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 〈I1,I2,I3〉 is discrete and faithful if and only if I1I3I2I3 is not elliptic. Our result answers a conjecture of Schwartz for complex hyperbolic (3,n,∞) triangle groups.