Just infinite quotients of finitely generated subgroups of $PL^+[0,1]$

TL;DR

The paper investigates whether finitely generated subgroups of Thompson's group $F$ have every just infinite quotient virtually abelian, and extends the analysis to $\textup{PL}^+[0,1]$ and $\textup{H}(\mathbf{R})$. It develops a constructive argument using interval decompositions and supports to lift derived subgroups along a surjection to a just infinite quotient, culminating in an abelian (hence virtually abelian) image. The results establish the LEJI property for $\textup{PL}^+[0,1]$, Thompson's group $F$ (as a subgroup), and the nonamenable group $G_0$ (LodhaMoore), yielding numerous nonamenable LEJI examples and answering a question of Grigorchuk. Overall, the work reveals rigidity of just infinite quotients in these piecewise-linear and piecewise-projective groups and expands the landscape of nonamenable LEJI groups, while leaving open questions about potential LEJI groups of intermediate growth.

Abstract

We show that just infinite quotients of finitely generated subgroups of Richard Thompson's group F are virtually abelian, answering a question of Grigorchuk. We show the same holds for the group of piecewise linear orientation preserving homeomorphisms of the interval, and the group of piecewise projective homeomorphisms of the real line. The latter provides a plethora of examples of nonamenable groups with this property.

Theorems & Definitions (2)

• Theorem 1
• proof : Proof of Theorem \ref{['main']}