Finite presentability of S-arithmetic groups. Compact presentability of solvable groups
type: Item
instance of: scholarly article
title: Finite presentability of S-arithmetic groups. Compact presentability of solvable groups
zbMATH Open document ID: 0621.20015
published in: Lecture Notes in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: This monograph gives a self-contained proof of what is essentially a single result (with a considerable number of consequences). The proof has not appeared elswhere in the literature although a discussion of the work (without proofs) appears in the Proceedings of the 1985 conference at St. Andrews [Lond. Math. Soc. Lect. Note Ser. 121, 128-134 (1986; Zbl 0607.20027)]. The author is interested in the following general problems. Let G denote a linear algebraic group. Suppose G is defined over an algebraic number field k, S is a finite set of primes of k and \(\Gamma\) is an S-arithmetic subgroup of G. How can it be determined whether \(\Gamma\) has a finite presentation? Suppose that K is a p-adic field (p a prime in S, say) so that the K-points \(G_ K\) of G are a subgroup of the general linear group over K. Since the latter is a topological group, it makes sense to refer to a compact generating set or even compact presentation for \(G_ K\). How can we determine whether \(G_ K\) has a compact presentation? These are well studied problems. \textit{A. Borel} and \textit{Harish-Chandra} [Ann. Math., II. Ser. 75, 485-535 (1962; Zbl 0107.148)] have shown that arithmetic groups are always finitely presented and \textit{H. Behr} [Invent. Math. 4, 265-274 (1967; Zbl 0157.367)] has shown that if G is reductive, then \(\Gamma\) is always finitely presented, whatever the set S of primes. In the case that K is a p-adic field, \textit{A. Borel} and \textit{J. Tits} [Publ. Math., Inst. Hautes Etud. Sci. 27, 659-755 (1965; Zbl 0145.174)] have shown that G will always have a maximal K-split solvable subgroup H so that \(G_ K/H_ K\) is compact and have given a criterion for solving the question of whether G is compactly generated. The connection between the two clauses of the title is given by a Theorem of \textit{M. Kneser} [J. Reine Angew. Math. 214/215, 345-349 (1964; Zbl 0141.026)]. Kneser shows that \(\Gamma\) is finitely generated (finitely presented) if and only if, for each p…
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 20G25
Mathematics Subject Classification ID: 20-02
zbMATH DE Number: 4006522
zbMATH Keywords: linear algebraic group
zbMATH Keywords: algebraic number field
zbMATH Keywords: S-arithmetic subgroup
zbMATH Keywords: finite presentation
zbMATH Keywords: p-adic field
zbMATH Keywords: compact generating set
zbMATH Keywords: compact presentation
zbMATH Keywords: arithmetic groups
zbMATH Keywords: finitely presented
zbMATH Keywords: solvable subgroup
zbMATH Keywords: finitely generated
zbMATH Keywords: group of K-points
zbMATH Keywords: K-split soluble linear algebraic group
zbMATH Keywords: unipotent radical
zbMATH Keywords: Lie algebra
zbMATH Keywords: Abelianisation
zbMATH Keywords: weights
author: Herbert Abels
reviewed by: J. R. J. Groves
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4006522