Abstract
Let \(N=2^\alpha 3^\beta\). The normalizer \(\Gamma _B(N)\) of \(\Gamma _0(N)\) in \(PSL(2,\mathbb {R})\) is the triangle group \((2,4,\infty )\) for \(\alpha =1,3,5,7\); \(\beta =0,2\) and the triangle group \((2,6,\infty )\) for \(\alpha =0,2,4,6\); \(\beta =1,3\). In this paper we examine relationship between the normalizer and the regular maps. We define a family of subgroups of the normalizer and then we study maps with quadrilateral and hexagonal faces using these subgroups and calculating the associated arithmetic structure.
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References
Akbaş, M., Singerman, D.: The normalizer of \(\Gamma _0(N)\) in \(PSL(2,\mathbb{R})\). Glas. 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)
Bars, F.: The group structure of the normalizer of \(\Gamma _0(N)\). Commun. Algebra 36, 2160–2170 (2008)
Chua, K.S., Lang, M.L.: Congruence subgroups associated to the monster. Exp. Math. 13(3), 343–360 (2004)
Conway, J.H., Norton, S.P.: Monstrous moonshine. Bull. Lond. Math. Soc. 11(3), 308–339 (1979)
Ivrissimtzis, I.P., Singerman, D.: Regular maps and principal congruence subgroups of Hecke groups. Eur. J. Comb. 26, 437–456 (2005)
Jones, G.A., Singerman, D.: Theory of maps on orientable surfaces. Proc. Lond. Math. Soc. 37(3), 273–307 (1978)
Jones, G.A., Singerman, D.: Complex Functions: An Algebraic and Geometric Viewpoint. Cambridge Univ. Press, Cambridge (1987)
Kattan, D., Singerman, D.: The diameter of some heckefarey maps. Albanian J. Math. 15(1), 39–60 (2019)
Kattan, D., Singerman, D.: Universal q-gonal tessellations and their petrie paths, AMS Contemp. Math. Ser., Automorphsms of Riemann surfaces, subgroups of mapping class groups and related topics (2021)
Keskin, R.: Suborbital graphs for the normalizer \(\Gamma _0(m)\). Eur. J. Comb. 27(2), 193–206 (2006)
Keskin, R., Demirtürk, B.: On suborbital graphs for the normalizer \(\Gamma _0(N)\). Electron. J. Comb. 16, 1–10 (2009)
Macbeath, A.: On a theorem of Hurwitz. Proc. Glasg. Math. Assoc. 5(2), 90–96 (1961)
Machlaclan, C.: Groups of units of zero ternary quadratic forms. Proc. R. Soc. Edinb. 88(A), 141–157 (1981)
Singerman, D.: Universal tesselations. Rev. Mat. Univ. Complut. Madrid 1, 111–123 (1988)
Yazıcı Gözütok, N., Gözütok, U., Güler, B.Ö.: Maps corresponding to the subgroups \(\Gamma _0(N)\) of the modular group. Graphs Comb. 35(1695–1705), 408–425 (2019)
Yazıcı Gözütok, N., Güler, B.Ö.: Quadrilateral cell graphs of the normalizer with signature \((2,4,\infty )\). Stud. Sci. Math. Hung. 57(3), 408–425 (2020)
Acknowledgements
The authors would like to thank Prof. Mehmet Akbaş and Prof. David Singerman for their valuable suggestions that improved the earlier versions of the paper. Also, the authors would like to express their sincere gratitude to Prof. Jack Koolen for his support during the review process.
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Yazıcı Gözütok, N., Güler, B.Ö. Quadrilateral and Hexagonal Maps Corresponding to the Subgroups \(\Gamma _0(N)\) of the Modular Group. Graphs and Combinatorics 38, 99 (2022). https://doi.org/10.1007/s00373-022-02503-0
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DOI: https://doi.org/10.1007/s00373-022-02503-0