On Normal Subgroups of Generalized Hecke Groups

We consider the generalized Hecke groups H-p,H-q generated by X(z) = -(z - lambda(p))(-1), Y(z) = -(z + lambda(q))(-1) with lambda(p) = 2 cos(pi/p) and lambda(q) = 2 cos(pi/q) where 2 <= p <= q < infinity, p + q > 4. In this work we study the structure of genus 0 normal subgroups of generalized Hecke groups. We construct an interesting genus 0 subgroup called even subgroup, denoted by H-Ep,H-q. We state the relation between commutator subgroup H'(p,q) of H-p,H-q defined in [1] and the even subgroup. Then we extend this result to extended generalized Hecke groups (H) over bar (p,q).