A uniformizable spherical CR structure on a two-cusped hyperbolic 3-manifold

TL;DR

This paper proves a conjecture about complex hyperbolic triangle groups and constructs a spherical CR structure on a specific two-cusped hyperbolic 3-manifold, linking group discreteness to geometric structures.

Contribution

It confirms Schwartz's conjecture for a class of complex hyperbolic triangle groups and establishes a uniformizable spherical CR structure on a two-cusped hyperbolic 3-manifold.

Findings

Discreteness of the triangle group is characterized by a specific element being nonelliptic.

When the element is parabolic, a spherical CR structure on the manifold is constructed.

The even subgroup provides the holonomy representation of this CR structure.

Abstract

Let $\langle I_{1}, I_{2}, I_{3}\rangle$ be the complex hyperbolic $(4,4,\infty)$ triangle group. In this paper we give a proof of a conjecture of Schwartz for $\langle I_{1}, I_{2}, I_{3}\rangle$. That is $\langle I_{1}, I_{2}, I_{3}\rangle$ is discrete and faithful if and only if $I_1I_3I_2I_3$ is nonelliptic. When $I_1I_3I_2I_3$ is parabolic, we show that the even subgroup $\langle I_2 I_3, I_2I_1 \rangle$ is the holonomy representation of a uniformizable spherical CR structure on the two-cusped hyperbolic 3-manifold $s782$ in SnapPy notation.