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\title[New examples of elements in the kernel of 
the Magnus representation] 
{New examples of elements in the kernel \\
of the Magnus representation of the Torelli group}

\author[M. Suzuki]{Masaaki Suzuki}

\address{
Department of Mathematics\\
Akita University \\
1-1 Tegata-Gakuenmachi, Akita, 010-8502 \\
Japan}

\rcvdate{Xxx xx, 2007}
\rvsdate{Xxx xx, 2007}
%\rcvdate{April 1, 2007}% to be supplied by the volume editor
%\rvsdate{May 1, 2007}% to be supplied by the volume editor

\thanks{The research is supported in part by the 
Grant-in-Aid for Scientific Research 
(No.\ 18840008), the Ministry of Education, Culture, 
Sports, Science and Technology, Japan.}

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\begin{document}

\begin{abstract}
From our previous paper, it is known that 
the Magnus representation of the Torelli group is not faithful. 
In this paper, we show a certain kind of elements in  
the kernel of the representation. 
\end{abstract}

\maketitle

\section{Introduction}\label{sect-intro}

Let $\sig$ be an oriented closed surface 
and $\sigone$ a compact surface 
removed an open disk from $\sig$. 
We denote by $\mgone$ 
the mapping class group of $\sigone$ relative to the boundary, 
that is the group of isotopy classes 
of orientation preserving diffeomorphisms of $\sigone$ 
which restrict to the identity on the boundary. 
Let $\igone$ be the Torelli group of $\sigone$, 
which is the normal subgroup of $\mgone$ consisting of all the 
elements which act trivially 
on the first homology group of $\sigone$. 

From our previous paper \cite{szk1}, 
the Magnus representation of the Torelli group 
\[
r_1: \igone \to GL(2g;{\mathbb Z} [H])
\]
is not faithful for $g \geq 2$, 
where $H= H_1(\sigone;{\mathbb Z})$. 
Then this representation cannot determine the linearity of 
the Torelli group. 
In \cite{szk4}, we characterize the kernel of $r_1$ 
for the commutator of two BSCC maps, 
where the Dehn twist along a bounding simple closed curve 
is called BSCC map. 

In Section \ref{sect-def}, 
we review the definition and the irreducible decomposition of 
the Magnus representation of the Torelli group. 

\section{Definition and irreducible decomposition 
of the Magnus representation of the Torelli group}
\label{sect-def}

In this section, 
we recall the definition and the irreducible decomposition of 
the Magnus representation of the Torelli group 
from \cite{mrt9}, \cite{szk1} and \cite{szk2}. 

\begin{definition}
We call the mapping 
\[
\begin{array}{cccc}
r:& \mgone & \longrightarrow & GL(2g;{\mathbb Z} [\Gamma_0])\\
& \varphi & \longmapsto & 
~~ \left(
\overline{
\frac{\partial \varphi(\gamma_j)}{\partial \gamma_i}}\right)_{i,j}
\end{array}
\]
the Magnus representation of the mapping class group, 
where $\frac{\partial}{\partial \gamma_i}:
{\mathbb Z} [\Gamma_0] \to {\mathbb Z} [\Gamma_0]$ 
is the Fox derivation of the integral group ring 
${\mathbb Z}[\Gamma_0]$ and 
$\bar{~~}:{\mathbb Z} [\Gamma_0] \to {\mathbb Z} [\Gamma_0]$ 
is the antiautomorphism induced by the mapping 
$\gamma \mapsto \gamma^{-1}$. 
\end{definition}

The mapping $r$ is not a homomorphism but a crossed homomorphism. 
Namely, the Magnus representation of the mapping class group 
is not a genuine representation. 

\begin{proposition}[Morita \cite{mrt9}]\label{magnusproduct}
For arbitrary two elements $\varphi, \psi \in \mgone$, we have 
\[
r(\varphi \psi) = r(\varphi) \cdot {}^{\varphi} r(\psi)
\] 
where ${}^{\varphi} r(\psi)$ denotes the matrix obtained 
from $r(\psi)$ by applying the automorphism 
$\varphi : {\mathbb Z}[\Gamma_0] \to {\mathbb Z}[\Gamma_0]$ 
on each entry. 
\end{proposition}

\begin{proof}
It follows from the chain rule of the Fox derivation. 
\end{proof}

\begin{thebibliography}{99}
\bibitem[1]{bi}J. Birman,
Braids, Links and Mapping Class Groups,
Annals of Math. Studies, No. {\bf 82},
Princeton University Press, 1975.
\bibitem[2]{fox2}R. H. Fox, 
Free Differential Calculus I,
Annals of Math., {\bf 57} (1953), 547-560. 
\bibitem[3]{jo4}D. Johnson,
A survey of the Torelli group,
Contemporary Math., {\bf 20} (1983), 165-179. 
\bibitem[4]{mrt9}S. Morita,
Abelian quotients of subgroups of 
the mapping class group of surfaces,
Duke Math. J., {\bf 70} (1993), 699-726. 
\bibitem[5]{mrt0}S. Morita,
Structure of the mapping class groups of surfaces:
a survey and a prospect,
Geom. Topol. Monogr., {\bf 2} 
Proceedings of the Kirbyfest (1999), 349-406.
\bibitem[6]{szk1}M. Suzuki,
The Magnus representation of the Torelli group $\igone$ is not 
faithful for $g \geq 2$,
Proc. Amer. Math. Soc., {\bf 130} (2002), 909-914. 
\bibitem[7]{szk2}M. Suzuki,
Irreducible decomposition of 
the Magnus representation of the Torelli group, 
Bull. Austral. Math. Soc., {\bf 67} (2003), 1-14. 
\bibitem[8]{szk3}M. Suzuki,
A class function on the Torelli group, 
Kodai Math. J., {\bf 26} (2003), 304-316. 
\bibitem[9]{szk4}M. Suzuki,
On the kernel of the kernel of 
the Magnus representation of the Torelli group, 
Proc. Amer. Math. Soc., {\bf 133} (2005), 1865-1872. 
\bibitem[10]{szk5}M. Suzuki,
A geometric interpretation of 
the Magnus representation of the mapping class group, 
to appear in Kobe J. Math. 
\end{thebibliography}

\end{document}

