# Adjoining Roots and Rational Powers of Generators in PSL(2,\RR) and Discreteness

## TL;DR

This paper investigates when adjoining roots to hyperbolic generators in PSL(2, R) preserves discreteness, providing necessary and sufficient conditions for certain generator pairs and an algorithmic approach for others.

## Contribution

It offers a complete characterization of adjoining roots for hyperbolic generator pairs with disjoint axes in PSL(2, R), including necessary and sufficient conditions and an algorithmic solution.

## Key findings

- Necessary and sufficient conditions for adjoining roots in specific generator pairs.
- Algorithmic methods for cases beyond the main theoretical results.
- Geometric and computational approaches to the discreteness problem.

## Abstract

Let $G$ be a finitely generated group of isometries of $\HH^m$, hyperbolic $m$-space, for some positive integer $m$. %or equivalently elements of $PSL(2,\CC)$. The discreteness problem is to determine whether or not $G$ is discrete. Even in the case of a two generator non-elementary subgroup of $\HH^2$ (equivalently $PSL(2,\mathbb{R})$) the problem requires an algorithm \cite{GM,JGtwo}. If $G$ is discrete, one can ask when adjoining an $n$th root of a generator results in a discrete group. In this paper we address the issue for pairs of hyperbolic generators in $PSL(2, \RR)$ with disjoint axes and obtain necessary and sufficient conditions for adjoining roots for the case when the two hyperbolics have a hyperbolic product and are what as known as {\sl stopping generators} for the Gilman-Maskit algorithm \cite{GM}. We give an algorithmic solution in other cases. It applies to all other types of pair of generators that arise in what is known as the {\sl intertwining case}. The results are geometrically motivated and stated as such, but also can be given computationally using the corresponding matrices.