The normalizer of modular subgroups

In Chapter 1, we discuss the structure of non-Euclidean Crystallographic (NEC, for short) groups and calculate the parabolic class number of Γo(N) by using a number theoretical method. We mention the modular group Γ and the extended modular group hat Γ, give two kinds of modular subgroups, mainly Γ(N) and Γo(N) and their extensions by the involution T(z) = -=z. We endow the quotient space U^*/Γo(N) with a topological structure so that U^*/Γo(N) is a compact Riemann surface. By introducing the idea of commensurability of groups, we conclude that U^*/Λ is compact if Γo(N) and Λ are commensurable. In Chapter 2, the normalizer ΓB(N) of Γo(N) in PSL(2,R) is discussed. The property of an integer h is given. Then we find the order of the quotient group B(N) = ΓB(N)/Γo(N) and in two different ways we show that B(N) is generated by the Atkin-Lehner involutions W(N) and a special matrix s. We introduce the special cusps of Γo(N) and then find their number. Next, we conclude that B(N) acts on the special cusps and from this result /B(N)/ is determined. Furthermore, the structures of B(P^α) are given for prime powers p^α. Finally we find the structure of B(N) in general and give the necessary and sufficient conditions for B(N) to be a direct product of its subgroups. In Chapter 3, we find the parabolic class number π(N) of the normalizer ΓB(N), where N is any positive integer, and we give the necessary and sufficient conditions for π(N) to be one. We also find the signature of ΓB(N), where N ≤ 50, and all possible triangle groups ΓB(N). Finally in Chapter 4, the boundary components of three kinds of NEC groups, mainly hat Γ(N), hat Γo(N) and ΓF(N) are investigated by using the Hoare-Uzzel theorem and the fixed point set F(T) of T where T(z) will be -=z or 1 over Nbar z, where N will be any positive integer except for the group ΓF(N) where N will be a square. (DX90500)