Decidability of fundamental groups of Einstein 4-manifolds

Determine whether the fundamental groups of general Einstein 4-manifolds form a sufficiently special class for the isomorphism problem to be decidable.

Background

The classification problem for smooth compact 4-manifolds with arbitrary fundamental group is described as formally undecidable because every finitely presented group occurs as the fundamental group of such a space, and no algorithm can always determine whether two finitely presented groups are isomorphic.

Certain geometrically defined groups, including fundamental groups of locally symmetric spaces, avoid this general undecidability obstruction. The unresolved issue is whether fundamental groups arising from Einstein 4-manifolds are similarly constrained.