  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of this paper partly overlaps with the content of the unpublished preprint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]).
