# The realization problem for J{\o}rgensen numbers

## TL;DR

This paper investigates the Jørgensen number for non-elementary Kleinian groups, proving that any value greater than or equal to 1 can be realized, and provides computational visualizations of these numbers.

## Contribution

It demonstrates that for any r >= 1, there exists a non-elementary Kleinian group with Jørgensen number r, answering a previously posed question.

## Key findings

- Existence of Kleinian groups with any Jørgensen number >= 1
- Computational estimates of Jørgensen numbers in Schottky space
- Visualization of Jørgensen numbers through computer-generated images

## Abstract

Let G be a two generator subgroup of PSL(2,C). The Jorgensen number J(G) of G is defined by J(G)=inf{ |tr^2 A-4|+|tr[A,B]-2| ; G=<A,B>}. If G is a non-elementary Kleinian group, then J(G) >= 1. This inequality is called Jorgensen's inequality. In this paper, we show that, for any r >= 1, there exists a non-elementary Kleinian group whose Jorgensen number is equal to r. This answers a question posed by Oichi and Sato. We also present our computer generated picture which estimates Jorgensen numbers from above in the diagonal slice of Schottky space.