On the finiteness length of some soluble linear groups

Abstract

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 ${\mathbf{B}}_2^{\circ }(R)=\left(\begin{array}{cc}\ast & \ast \\ 0& \ast \end{array}\right)\le {\mathrm{SL}}_2(R)$ 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 We also give an alternative proof of an unpublished theorem due to Strebel, characterizing finite presentability of Abels' groups ${\mathbf{A}}_n(R)\le {\mathrm{GL}}_n(R)$ in terms of $n$ and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.

35 pages. v3: Incorporated referees' suggestions. Final version, to appear in the Canadian Journal of Mathematics