Commutator Subgroups of Generalized Hecke and Extended Generalized Hecke Groups

Let p and q be integers such that 2 ≤ p ≤ q; p + q > 4 and let H_p,q be the generalized Hecke group associated to p and q: The generalized Hecke group Hp,q is generated by X(z) = -(z-λ_p)^-1 and Y (z) = -(z+ λ_q)^-1 where λ_p = 2 cos ≤ π/p and λ_q = 2 cos π/q . The extended generalized Hecke group H̅_p,q is obtained by adding the reection R(z) = 1/z̅ to the generators of generalized Hecke group H_p,q: In this paper, we study the commutator subgroups of generalized Hecke groups H_p,q and extended generalized Hecke groups H̅_p,q.