Three-dimensional complex reflection groups via Ford domains

Abstract: We initiate the study of deformations of groups in three-dimensional complex hyperbolic geometry. Let $$G=\left\langle \iota_1, \iota_2, \iota_3, \iota_4 \Bigg| \begin{array}{c} \iota_1^2= \iota_2^2 = \iota_3^2=\iota_4^2=id,\ (\iota_1 \iota_3)^{2}=(\iota_1 \iota_4)^{3}=(\iota_2 \iota_4)^{2}=id \end{array}\right\rangle$$ be an abstract group. We study representations $\rho: G \rightarrow \mathbf{PU}(3,1)$, where $\rho( \iota_{i})=I_{i}$ is a complex reflection fixing a complex hyperbolic plane in ${\bf H}^{3}_{\mathbb C}$ for $1 \leq i \leq 4$, with the additional condition that $I_1I_2$ is parabolic. When we assume two pairs of hyper-parallel complex hyperbolic planes have the same distance, then the moduli space $\mathcal{M}$ is parameterized by $(h,t) \in [1, \infty) \times [0, \pi]$ but $t \leq \operatorname{arccos}(-\frac{3h^2+1}{4h^2})$. In particular, $t=0$ and $t=\operatorname{arccos}(-\frac{3h^2+1}{4h^2})$ degenerate to ${\bf H}^{3}_{\mathbb R}$-geometry and ${\bf H}^{2}_{\mathbb C}$-geometry respectively. Using the Ford domain of $\rho_{(\sqrt{2},\operatorname{arccos}(-\frac{7}{8}))}(G)$ as a guide, we show $\rho_{(h,t)}$ is a discrete and faithful representation of $G \rightarrow \mathbf{PU}(3,1)$ when $(h,t) \in \mathcal{M}$ is near to $(\sqrt{2}, \operatorname{arccos}(-\frac{7}{8}))$. This is the first nontrivial example of the Ford domain of a subgroup in $\mathbf{PU}(3,1)$ that has been studied.