Solvable subgroup theorems, length functions and topological entropy

We prove a general solvable subgroup theorem in terms of length functions. As applications, we obtain a solvable subgroup theorem in dynamical systems: any solvable group of finite Hirsch length acting on a smooth manifold with uniformly positive topological entropies must be virtually [Formula: see text]. Some applications to group actions on cubical complexes and injective spaces, interval exchange transformations, and RFRS groups are also discussed.