On the parametrized modular group

Abstract

Generalizing the modular and Hecke groups, we consider the subgroup Π of SL(2, ℤ[ξ]) generated by the parabolic element \(\left( {_{01}^{1\xi } } \right)\) and the elliptic element \(\left( {_{10}^{0 - 1} } \right)\), where ℤ[ξ] is the ring of polynomials in the variable ξ. For ζ ∈ ℂ and W ∈ Π, we denote by W (ζ) the matrix in SL(2, ℂ) obtained when evaluating the parameter ξ at ζ. We enumerate the elements of Π and study the relators, defined as those W ∈ ℂ for which there exists ζ ∈ ℂ with W(ζ) = ±I. Then, for W ∈ Π, we investigate the sets of ζ for which W(ζ) is not loxodromic; their union is the singular set S(Π) ⊂ ℂ. The closure of the singular set for the two-parabolic group, which is isomorphic to a free subgroup Π of index 4, has been studied extensively.