NON-ARITHMETIC POLYCYCLIC GROUPS

We prove that for every n > 2 there are semi-direct products of a free Abelian group of rank n and an infinite cyclic group which are not ar ithmetic groups. In particular, we show that there are non-arithmetic polycyclic groups of every Hirsch number > 3. We also show that all po lycyclic groups of Hirsch number I 3 are arithmetic. The proofs are ba sed on our previous results about solvable arithmetic groups and on ar ithmetic properties of algebraic tori. (C) Academie des Sciences/Elsev ier, Paris.