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In this paper we use the Lubotzky alternative for finitely generated lineargroups to determine which 4-manifolds admitting a free circle action can beendowed with a symplectic structure with trivial canonical class. The contentof this paper partly overlaps with the content of the unpublished preprint"Symplectic 4-manifolds with a free circle action" (arXiv:0801.1313 [math.GT]).

Symplectic 4--manifolds with K = 0 and the Lubotzky alternative