On the parametrized modular group

Abstract

Generalizing the modular and Hecke groups, we consider the subgroup ΙΙ of SL(2; Z[ξ]) generated by the parabolic (1/0 ξ/1) and the elliptic (0/1 (-1)/0)where Z[ξ] is the ring of polynomials in the variable ξ. For Ϛ Є C and W Є ΙΙ, W (Ϛ) means the matrix in SL (2, C) obtained when we evaluate the parameter ξ at Ϛ We enumerate the elements of ΙΙ and study the relators, defined as those W Є ΙΙ, for which there exists Ϛ Є C with W(ξ ) = ± I. Then, for W Є ΙΙ , we investigate the sets of for which W(Ϛ) is not loxodromic; their union is the singular set S(ΙΙ) C C. Its closure has been much studied for the two-parabolic group, which is a free subgroup of ΙΙ of index 4.