Derived length and nilpotency class of zero entropy groups acting on compact Kahler manifolds.

Let X be a compact Kahler manifold of dimension n and of Kodaira dimension kappa(X). Let G be a group of zero entropy automorphisms of X and G_0 the set of elements in G which are isotopic to the identity. We prove that replacing G by a suitable finite-index subgroup, G/G_0 is a unipotent group of derived length at most n-max(kappa(X),1) and the derived length of G is at most n-kappa(X). Up to taking a finite-index subgroup, we conjecture that the nilpotency class c(G/G_0) of G/G_0 is at most n-1. The conjecture is proved to be true for all complex tori. Assuming this conjecture for all compact Kahler manifolds, we show that c(G) is at most n-kappa(X). The algebro-geometric structure of X is studied when c(G)=n or c(G/G_0)=n-1. We also give an optimal upper bound of the size of the Jordan blocks of the unipotent automorphisms of H^2(X,C) induced by automorphisms of X.