On the finiteness length of some soluble linear groups

Given a commutative unital ring $R$, we show that the finiteness length of a group $G$ is bounded above by the finiteness length of the Borel subgroup of rank one $$\mathbf{B}$_2^\circ®=\left( \begin{smallmatrix} * & * \ 0 & * \end{smallmatrix} \right)\leq$\mathrm{S}\mathrm{L}$_2®$whenever$G$admitscertain$R$-representations with metabelian image. Combined with results due to Bestvina–Eskin–Wortman and Gandini, this gives a new proof of (a generalization of) Bux’s equality on the finiteness length of $S$-arithmetic Borel groups. We also give an alternative proof of an unpublished theorem due to Strebel, characterizing finite presentability of Abels’ groups ${\mathbf{A}}_n(R)\le {\mathrm{G}\mathrm{L}}_n(R)$ in terms of $n$ and ${\mathbf{B}}_2^{\circ }(R)$. This generalizes earlier results due to Remeslennikov, Holz, Lyul’ko, Cornulier–Tessera, and points out to a conjecture about the finiteness length of such groups.