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Echoes in genus three of Teichmüller curves in genus two

Thomas Le Fils

Abstract

We classify the Teichmüller curves in the moduli space of genus three Riemann surfaces $\mathcal M_3$ that are obtained by a covering construction from a primitive Teichmüller curve in $\mathcal M_2$. We describe the action on homology modulo two of the affine groups of translation surfaces generating these primitive curves. We also classify the $\mathrm{SL}_2(\mathbb Z)$-orbits of square-tiled surfaces in $\mathcal H(2, 2)$ that are a cover of a genus two one.

Echoes in genus three of Teichmüller curves in genus two

Abstract

We classify the Teichmüller curves in the moduli space of genus three Riemann surfaces that are obtained by a covering construction from a primitive Teichmüller curve in . We describe the action on homology modulo two of the affine groups of translation surfaces generating these primitive curves. We also classify the -orbits of square-tiled surfaces in that are a cover of a genus two one.
Paper Structure (60 sections, 46 theorems, 58 equations, 6 figures, 3 tables)

This paper contains 60 sections, 46 theorems, 58 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Teichmüller curves
  3. The flat viewpoint
  4. Square-tiled surfaces
  5. Monodromy representation
  6. Pure echoes
  7. Organisation of the article
  8. Acknowledgements
  9. Abelian differentials
  10. Teichmüller spaces
  11. The classical Teichmüller space.
  12. The Teichmüller space of abelian differentials
  13. Linear charts
  14. Mapping class group and moduli space
  15. Strata of abelian differentials
  16. ...and 45 more sections

Key Result

Theorem 1.1

Let $V\subset \mathcal{M}_2$ be a Weierstrass curve of discriminant $D$. If $D\equiv 5\mod 8$ then $V$ has three echoes in $\mathcal{M}_3$, and five otherwise.

Figures (6)

  • Figure 1: Translation surfaces, in $\widetilde{\mathcal{H}}(2)^\mathrm{odd}$ on the left, and in $\widetilde{\mathcal{H}}(2)^\mathrm{hyp}$ on the right.
  • Figure 2: Translation surface $L(b, e)$.
  • Figure 3: Translation surface $L(b, e)$ rotated.
  • Figure 4: Cylinder decomposition with slope $s=2/b$.
  • Figure 5: Cylinder decomposition of $L(6, 1)$ with slope $s=1/2$.
  • ...and 1 more figures

Theorems & Definitions (86)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Theorem 2.2: McMullen MCM06
  • ...and 76 more