= Unipotent torus-bundle group
{title2=$\Gamma_b=\mathbb Z^2\rtimes_{\left(\begin{smallmatrix}1&b\\0&1\end{smallmatrix}\right)}\mathbb Z$}
The <mapping torus> of a <torus> map with this monodromy has <fundamental group> generated by $x,y,t$ with $x$ central and $tyt^{-1}=x^by$. Normal forms $x^my^nt^k$ multiply by
$$
(m,n,k)(m',n',k')=(m+m'+bkn',\ n+n',\ k+k').
$$
This proves torsion-freeness and nilpotence of class at most two. For $b\ne0$ its center is $\langle x\rangle$ and commutator subgroup $\langle x^b\rangle$; for $b=0$ it is $\mathbb Z^3$. These realize all noncyclic torsion-free nilpotent closed orientable three-manifold groups.
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