Groups generated by two elliptic elements in PU(2,1)

TL;DR

This paper investigates conditions under which groups generated by two elliptic elements in PU(2,1) are discrete, non-elementary, and free products, based on the distance between their fixed points or lines.

Contribution

It establishes a criterion relating the distance between fixed points or lines of elliptic elements to the discreteness and free product structure of the generated group.

Findings

Groups are discrete and free products when fixed points are sufficiently far apart.

Provides a threshold distance for discreteness and non-elementarity.

Shows the generated group is isomorphic to Z_m * Z_n under certain conditions.

Abstract

Let $f$ and $g$ be two elliptic elements in $\mathbf{PU}(2,1)$ of order $m$ and $n$ respectively, where $m\geq n>2$. We prove that if the distance $\delta(f,g)$ between the complex lines or points fixed by $f$ and $g$ is large than a certain number, then the group $< f, g >$ is discrete nonelementary and isomorphic to the free product $\mathbf{Z}_{m}*\mathbf{Z}_{n}$.