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

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+l), we associate an invariant Psi(k)(f) epsilon End(H-1(S, Z)) which is constructed from its action on Gamma/Gamma(k+2). We show that if the characteristic polynomial Of Psi(k)(f) is irreducible over Z, then f must be pseudo-Anosov. Some explicit mapping classes are then shown to be pseudo-Anosov.