= Borromean link group
{c}
{title2=$\langle x,y,z\mid[x,[y^{-1},z]]=[y,[z^{-1},x]]=1\rangle$}
With <group commutator> convention $[a,b]=a^{-1}b^{-1}ab$, the displayed <group presentation> describes the <fundamental group> of the <Borromean rings> complement. Its <abelianization> is $\mathbb Z^3$, since each relator is a <group commutator> and hence imposes no abelian relation. It embeds discretely in the <Gaussian Bianchi group> by sending the meridians to $\begin{pmatrix}1&1\\0&1\end{pmatrix}$, $\begin{pmatrix}1&0\\2i&1\end{pmatrix}$ and $\begin{pmatrix}i&1\\2i&2-i\end{pmatrix}$.
Back to article page
