A structure theorem and left-orderability of a quotient of quasi-isometry group of the real line

Abstract

It is well-known that $QI(\mathbb{R})\cong(QI(\mathbb{R}_{+})\times QI(\mathbb{R}_{-}))\rtimes $, where $QI(\mathbb{R})$(resp. $QI(\mathbb{R}_{+})(\cong QI(\mathbb{R_-}))$) is the group of quasi-isometries of the real line (resp. $[0,\infty)$). We introduce an invariant for the elements of $QI(\mathbb{R_{+}})$ and split it into smaller units. We give an almost characterization of the elements of these units. We also show that a quotient of $QI(\mathbb{R_{+}})$ gives an example of a left-orderable group which is not locally indicable.