= Scaled Heisenberg lattice
{c}
{title2=$N_k=\{(x,y,z):x,y,z\in k\mathbb Z\}$}
In the <real Heisenberg group> with multiplication $(x,y,z)(x',y',z')=(x+x',y+y',z+z'+xy')$, the displayed set is a discrete cocompact subgroup. Writing $a=(k,0,0)$, $b=(0,k,0)$, $c=(0,0,k)$ gives $[a,b]=c^k$ and $c$ central. Hence its <abelianization> is $\mathbb Z^2\oplus\mathbb Z/k$. The right quotient maps to $\mathbb R^2/k\mathbb Z^2$ by its first two coordinates; each fibre is the central circle $\mathbb R/k\mathbb Z$.
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