{ "id": "1901.06704", "version": "v1", "published": "2019-01-20T17:29:26.000Z", "updated": "2019-01-20T17:29:26.000Z", "title": "On the finiteness length of some soluble linear groups", "authors": [ "Yuri Santos Rego" ], "comment": "32 pages", "categories": [ "math.GR" ], "abstract": "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 We also give an alternative proof of an unpublished theorem due to Strebel, characterizing---in terms of $n$ and $\\mathbf{B}_2^\\circ(R)$---the finite presentability of Abels' soluble groups $\\mathbf{A}_n(R)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.", "revisions": [ { "version": "v1", "updated": "2019-01-20T17:29:26.000Z" } ], "analyses": { "subjects": [ "20F65", "20H25", "20F05", "20G30", "57M07" ], "keywords": [ "finiteness length", "soluble linear groups", "arithmetic borel subgroups", "generalizes earlier results", "buxs equality" ], "note": { "typesetting": "TeX", "pages": 32, "language": "en", "license": "arXiv", "status": "editable" } } }