Deformations of nilpotent Lie algebras and nilmanifolds

Milnor asked in 1977 whether a torsion-free and virtually polycyclic group Γ can be the fundamental group of a complete compact affine manifold. In 1993 Benoist constructed a finitely generated torsion-free nilpotent group of rank 11 that is not the fundamental group of any compact complete affine manifold, later Burde and Grunewald added new counter-examples of ranks 10 and 12 respectively. We will discuss the relations of the deformation theory of the positive part W₊ ⁿ of the Virasoro algebra and possible new counter-examples for an arbitrary rank r ≥ 13.