Commensurations and subgroups of finite index of Thompson's group $F$

Abstract

We determine the abstract commensurator $Com\left(F\right)$ of Thompson’s group $F$ and describe it in terms of piecewise linear homeomorphisms of the real line. We show $Com\left(F\right)$ is not finitely generated and determine which subgroups of finite index in $F$ are isomorphic to $F$. We also show that the natural map from the commensurator group to the quasi-isometry group of $F$ is injective.