#### Volume 7, issue 4 (2007)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Pseudo-Anosov homeomorphisms and the lower central series of a surface group

### Justin Malestein

Algebraic & Geometric Topology 7 (2007) 1921–1948
 arXiv: math.gt/0702608
##### 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 ${\Psi }_{k}\left(f\right)\in End\left({H}_{1}\left(S,ℤ\right)\right)$ 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 $ℤ$, then $f$ must be pseudo-Anosov. Some explicit mapping classes are then shown to be pseudo-Anosov.

##### Keywords
pseudo-Anosov, lower central series, Torelli group, Johnson filtration
##### Mathematical Subject Classification 2000
Primary: 57M60, 37E30