Discreteness criteria for \(\mathcal{RP}\) groups
After the classification of the discrete two-generator subgroups of \(\text{PSL}(2,\mathbb{R})\) was finished [see for instance \textit{B. Fine} and \textit{G. Rosenberg}, Groups '93, Galway/St. Andrews, Vol. 1, Lond. Math. Soc. Lect. Note Ser. 211, 205-232 (1995; Zbl 0847.20044)], Gehring, Gilman and Martin introduced the class of two-generator subgroups of \(\text{PSL}(2,\mathbb{C})\) with real parameters \[ \{\langle f,g\rangle\mid\text{tr}^2f-4,\text{tr}^2g-4,\text{tr}(fgf^{-1}g^{-1})-2\in\mathbb{R}\}\subset\text{PSL}(2,\mathbb{C}). \] Here discreteness criteria are given for \(f\) hyperbolic, that is, \(\text{tr }f\in\mathbb{R}\), \(|\text{tr }f|>2\), and \(g\) elliptic of even order, that is, \(\text{tr }g\in\mathbb{R}\), \(\text{tr }g=2\cos\tfrac{r\pi}n\), \(2\leq n\) even, \(\text{gcd}(r,n)=1\).
scientific article; zbMATH DE number 1819989
scientific article