Free product of two elliptic quaternionic Möbius transformations (Q2360867)

Let \(f\) and \(g\) be two elliptic elements of PSL\((2, \mathbb{C})\) of orders \(m\geq 2\) and \(n\geq 2\) respectively, and \(\max\{m,n\}\geq 3\). Let \(\delta(f,g)\) be the hyperbolic distance between fix\((f)\) and fix\((g)\). In [\textit{F. W. Gehring} et al., Mitt. Math. Semin. Gießen 228, 9--15 (1996; Zbl 0869.30035)] it was proved that if \[ \cosh{\delta(f,g)}\geq \frac{\cos{\frac{\pi}{m}} \cos{\frac{\pi}{n}} +1}{\sin{\frac{\pi}{m}} \sin{\frac{\pi}{n}}}, \] then \(\langle f,g \rangle\) is discrete, non-elementary and isomorphic to the free product \(\langle f \rangle \ast \langle g \rangle\). In the paper under review an analogous theorem in the group PSp(1,1) is proved. Let us denote by \(\mathbb{H}\) the quaternions. The author takes \(\mathfrak{I}(\mathbb{H})\times \mathbb{R}^+\) as a model of the space \(\mathbf{H}^4\). The group Isom\((\mathbf{H}^4)\) is identified with PSp(1,1). In order to prove the above result, the author studies some properties of elliptic elements of PSp(1,1). Since an elliptic element in PSp(1,1) can be a boundary elliptic element or a regular elliptic element, the proof of the theorem is divided in three cases according to whether \(f\) and \(g\) are two boundary elements, a boundary element and a regular element, or two regular elements, respectively.