Distorted and undistorted subgroups of the Lodha-Moore group

TL;DR

The paper investigates the large-scale geometry of the Lodha--Moore group $G_0$ by examining distortions of natural subgroups. It develops two lower-bound tools for word length via $G_0$’s realizations on $\mathbb{R}$ and on the Cantor set, and applies them to show that $BS(1,2)$ is undistorted in $G_0$ while Thompson's group $F$ is exponentially distorted. The work leverages explicit element constructions—including a Cantor-set realization and explicit subgroups isomorphic to $BS(1,2)$—to relate intrinsic word lengths to ambient presentations. These results illuminate how $G_0$ differs geometrically from $F$ and contribute to understanding distortion phenomena in groups closely related to Thompson's groups, with implications for the geometry of subgroups and potential extensions to $G_0(n)$.

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 $G_0$.

Figures (4)

• Figure 1: Two generators of $F$.
• Figure 2: Tree diagrams of $g_1$ and $g_2$.
• Figure 3: The reduced pair of $x_0x_1^2x_0^{-1}x_1^{-1}x_0x_1^{-1}x_0^{-1}$.
• Figure 4: The reduced pair of $a_n$.

Theorems & Definitions (13)

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