1-Lefschetz contact solvmanifolds

We study the contact Lefschetz condition on compact contact solvmanifolds, as introduced by B.\ Cappelletti-Montano, A.\ De Nicola and I.\ Yudin. We seek to fill the gap in the literature concerning Benson-Gordon type results, characterizing $1$-Lefschetz contact solvmanifolds. We prove that the $1$-Lefschetz condition on Lie algebras is preserved via $1$-dimensional central extensions by a symplectic cocycle, thereby establishing that a unimodular symplectic Lie algebra $(\mathfrak{h}, \omega)$ is $1$-Lefschetz if and only if its contactization $(\mathfrak{g}, \eta)$ is $1$-Lefschetz. We achieve this by showing an explicit relation for the relevant cohomology degrees of $\mathfrak{h}$ and $\mathfrak{g}$. Using this, we show how the commutators $[\mathfrak{h},\mathfrak{h}]$ and $[\mathfrak{g},\mathfrak{g}]$ are related, especially when the $1$-Lefschetz condition holds. By specializing to the nilpotent setting, we prove that $1$-Lefschetz contact nilmanifolds equipped with an invariant contact form are quotients of a Heisenberg group, and deduce that there are many examples of compact $K$-contact solvmanifolds not admitting compatible Sasakian structures. We also construct examples of completely solvable $1$-Lefschetz solvmanifolds, some having the $2$-Lefschetz property and some failing it.