Complex hyperbolic (3,3,n)-triangle groups.

Abstract

Let p,q,rp,q,r be positive integers. Complex hyperbolic (p,q,r)(p,q,r) triangle groups are representations of the hyperbolic (p,q,r)(p,q,r) reflection triangle group to the holomorphic isometry group of complex hyperbolic space H2CHℂ2, where the generators fix complex lines. In this paper, we obtain all the discrete and faithful complex hyperbolic (3,3,n)(3,3,n) triangle groups for n≥4n≥4. Our result solves a conjecture of Schwartz in the case when p=q=3p=q=3.