Abelianization of a semidirect product by the integers
= Abelianization of a semidirect product by the integers
Let $B\in\operatorname{GL}_d(\mathbb Z)$ and $\Gamma_B=\mathbb Z^d\rtimes_B\mathbb Z$. If $t$ generates the second factor, then the elements $tnt^{-1}n^{-1}$ fill $(B-I)\mathbb Z^d$, and hence
$$
\Gamma_B^{\mathrm{ab}}\cong
\mathbb Z\oplus\mathbb Z^d/(B-I)\mathbb Z^d.
$$
If $1$ is not an <eigenvalue> of $B$, the second summand is finite.