Hexagonal cell graphs of the normalizer with signature $(2, 6, \infty)$

Year 2022, , 666 - 679, 01.06.2022

Nazlı Yazıcı Gözütok , Bahadır Özgür Güler

https://doi.org/10.15672/hujms.824436

Abstract

In this paper, we investigate suborbital graphs $G_{u,n}$ of the normalizer $\Gamma_B(N)$ of $\Gamma_0(N)$ in $PSL(2,\mathbb{R})$ for $N= 2^\alpha 3^\beta$, where $\alpha=0,2,4,6$ and $\beta =1,3$. In each of these cases, the normalizer becomes a triangle group and the graph arising from the action of the normalizer contains hexagonal circuits. In order to obtain graphs, we first define an imprimitive action of $\Gamma _B(N)$ on $\widehat{\mathbb{Q}}$ using the group $H_B(N)$ and then we obtain some properties of the graphs arising from this action.

Keywords

normalizer, suborbital graph, hexagon

References

• [1] M. Akbaş and D. Singerman, The signature of the normalizer of $\Gamma _{0}(N)$ in $PSL(2,R)$,London Math. Soc. 165, 77–86, 1992.
• [2] N.L. Biggs and A.T. White, Permutation groups and combinatorial structures, London Math. Soc. Lec. Not. Ser., 33rd ed. CUP, Cambridge, 1979.
• [3] I.N. Cangül and D. Singerman, Normal subgroups of Hecke groups and regular maps, Math. Proc. Camb. Phil. Soc. 123, 59–74, 1998.
• [4] K.S. Chua and M.L Lang, Congruence subgroups associated to the monster, Experiment. Math. 13 (3), 343–360, 2004.
• [5] J.H. Conway and S.P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11, 308–339, 1977.
• [6] H.S.M. Coxeter and W.O.F. Moser, Generators and Relations for Discrete Groups, 4th ed. Springer-Verlag, 1984.
• [7] H.M. Farkas and I. Kra, Theta constants, Riemann surfaces and the modular group, Graduate Texts in Mathematics, 37, AMS, 2001.
• [8] B.Ö. Güler, M. Beşenk and S. Kader, On congruence equations arising from suborbital graphs, Turk. J. Math. 43 (5), 2396–2404, 2019.
• [9] I. Ivrissimtzis, D. Singerman and J. Strudwick, From farey fractions to the Klein quartic and beyond, Ars Math. Comtemp. 20 (1), 37–50, 2021.
• [10] G.A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. 37 (3), 273–307, 1978.
• [11] I. Ivrissimtzis and D. Singerman, Regular maps and principal congruence subgroups of Hecke groups, Eur. J. Comb. 26, 437–456, 2005.
• [12] G.A. Jones and D. Singerman, Complex Functions, an Algebraic and Geometric Viewpoint, CUP, 1987.
• [13] S. Kader, Circuits in suborbital graphs for the normalizer, Graphs Combin. 33 (6), 1531–1542, 2017.
• [14] C. Maclachlan, Groups of units of zero ternary quadratic forms, Proc. Royal Soc. 88(A), 141-157, Edinburgh, 1981.
• [15] C.C. Sims, Graphs and finite permutation groups, Math. Z. 95, 76-86, 1967.
• [16] D. Singerman, Universal tessellations, Rev. Mat. Univ. Complut. 1, 111–123, 1988.
• [17] D. Singerman and J. Strudwick, Petrie polygons, Fibonacci sequences and Farey maps, Ars Math. Contemp. 10 (2), 349–357, 2016.
• [18] D. Singerman and J. Strudwick, The Farey maps modulo n, Acta Math. Univ. Comen. 89 (1), 39–52, 2020.
• [19] J. Siran, How symmetric can maps on surfaces be?, Surveys in Combinatorics, 161– 238, London Math. Soc. Lec. Not. Ser. 409, CUP, Cambridge, 2013.
• [20] N. Yazıcı Gözütok, U. Gözütok and B.Ö. Güler, Maps corresponding to the subgroups $\Gamma_0(N)$ of the modular group, Graphs Combin. 35 (6), 1695–1705, 2019.
• [21] N. Yazıcı Gözütok and B.O. Güler, Quadrilateral cell graphs of the normalizer with signature $(2,4,\infty)$, Stud. Sci. Math. Hung. 57 (3), 408–425, 2020.