The Galois Action and a Spin Invariant for Prym-Teichmüller Curves in Genus 3

Abstract

Given a Prym-Teichmüller 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 Teichmüller curves, extending the results of Bouw and Möller in genus $2$. We use this to show that the components of the genus $3$ Prym-Teichmüller curves are homeomorphic.

arXiv comment10 pages, 1 figure; v2: Minor improvements