On the finiteness length of some soluble linear groups
| Autor: | Rego, Yuri Santos |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Canad. J. Math., vol. 74, no. 5, 2022 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.4153/S0008414X21000213 |
| Popis: | 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 $\mathbf{B}_2^\circ(R)=\left( \begin{smallmatrix} * & * \\ 0 & * \end{smallmatrix} \right)\leq\mathrm{SL}_2(R)$ 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 We also give an alternative proof of an unpublished theorem due to Strebel, characterizing finite presentability of Abels' groups $\mathbf{A}_n(R) \leq \mathrm{GL}_n(R)$ in terms of $n$ and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. Comment: 35 pages. v3: Incorporated referees' suggestions. Final version, to appear in the Canadian Journal of Mathematics |
| Databáze: | arXiv |
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