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Diameter bounds for $SL(2,\mathbb{Z})$-orbits of origamis in $\mathcal{H}(2)$ and the Prym loci in $\mathcal{H}(4)$ and $\mathcal{H}(6)$

Luke Jeffreys, Carlos Matheus

TL;DR

This work provides explicit diameter bounds for the SL(2,\mathbb{Z})-orbit graphs of primitive origamis in the minimal stratum \mathcal{H}(2) and extends the methodology to Prym loci in \mathcal{H}(4) and \mathcal{H}(6). Leveraging Hubert--Lelièvre's height-reduction framework and McMullen's butterfly-move theory, the authors derive a unified bound of \mathcal{O}(n^{2/3}\log n) in terms of orbit size, and further refine the analysis for Prym loci by translating the McMullen construction to Lanneau--Nguyen's Prym prototypes. The paper provides detailed case analyses (two- to one-cylinder reductions, a,b,c-saddle decompositions, and discriminant-dependent prototype connectivity) to establish the asymptotic diameter bounds \mathcal{O}(n^{2}\log n) in several settings, while also discussing potential improvements that could drive bounds toward optimal values under conjectural hypotheses. Overall, the results connect arithmetic invariants, degeneration techniques, and explicit combinatorial moves to quantify orbit-graph diameters across genus-two and Prym-extended strata with concrete implications for the geometry of arithmetic Teichmüller curves.

Abstract

Using algorithms implicit in the classification of $SL(2,\mathbb{Z})$-orbits of primitive origamis in the stratum $\mathcal{H}(2)$ due to Hubert-Lelièvre and McMullen, we give diameter bounds on the resulting orbit graphs. Since the machinery of McMullen from $\mathcal{H}(2)$ is generalised and reused in Lanneau and Nguyen's classification of the orbits of Prym eigenforms in $\mathcal{H}(4)$ and $\mathcal{H}(6),$ we are also able to obtain diameter bounds for the orbit graphs in this setting as well. In each stratum, we obtain diameter bounds of the form $O(N^{2/3}\log N)$, where $N$ is the size of the orbit graph.

Diameter bounds for $SL(2,\mathbb{Z})$-orbits of origamis in $\mathcal{H}(2)$ and the Prym loci in $\mathcal{H}(4)$ and $\mathcal{H}(6)$

TL;DR

This work provides explicit diameter bounds for the SL(2,\mathbb{Z})-orbit graphs of primitive origamis in the minimal stratum \mathcal{H}(2) and extends the methodology to Prym loci in \mathcal{H}(4) and \mathcal{H}(6). Leveraging Hubert--Lelièvre's height-reduction framework and McMullen's butterfly-move theory, the authors derive a unified bound of \mathcal{O}(n^{2/3}\log n) in terms of orbit size, and further refine the analysis for Prym loci by translating the McMullen construction to Lanneau--Nguyen's Prym prototypes. The paper provides detailed case analyses (two- to one-cylinder reductions, a,b,c-saddle decompositions, and discriminant-dependent prototype connectivity) to establish the asymptotic diameter bounds \mathcal{O}(n^{2}\log n) in several settings, while also discussing potential improvements that could drive bounds toward optimal values under conjectural hypotheses. Overall, the results connect arithmetic invariants, degeneration techniques, and explicit combinatorial moves to quantify orbit-graph diameters across genus-two and Prym-extended strata with concrete implications for the geometry of arithmetic Teichmüller curves.

Abstract

Using algorithms implicit in the classification of -orbits of primitive origamis in the stratum due to Hubert-Lelièvre and McMullen, we give diameter bounds on the resulting orbit graphs. Since the machinery of McMullen from is generalised and reused in Lanneau and Nguyen's classification of the orbits of Prym eigenforms in and we are also able to obtain diameter bounds for the orbit graphs in this setting as well. In each stratum, we obtain diameter bounds of the form , where is the size of the orbit graph.

Paper Structure

This paper contains 45 sections, 14 theorems, 83 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Translation surfaces and their moduli spaces
  4. Origamis and their $\boldsymbol{\mathop{\mathrm{\text{SL}}}\nolimits(2,\mathbb{Z})}$-orbits
  5. Surface parameters
  6. Prototypes and butterfly moves
  7. Applying butterfly moves to square-tiled surfaces
  8. Bounds via the algorithm of Hubert-Lelièvre
  9. From two cylinders cusps to one cylinder cusps
  10. Reduction of Case $(I)$
  11. Reduction of Case $(II)$
  12. Reduction of Case $(III)$
  13. Reduction of case $(IV)$
  14. Two-cylinder counts:
  15. Connecting one cylinder cusps among themselves
  16. ...and 30 more sections

Key Result

Theorem 1.1

If $X$ is a primitive origami in the stratum $\mathcal{H}(2)$ of the moduli space of translation surfaces of genus two with a single conical singularity, then the graph $\mathcal{G}(X)$ associated to its $\mathop{\mathrm{\text{SL}}}\nolimits(2,\mathbb{Z})$-orbit has a diameter $O(|V|^{2/3}\log |V|)

Figures (22)

  • Figure 2.1: Two-cylinder surface parameters in $\mathcal{H}(2)$.
  • Figure 2.2: A one-cylinder cusp representative in $\mathcal{H}(2)$.
  • Figure 2.3: The surface associated to the prototypical splitting $(a,b,c,e)$. The splitting realises the surface $M$ as $E_{1}\underset{I}{\#}E_{2}$. A butterfly move changes the splitting to $F_{1}\underset{J}{\#}F_{2}$ with $J = [(b,0) + q(a,c)]$.
  • Figure 2.4: The origami $O$ and its cusp representative. The direction corresponding to the butterfly move $B_{2}$ is also shown, with the resulting torus $F_{1}$ shaded in blue.
  • Figure 2.5: Realising the prototype corresponding to $O$. The direction corresponding to the butterfly move $B_{2}$ is also shown.
  • ...and 17 more figures

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.2
  • Remark 2.1
  • Theorem 4.1: McM
  • Lemma 4.2
  • proof
  • Theorem 5.1: LN14
  • Proposition 5.2: LN14
  • Theorem 5.3: Follows from LN14
  • Theorem 5.4: LN14
  • ...and 9 more