Rigidity and flexibility of triangle groups in complex hyperbolic geometry

We show that the Teichmüller space of the triangle groups of type (p,q,∞) in the automorphism group of the two-dimensional complex hyperbolic space contains open sets of 0, 1 and two real dimensions. In particular, we identify the Teichmüller space near embeddings of the modular group preserving a complex geodesic.