###
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

####
To appear in American Journal of Mathematics.

This is a substantial rewrite of our previous Arxiv article 0809.3704

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.
Moreover, there is an algorithm that will construct a finite presentation for S

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 presentable

subgroups of residually free groups.
We also prove that there is an algorithm that, given a

finite generating
set for such a subgroup, will construct a finite presentation.

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.