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Let $G$ be a finitely generated group of isometries of $\HH^m$, hyperbolic$m$-space, for some positive integer $m$. %or equivalently elements of$PSL(2,\CC)$.  The discreteness problem is to determine whether or not $G$ is discrete. Evenin the case of a two generator non-elementary subgroup of $\HH^2$ (equivalently$PSL(2,\mathbb{R})$) the problem requires an algorithm \cite{GM,JGtwo}. If $G$is discrete, one can ask when adjoining an $n$th root of a generator results ina discrete group.  In this paper we address the issue for pairs of hyperbolic generators in$PSL(2, \RR)$ with disjoint axes and obtain necessary and sufficient conditionsfor adjoining roots for the case when the two hyperbolics have a hyperbolicproduct and are what as known as {\sl stopping generators} for theGilman-Maskit algorithm \cite{GM}. We give an algorithmic solution in othercases. It applies to all other types of pair of generators that arise in whatis known as the {\sl intertwining case}. The results are geometricallymotivated and stated as such, but also can be given computationally using thecorresponding matrices.

Adjoining Roots and Rational Powers of Generators in PSL(2,\RR) and Discreteness