All Discrete RP Groups Whose Generators have Real Traces

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 H3/Γ353 has real parameters. Mathematics Subject Classification (2000): Primary: 30F40; Secondary: 20H10, 22E40, 57M60, 57S30.