    C.T. McMullen. Teichmüller curves in genus two: torsion divisors and ratios of sines. Invent. Math., (3)165 (2006), 651–672MathSciNetCrossRefGoogle Scholar
  11. McM07.
    C.T. McMullen. Dynamics of \({\rm SL}_2(\mathbb{R})\) over moduli space in genus two. Ann. of Math. (2), (2)165 (2007), 397–456MathSciNetCrossRefGoogle Scholar
  12. Wri15a.
    A. Wright. Cylinder deformations in orbit closures of translation surfaces. Geom. Topol.. (1)19 (2015), 413–438MathSciNetCrossRefGoogle Scholar
  13. Wri15b.
    A. Wright. Translation surfaces and their orbit closures: An introduction for a broad audience. EMS Surv. Math. Sci., (1)2 (2015), 63–108MathSciNetCrossRefGoogle Scholar
  14. Zor06.
    A. Zorich. Flat surfaces. Frontiers in number theory, physics, and geometry. I, Springer, Berlin, (2006), pp. 437–583Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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