Three-dimensional complex reflection groups via Ford domains

TL;DR

This paper explores the deformation space of certain complex hyperbolic groups in three dimensions, establishing conditions for discreteness and faithfulness of representations, and providing the first detailed Ford domain analysis in PU(3,1).

Contribution

It introduces a new family of complex hyperbolic groups in three dimensions, analyzes their deformation space, and constructs explicit Ford domains to study their geometric properties.

Findings

The moduli space is parameterized by (h,t) with specific bounds.

Discreteness and faithfulness are confirmed near a particular parameter point.

First detailed Ford domain example for a subgroup in PU(3,1).

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.