The moduli space of the modular group in three-dimensional complex hyperbolic geometry

TL;DR

This paper fully constructs the moduli space of certain representations of the modular group into PU(3,1), revealing its structure as unions of parameterized spaces and connecting to previous geometrical studies.

Contribution

It provides the first complete construction of the PU(3,1)-representation space for the modular group, including explicit parameterizations and connections to lower-dimensional cases.

Findings

The moduli space is a union of three parts with explicit parameterizations.

Two subspaces are parameterized by a square, linking to PU(2,1) representations.

The constructed spaces interpolate previous geometrical models.

Abstract

We study the moduli space of discrete, faithful, type-preserving representations of the modular group $\mathbf{PSL}(2,\mathbb{Z})$ into $\mathbf{PU}(3,1)$. The entire moduli space $\mathcal{M}$ is a union of $\mathcal{M}(0,\frac{2\pi}{3},\frac{4\pi}{3})$, $\mathcal{M}(\frac{2\pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$ and some isolated points. This is the first Fuchsian group such that its $\mathbf{PU}(3,1)$-representations space has been entirely constructed. Both $\mathcal{M}(0,\frac{2\pi}{3},\frac{4\pi}{3})$ and $\mathcal{M}(\frac{2\pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$ are parameterized by a square, where two opposite sides of the square correspond to representations of $\mathbf{PSL}(2,\mathbb{Z})$ into the smaller group $\mathbf{PU}(2,1)$. In particular, both sub moduli spaces $\mathcal{M}(0,\frac{2\pi}{3},\frac{4\pi}{3} )$ and $\mathcal{M}(\frac{2\pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$ interpolate the geometries studied in \cite{FalbelKoseleff:2002} and \cite{Falbelparker:2003}.