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Conformal Geometry and Dynamics

ISSN 1088-4173



Structural properties of quotient surfaces of a Hecke group

Author: K. Farooq
Journal: Conform. Geom. Dyn. 23 (2019), 262-282
MSC (2010): Primary 20H10, 30B70, 57M50; Secondary 11K60
Published electronically: December 3, 2019
MathSciNet review: 4038021
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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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Additional Information

K. Farooq
Affiliation: WMI, Univesity of Warwick, Coventry, CV4 7AL, United Kingdom
Address at time of publication: Department of Sciences and Humanities, National University of Emerging Sciences, FAST, A.K. Brohi Road, H-11/4, Islamabad, Pakistan

Keywords: Quotient surfaces, Hecke groups, symbolic sequences
Received by editor(s): September 4, 2014
Received by editor(s) in revised form: July 26, 2017, and July 13, 2019
Published electronically: December 3, 2019
Additional Notes: The author was supported by WPRS grant from the University of Warwick, and HEC Partial Support by the Government of Pakistan
Article copyright: © Copyright 2019 American Mathematical Society