On abstract commensurators of groups

We prove, under a technical assumption, that the abstract commensurator of a group that splits over a cyclic subgroup is not finitely generated. This applies in particular to free groups, surface groups, and more generally to all torsion-free word-hyperbolic groups with infinite outer automorphism group and abelianization of rank at least 2. On the other hand, we remark that the condition on the outer automorphism group cannot be removed.