  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. Even
in 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 in
a 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 conditions
for adjoining roots for the case when the two hyperbolics have a hyperbolic
product and are what as known as {\sl stopping generators} for the
Gilman-Maskit algorithm \cite{GM}. We give an algorithmic solution in other
cases. It applies to all other types of pair of generators that arise in what
is known as the {\sl intertwining case}. The results are geometrically
motivated and stated as such, but also can be given computationally using the
corresponding matrices.
