On the center of the group of quasi-isometries of the real line

TL;DR

This paper investigates the structure of the group of all quasi-isometries of the real line, revealing its center is trivial and describing its subgroup decomposition related to end-fixing properties.

Contribution

It establishes an isomorphism between the subgroup fixing both ends and a direct product, and proves the center of the entire group is trivial.

Findings

The subgroup fixing both ends is isomorphic to a direct product of two subgroups.

The center of the group of all quasi-isometries of the real line is trivial.

Provides structural insights into the group of quasi-isometries of the real line.

Abstract

Let $QI(\mathbb{R})$ denote the group of all quasi-isometries $f:\mathbb{R}\to \mathbb{R}.$ Let $Q_+( \text{and}~ Q_-)$ denote the subgroup of $QI(\mathbb{R})$ consisting of elements which are identity near $-\infty$ (resp. $+\infty$). We denote by $QI^+(\mathbb R)$ the index $2$ subgroup of $QI(\mathbb R)$ that fixes the ends $+\infty, -\infty$. We show that $QI^+(\mathbb R)\cong Q_+\times Q_-$. Using this we show that the center of the group $QI(\mathbb{R})$ is trivial.