Complexification of an infinite volume Coxeter tetrahedron

TL;DR

This paper explores the space of certain geometric group representations of an infinite volume Coxeter tetrahedron in hyperbolic space, revealing a continuous family of discrete, faithful representations in complex hyperbolic space.

Contribution

It provides a complete study of the moduli space of type-preserving representations of a Coxeter group into PU(3,1), including discreteness and faithfulness for all parameters in a specific range.

Findings

The moduli space is parameterized by an angle θ in [5π/6, π].

Representations are discrete and faithful for all θ in the parameter range.

Degenerations at the endpoints correspond to complex hyperbolic and real hyperbolic geometries.

Abstract

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.