The action of mapping classes on nilpotent covers of surfaces

Let $\mathrm{\Sigma }$ be a surface whose interior admits a hyperbolic structure of finite volume. In this paper, we show that any infinite order mapping class acts with infinite order on the homology of some universal $k$–step nilpotent cover of $\mathrm{\Sigma }$. We show that a Torelli mapping class either acts with infinite order on the homology of a finite abelian cover, or the suspension of the mapping class is a 3–manifold whose fundamental group has positive homology gradient. In the latter case, it follows that the suspended 3–manifold has a large fundamental group. It follows that every element of the Magnus kernel suspends to give a 3–manifold with a large fundamental group.