Addendum to "Commensurations and Subgroups of Finite Index of Thompson's Group F"

TL;DR

This paper characterizes the structure of the abstract commensurator of Thompson's group F, revealing it as composed of specific algebraic components, and proves the simplicity of a related group of piecewise linear homeomorphisms.

Contribution

It provides a detailed description of the abstract commensurator of Thompson's group F and proves the simplicity of a particular group of homeomorphisms, advancing understanding of these algebraic structures.

Findings

The abstract commensurator of F consists of four specific algebraic components.

The simplicity of a certain group of piecewise linear homeomorphisms is established.

The structure of the commensurator includes simple groups, rationals, and a cyclic group.

Abstract

We show that the abstract commensurator of Thompson's group F is composed of four building blocks: two isomorphism types of simple groups, the multiplicative group of the positive rationals and a cyclic group of order two. The main result establishes the simplicity of a certain group of piecewise linear homeomorphisms of the real line.