Finiteness properties of certain arithmetic groups in the function field case
type: Item
instance of: scholarly article
title: Finiteness properties of certain arithmetic groups in the function field case
zbMATH Open document ID: 0819.20051
DOI: 10.1007/BF02782847
author: Herbert Abels
published in: Israel Journal of Mathematics
publication date: 1992-12-09T00:00:00Z
review text: Arithmetic subgroups of reductive algebraic groups over number fields are finitely presentable, but in the case of global function fields the situation becomes more complicated. To attack this problem cohomological methods are often applied. A group \(\Gamma\) is called of type \(FP_ n\) if the \(Z\Gamma\)-module \(Z\) (with trivial \(\Gamma\)-module-structure, i.e. \(\gamma\cdot n = n\) for \(\gamma \in \Gamma\), \(n \in Z\)) admits a projective resolution which is finitely generated in dimensions \(\leq n\). The group \(\Gamma\) is of type \(FP_ 1\) if and only if \(\Gamma\) is finitely generated, and \(\Gamma\) is of type \(FP_ 2\) if it is finitely presented. Since the converse of the last assertion is not known one defines a group to be of type \(F_ n\) for \(n \geq 2\), if it is finitely presented and of type \(FP_ n\). The largest \(n\) such that \(\Gamma\) is of type \(F_ n\) is called the finiteness length of \(\Gamma \) and is denoted by \(\Phi(\Gamma)\). The main result of the paper under review is the following Theorem: \(\Phi(\text{SL}_ n(F_ q[t])) = n - 2\) if \(n \geq 2\) and \(q \geq 2^{n-2}\) (\(F_ q\) the finite field with \(q\) elements). The proof of the theorem uses a careful study of the action of \(\Gamma\) on the Bruhat-Tits building \(X\) of \(\text{GL}_{n+1}\) and a general result of K. Brown on the cellular action of groups on contractible CW- complexes.
reviewed by: Hans-Jochen Bartels
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E42
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 78197
zbMATH Keywords: arithmetic subgroups
zbMATH Keywords: reductive algebraic groups
zbMATH Keywords: global function fields
zbMATH Keywords: projective resolutions
zbMATH Keywords: finitely generated groups
zbMATH Keywords: finitely presented groups
zbMATH Keywords: finiteness length
zbMATH Keywords: Bruhat-Tits buildings
zbMATH Keywords: cellular action of groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 78197