A reflection triangle \((p,q,r)\) group \(\Delta_{p,q,r}\) is the group with generators \(\sigma_1,\sigma_2,\sigma_3\) and relations \(\sigma_1^2=\sigma_2^2=\sigma_3^2=(\sigma_2\sigma_3)^p=(\sigma_3\sigma_1)^q=(\sigma_1\sigma_2)^r=id\). This group may be realised as generated by the reflections on the sides of a triangle with internal angles \(\pi/p,\pi/q,\pi/r\). The triangle is spherical, Euclidean or hyperbolic, according to \(1/p+1/q+1/r\) being greater than, equal to, or less than, 1, respectively.
The paper under review considers representations of \(\Delta_{p,q,r}\) to \(SU(2,1)\), the group of holomorphic isometries of the complex hyperbolic space \(H_\mathbb C^2\). Such a representation is called a complex hyperbolic triangle group, and the generators of the group fix complex lines. R. E. Schwartz studied these groups in his Beijing 2002 ICM survey [Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20-28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press; Singapore: World Scientific. 339-349 (2002; Zbl 1022.53034)]. He made there a conjecture on necessary and sufficient conditions in order for any complex hyperbolic representation of \(\Delta_{p,q,r}\) to be discrete and faithful.
The main result of the present paper is Theorem 1.6, which solves the conjecture of Schwartz for \(p=q=3\), in the following terms: Theorem. Let \(n\geq 4\), and \(\Gamma=\langle I_1,I_2,I_3\rangle\) be a complex hyperbolic \((3,3,n)\) triangle group. Then \(\Gamma\) is a discrete and faithful representation of \(\Delta_{3,3,n}\) if and only if the product \(I_1I_3I_2I_3\) is non-elliptic.