The moduli space of the modular group in complex hyperbolic geometry.

We construct the space of discrete, faithful, type-preserving representations of the modular group into the isometry group of complex hyperbolic 2-space up to conjugacy. This is the first Fuchsian group for which the entire complex hyperbolic deformation space has been constructed. We also show how the ³-spheres of Falbel-Zocca are related to the Â-spheres (hybrid spheres) of Schwartz