Discreteness of the complex hyperbolic ultra-parallel triangle groups

Wei Liao; Baohua Xie

paper

Discreteness of the complex hyperbolic ultra-parallel triangle groups

arXiv:2504.04407

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.

30 pages, 7 figures, to appear in Algebraic and Geometric Topology