# Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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## Structural properties of quotient surfaces of a Hecke groupHTML articles powered by AMS MathViewer

by K. Farooq
Conform. Geom. Dyn. 23 (2019), 262-282 Request permission

## 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 $q>3$.
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