On the parametrized modular group (Q891349)

The modular group is the most well-known discrete group. In 1936, E. Hecke defined the so-called Hecke groups, which are generalisations of the modular group. Further generalizing the modular group and the Hecke groups, the authors consider the subgroups \(\Pi\) of \(\mathrm{SL}(2,Z[\xi])\) generated by a parabolic element and an elliptic element, where \(Z[\xi]\) is the ring of polynomials of the variable \(\xi\). The elements of \(\Pi\) are classified, and the relators are determined. The authors also investigate the set of all \(\zeta\) for which \(W(\zeta)\) is not loxodromic.