On manifolds supporting quasi Anosov diffeomorphisms

Let $M$ be an $n$-dimensional manifold supporting a quasi Anosov diffeomorphism. If $n=3$ then either $M={\mathbb T}^3$, in which case the diffeomorphisms is Anosov, or else its fundamental group contains a copy of ${\mathbb Z} ^6$. If $n=4$ then $\Pi_1(M)$ contains a copy of ${\mathbb Z} ^4$, provided that the diffeomorphism is not Anosov.