Pseudo-Anosov homeomorphisms and the lower central series of a surface group

Abstract

Let ${\Gamma }_k$ be the lower central series of a surface group $\Gamma$ of a compact surface $S$ with one boundary component. A simple question to ponder is whether a mapping class of $S$ can be determined to be pseudo-Anosov given only the data of its action on $\Gamma ∕{\Gamma }_k$ for some $k$. In this paper, to each mapping class $f$ which acts trivially on $\Gamma ∕{\Gamma }_{k+1}$, we associate an invariant ${\Psi }_k\left(f\right)\in End\left(H_1\left(S,\mathbb{Z}\right)\right)$ which is constructed from its action on $\Gamma ∕{\Gamma }_{k+2}$ . We show that if the characteristic polynomial of ${\Psi }_k\left(f\right)$ is irreducible over $\mathbb{Z}$, then $f$ must be pseudo-Anosov. Some explicit mapping classes are then shown to be pseudo-Anosov.