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We study the moduli space of discrete, faithful, type-preservingrepresentations of the modular group $\mathbf{PSL}(2,\mathbb{Z})$ into$\mathbf{PU}(3,1)$. The entire moduli space $\mathcal{M}$ is a union of$\mathcal{M}(0,\frac{2\pi}{3},\frac{4\pi}{3})$,$\mathcal{M}(\frac{2\pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$ and some isolatedpoints. This is the first Fuchsian group such that its$\mathbf{PU}(3,1)$-representations space has been entirely constructed. Both$\mathcal{M}(0,\frac{2\pi}{3},\frac{4\pi}{3})$ and$\mathcal{M}(\frac{2\pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$ are parameterizedby a square, where two opposite sides of the square correspond torepresentations of $\mathbf{PSL}(2,\mathbb{Z})$ into the smaller group$\mathbf{PU}(2,1)$. In particular, both sub moduli spaces$\mathcal{M}(0,\frac{2\pi}{3},\frac{4\pi}{3} )$ and$\mathcal{M}(\frac{2\pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$ interpolate thegeometries studied in \cite{FalbelKoseleff:2002} and \cite{Falbelparker:2003}.

The moduli space of the modular group in three-dimensional complex hyperbolic geometry