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Arithmeticity of the Kontsevich--Zorich monodromies of certain families of square-tiled surfaces II

Manuel Kany, Carlos Matheus

Abstract

In this note, we extend the scope of our previous work joint with Bonnafoux, Kattler, Niño, Sedano-Mendoza, Valdez and Weitze-Schmithüsen by showing the arithmeticity of the Kontsevich--Zorich monodromies of infinite families of square-tiled surfaces of genera four, five and six.

Arithmeticity of the Kontsevich--Zorich monodromies of certain families of square-tiled surfaces II

Abstract

In this note, we extend the scope of our previous work joint with Bonnafoux, Kattler, Niño, Sedano-Mendoza, Valdez and Weitze-Schmithüsen by showing the arithmeticity of the Kontsevich--Zorich monodromies of infinite families of square-tiled surfaces of genera four, five and six.
Paper Structure (25 sections, 13 theorems, 92 equations, 8 figures)

This paper contains 25 sections, 13 theorems, 92 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Square-tiled surfaces and their Kontsevich--Zorich monodromies
  4. Galois groups as permutation group
  5. Galois groups of polynomials of degree four and five
  6. Galois group of certain reciprocal polynomials
  7. Description of the Galois group
  8. Action of the hyperoctahedral group on the splitting field
  9. Real roots of cubic, quartic and quintic polynomials
  10. Real roots for cubic polynomials
  11. Real roots for quartic polynomials
  12. Real roots for quintic polynomials
  13. Genus four stairs
  14. Dehn twists in genus four
  15. Zariski density and arithmeticity for a genus four family
  16. ...and 10 more sections

Key Result

Theorem 1.0.1

Let $\mathcal{H}(2g-2)$ be the minimal stratum of the moduli space of translation surfaces of genus $g$. For each $g\in\{4,5,6\}$, there are infinitely many square-tiled surfaces in $\mathcal{H}(2g-2)$ whose Kontsevich--Zorich monodromies are arithmetic.

Figures (8)

  • Figure 1: The origami $\mathcal{O}_{M,M}^{(4)}$.
  • Figure 2: Origami $\mathcal{O}_{N,M}^{(5)}$ with horizontal waist curves $\sigma_1,\sigma_2,\sigma_3,\sigma_4,\sigma_N$ and vertical waist curves $\zeta_1,\zeta_2,\zeta_3,\zeta_4,\zeta_M$.
  • Figure 3: Cylinder decomposition in direction $(1,2)$ and direction $(1,-2)$ of the origami $\mathcal{O}_{N,M}^{(5)}$. Here $\gamma_1$ is the waist curve of the blue cylinder in direction $(1,2)$ and $\alpha_1$ is the waist curve of the blue cylinder in direction $(1,-2)$.
  • Figure 4: Origami $\mathcal{O}^{(5)}_{N,M}$ with cylinder decomposition in direction $(1,4)$. Here $\chi_1$ is the waist curve of the blue cylinder.
  • Figure 5: Origami $\mathcal{O}^{(5)}_{N,M}$ with cylinder decomposition in vertical and horizontal direction.
  • ...and 3 more figures

Theorems & Definitions (26)

  • Theorem 1.0.1
  • Remark 1.0.2
  • Remark 1.0.3
  • Theorem 2.2.1: Dedekind
  • Theorem 2.3.1
  • Remark 2.3.2
  • Theorem 2.3.3
  • Remark 2.3.4
  • Lemma 2.4.1
  • proof
  • ...and 16 more