Transcendence of simple geodesics on finite modular covers

Abstract

The real projective line $\mathbb{R}{\mathbf{P}}^1$ is the boundary of $\mathbf{HP}=\{z\in \mathbb{C}:\Im (z)>0\}$, 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 $\Gamma ={\mathrm{PSL}}_2(\mathbb{Z})$ acts on $\mathbf{HP}$ with quotient the modular orbifold $\mathbf{M}=\Gamma \backslash \mathbf{HP}$. Consider a finite-index subgroup of the modular group ${\Gamma }^{\prime }\subset \Gamma ={\mathrm{PSL}}_2(\mathbb{Z})$ corresponding to a finite cover $\mathbf{M}\to {\mathbf{M}}^{\prime }$. A geodesic $({\xi }^{-},{\xi }^{+})\in \mathcal{G}(\mathbf{HP})$ projects $\mathrm{mod}{\Gamma }^{\prime }$ 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 ${\Gamma }^{\prime }=[\Gamma ,\Gamma ]$.

27 pages, 7 figures