Free product of two elliptic quaternionic Möbius transformations

Abstract

Suppose that $f$ and $g$ are two elliptic quaternionic Möbius transformations of orders $m$ and $n$ respectively. If the hyperbolic distance $\delta(f,g)$ between ${\rm fix}(f)$ and ${\rm fix}(g)$ satisfies $$\cosh \delta(f,g) \geq\frac{\cos\frac{\pi}{m}\cos\frac{\pi}{n}+1}{\sin\frac{\pi}{m}\sin\frac{\pi}{n}},$$ then the group $\langle f , g\rangle$ is discrete non-elementary and isomorphic to the free product $\langle f \rangle* \langle g\rangle$.