Special structures on almost abelian solvmanifolds

We characterize every almost abelian Lie algebra endowed with an integrable complex structure by a triple, called presentation, consisting of a real number, an element in some vector space and an endomorphism of that vector space. We then classify in terms of presentations the almost abelian Lie algebras admitting $p$-K\"ahler or $p$-pluriclosed structures, and in particular those carrying K\"ahler, balanced, pluriclosed and Gauduchon metrics. Furthermore, we characterize the existence of LCK and Bismut-Ricci flat metrics in this setting.