View More View Less
  • 1 Department of Mathematics, Faculty of Science, Karadeniz Technical University, Trabzon, Turkey
Restricted access

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

Abstract

In this study, we investigate suborbital graphs G u,n of the normalizer ΓB (N) of Γ0 (N) in PSL(2, ℝ) for N = 2α3β where α = 1, 3, 5, 7, and β = 0 or 2. In these cases the normalizer becomes a triangle group and graphs arising from the action of the normalizer contain quadrilateral circuits. In order to obtain graphs, we first define an imprimitive action of ΓB (N) on using the group ГΒ+ (N) and then obtain some properties of the graphs arising from this action.

  • [1]

    AKBAŎ, M. and SINGERMAN, D., The signature of the normalizer of Γ0(N) in PSL(2, R), London Math. Soc, 165 (1992), 77-86.

  • [2]

    BIGGS, N. L. and WHITE, A. T., Permutation groups and combinatorial structures, London Mathematical Society Lecture Note Series, 33rd edn. CUP, Cambridge, 1979.

    • Search Google Scholar
    • Export Citation
  • [3]

    CANGÜL, I. N. and SlNGERMAN D., Normal subgroups of Hecke groups and regular maps, Math. Proc. Camb. Phil. Soc, 123 (1998), 59-74.

  • [4]

    CHUA, K. S. and LANG, M. L., Congruence subgroups associated to the monster, Experiment. Math., 13(3) (2004), 343-360.

  • [5]

    CONWAY, J. H. and NORTON, S. P., Monstrous Moonshine, Bull. London Math. Soc., 11 (1977), 308-339.

  • [6]

    COXETER, H. S. M. and MOSER, W. O. F., Generators and Relations for Discrete Groups, fourth ed., Springer-Verlag, 1984.

  • [7]

    FARKAS, H. M. and KRA, I., Theta constants, Riemann surfaces and the modular group, Graduate Texts in Mathematics, vol. 37, American Mathematical Society, 2001.

    • Search Google Scholar
    • Export Citation
  • [8]

    GULER, B. O., BESENK, M. and KADER, S., On congruence equations arising from suborbital graphs, Turkish J. Math., 43(5) (2019), 2396-2404.

    • Search Google Scholar
    • Export Citation
  • [9]

    IVRISSIMTZIS, I., SINGERMAN, D. and STRUDWICK, J., From farey fractions to the Klein quartic and beyond, arXiv:1909.08568 [math.GR].

  • [10]

    IVRISSIMTZIS, I. P. and SINGERMAN, D., Regular maps and principal congruence subgroups of Hecke groups, European J. Combinatorics, 26 (2005), 437-456.

    • Search Google Scholar
    • Export Citation
  • [11]

    JONES, G. A. and SINGERMAN, D., Theory of maps on orientable surfaces, Proc. London Math. Soc, 37(3) (1978), 273-307.

  • [12]

    JONES, G. A. and SINGERMAN, D., Complex Functions, an Algebraic and Geometric Viewpoint, Cambridge University Press, 1987.

  • [13]

    KADER, S., Circuits in suborbital graphs for the normalizer, Graphs Combin., 33(6) (2017), 1531-1542.

  • [14]

    MACLACHLAN, C., Groups of units of zero ternary quadratic forms, Proc. Royal Soc. Edinburgh., 88(A) (1981), 141-157.

  • [15]

    SIMS, C. C, Graphs and finite permutation groups, Mathematische Zeitschrift, 95 (1967), 76-86.

  • [16]

    SINGERMAN D. and STRUDWICK, J., The Farey maps modulo n, Acta Mathematica Universitatis Comenianae, 89(1) (2020), 39-52.

  • [17]

    SINGERMAN D., Universal tessellations, Rev. Mat. Univ. Complut., 1 (1988), 111123.

  • [18]

    SINGERMAN, D. and STRUDWICK, J., Petrie polygons, Fibonacci sequences and Farey maps, Ars Math. Contemp., 10(2) (2016), 349-357.

  • [19]

    YAZICI GÖZÜTOK, N., GÖZÜTOK, U. and GÜLER, B. O., Maps corresponding to the subgroups Γ0(N) of the modular group, Graphs Combin., 35(6) (2019), 1695-1705.

    • Search Google Scholar
    • Export Citation