Complete hyperbolic structures in the complement of the Borromean rings

TL;DR

This paper proves that the complement of the Borromean rings in 3-sphere admits a unique hyperbolic structure by analyzing its fundamental group's representations into PSL(2,C).

Contribution

It establishes the exact number of faithful, discrete, parabolic-meridian representations of the Borromean rings' fundamental group, confirming the uniqueness of its hyperbolic structure.

Findings

Exactly two such representations exist.

Borromean rings have a unique hyperbolic structure.

The fundamental group's representations determine hyperbolic geometry.

Abstract

In this note, we show the fundamental group of the complement of the Borromean rings in $\Bbb{S}^3$ has exactly two representations in ${\rm PSL}(2,\Bbb{C})$ which are faithful, discrete and send meridians into parabolic elements. Using this result we are able to show that Borromean rings admit exactly one hyperbolic structure.