An infinite family of non-cyclic 1-cylinder pillowcase-tiled surfaces

TL;DR

The paper addresses the Apisa-Wright conjecture that all 1-cylinder pillowcase-tiled surfaces are cyclic covers. It develops a purely algebraic 1-cylinder criterion via monodromy triples in $\mathrm{Sym}(n)^3/\mathrm{Sym}(n)$ and leverages the braid-group action, equivalent to a $\text{PSL}(2,\mathbb{Z})$ action, to distinguish cylinder structures. The authors construct an infinite family of counterexamples for odd $n\ge 5$ with non-cyclic 1-cylinder PTS and prove nonexistence for even $n$, using explicit $n$-cycle triples with $abc=1$ and a parity argument; they also identify the monodromy group as $A_n$ in the examples. These results advance the understanding of cylinder-free affine-invariant subvarieties and offer new tools for exploring Teichmüller dynamics and Shimura-Teichmüller curves in this setting.

Abstract

We provide infinitely many counterexamples to a conjecture of Apisa-Wright.

Figures (2)

• Figure 1: Generators of the fundamental group of the pillowcase.
• Figure 2: The actions of $T$ on the pillowcase.

Theorems & Definitions (11)

• Theorem 1: Main result
• Theorem 2
• Lemma 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• proof : Proof of Thm \ref{['onecyl']}