Duke Mathematical Journal

GL2R orbit closures in hyperelliptic components of strata

Paul Apisa

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The object of this article is to study GL2R orbit closures in hyperelliptic components of strata of Abelian differentials. The main result is that all higher-rank affine invariant submanifolds in hyperelliptic components are branched covering constructions; that is, every translation surface in the affine invariant submanifold covers a translation surface in a lower genus hyperelliptic component of a stratum of Abelian differentials. This result implies the finiteness of algebraically primitive Teichmüller curves in all hyperelliptic components for genus greater than two. A classification of all GL2R orbit closures in hyperelliptic components of strata (up to computing connected components and up to finitely many nonarithmetic rank one orbit closures) is provided. Our main theorem resolves a pair of conjectures of Mirzakhani in the case of hyperelliptic components of moduli space.

Article information

Source
Duke Math. J., Volume 167, Number 4 (2018), 679-742.

Dates
Received: 26 March 2016
Revised: 2 June 2017
First available in Project Euclid: 30 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1517281370

Digital Object Identifier
doi:10.1215/00127094-2017-0043

Mathematical Reviews number (MathSciNet)
MR3769676

Zentralblatt MATH identifier
06857028

Subjects
Primary: 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems
Secondary: 32G

Keywords
translation surfaces Abelian differentials affine invariant submanifolds Teichmüller theory

Citation

Apisa, Paul. $\operatorname{GL}_{2}\mathbb{R}$ orbit closures in hyperelliptic components of strata. Duke Math. J. 167 (2018), no. 4, 679--742. doi:10.1215/00127094-2017-0043. https://projecteuclid.org/euclid.dmj/1517281370


Export citation

References

  • [1] P. Apisa, $\operatorname{GL}_{2}{\mathbb{R}}$-invariant measures in marked strata: Generic marked points, Earle–Kra for strata, and illumination, preprint, arXiv:1601.07894v1 [math.DS].
  • [2] D. Aulicino and D.-M. Nguyen, Rank two affine submanifolds in $\mathcal{H}(2,2)$ and $\mathcal{H}(3,1)$, Geom. Topol. 20 (2016), 2837–2904.
  • [3] A. Avila, A. Eskin, and M. Möller, Symplectic and isometric $\operatorname{SL}(2,\mathbb{R})$-invariant subbundles of the Hodge bundle, J. Reine Angew. Math., published electronically 14 April 2015.
  • [4] M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol. 11 (2007), 1887–2073.
  • [5] K. Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc. 17 (2004), 871–908.
  • [6] A. Eskin, S. Filip, and A. Wright, The algebraic hull of the Kontsevich–Zorich cocycle, preprint, arXiv:1702.02074v1 [math.DS].
  • [7] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $\operatorname{SL}(2,\mathbb{R})$ action on moduli space, preprint, arXiv:1302.3320v4 [math.DS].
  • [8] A. Eskin, M. Mirzakhani, and A. Mohammadi, Isolation, equidistribution, and orbit closures for the $\operatorname{SL}(2,\mathbb{R})$ action on moduli space, Ann. of Math. (2) 182 (2015), 673–721.
  • [9] S. Filip, Semisimplicity and rigidity of the Kontsevich–Zorich cocycle, Invent. Math. 205 (2016), 617–670.
  • [10] S. Filip, Splitting mixed Hodge structures over affine invariant manifolds, Ann. of Math. (2) 183 (2016), 681–713.
  • [11] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), 631–678.
  • [12] E. Lanneau, D.-M. Nguyen, and A. Wright, Finiteness of Teichmüller curves in non-arithmetic rank $1$ orbit closures, to appear in Amer. J. Math., preprint, arXiv:1504.03742v2 [math.DS].
  • [13] K. A. Lindsey, Counting invariant components of hyperelliptic translation surfaces, Israel J. Math. 210 (2015), 125–146.
  • [14] B. Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381–386.
  • [15] C. Matheus and A. Wright, Hodge–Teichmüller planes and finiteness results for Teichmüller curves, Duke Math. J. 164 (2015), 1041–1077.
  • [16] C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003), 857–885.
  • [17] C. T. McMullen, Teichmüller curves in genus two: Discriminant and spin, Math. Ann. 333 (2005), 87–130.
  • [18] C. T. McMullen, Teichmüller curves in genus two: Torsion divisors and ratios of sines, Invent. Math. 165 (2006), 651–672.
  • [19] C. T. McMullen, Dynamics of $\operatorname{SL}_{2}(\mathbb{R})$ over moduli space in genus two, Ann. of Math. (2) 165 (2007), 397–456.
  • [20] C. T. McMullen, Braid groups and Hodge theory, Math. Ann. 355 (2013), 893–946.
  • [21] C. T. McMullen, Navigating moduli space with complex twists, J. Eur. Math. Soc. (JEMS) 15 (2013), 1223–1243.
  • [22] C. T. McMullen, R. E. Mukamel, and A. Wright, Cubic curves and totally geodesic subvarieties of moduli space, Ann. of Math. (2) 185 (2017), 957–990.
  • [23] M. Mirzakhani and A. Wright, The boundary of an affine invariant submanifold, Invent. Math. 209 (2017), 927–984.
  • [24] M. Möller, Periodic points on Veech surfaces and the Mordell–Weil group over a Teichmüller curve, Invent. Math. 165 (2006), 633–649.
  • [25] M. Möller, Finiteness results for Teichmüller curves, Ann. Inst. Fourier (Grenoble) 58 (2008), 63–83.
  • [26] R. E. Mukamel, Orbifold points on Teichmüller curves and Jacobians with complex multiplication, Geom. Topol. 18 (2014), 779–829.
  • [27] D.-M. Nguyen and A. Wright, Non-Veech surfaces in $\mathcal{H}^{\mathrm{hyp}}(4)$ are generic, Geom. Funct. Anal. 24 (2014), 1316–1335.
  • [28] J. Smillie and B. Weiss, Minimal sets for flows on moduli space, Israel J. Math. 142 (2004), 249–260.
  • [29] A. Wright, Cylinder deformations in orbit closures of translation surfaces, Geom. Topol. 19 (2015), 413–438.
  • [30] A. Wright, Translation surfaces and their orbit closures: An introduction for a broad audience, EMS Surv. Math. Sci. 2 (2015), 63–108.
  • [31] A. Zorich, “Flat surfaces” in Frontiers in Number Theory, Physics, and Geometry, I, Springer, Berlin, 2006, 437–583.