  In this work we ask when a group is a 3-manifold group, or more specifically,
when does a group presentation come naturally from a Heegaard diagram for a
3-manifold? We will give some conditions for partial answers to this form of
the Isomorphism Problem by addressing how the presentation associated to a
diagram for a splitting is related to the fundamental group of a 3-manifold. In
the process, we determine an invariant of groups (by way of group
presentations) for how far such presentations are from 3-manifolds.
