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

Swarup Bhowmik, Prateep Chakraborty

Published 2023-03-30Version 1

It is well-known that $QI(\mathbb{R})\cong(QI(\mathbb{R}_{+})\times QI(\mathbb{R}_{-}))\rtimes <t>$, where $QI(\mathbb{R}_{+})(\cong QI(\mathbb{R_-}))$ is the group of quasi-isometries of the real line. 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.