On the monodromy and spin parity of single-cylinder origamis in the minimal stratum

TL;DR

This work analyzes minimal $[1,1]$ origamis in the minimal stratum $\,\mathcal{H}(2g-2)\,$, combining AMN’s factorially many constructions with new generalisations to control spin parity. It shows that odd-genus origamis all lie in the odd spin component, while even-genus origamis split between odd and even components with a 3:1 asymptotic distribution, and it provides generalized variants yielding minimal origamis in the even component as well. The monodromy of primitive minimal $[1,1]$ origamis is studied systematically, proving that these monodromy groups are typically simple and often isomorphic to Alt$_{2g-1}$, PSL-type groups, or sporadic cases, with a near-complete dichotomy driven by two $n$-cycles whose commutator is an $n$-cycle. In addition, the paper identifies orientation-double-cover phenomena among the odd-genus AMN family, analyzes SL$(2,\mathbb{Z})$-orbit structure and conjectures precise orbit counts, and proves a key “spin-invariance” lemma that underpins the parity calculations. The results advance understanding of the arithmetic and geometric structure of minimal stratum origamis, their $SL(2,\mathbb{Z})$-orbits, and the possible monodromy groups arising from primitive square-tiled surfaces.

Abstract

In a paper with Menasco-Nieland, the first author constructed factorially many origamis in the minimal stratum of the moduli space of translation surfaces having simultaneously a single vertical cylinder and a single horizontal cylinder. Moreover, these origamis were constructed using the minimal number of squares required for origamis in the minimal stratum. We shall call such origamis minimal $[1,1]$-origamis. In this work, by calculating their spin parities, we determine that the odd genus origamis in this construction all lie in the odd component of the minimal stratum, while the even genus origamis are contained in both the odd and even components with an asymptotic ratio of 3:1. Noticing that the even component is missed in the odd genus case, we provide a generalisation of the odd genus construction that gives rise to factorially many minimal $[1,1]$-origamis lying in the even component. Motivated by understanding the $SL(2,\mathbb{Z})$-orbits of these origamis, we investigate their monodromy groups (a weak $SL(2,\mathbb{Z})$-invariant). We also prove that, with one exception, the monodromy group of any primitive minimal $[1,1]$-origami in the minimal stratum must be simple.

Figures (16)

• Figure 2.1: Two origamis in $\mathcal{H}(4)$. The side labels indicate the identifications by translation. The smaller numbers are square numberings.
• Figure 3.1: A realisation of the origami $O$ from Example \ref{['ex:ori:AMNodd']}.
• Figure 3.2: The surgery-like construction used to build the even genus AMN origami from Example \ref{['ex:ori:AMNeven']}.
• Figure 5.1: The generating set for $H_{1}(O,\mathbb{Z}/2\mathbb{Z})$ inside the origami $O$.
• Figure 5.2: Realisations of the origamis $O$ and $O'$ in $\mathcal{H}(10)$.

Theorems & Definitions (73)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5
• Theorem 1.6
• Conjecture 1.7
• Theorem 1.8
• Remark 2.1: Order of multiplication
• Example 3.1
• Example 3.2