On the finiteness length of some soluble linear groups

Yuri Santos Rego

Published 2019-01-20Version 1

Let $R$ be a commutative ring with unity. We prove that the finiteness length of a group $G$ is bounded above by the finiteness length of the Borel subgroup of rank one $\mathbf{B}_2^\circ(R) = \left( \begin{smallmatrix} * & * \\ 0 & * \end{smallmatrix} \right) \leq \mathrm{SL}_2(R)$ whenever $G$ admits certain representations over $R$ with soluble image. We combine this with results due to Bestvina--Eskin--Wortman and Gandini to give a new proof of (a generalization of) Bux's equality on the finiteness length of $S$-arithmetic Borel subgroups. We also give an alternative proof of an unpublished theorem due to Strebel, characterizing---in terms of $n$ and $\mathbf{B}_2^\circ(R)$---the finite presentability of Abels' soluble groups $\mathbf{A}_n(R) \leq \mathrm{GL}_n(R)$. This generalizes earlier results due to Remeslennikov, Holz, Lyul'ko, Cornulier--Tessera, and points out to a conjecture about the finiteness lengths of such groups.