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

Malestein, Justin

doi:10.2140/agt.2007.7.1921

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

Justin Malestein

Algebraic & Geometric Topology 7 (2007) 1921–1948

DOI: 10.2140/agt.2007.7.1921

arXiv: math.gt/0702608

Abstract

Let $Γ_{k}$ 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 $Γ ∕ Γ_{k}$ for some $k$ . In this paper, to each mapping class $f$ which acts trivially on $Γ ∕ Γ_{k + 1}$ , we associate an invariant $Ψ_{k} (f) \in End (H_{1} (S, ℤ))$ which is constructed from its action on $Γ ∕ Γ_{k + 2}$ . We show that if the characteristic polynomial of $Ψ_{k} (f)$ 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

References

Publication

Received: 7 March 2007

Revised: 17 July 2007

Accepted: 24 August 2007

Published: 18 December 2007

Authors

Justin Malestein
Department of Mathematics University of Chicago 5734 S University Ave Chicago IL 60637 USA
http://www.math.uchicago.edu/~justinm/

Volume 7, issue 4 (2007)