Akbas̨, Mehmet
(1989)
The normalizer of modular subgroups.
*University of Southampton, Doctoral Thesis*.

## Abstract

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 Γ_{o}(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)

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