Finitely presented residually free groups
  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 ${\rm{FP}}_n$ if and only if it is of type ${\rm{F}}_n$.   New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups that are neither ${\rm{FP}}_\infty$ 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. 
Martin R. Bridson, James Howie, Charles F.Miller III, Hamish Short