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

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.