Lacunary free actions on the real line
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.
noncyclic subgroups of the homeomorphism group of the real line
scientific article; zbMATH DE number 3909322