Betti and Hodge numbers of solvmanifolds arising from integer polynomials

We study the de Rham cohomology of three families of completely solvable almost abelian solvmanifolds (called basic, complex, and hypercomplex) constructed from a monic integer polynomial with positive distinct roots whose product equals 1, following the work of Andrada and Barberis. Under two algebraic restrictions on such polynomials (the full rank and quasi full rank conditions) we compute the Betti numbers and Poincar\'e polynomials of these manifolds. Moreover, we study the Dolbeault cohomology of the complex solvmanifolds by identifying them with generalized Nakamura manifolds recently introduced by Cattaneo and Tomassini. Assuming a suitable condition on the lattice, we compute their Hodge numbers, which exhibit a combinatorial structure related to Pascal's triangle in the full rank setting, and are described by explicit generating polynomials in the quasi full rank case.