Structural properties of quotient surfaces of a Hecke group

Abstract:

We study the properties of the surface $\Sigma _q$, which is a $2q$-fold cover of $\mathbb H/G_q$, where $G_q$ is a Hecke group and $q$ is an integer greater than $3$. We have slightly different situations for the even and odd values of $q$. For odd values of $q$ the surface $\Sigma _q$ is a $\frac {q-1}{2}$ genus surface with a cusp, whereas, for even values it is a $\frac {q-2}{2}$ genus surface with two cusps. We prove that there exist $g$ embedded tori with a hole on $\Sigma _q$, where $g=\frac {q-1}{2}$ when $q$ is an odd integer and $g=\frac {q-2}{2}$ when $q$ is even, with $g$ boundary geodesics at different heights. These boundary geodesics are the separating geodesics intersecting each other transversally. We also prove that the surface $\Sigma _q$ is a hyper-elliptic surface for every integer $q>3$.