On abstract commensurators of groups

TL;DR

This paper investigates the structure of abstract commensurators in certain groups, proving they are generally not finitely generated for many classes, but also constructs a specific example with a finitely generated commensurator.

Contribution

It establishes the non-finite generation of abstract commensurators for broad classes of groups and provides a counterexample with a finitely generated commensurator.

Findings

Abstract commensurators of nonabelian free groups are not finitely generated.

Groups splitting over cyclic subgroups have non-finitely generated commensurators.

A finitely generated, torsion-free group with a finitely generated commensurator is constructed.

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.