  Let $T$ be an infinite volume Coxeter tetrahedron in three dimensional real
hyperbolic space ${\bf H}^{3}_{\mathbb R}$ with two opposite right-angles and
the other angles are all zeros. Let $G$ be the Coxeter group of $T$, so
$$G=\left\langle \iota_1, \iota_2, \iota_3, \iota_4 \Bigg| \begin{array} {c}
\iota_1^2= \iota_2^2 = \iota_3^2=\iota_4^2=id, \\ (\iota_1
\iota_3)^{2}=(\iota_2 \iota_4)^{2}=id \end{array}\right\rangle$$ as an abstract
group. We study type-preserving representations $\rho: G \rightarrow
\mathbf{PU}(3,1)$, where $\rho( \iota_{i})=I_{i}$ is a complex reflection
fixing a complex hyperbolic plane in three dimensional complex hyperbolic space
${\bf H}^{3}_{\mathbb C}$ for $1 \leq i \leq 4$. 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
${\bf H}^{2}_{\mathbb C}$-geometry and ${\bf H}^{3}_{\mathbb R}$-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.

Geometric Topology;Group Theory