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

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

2007-02-21

arxiv.org/abs/math/0702608v1

Algebr. Geom. Topol. {\bf 7} (2007), pp. 1921--1948

Mathematics

Geometric Topology

23 pages, 8 figures; previous versions of this paper had the title: An algebraic criterion to detect pseudo-Anosovs

Scientific paper

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(f) in 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.

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