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 paper establishes diameter bounds for $SL(2, Z)$-orbit graphs of origamis in specific strata, extending previous classifications and providing bounds of order $O(N^{2/3}\,log N)$.

Contribution

It generalizes existing diameter bounds from $ ext{H}(2)$ to Prym loci in $ ext{H}(4)$ and $ ext{H}(6)$, using and extending McMullen's machinery.

Findings

Diameter bounds of $O(N^{2/3}\,log N)$ for orbit graphs.

Extension of bounds from $ ext{H}(2)$ to Prym loci in $ ext{H}(4)$ and $ ext{H}(6)$.

Unified approach using algorithms from orbit classifications.

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-Leli\`evre 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.