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.