UC Riverside

Symplectic 4-manifolds are coarsely classified by Kodaira dimension; those of Kodaira dimension 0 are characterized by torsion canonical class. A symplectic Calabi-Yau 4-manifold (SCY) is a symplectic 4-manifold with trivial canonical class. SCYs satisfy strict homological constraints: the (virtual) first Betti number is 0, 2, 3, or 4. The fundamental group of an SCY satisfies the same constraints and is called an SCY group. A symplectic manifold is almost complex and so admits a canonical spin^c structure, permitting access to Seiberg-Witten theory by which it is shown that linear SCY groups with virtual first Betti number 4 are virtually solvable and hence elementary amenable; Hirsch length calculations force such a group to be virtually Z^4.