= Nilpotence criterion for an abelian-by-cyclic group
{title2=$\mathbb Z^r\rtimes_B\mathbb Z\text{ is nilpotent}\iff B-I\text{ is nilpotent}$}
For $tvt^{-1}=Bv$, the <group commutator> convention $[x,y]=x^{-1}y^{-1}xy$ gives $[t,v]=(I-B^{-1})v$ in additive notation. Its image is $(B-I)\mathbb Z^r$, since $B$ is an integral <automorphism>. The lower central series from its second term is $(B-I)\mathbb Z^r,(B-I)^2\mathbb Z^r,\ldots$. It terminates exactly when $B-I$ is a <nilpotent matrix>. For rank two this means $(B-I)^2=0$, equivalently $\det B=1$ and $\operatorname{tr}B=2$. Nilpotence of a finite-index <subgroup> alone gives virtual nilpotence of the whole group.
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