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

#### Martin R. Bridson, James Howie, Charles F. Miller III,
Hamish Short

#### 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.