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

Abstract

Let p1, • • • , pn be integers where n ≥ 2 and each pi ≥ 2 . Let also H(p1, • • • , pn) be the generalized Hecke group associated to all pi ≥ 2. In this paper, we study the commutator subgroups H ′ (p1, • • • , pn) and H ′ (p1, • • • , pn) of the generalized Hecke group H(p1, • • • , pn) and the extended generalized Hecke group H(p1, • • • , pn) . We give the generators and the signatures of H ′ (p1, • • • , pn) and H ′ (p1, • • • , pn) .

A Classification of Surds of Non-Arithmetical Groups

“…The matrix elements of Hecke groups are algebraic integer numbers in the totally real field Q = 2cos(π/n). The isomorphisms φk of this field are written as α such that (9) ∀k odd integers, s.t. k < n, coprime with n.…”

“…More in detail, the desymmetrized non-arithmetical domain, the Γ2 congruence subgroup and generalized Hecke groups have been analyzed, for which reduced surds have been defined. The choice of the definition of the commutator subgroups of the generalized Hecke groups in [9] can be compared with the generalized Hecke groups analyzed in the present paper. In particular, the commutator subgroups of the generalized Hecke groups chosen in [9] is one containing the reflection c of Eq.…”

“…The choice of the definition of the commutator subgroups of the generalized Hecke groups in [9] can be compared with the generalized Hecke groups analyzed in the present paper. In particular, the commutator subgroups of the generalized Hecke groups chosen in [9] is one containing the reflection c of Eq. 's (6).…”

“…Pseudomodular reflection groups have been studied in [8]. Commutator Subgroups of Generalized Hecke groups and Extended Generalized Hecke groups have been studied in [9] and further generalized in [10]. In particular, the choices of the reflection on the non-degenerate circumference will be discussed and compared with the choice adopted in the present paper.…”