Free products of cyclic subgroups in the group \(\text{PSL}_2(\mathbb{C})\) (Q5942065)

The authors consider sufficient conditions under which a 2-generated subgroup \(\langle A,B\rangle\) of the \(\text{PSL}(2,\mathbb{C})\) is a free product of its cyclic subgroups \(\langle A\rangle\) and \(\langle B\rangle\). For instance, if \(\text{tr }A=2\cos{\pi\over n}\), \(n\geq 2\), \(\text{tr }B=2\cos{\pi\over m}\), \(m\geq 3\), and \[ \Biggl|\text{tr}(AB)-\cos\Biggl({\pi\over n}+{\pi\over m}\Biggr)\Biggr|\geq 4, \] then \(\langle A,B\rangle=\langle A\rangle*\langle B\rangle\cong\mathbb{Z}_n*\mathbb{Z}_m\).