Algebraic & Geometric Topology

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

Justin Malestein

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

Article information

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

Received: 7 March 2007
Revised: 17 July 2007
Accepted: 24 August 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M60: Group actions in low dimensions 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces

pseudo-Anosov lower central series Torelli group Johnson filtration


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.

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