On parameters and discreteness of Maskit subgroups in $\mathrm {PSL} (2, \mathbb{C})$

In 1989 B. Maskit formulated the following problem. Let $G$ be the subgroup of ${\rm PSL} (2, \mathbb{C})$ generated by the elements $f$ and $g$, where $f$ has two fixed points in $\overline{\mathbb{C}}$, and $g$ maps one fixed point of $f$ onto the other; when is $G$ discrete? Partial solutions of the problem were found by B. Maskit and E. Klimenko, but complete solution is not known. In this paper, the trace parameters for such groups are considered. Properties of the parameters are used to find new necessary and sufficient discreteness conditions for the groups.