The Galois Action and a Spin Invariant for Prym-Teichm\"uller Curves in Genus 3

TL;DR

This paper introduces a new invariant for Prym-Teichmüller curves in genus 3 that classifies cusp prototypes and explores the Galois action on these curves, extending known results from genus 2.

Contribution

It defines a novel invariant for genus 3 Prym-Teichmüller curves and analyzes the Galois action on their cusps, showing the homeomorphism of their components.

Findings

The invariant sorts cusp prototypes by component.

The Galois action on cusps is characterized.

Components of genus 3 Prym-Teichmüller curves are homeomorphic.

Abstract

Given a Prym-Teichm\"uller curve in $\mathcal{M}_3$, this note provides an invariant that sorts the cusp prototypes of Lanneau and Nguyen by component. This can be seen as an analogue of McMullen's genus $2$ spin invariant, although the source of this invariant is different. Moreover, we describe the Galois action on the cusps of these Teichm\"uller curves, extending the results of Bouw and M\"oller in genus $2$. We use this to show that the components of the genus $3$ Prym-Teichm\"uller curves are homeomorphic.