Gehring–Martin–Tan numbers and Tan numbers of elementary subgroups of PSL(2,ℂ)

Abstract

The Gehring–Martin–Tan number and the Tan number are real quantities defined for two-generated subgroups of the group PSL(2,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 ≥ 1, there are infinitely many elementary discrete groups with the Gehring–Martin–Tan number equal to r and the Tan number equal to r.