On Normal Subgroups of Generalized Hecke Groups

Abstract

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.

Commutator subgroups of generalized Hecke and extended generalized Hecke groups,II

“…If we use the Reidemeister-Schreier method and make the required calculations, then we get one generator of the form A 1 A 2 A 1 A 2 ; one generator of the form B 2 1 ; four generators of the form 2,3,4,5). Also, the signature of H ′ (2, 2, 3, 4, 5) is (1; 2 (4) , 3 (8) , 5 (8) , ∞ (4) ) .…”

“…1 . Therefore, there are seven generators of H ′ (5,5,7,8). Also, we obtain the signature of H ′ (5, 5, 7, 8) as (0; 5 (4) , 7 (2) , 4, ∞).…”