The group of quasi-isometries of the real line cannot act effectively on the line

TL;DR

The paper proves that the group of orientation-preserving quasi-isometries of the real line is left-orderable and non-simple but cannot act effectively on the real line, revealing limitations on its dynamical actions.

Contribution

It establishes that this specific group cannot have an effective action on the real line, despite being left-orderable and non-simple, highlighting new constraints on such groups.

Findings

The group is left-orderable.

The group is non-simple.

The group cannot act effectively on the real line.

Abstract

We prove that the group $\mathrm{QI}^{+}(\mathbb{R})$ of orientation-preserving quasi-isometries of the real line is a left-orderable, non-simple group, which cannot act effectively on the real line $\mathbb{R}.$