Uniqueness of the Veech 14-gon

TL;DR

This paper proves that the Veech 14-gon uniquely generates the only algebraically primitive Teichmüller curve in a specific genus 3 hyperelliptic stratum, providing a complete classification in this context.

Contribution

It provides the first complete classification of algebraically primitive Teichmüller curves in a genus greater than 2 stratum, identifying the Veech 14-gon as unique in this setting.

Findings

Veech 14-gon generates the unique algebraically primitive Teichmüller curve in the hyperelliptic component of AA M_3(2,2)

Complete classification result for algebraically primitive Teichmüller curves in this stratum

First such classification in genus > 2

Abstract

We obtain the first complete classification result for algebraically primitive Teichm\"{u}ller curves in a stratum component in genus greater than 2. Specifically, we show that the Veech 14-gon generates the unique algebraically primitive Teichm\"{u}ller curve in the hyperelliptic component of the stratum $\Omega\mathcal{M}_3(2,2)$.