On Gehring-Martin-Tan groups with an elliptic generator

TL;DR

This paper introduces a method to construct Gehring-Martin-Tan groups with elliptic generators of order four, providing new examples of Kleinian groups that satisfy the equality in the Gehring-Martin-Tan inequality, and relates them to hyperbolic 3-orbifolds.

Contribution

It presents a novel construction method for Gehring-Martin-Tan groups with elliptic generators of order four and provides explicit examples linked to hyperbolic 3-orbifolds.

Findings

Constructed new Gehring-Martin-Tan groups with order four elliptic generators

Connected these groups to finite volume hyperbolic 3-orbifolds

Expanded understanding of equality cases in the Gehring-Martin-Tan inequality

Abstract

The Gehring-Martin-Tan inequality for 2-generator subgroups of PSL(2,C) is one of the best known discreteness conditions. A Kleinian group $G$ is called a Gehring-Martin-Tan group if the equality holds for the group $G$. We give a method for constructing Gehring-Martin-Tan groups with a generator of order four and present some examples. These groups arise as groups of finite volume hyperbolic 3-orbifolds.