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Conformal Geometry and Dynamics

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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
DOI: https://doi.org/10.1090/ecgd/341
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
Email: khadija.farooq@nu.edu.pk

DOI: https://doi.org/10.1090/ecgd/341
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