Abstract
The normalizer \(\varGamma _B(N)\) of \(\varGamma _0(N)\) in \(PSL(2,{\mathbb {R}})\) is the triangle group \((2,3,\infty )\) for \(N=2^\alpha 3^\beta \) where \(\alpha =0,2,4,6\); \(\beta =0,2\). In this paper we examine relationship between the normalizer and the regular maps for these values. We define a family of subgroups of the normalizer and then we study maps with triangular faces using these subgroups and calculating the associated arithmetic structure.
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References
Akbas, M., Singerman, D.: The normalizer of \(\varGamma _0(N)\) in \(PSL(2,\mathbb{R})\). Glasgow Math. J. 32(3), 317–327 (1990)
Akbaş, M., Singerman, D.: The signature of the normalizer of \(\Gamma _0(N)\). Lond. Math. Soc. Lect. Note Ser. 165, 77–86 (1992)
Conway, J., Norton, S.: Monstrous moonshine. Bull. Lond. Math. Soc. 11(3), 308–339 (1979). https://doi.org/10.1112/blms/11.3.308
Ivrissimtzis, I., Singerman, D.: Regular maps and principal congruence subgroups of Hecke groups. Eur. J. Comb. 26, 437–456 (2005). https://doi.org/10.1016/j.ejc.2004.01.010
Jones, G., Singerman, D.: Theory of maps on orientable surfaces. Proc. Lond. Math. Soc. 37(3), 273–307 (1978). https://doi.org/10.1112/plms/s3-37.2.273
Jones, G., Singerman, D.: Complex Functions: An algebraic and geometric viewpoint. Cambridge University Press, Cambridge (1987)
Macbeath, A.: On a theorem of Hurwitz. Proc. Glasg. Math. Assoc. 5(2), 90–96 (1961). https://doi.org/10.1017/S2040618500034365
Singerman, D.: Universal tessellations. Rev. Mat. Univ. Comput. Madrid 1, 111–123 (1988). https://doi.org/10.5209/rev_REMA.1988.v1.n1.18188
Singerman, D., Strudwick, J.: Petrie polygons, Fibonacci sequences and Farey maps. Ars Math. Contemp. 10(2), 349–357 (2016)
Acknowledgements
The authors would like to thank the anonymous reviewers for their helpful and constructive comments that greatly contributed to improving the final version of the paper. They would also like to thank the Editors for their generous comments and support during the review process.Finally, the first and second authors would like to thank TUBITAK (The Scientific and Technological Research Council of Turkey) for their financial supports during their doctorate studies.
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Yazıcı Gözütok, N., Gözütok, U. & Güler, B.Ö. Maps Corresponding to the Subgroups \(\varGamma _0(N)\) of the Modular Group. Graphs and Combinatorics 35, 1695–1705 (2019). https://doi.org/10.1007/s00373-019-02080-9
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DOI: https://doi.org/10.1007/s00373-019-02080-9