Discreteness of the complex hyperbolic ultra-parallel triangle groups

Abstract: 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.