Lifting free subgroups of PSL(2, R) to free groups

Let F = < a, b > be a rank two free group, let G = < A, B > be a two generator subgroup of PSL(2, R) and let rho be a faithful representation of F with rho(a) = A and rho(b) = B. If G is discrete and free, many results about the primitive elements of G are proved using the geometry that G inherits from PSL(2, R), the group of orientation preserving isometries of the hyperbolic plane. Some of these results can be lifted to F modulo the replacement of a and/or b by their inverse and the interchange of a and b. In this paper we lift these results and obtain results that are independent of any replacement by inverses or interchange of generators and independent of the given representation.