A free group acting on \(\mathbb{Z}^2\) without fixed points
type: Item
instance of: scholarly article
title: A free group acting on \(\mathbb{Z}^2\) without fixed points
zbMATH Open document ID: 0989.20025
author: Kenzi Satô
published in: L'Enseignement Mathématique. 2e Série
publication date: 2001-01-30T00:00:00Z
review text: Let \(\text{SA}_2(\mathbb{Z})\) be the group of all transformations of \(\mathbb{Z}^2\) of the type \({\mathbf x}\mapsto A{\mathbf x}+{\mathbf a}\), where \(A\in\text{SL}_2(\mathbb{Z})\) and \({\mathbf a}\in\mathbb{Z}^2\). The author proves that \(\text{SA}_2(\mathbb{Z})\) contains a free group of rank 2 which acts on \(\mathbb{Z}^2\) without non-trivial fixed points. This is related to the problem of determining whether or not the more general group \(\text{SA}_2(\mathbb{R})\), which acts on \(\mathbb{R}^2\), contains a free subgroup \(F\) of rank 2, with the property that the stabilizer in \(F\) of every point of \(\mathbb{R}^2\) is commutative. Questions like these arise from the so-called Banach-Tarski paradox for bounded subsets of \(\mathbb{R}^3\).
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 20F38
zbMATH DE Number: 1558734
zbMATH Keywords: free groups
zbMATH Keywords: fixed points
reviewed by: A. W. Mason
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1558734