Finally, The number of boundary components in the signature of some groups $\hat{\Gamma}_{0,p}(p)$ and $\hat{\Gamma}_{0,p}(p^2), p$ is a prime number, and the number of link periods were found.
📌 3 Maddede Bu Makale
🎯
Amaç & Hipotez:In this paper, we established the group $\hat{\Gamma}_{0,n}(N)$ by group $\Gamma_{0,n}(N)$ extending with reflection.
📊
Temel Bulgular & Veri:Then, we obtain boundary components in signature of the group and we get some calculation for link periods $2, 3, \infty$.
💡
Sonuç & Katkı:Finally, The number of boundary components in the signature of some groups $\hat{\Gamma}_{0,p}(p)$ and $\hat{\Gamma}_{0,p}(p^2), p$ is a prime number, and the number of link periods were found.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Signature · Boundary component · Extended Modular Group
Abstract
In this paper, we established the group $\hat{\Gamma}_{0,n}(N)$ by group $\Gamma_{0,n}(N)$ extending with reflection. Then, we obtain boundary components in signature of the group and we get some calculation for link periods $2, 3, \infty$. And then, we constitute chain of reflections with fixed points via extended Hoore-Uzzell Theorem in the group. Finally, The number of boundary components in the signature of some groups $\hat{\Gamma}_{0,p}(p)$ and $\hat{\Gamma}_{0,p}(p^2), p$ is a prime number, and the number of link periods were found.
ÜNLÜYOL, E., BÜYÜKKARAGÖZ, A. (2020). The Signatures and Boundary Components of the Groups $\hat{\Gamma}_{0,n}(N)$. Turkish Journal of Science, Cilt: 5 - Sayı: 3, 268–279. https://ajindex.com/articles/809485
Akba\c{s} M. The Normalizer of Modular Subgroup. Ph. D. Theis, Facult of Mathematical Studies, University of Southampton, Southampton 1989
Ba\c{s}kan T. Ayrık Gruplar H. \"{U}., Fen Fakültesi Basımevi, Ders Kitapları Dizisi: 11, Ankara 1980
Bujalance E., Etayo J. J., Gamboa J. M. Gromadzki G. Automorphism Groups of Compact Bordered Klein Surfaces. Springer Verlag, Berlin 1990
Jones G. A., Singerman D. Complex functons: an algebraic and geometric viewpoint. Cambridge University Press, Cambridge 1987
Jones G. A., Singerman D. Wicks K. The Modular Group and Generalized Farey Graphs. London Math. Soc. Lecture Notes, CUP, Cambridge, 160, 1991, 316--338
Macbeath A. M. The classification of non-euclidean crystallographic groups. Can. J. Math. 19, 1967, 1192--1205
Schoeneberg B. Elliptic Modular Functions. Springer Verlag, Berlin 1974
Wilkie H. C. On non-euclidean crystallographic groups. Math. Zeitschr. 91, 1966, 87--102.