Flexibility of ideal triangle groups in complex hyperbolic geometry

We show that the Teichmüller space of the ideal triangle group in the automorphism group of complex hyperbolic space contains a real four-dimensional ball. This implies the existence of a four-dimensional family of spherical CR structures on the trivial circle bundle over the sphere minus three points. The proof is an explicit construction of fundamental domains whose boundaries are special hypersurfaces foliated by complex geodesics.