Complete hyperbolic structures in the complement of the Borromean rings

Abstract

In this note, we show the fundamental group of the complement of the Borromean rings in ${\mathbb{S}}^3$ has exactly two representations in $\mathrm{PSL}(2,\mathbb{C})$ 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.