Lacunary free actions on the real line
type: Item
instance of: scholarly article
title: Lacunary free actions on the real line
zbMATH Open document ID: 0569.54039
DOI: 10.1016/0166-8641(85)90073-2
author: Rainer Löwen
published in: Topology and its Applications
publication date: 1985-01-01T00:00:00Z
review text: This paper determines, up to topological conjugacy, all noncyclic subgroups of the homeomorphism group of the real line \({\mathbb{R}}\) whose action on \({\mathbb{R}}\) is free. Such a subgroup G has, in a natural way, the structure of an Archimedean ordered group (as has been observed by \textit{H. Salzmann}, Arch. Math. 9, 447-454 (1958; Zbl 0082.358)). Therefore, as a group, G is isomorphic to a subgroup of the additive group \({\mathbb{R}}^+\) (by a theorem of Hölder). The action of G on \({\mathbb{R}}\), however, need not be equivalent to the standard action of the corresponding subgroup of \({\mathbb{R}}^+\). It is shown that the action is in fact standard if there is a dense orbit. Actions without such a dense orbit are termed lacunary; it is at first sight surprising that also every noncyclic countable subgroup of \({\mathbb{R}}^+\) (even though a dense subgroup) has such lacunary free actions (the countability is necessary, of course). These actions can all be explicitly constructed and completely classified by a method which roughly consists in ''blowing up'' the points of countably many orbits of the standard action to become closed intervals. In applications it is convenient to have criteria for the action to be standard. With this aim, it is shown that lacunary free actions of noncyclic groups on \({\mathbb{R}}\) have both discrete and nondiscrete orbits. It follows for instance that a transitive action of a group H as a group of homeomorphisms of \({\mathbb{R}}\) cannot induce a lacunary free action of a noncyclic normal subgroup. The methods of this paper could also be of interest in the domain of free topological or differentiable actions of \({\mathbb{Z}}\) on the circle [see, e.g., \textit{M. R. Herman}, Publ. Math., Inst. Hautes Étud. Sci. 49, 5- 233 (1979; Zbl 0448.58019)]; by lifting them to actions of \({\mathbb{Z}}\times {\mathbb{Z}}\) on \({\mathbb{R}}\) one could avail oneself of the order-theoretical methods used here.
Mathematics Subject Classification ID: 54H15
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 06A05
zbMATH DE Number: 3909322
zbMATH Keywords: noncyclic subgroups of the homeomorphism group of the real line
zbMATH Keywords: dense orbit
zbMATH Keywords: lacunary free actions
zbMATH Keywords: transitive action
reviewed by: Hermann Hähl
MaRDI profile type: Publication
full work available at URL: 0166-8641(85)90073-2
OpenAlex ID: W1981734474
cites work: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. (On smooth conjugacy of diffeomorphisms of the circle with rotations)
cites work: Q3223453
cites work: Kompakte zweidimensionale projektive Ebenen
cites work: Q5672987
cites work: Q3843292
Wikidata QID: Q126437499
description: scientific article; zbMATH DE number 3909322