Petrie Paths in Triangular Normalizer Maps
Abstract
This study is devoted to investigate the Petrie paths in the normalizer maps and regular triangular maps corresponding to the subgroups $\Gamma_0(N)$ of the modular group $\Gamma$. We show that each regular triangular map admits a closed Petrie path. Thus, for each regular map, we find the Petrie length of the corresponding map.
Keywords
References
- [1] B. Schoeneberg, Elliptic Modular Functions, Springer, Berlin, 1974.
- [2] N. Yazıcı Go¨zu¨tok, B. O¨ . Gu¨ler, Suborbital graphs for the group GC(N), Bull. Iran. Math. Soc., 45 (2019), 593-605.
- [3] Y. Kesicio˘glu, M. Akbas¸, On suborbital graphs for the group G3, Bull. Iran. Math. Soc., 46 (2020), 1731-1744.
- [4] B. O¨ . Gu¨ler, M. Bes¸enk, A.H. Deg˘er, S. Kader, Elliptic elements and circuits in suborbital graphs, Hacettepe J. Math. Stat., 40 (2011), 203-210.
- [5] P. Jaipong, W. Promduang, K. Chaichana, Suborbital graphs of the congruence subgroup G0(N), Beitr. Algebra Geom., 60 (2019), 181-192.
- [6] P. Jaipong, W. Tapanyo, Generalized classes of suborbital graphs for the congruence subgroups of the modular group, Algebra Discret. Math., 27 (2019), 20-36.
- [7] M. Akbas¸, D. Singerman, Onsuborbital graphs for the modular group, Bull. London Math. Soc., 33 (2001), 647-652.
- [8] N. Yazıcı Go¨zu¨tok, U. Go¨zu¨tok, B. O¨ . Gu¨ler, Maps corresponding to the subgroups G0(N) of the modular group, Graphs Combin., 35 (2019), 1695-1705.
- [9] N. Yazıcı G¨oz¨utok, Normalizer maps modulo N, Mathematics, 10 (2022), 1046.
- [10] J. H. Conway, S. P. Norton, Monstrous moonshine, Bull. London Math. Soc., 11 (1977), 308-339.
