## Algebraic & Geometric Topology

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

Justin Malestein

#### Abstract

Let $Γk$ be the lower central series of a surface group $Γ$ 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 $Γ∕Γk$ for some $k$. In this paper, to each mapping class $f$ which acts trivially on $Γ∕Γk+1$, we associate an invariant $Ψk(f)∈ End(H1(S,ℤ))$ which is constructed from its action on $Γ∕Γk+2$ . We show that if the characteristic polynomial of $Ψk(f)$ is irreducible over $ℤ$, then $f$ must be pseudo-Anosov. Some explicit mapping classes are then shown to be pseudo-Anosov.

#### Article information

Source
Algebr. Geom. Topol., Volume 7, Number 4 (2007), 1921-1948.

Dates
Revised: 17 July 2007
Accepted: 24 August 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796776

Digital Object Identifier
doi:10.2140/agt.2007.7.1921

Mathematical Reviews number (MathSciNet)
MR2366181

Zentralblatt MATH identifier
1197.37056

#### Citation

Malestein, Justin. Pseudo-Anosov homeomorphisms and the lower central series of a surface group. Algebr. Geom. Topol. 7 (2007), no. 4, 1921--1948. doi:10.2140/agt.2007.7.1921. https://projecteuclid.org/euclid.agt/1513796776

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