Compact Kähler manifolds admitting large solvable groups of automorphisms

Let G be a group of automorphisms of a compact Kähler manifold X of dimension n   and N(G)N(G) the subset of null-entropy elements. Suppose G admits no non-abelian free subgroup. Improving the known Tits alternative, we obtain that, up to replace G   by a finite-index subgroup, either G/N(G)G/N(G) is a free abelian group of rank ≤n−2≤n−2, or G/N(G)G/N(G) is a free abelian group of rank n−1n−1 and X is a complex torus, or G   is a free abelian group of rank n−1n−1. If the last case occurs, X is G-equivariant birational to the quotient of an abelian variety provided that X   is a projective manifold of dimension n≥3n≥3 and is not rationally connected. We also prove and use a generalization of a theorem by Fujiki and Lieberman on the structure of Aut(X)Aut(X).