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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

arXiv: math.gt/0702608

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 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.

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/