J{\o}rgensen Number and Arithmeticity

TL;DR

This paper characterizes Jørgensen groups, identifies all non-cocompact arithmetic cases, and computes their Jørgensen numbers, advancing understanding of Kleinian groups and their arithmetic properties.

Contribution

It uniquely identifies the torsion-free Jørgensen group as the figure-eight knot group and classifies all non-cocompact arithmetic Jørgensen groups.

Findings

The only torsion-free Jørgensen group is the figure-eight knot group.

All non-cocompact arithmetic Jørgensen groups are identified.

Jørgensen numbers are computed for several non-cocompact Kleinian groups.

Abstract

A J{\o}rgensen group is a non-elementary Kleinian group that can be generated by two elements for which equality holds in J{\o}rgensen's Inequality. This paper shows that the only torsion-free J{\o}rgensen group is the figure-eight knot group, identifies all non-cocompact arithmetic J{\o}rgensen groups, and establishes a characterization of cocompact arithmetic J{\o}rgensen groups. The paper also defines and computes the J{\o}rgensen number of several non-cocompact Kleinian groups including some two-bridge knot and link groups.