Teichmueller curves, Galois actions and GT-relations

Teichmueller curves are geodesic discs in Teichmueller space that project to algebraic curves $C$ in the moduli space $M_g$. Some Teichm"uller curves can be considered as components of Hurwitz spaces. We show that the absolute Galois group $G_{\text{QQ}}$ acts faithfully on the set of these embedded curves. We also compare the action of $G_{\text{QQ}}$ on ${\pi }_1(C)$ with the one on ${\pi }_1(M_g)$ and obtain a relation in the Grothendieck-Teichmueller group, seemingly independent of the known ones.