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 C-spheres of Falbel-Zocca are related to the R-spheres (hybrid spheres) of Schwartz.
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