Commutator subgroups of the power subgroups of generalized Hecke groups

Abstract

Let p, q >= 2 be relatively prime integers and let H p, q be the generalized Hecke group associated to p and q. The generalized Hecke group H-p,H-q is generated by X (z) = - (z - lambda(p))(-1) and Y (z) = -(z + lambda(q))(-1) where lambda(p) = 2cos pi/p and lambda(q) = 2cos pi/q. In this paper, for positive integer m, we study the commutator subgroups (H-p,q(m))' of the power subgroups H-p,q(m) of generalized Hecke groups H-p,H-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.