  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.
