On abstract commensurators of groups

Abstract

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.

13 pages, no figure

References in corpus (1)

• Commensurability classes of 2-bridge knot complements

Cited by in corpus (1)

• A note on separability in outer automorphism groups