Signature and congruance equations

In this thesis, the numbers of link periods in the signatures of some NEC groups and the numbers of ∞ in the link periods are calculated. In addition, by the aid of imprimitive action of some modular subgroups, the solutions of congruence equations are given. In the first part, the structure of Non-Euclidean crystallized groups is examined. Required basic definitions, theorems and conclusions related with PSL(2,R), the modular group Γ, some properties of congruence subgroups, discrete groups, fundamental regions and imprimitive action are given. In the second part, the numbers of link periods in the signatures of the group Γ ̂_0^2 (N) for all N∈N and the group Γ ̂_0^3 (N) when N is odd or 2||N are calculated by using the Hoare-Uzzel Theorem. Moreover, the index of Λ_n (N) group in the congruence subgroup Γ_0 (N) is calculated. In addition, if p is a prime number with p≡1 mod 4 and β∈N, for any integer a coprime with p, the integer solutions x of the congruence equation a^2+x^2≡0 mod p^β are obtained by using a special imprimitive transitive action of modular subgroups.