Geodesic
Gehring--Martin--Tan Numbers and Tan Numbers of Elementary Subgroups of~$\operatorname{PSL}(2,\mathbb{C})$
Matematičeskie zametki, Tome 102 (2017) no. 2, pp. 255-269

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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 $r$.
Keywords: hyperbolic space, discrete group. 
@article{MZM_2017_102_2_a7,
     author = {A. V. Maslei},
     title = {Gehring--Martin--Tan {Numbers} and {Tan} {Numbers} of {Elementary} {Subgroups} of~$\operatorname{PSL}(2,\mathbb{C})$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {255--269},
     publisher = {mathdoc},
     volume = {102},
     number = {2},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2017_102_2_a7/}
}
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A. V. Maslei. Gehring--Martin--Tan Numbers and Tan Numbers of Elementary Subgroups of~$\operatorname{PSL}(2,\mathbb{C})$. Matematičeskie zametki, Tome 102 (2017) no. 2, pp. 255-269. http://geodesic.mathdoc.fr/item/MZM_2017_102_2_a7/