On the limit set of a spherical CR uniformization

TL;DR

This paper investigates the geometric structure of the limit set in a specific spherical CR uniformization of a cusped hyperbolic manifold, revealing its complex topology and properties of associated cusps.

Contribution

It establishes the structure of the limit set as a union of r-circles, proves its connectedness, and analyzes the fundamental group of its complement, advancing understanding of spherical CR uniformizations.

Findings

Limit set is the closure of a countable union of r-circles.

The limit set is connected and contains a Hopf link with three components.

The fundamental group of the complement in S^3 is not finitely generated.

Abstract

We explore the limit set of a particular spherical CR uniformization of a cusped hyperbolic manifold. We prove that the limit set is the closure of a countable union of $\mathbb{R}$-circles, is connected, and contains a Hopf link with three components; we also show that the fundamental group of its complement in $S^3$ is not finitely generated. Additionally, we prove that rank-one spherical CR cusps are quotients of horotubes.