Discreteness of the complex hyperbolic ultra-parallel triangle groups

2025/04/06 by Wei Liao, Liao, Wei, Baohua Xie +1
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2504.04407

openalex publication_date 2025/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that a family of complex hyperbolic ultra-parallel [m₁, m₂, m₃]-triangle group representations, where \( m₃ > 0 \), is discrete and faithful if and only if the isometry \( R₁(R₂R₁)ⁿR₃ \) is non-elliptic for some positive integer \( n \). Additionally, we investigate the special case where \( m₃ = 0 \) and provide a substantial improvement upon the main result by Monaghan, Parker, and Pratoussevitch.

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• Complex Hyperbolic Triangle Groups of Type [m,m,0; n₁, n₂, 2]
• Complex Hyperbolic Triangle Groups with 2-fold Symmetry
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• Notes on complex hyperbolic triangle groups of type (m,n,infinity)