Genişletilmiş Modüler Grubun Bazı Alt Gruplarının Simgeleri ve Graf Bağlantıları

Abstract

Bu tezde, bazı NEC gruplarının simgeleri, temel b¨olgeleri, grafları ve ¨ozellikleri incelenmi¸stir. Birinci b¨olu¨mde Ayrık Gruplar Teorisi’ nin tarihsel su¨reci ve literatu¨r ¨ozeti verilmi¸stir. ˙Ikinci b¨olu¨mde topolojik gruplar, Sayılar Teorisi, M¨obius d¨onu¨¸su¨mleri, hiperbolik geometri, yu¨zeyler, temel b¨olgeler ve simgeler hakkında genel bilgiler ifade edilmi¸stir. ¨ Ozellikle Γ modu¨ler grubu ve ˆΓ geni¸sletilmi¸s modu¨ler grubu ayrıntılı bir¸sekilde ¸calı¸sılmı¸stır. Ayrıca imprimitif hareket ve Graf Teori unsurları tanıtılmı¸stır. ¨U¸cu¨ncu¨ b¨olu¨mde yapılan ¸calı¸smalar ise tezin ¨ozgu¨n kısmını olu¸sturmaktadır. Burada Γ(N),Γθ ve Γ0(N) kongru¨ans alt grupları ara¸stırılmı¸stır. Γ nın Γ0,n(N) ve Λn(N) alt gruplarının ¨ozellikleri verilmi¸stir. Λn(N) nin Γ0(N) deki indeksi elde edilmi¸stir. Yine Λn(N) nin ˆ Qn(N) deki Fu,n,N alt y¨oru¨ngesel graflarında uygulamalar yapılmı¸s ve ¨onemli sonuc¸lara ula¸sılmı¸stır. Ayrıca ˆ Γ0,n(N) nin ˆ Q(N) deki F∗u,N alt y¨oru¨ngesel grafında kenar ko¸sulları ve orman olma durumları incelenmi¸stir. Bazı ˆ Γ0,n(N) nin simgesindeki sınır bile¸senlerinin bir takım sonuc¸ları ve son olarak ¨ozel ΓF(N) Fricke gruplarının sınır bile¸senleri elde edilmi¸stir. D¨ordu¨ncu¨ b¨olu¨mde ise ¸calı¸sılan konunun sonu¸cları ortaya konularak ¨oneriler sunulmu¸stur.,In this thesis, the signature of some Non-Euclidean Crystallographic, NEC group for short, fundamental domains and suborbital graphs and their properties are investigated. In Chapter 1, Discrete groups and historical background in the literature are given. In Chapter 2, Topological Groups, Numbers Theory, Mobius transformations, hyperbolic geometry, surfaces, fundamental domains and signature of discrete groups, in general, are expressed. Especially the modular group Γ and its extension ˆΓ by the reflection z →−¯ z, imprimitive action of a group, and graph theory are studied in detail. In Chapter 3, which is the original part of the thesis, main calculations on the groups Γ(N),Γθ and Γ0(N) are discussed. The two subgroups Γ0,n(N) and Λn(N) of the modular group Γ are defined and their indexes in related groups are obtained. The suborbital graphs Fu,n,N of the Λn(N) on the set ˆ Qn(N) are investigated and some important results, the edge condition and being forest of the suborbital graph F∗u,N of the ˆ Γ0,n(N) on ˆ Q(N) are given. And, some results of boundary components in the signature of some ˆ Γ0,n(N) and furthermore, boundary components of very special Fricke groups ΓF(N) are obtained. In Chapter 4, conclusions of the thesis and some suggestions to the readers are expressed.

In this thesis, the signature of some Non-Euclidean Crystallographic, NEC group for short, fundamental domains and suborbital graphs and their properties are investigated. In Chapter 1, Discrete groups and historical background in the literature are given. In Chapter 2, Topological Groups, Numbers Theory, Mobius transformations, hyperbolic geometry, surfaces, fundamental domains and signature of discrete groups, in general, are expressed. Especially the modular group Γ and its extension ˆΓ by the reflection z →−¯ z, imprimitive action of a group, and graph theory are studied in detail. In Chapter 3, which is the original part of the thesis, main calculations on the groups Γ(N),Γθ and Γ0(N) are discussed. The two subgroups Γ0,n(N) and Λn(N) of the modular group Γ are defined and their indexes in related groups are obtained. The suborbital graphs Fu,n,N of the Λn(N) on the set ˆ Qn(N) are investigated and some important results, the edge condition and being forest of the suborbital graph F∗u,N of the ˆ Γ0,n(N) on ˆ Q(N) are given. And, some results of boundary components in the signature of some ˆ Γ0,n(N) and furthermore, boundary components of very special Fricke groups ΓF(N) are obtained. In Chapter 4, conclusions of the thesis and some suggestions to the readers are expressed.