ALL DISCRETE $\mathcal{RP}$ GROUPS WHOSE GENERATORS HAVE REAL TRACES

Abstract

In this paper we give necessary and sufficient conditions for discreteness of a subgroup of PSL(2,ℂ) generated by a hyperbolic element and an elliptic one of odd order with non-orthogonally intersecting axes. Thus we completely determine two-generator non-elementary Kleinian groups without invariant plane with real traces of the generators and their commutator. We also give a list of all parameters that correspond to such groups. An interesting corollary of the result is that the group of the minimal known volume hyperbolic orbifold ℍ 3 /Γ 353 has real parameters.