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Year 2022, Volume: 51 Issue: 3, 666 - 679, 01.06.2022
https://doi.org/10.15672/hujms.824436

Abstract

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.

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

Year 2022, Volume: 51 Issue: 3, 666 - 679, 01.06.2022
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.

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.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Nazlı Yazıcı Gözütok 0000-0002-3645-0623

Bahadır Özgür Güler 0000-0003-3131-3643

Publication Date June 1, 2022
Published in Issue Year 2022 Volume: 51 Issue: 3

Cite

APA Yazıcı Gözütok, N., & Güler, B. Ö. (2022). Hexagonal cell graphs of the normalizer with signature $(2, 6, \infty)$. Hacettepe Journal of Mathematics and Statistics, 51(3), 666-679. https://doi.org/10.15672/hujms.824436
AMA Yazıcı Gözütok N, Güler BÖ. Hexagonal cell graphs of the normalizer with signature $(2, 6, \infty)$. Hacettepe Journal of Mathematics and Statistics. June 2022;51(3):666-679. doi:10.15672/hujms.824436
Chicago Yazıcı Gözütok, Nazlı, and Bahadır Özgür Güler. “Hexagonal Cell Graphs of the Normalizer With Signature $(2, 6, \infty)$”. Hacettepe Journal of Mathematics and Statistics 51, no. 3 (June 2022): 666-79. https://doi.org/10.15672/hujms.824436.
EndNote Yazıcı Gözütok N, Güler BÖ (June 1, 2022) Hexagonal cell graphs of the normalizer with signature $(2, 6, \infty)$. Hacettepe Journal of Mathematics and Statistics 51 3 666–679.
IEEE N. Yazıcı Gözütok and B. Ö. Güler, “Hexagonal cell graphs of the normalizer with signature $(2, 6, \infty)$”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 3, pp. 666–679, 2022, doi: 10.15672/hujms.824436.
ISNAD Yazıcı Gözütok, Nazlı - Güler, Bahadır Özgür. “Hexagonal Cell Graphs of the Normalizer With Signature $(2, 6, \infty)$”. Hacettepe Journal of Mathematics and Statistics 51/3 (June 2022), 666-679. https://doi.org/10.15672/hujms.824436.
JAMA Yazıcı Gözütok N, Güler BÖ. Hexagonal cell graphs of the normalizer with signature $(2, 6, \infty)$. Hacettepe Journal of Mathematics and Statistics. 2022;51:666–679.
MLA Yazıcı Gözütok, Nazlı and Bahadır Özgür Güler. “Hexagonal Cell Graphs of the Normalizer With Signature $(2, 6, \infty)$”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 3, 2022, pp. 666-79, doi:10.15672/hujms.824436.
Vancouver Yazıcı Gözütok N, Güler BÖ. Hexagonal cell graphs of the normalizer with signature $(2, 6, \infty)$. Hacettepe Journal of Mathematics and Statistics. 2022;51(3):666-79.