Symplectic 4--manifolds with K = 0 and the Lubotzky alternative
  In this paper we use the Lubotzky alternative for finitely generated linear groups to determine which 4-manifolds admitting a free circle action can be endowed with a symplectic structure with trivial canonical class. The content of this paper partly overlaps with the content of the unpublished preprint "Symplectic 4-manifolds with a free circle action" (arXiv:0801.1313 [math.GT]). 
Stefan Friedl, Stefano Vidussi
math.GT math.SG