Distorted and undistorted subgroups of the Lodha-Moore group

TL;DR

This paper investigates the geometric properties of subgroups within the Lodha-Moore group, demonstrating that some are distorted while others are undistorted, revealing complex subgroup structures.

Contribution

It provides explicit bounds for subgroup distortions in the Lodha-Moore group, notably showing $BS(1,2)$ is undistorted and $F$ is distorted.

Findings

$BS(1,2)$ is undistorted in $G_0$

Thompson's group $F$ is distorted in $G_0$

Explicit lower bounds for word length in $G_0$

Abstract

We show that the Baumslag-Solitar group $BS(1,2)$ is undistorted in the Lodha-Moore group $G_0$ using an explicit lower bound for the word length of $G_0$. We also show that Thompson's group $F$ is distorted in $G_0$.