Commutator subgroups of the power subgroups of generalized Hecke groups

Abstract

Let \(p\), \(q\geq 2\) be relatively prime integers and let \(H_{p,q}\) be the generalized Hecke group associated to \(p\) and \(q\). The generalized Hecke group \(H_{p,q}\) is generated by \(X(z)=-(z-\lambda _{p})^{-1}\) and \(Y(z)=-(z+\lambda_{q})^{-1}\) where \(\lambda _{p}=2\cos \frac{\pi }{p}\) and \(\lambda_{q}=2\cos \frac{\pi }{q}\). 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 \(H_{p,q}\). 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 \(n\).