An Interesting Family of Groups of Homeomorphisms of the Real Line

Abstract

This paper is meant to serve as an exposition on the 2019 paper [3] by Hyde and Lodha where they managed to resolve the question posed by Rhemtulla in 1980. The authors in the paper offer a construction of families of orientation preserving homeomorphisms of the real line. Each of these families can be viewed as a finitely generated, simple, left-orderable group. Hence, each of these families satisfies the criteria that Rhemtulla laid out. In this paper, we primarily intend to offer a simplified construction of a family of groups Gρ that the authors have constructed in the first part of their paper. The mathematics involved with the justification that these groups truly do follow the criteria laid out by Rhemtulla is not discussed in this exposition. The reader, if interested in these details, should refer to the original paper. We begin by stating the question that Rhemtulla posed in 1980 and by explaining what it means. This is followed by a brief discussion of some preliminary concepts and tools needed for the rest of the paper. The construction is then laid out. The family of groups thus constructed contains ℵ0´many groups that satisfy the criteria of Rhemtulla. The authors of the original paper use this family to construct a much bigger family of such groups where they are able to find continuum many groups that satisfy the Rhemtulla criteria. The reader, if interested should again refer to the original paper for this construction. The construction of Gρ in this paper is simplified to a great degree for the reader’s ease and we hope that therefore it would serve as a good introduction to the original paper.