Commutator subgroups of the power subgroups of generalized Hecke groups

Abstract

Let p, q > 2 be relatively prime integers and let Hp,q be the generalized Hecke group associated to p and q. The generalized Hecke group Hp,q is generated by X(z) = −(z − λp)−1 and Y (z) = −(z + λq) −1 where λp = 2 cos π p and λq = 2 cos π q.In this paper, for positive integer m, we study the commutator subgroups (Hm p,q) ′ of the power subgroups Hm p,q of generalized Hecke groups Hp,q. We give an application related with the derived series for all triangle groups of the form (0; p, q, n), for distinct primes p, q and for positive integer n.