Anosov diffeomorphisms on nilmanifolds modelled on a free nilpotent Lie group

Abstract

In this article we give an elementary proof, using standard arguments from algebraic number theory, of the fact that a nilmanifold modelled on a free c-step nilpotent Lie group on n generators admits an Anosov diffeomorphism if and only if n > c. In fact, we need to show that for any integer n > 1, there exists a matrix A ∈ GL(n, ℤ), such that any product of less than n eigenvalues of A is of modulus ≠ 1.