On homotopy invariance for homology of rank two groups.
type: Item
instance of: scholarly article
title: On homotopy invariance for homology of rank two groups.
zbMATH Open document ID: 1267.20062
author: Matthias Wendt
published in: Journal of Pure and Applied Algebra
publication date: 2013-01-30T00:00:00Z
review text: Homotopy invariance fails for homology of elementary groups of rank two over integral domains which are not fields. The proof is an adaptation of the argument used by Behr to show that rank two groups are not finitely presentable. As a by-product, the author obtains examples of rings where the Steinberg group \(\text{St}_3\) is not a central extension of the elementary group \(E_3\). It is also shown that homotopy invariance works for the Steinberg groups of rank two groups over integral domains with many units.
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20G35
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 19C09
Mathematics Subject Classification ID: 55N35
zbMATH DE Number: 6132796
zbMATH Keywords: homotopy invariance
zbMATH Keywords: homology of linear groups
zbMATH Keywords: elementary subgroups
zbMATH Keywords: polynomial rings
zbMATH Keywords: buildings
zbMATH Keywords: Chevalley groups
zbMATH Keywords: Steinberg groups
reviewed by: L. N. Vaserstein
MaRDI profile type: Publication
full work available at URL: j.jpaa.2012.03.004
OpenAlex ID: W2061350790
cites work: Whitehead groups of chevalley groups over polynomial rings
cites work: Buildings
cites work: Q3872657
cites work: Solution of the congruence subgroup problem for \(\text{SL}_ n\) \((n\geq 3)\) and \(\text{Sp}_{2n}\) \((n\geq 2)\)
cites work: Q3999894
cites work: Reductive groups over a local field
cites work: Finiteness properties of arithmetic groups over function fields.
cites work: On symplectic groups over polynomial rings
cites work: Free quotients of 𝑆𝐿₂(𝑅[𝑥])
cites work: Homology of linear groups
cites work: An algorithmic proof of Suslin's stability theorem for polynomial rings
cites work: Q3889265
cites work: Q3877860
cites work: Generators, Relations and Coverings of Chevalley Groups Over Commutative Rings
cites work: Q4124941
cites work: Groups of cohomological dimension one
cites work: Q4155073
DOI: 10.1016/J.JPAA.2012.03.004
description: scientific article; zbMATH DE number 6132796