### ON THE FINITE PRESENTATION OF SUBDIRECT PRODUCTS AND THE NATURE OF RESIDUALLY FREE GROUPS

#### Preprint, September 2008. Submitted for publication.

We establish virtual surjection to pairs (VSP) as a general criterion for the finite presentability of
subdirect products of groups: if G_1,...,G_n are finitely presented groups and S is a subgroup of their direct product
that projects to a subgroup of finite index in each G_i\times G_j, then S is finitely presented.

We use the VSP criterion to characterize finitely
presented residually free groups. We prove that the class of such groups is recursively enumerable. We describe
an algorithm that, given a finite presentation of a group in the class, will construct a canonical embedding into a direct product of
finitely many limit groups. We solve the (multiple) conjuagacy and membership problems for finitely presented residually free groups.

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.