Free isometric actions on the affine space \(\mathbb{Q}^n\).
type: Item
instance of: scholarly article
title: Free isometric actions on the affine space \(\mathbb{Q}^n\).
zbMATH Open document ID: 1033.20025
DOI: 10.1016/S0019-3577(02)80017-9
author: Kenzi Satô
published in: Indagationes Mathematicae. New Series
publication date: 2003-06-01T00:00:00Z
review text: Let \(G\) be a group of isometries on a metric space \(X\). The action of \(G\) is said to be `locally commutative' if any pair of elements with a common fixed point commutes. For example if \(G\) acts without fixed points its action is locally commutative. This paper is concerned with the action of the special orthogonal group \(\text{SO}_n(\mathbb{Q})\) on the space \(\mathbb{Q}^n\), where \(\mathbb{Q}\) is the set of rational numbers. Using previous results it is proved that, for all \(n \geq 3\), the group \(\text{SO}_n(\mathbb{Q})\) has a subgroup, isomorphic to \(F_2\), which acts on the non-zero elements of \(\mathbb{Q}^n\) in a locally commutative way. It follows that any subgroup of such a group which fixes a non-zero element of \(\mathbb{Q}^n\) is necessarily cyclic. Results of this type have a number of important geometric implications, going back to work of F. Hausdorff. If the group \(F_2\) acts on a space \(X\) in this way, then it often follows that \(X\) can be partitioned by means of subsets which are permuted by elements of the group.
reviewed by: A. W. Mason
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 51F25
zbMATH DE Number: 1893652
zbMATH Keywords: free groups of isometries
zbMATH Keywords: metric spaces
zbMATH Keywords: locally commutative actions
zbMATH Keywords: fixed point free actions
zbMATH Keywords: special orthogonal groups
MaRDI profile type: Publication
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full work available at URL: s0019-3577(02)80017-9
OpenAlex ID: W2094788291
description: scientific article; zbMATH DE number 1893652