Formality and hard Lefschetz property of aspherical manifolds

For a Lie group $G=\R^{n}\ltimes_ϕ\R^{m}$ with the semi-simple action $ϕ:\R^{n}\to {\rm Aut}(\R^{m})$, we show that if $Γ$ is a finite extension of a lattice of $G$ then $K(Γ, 1)$ is formal. Moreover we show that a compact symplectic aspherical manifold with the fundamental group $Γ$ satisfies the hard Lefschetz property. By those results we give many examples of formal solvmanifolds satisfying the hard Lefschetz property but not admitting Kähler structures.

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