  We prove that the abstract commensurator of a nonabelian free group, an
infinite surface group, or more generally of a group that splits appropriately
over a cyclic subgroup, is not finitely generated.
  This applies in particular to all torsion-free word-hyperbolic groups with
infinite outer automorphism group and abelianization of rank at least 2.
  We also construct a finitely generated, torsion-free group which can be
mapped onto Z and which has a finitely generated commensurator.
