Discreteness criteria forRP groups

Recently Gehring, Gilman, and Martin introduced an important class of two-generator groups with real parameters:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\{ {\Gamma = \left\langle {f,g} \right\rangle \left| {f,g \in PSL(2,C);\beta ,\beta ^\prime ,\gamma \in R} \right.} \right\}$$ \end{document} whereβ=tr2f−4,β′=tr2g−4, and γ=tr(fgf−1g−1)−2. The groups that belong to this class we callRP groups. We find criteria for discreteness ofRP groups generated by a hyperbolic element and an elliptic one of even order with intersecting axes.