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

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 $\mathbf{Z}_{m}*\mathbf{Z}_{n}$.