Power and free normal subgroups of generalized Hecke groups

Let [Formula: see text] and [Formula: see text] be integers such that [Formula: see text] [Formula: see text] and let [Formula: see text] be generalized Hecke group associated to [Formula: see text] and [Formula: see text] Generalized Hecke group [Formula: see text] is generated by [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] In this paper, for positive integer [Formula: see text] we study the power subgroups [Formula: see text] of generalized Hecke groups [Formula: see text]. Also, we give some results about free normal subgroups of generalized Hecke groups [Formula: see text]

Maximal subgroups of the modular and other groups

In 1933 B. H. Neumann constructed uncountably many subgroups of {{\rm SL}_{2}(\mathbb{Z})} which act regularly on the primitive elements of {\mathbb{Z}^{2}} . As pointed out by Magnus, their images in the modular group {{\rm PSL}_{2}(\mathbb{Z})\cong C_{3}*C_{2}} are maximal nonparabolic subgroups, that is, maximal with respect to containing no parabolic elements. We strengthen and extend this result by giving a simple construction using planar maps to show that for all integers {p\geq 3} , {q\geq 2} the triangle group {\Gamma=\Delta(p,q,\infty)\cong C_{p}*C_{q}} has uncountably many conjugacy classes of nonparabolic maximal subgroups. We also extend results of Tretkoff and of Brenner and Lyndon for the modular group by constructing uncountably many conjugacy classes of such subgroups of Γ which do not arise from Neumann’s original method. These maximal subgroups are all generated by elliptic elements, of finite order, but a similar construction yields uncountably many conjugacy classes of torsion-free maximal subgroups of the Hecke groups {C_{p}*C_{2}} for odd {p\geq 3} . Finally, an adaptation of work of Conder yields uncountably many conjugacy classes of maximal subgroups of {\Delta(2,3,r)} for all {r\geq 7} .

Commutator Subgroups of Generalized Hecke and Extended Generalized Hecke Groups

Let p and q be integers such that 2 ≤ p ≤ q; p + q > 4 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 . The extended generalized Hecke group H̅p,q is obtained by adding the reection R(z) = 1/z̅ to the generators of generalized Hecke group Hp,q: In this paper, we study the commutator subgroups of generalized Hecke groups Hp,q and extended generalized Hecke groups H̅p,q.

On Normal Subgroups of Generalized Hecke Groups

We consider the generalized Hecke groups Hp,q generated by X(z) = -(z -λp)-1, Y (z) = -(z +λq)-1 with and where 2 ≤ p ≤ q < ∞, 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 . We state the relation between commutator subgroup H′p,q of Hp,q defined in [1] and the even subgroup. Then we extend this result to extended generalized Hecke groups H̅p,q.