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Euler characteristics of $SL(2,\mathbb{Z})$-orbit graphs of origamis

Luke Jeffreys, Carlos Matheus

Abstract

The $SL(2,\mathbb{Z})$-orbits of primitive $n$-squared origamis can be represented by finite four-regular graphs. It is a conjecture of McMullen that the orbit graphs of such origamis in the stratum $\mathcal{H}(2)$ form an expander family. We provide indirect evidence for this conjecture by proving that the absolute values of the Euler characteristics of the graphs in this family go to infinity with the number of squares $n$. This generalises previous work of the authors, in which we established eventual non-planarity for this family, and provides the strongest indirect evidence to date for McMullen's conjecture. We also prove that the same phenomenon holds for primitive origamis in the Prym loci of $\mathcal{H}(4)$ and $\mathcal{H}(6)$. Assuming conjectures of Zmiaikou and Delecroix--Lelièvre, we establish the same in $\mathcal{H}(1,1)$ and for two families of non-Prym origamis in $\mathcal{H}(4)$. Finally, assuming a stronger conjecture concerning orbit growth in low-complexity strata, we establish that for any family of $SL(2,\mathbb{Z})$-orbit graphs of primitive origamis in a stratum with one or two conical singularities, the absolute values of the Euler characteristics of the graphs go to infinity with the number of squares along a density one subsequence. By relating the genus of the elliptic generator orbit graphs to the genus of the associated arithmetic Teichmüller curves, we also recover results of Mukamel and extend results of Torres-Teigel--Zachhuber establishing that the genus of these Teichmüller curves in $\mathcal{H}(2)$ and the Prym loci of $\mathcal{H}(4)$ and $\mathcal{H}(6)$ also go to infinity. The proofs rely on counts of integral points on algebraic hypersurfaces using methods of Bombieri--Pila and Browning--Gorodnik, on counts of orbifold points on Teichmüller curves, and on counts of pseudo-Anosov diffeomorphisms with bounded dilatation.

Euler characteristics of $SL(2,\mathbb{Z})$-orbit graphs of origamis

Abstract

The -orbits of primitive -squared origamis can be represented by finite four-regular graphs. It is a conjecture of McMullen that the orbit graphs of such origamis in the stratum form an expander family. We provide indirect evidence for this conjecture by proving that the absolute values of the Euler characteristics of the graphs in this family go to infinity with the number of squares . This generalises previous work of the authors, in which we established eventual non-planarity for this family, and provides the strongest indirect evidence to date for McMullen's conjecture. We also prove that the same phenomenon holds for primitive origamis in the Prym loci of and . Assuming conjectures of Zmiaikou and Delecroix--Lelièvre, we establish the same in and for two families of non-Prym origamis in . Finally, assuming a stronger conjecture concerning orbit growth in low-complexity strata, we establish that for any family of -orbit graphs of primitive origamis in a stratum with one or two conical singularities, the absolute values of the Euler characteristics of the graphs go to infinity with the number of squares along a density one subsequence. By relating the genus of the elliptic generator orbit graphs to the genus of the associated arithmetic Teichmüller curves, we also recover results of Mukamel and extend results of Torres-Teigel--Zachhuber establishing that the genus of these Teichmüller curves in and the Prym loci of and also go to infinity. The proofs rely on counts of integral points on algebraic hypersurfaces using methods of Bombieri--Pila and Browning--Gorodnik, on counts of orbifold points on Teichmüller curves, and on counts of pseudo-Anosov diffeomorphisms with bounded dilatation.
Paper Structure (43 sections, 46 theorems, 121 equations, 13 figures, 1 table)

This paper contains 43 sections, 46 theorems, 121 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Conditional extensions to $\mathcal{H}(1,1)$ and $\mathcal{H}(4)$
  3. A conditional extension to low-complexity strata
  4. Applications to arithmetic Teichmüller curve genera
  5. Proof strategy
  6. Outline of the paper
  7. Acknowledgements
  8. Preliminaries
  9. Translation surfaces and their moduli spaces
  10. Origamis and their ${\mathop{\mathrm{\rm{SL}}}\nolimits(2,\mathbb{Z})}$-orbits
  11. Surface parameters and cusps
  12. Block systems for the action of $\mathop{\mathrm{\rm{SL}}}\nolimits(2,\mathbb{Z})$
  13. Block systems for orbits of origamis admitting anti-involutions
  14. The case ${l_{1}\neq l_{2}\neq l_{3}\neq l_{1}}$
  15. The case ${l_{1}\neq l_{2} = l_{3}}$
  16. ...and 28 more sections

Key Result

Theorem 1.2

Let $(\mathcal{G}_{n})_{n}$ be a family of $\mathop{\mathrm{\rm{SL}}}\nolimits(2,\mathbb{Z})$-orbit graphs of primitive $n$-squared origamis in $\mathcal{H}(2)$ or in the Prym loci of $\mathcal{H}(4)$ or $\mathcal{H}(6)$ with either the parabolic or elliptic generators. Then $|\chi(\mathcal{G}_{n})|

Figures (13)

  • Figure 2.1: The graph $\mathcal{G}_{3}$ with the paraboic generators is the undirected version of this graph.
  • Figure 2.2: Two-cylinder surface parameters in $\mathcal{H}(2)$.
  • Figure 3.1: Block system for $\{B_{i}\}_{i = 1}^{6}$.
  • Figure 3.2: Block system for $\{B_{1}\cup B_{6}, B_{2}\cup B_{5}, B_{3}\cup B_{4}\}$.
  • Figure 3.3: Block system for $\{B_{1}\cup B_{3}\cup B_{5},B_{2}\cup B_{4}\cup B_{6}\}$.
  • ...and 8 more figures

Theorems & Definitions (79)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof : Proof of the first part of Theorem \ref{['thm:main']} given Theorem \ref{['thm:submain-1']}
  • Conjecture 1.4: Zm
  • Theorem 1.5
  • Conjecture 1.6: Parity Conjecture
  • Theorem 1.7
  • Conjecture 1.8: MMY
  • Theorem 1.9
  • ...and 69 more