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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 )
∈ 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
Publication
Received: 7 March 2007
Revised: 17 July 2007
Accepted: 24 August 2007
Published: 18 December 2007