Finitely presented residually free groups

Abstract

We establish a general criterion for the finite presentability of subdirect products of groups and use this to characterize finitely presented residually free groups. We prove that, for all $n\in \mathbb{N}$, a residually free group is of type ${\mathrm{FP}}_n$ if and only if it is of type ${\mathrm{F}}_n$. New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups that are neither ${\mathrm{FP}}_{\infty }$ nor of Stallings-Bieri type. The template for these examples leads to a more constructive characterization of finitely presented residually free groups up to commensurability. We show that the class of finitely presented residually free groups is recursively enumerable and present a reduction of the isomorphism problem. A new algorithm is described which, given a finite presentation of a residually free group, constructs a canonical embedding into a direct product of finitely many limit groups. The (multiple) conjugacy and membership problems for finitely presented subgroups of residually free groups are solved.

38 pages

References in corpus (2)

• Limit groups are CAT(0)
• Isoperimetric functions for subdirect products and Bestvina-Brady groups

Cited by in corpus (1)

• Isoperimetric functions for subdirect products and Bestvina-Brady groups