Gehring–Martin–Tan Numbers and Tan Numbers of Elementary Subgroups of $\operatorname{PSL}(2,\mathbb{C})$

The Gehring–Martin–Tan number and the Tan number are real quantities defined for two-generated subgroups of the group $\operatorname{PSL}(2,\mathbb{C})$. It follows from the necessary discreteness conditions proved by Gehring and Martin and, independently, by Tan that, for discrete groups, these quantities are bounded below by $1$. In the paper, we find precise values of these numbers for the majority of elementary discrete groups and prove that, for every real $r \ge 1$, there are infinitely many elementary discrete groups with the Gehring–Martin–Tan number equal to $r$ and the Tan number equal to $r$.