Solvable lattices and their groups of automorphisms

The main goal of this paper is to construct an "algebraic" representation of a group automorphisms Ant Gamma for any elementary solvable group Gamma. "Algebraic" means that the image of a semisimple (unipotent) automorphism, in the sense of Wang, will be a sernisimple (unipotent) matrix Theorem 1. This gives an answer on the question asking by Wang. As a corollary of this theorem, we show that for any group of automorphisms Ant F of a lattice Gamma in a solvable connected group Lie G there exist a representation rho : Aut Gamma--> GL(n)(Z), such that rho(Aut Gamma) is an arithmetic subgroup in the Zariski closure rho(Aut Gamma).