Transcendence of simple geodesics on finite modular covers

Abstract

The real projective line $\mathbb{R}\mathbf{P}^1$ is the boundary of $\mathbf{HP}=\{z\in \mathbb{C}\colon \Im(z)>0\}$, a model of the hyperbolic plane whose space of geodesics identifies with $\mathcal{G}(\mathbf{HP})=\mathbb{R}\mathbf{P}^1 \times \mathbb{R}\mathbf{P}^1 \setminus \mathrm{diagonal}$. The modular group $\Gamma=\operatorname{PSL}_2(\mathbb{Z})$ acts on $\mathbf{HP}$ with quotient the modular orbifold $\mathbf{M}=\Gamma\backslash \mathbf{HP}$. Consider a finite-index subgroup of the modular group $\Gamma^\prime \subset \Gamma = \operatorname{PSL}_2(\mathbb{Z})$ corresponding to a finite cover $\mathbf{M} \to \mathbf{M}^\prime$. A geodesic $(\xi^-,\xi^+)\in \mathcal{G}(\mathbf{HP})$ projects $\bmod{\Gamma^\prime}$ to a geodesic $\xi^\prime \subset \mathbf{M}^\prime$. We show that if $\xi^\prime$ is simple, then $\xi^+$ is either rational or quadratic or transcendental. In the transcendental case, we obtain bounds on the Mahler measures and show that those can be improved for geodesics fixed by pseud-Anosov maps. Finally, we also explain in detail why all this was known for the modular torus cover associated to the derived subgroup $\Gamma^\prime = [\Gamma, \Gamma]$.