Complexification of an infinite volume Coxeter tetrahedron

Let $T$ be an infinite volume Coxeter tetrahedron in three dimensional real hyperbolic space ${\mathbf{H}}_{\mathbb{R}}^3$ with two opposite right-angles and the other angles are all zeros. Let $G$ be the Coxeter group of $T$, so $$G=⟨{\iota }_1,{\iota }_2,{\iota }_3,{\iota }_4\mid \begin{array}{c}{\iota }_1^2={\iota }_2^2={\iota }_3^2={\iota }_4^2=id,\\ ({\iota }_1{\iota }_3{)}^2=({\iota }_2{\iota }_4{)}^2=id\end{array}⟩$$ as an abstract group. We study type-preserving representations $\rho :G\to \mathbf{P}\mathbf{U}(3,1)$, where $\rho ({\iota }_i)=I_i$ is a complex reflection fixing a complex hyperbolic plane in three dimensional complex hyperbolic space ${\mathbf{H}}_{\mathbb{C}}^3$ for $1\le i\le 4$. The moduli space $\mathcal{M}$ of these representations is parameterized by $\theta \in [\frac{5\pi }{6},\pi ]$. In particular, $\theta =\frac{5\pi }{6}$ and $\theta =\pi$ degenerate to ${\mathbf{H}}_{\mathbb{C}}^2$-geometry and ${\mathbf{H}}_{\mathbb{R}}^3$-geometry respectively. Via Dirichlet domains, we show $\rho ={\rho }_{\theta }$ is a discrete and faithful representation of the group $G$ for all $\theta \in [\frac{5\pi }{6},\pi ]$. This is the first nontrivial moduli space in three dimensional complex hyperbolic space that has been studied completely.