Commutator subgroups of generalized hecke and extended generalized hecke groups

Abstract

Let p and q be integers such that 2 <= p <= q, p + q > 4 and let H-p,H-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) = 2 cos pi/p and lambda(q) = 2 cos pi/p. The extended generalized Hecke group (H) over bar (p,q) is obtained by adding the reflection R(z) = 1/(z) over bar to the generators of generalized Hecke group H-p,H-q. In this paper, we study the commutator subgroups of generalized Hecke groups H-p,H-q and extended generalized Hecke groups (H) over bar (p,q).