Complex hyperbolic (3,n,∞) triangle groups

Abstract Let n ≥ 5 be a positive integer. A complex hyperbolic ( 3 , n , ∞ ) triangle group is a representation from the hyperbolic ( 3 , n , ∞ ) triangle group into the holomorphic isometry group of complex hyperbolic plane, which maps the generators to complex reflections fixing complex lines. In this paper, we show that a complex hyperbolic ( 3 , n , ∞ ) triangle group 〈 I 1 , I 2 , I 3 〉 is discrete and faithful if and only if I 1 I 3 I 2 I 3 is not elliptic. Our result answers a conjecture of Schwartz for complex hyperbolic ( 3 , n , ∞ ) triangle groups.