Trace Minimization and Roots in PSL(2,R)

Suppose that A,B ∈ PSL(2,R) generate a discrete and free group of rank 2, and let m,n≥ 1. We consider subgroups R,S of PSL(2,R) generated by roots of A and B, i.e., by elements such that Rᵐ=A and Sⁿ=B. Depending on whether the commutator trace τ= tr([A,B]) is larger or smaller than 2, we describe necessary and sufficient conditions for R,S to be discrete and free of rank 2. For τ≤ -2, this can be checked with an explicit formula. For $τ> 2$, one has to use the Trace Minimization Algorithm. Besides an explicit formulation of this algorithm, we prove new formulas for the powers and roots of elements of PSL(2,R), their traces and their commutator traces. The case of positive rational exponents $m,n$ is treated, as well.