Discreteness of the complex hyperbolic ultra-parallel triangle groups

We prove that a family of complex hyperbolic ultra-parallel $[m_1, m_2, m_3]$-triangle group representations, where $m_3 > 0$, is discrete and faithful if and only if the isometry $R_1(R_2R_1)^nR_3$ is non-elliptic for some positive integer $n$. Additionally, we investigate the special case where $m_3 = 0$ and provide a substantial improvement upon the main result by Monaghan, Parker, and Pratoussevitch.