Commensurations and Subgroups of Finite Index of Thompson's Group F

TL;DR

This paper characterizes the abstract commensurator of Thompson's group F, showing its structure, properties, and relation to subgroups of finite index, with implications for understanding its automorphisms and quasi-isometries.

Contribution

It explicitly determines the structure of the commensurator of F, proves it is not finitely generated, and analyzes finite index subgroups isomorphic to F.

Findings

com(F) is described via piecewise linear homeomorphisms and tree pair diagrams

com(F) is not finitely generated

The natural map from com(F) to the quasi-isometry group of F is injective

Abstract

We determine the abstract commensurator com(F) of Thompson's group F and describe it in terms of piecewise linear homeomorphisms of the real line and in terms of tree pair diagrams. We show com (F) is not finitely generated and determine which subgroups of finite index in F are isomorphic to F. We show that the natural map from the commensurator group to the quasi-isometry group of F is injective.