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 $\textbf {B}_2^{\circ }(R)=\left ( \begin {smallmatrix} * & * \\ 0 & * \end {smallmatrix}\right )\leq \operatorname {\textrm {SL}}_2(R)$ whenever G admits certain 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 We also give an alternative proof of an unpublished theorem due to Strebel, characterizing finite presentability of Abels’ groups $\textbf {A}_n(R) \leq \operatorname {\textrm {GL}}_n(R)$ in terms of n and $\textbf {B}_2^{\circ }(R)$ . 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.

给定一个可交换的单位环R，我们证明了一个群G的有限长度是由第一阶的 Borel 子群的有限长度限制的 $\textbf {B}_2^{\circ }(R)=\left ( \begin {smallmatrix} * & * \\ 0 & * \end {smallmatrix}\right )\leq \operatorname {\textrm {SL}}_2(R)$ 每当G承认某些具有元贝尔图像的R表示时。结合 Bestvina–Eskin–Wortman 和 Gandini 的结果，这给出了 Bux 对S算术 Borel 群的有限长度相等性的（推广）新证明。由于 Strebel，我们还给出了一个未发表的定理的替代证明，表征了 Abel 群的有限表示性 $\textbf {A}_n(R) \leq \operatorname {\textrm {GL}}_n(R)$ 在n和 $\textbf {B}_2^{\circ }(R)$ 方面。这概括了 Remeslennikov、Holz、Lyul'ko、Cornulier-Tessera 的早期结果，并指出了关于此类群的有限长度的猜想。