Unipotent torus-bundle group (${\Gamma }_b={\mathbb{Z}}^2{⋊}_{\left(\begin{array}{cc}1& b\\ 0& 1\end{array}\right)}\mathbb{Z}$)

The mapping torus of a torus map with this monodromy has fundamental group generated by $x,y,t$ with $x$ central and $ty{t}^{-1}=x^{b}y$. Normal forms $x^m{y}^n{t}^k$ multiply byThis proves torsion-freeness and nilpotence of class at most two. For $b\ne 0$ its center is $⟨x⟩$ and commutator subgroup $⟨x^b⟩$; for $b=0$ it is ${\mathbb{Z}}^3$. These realize all noncyclic torsion-free nilpotent closed orientable three-manifold groups.