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\title[TRANSVERSE INTERSECTIONS OF FOLIATIONS]{TRANSVERSE 
INTERSECTIONS \\ OF FOLIATIONS IN THREE-MANIFOLDS}

\author[S. MATSUMOTO \and T. TSUBOI]{Shigenori \sn{Matsumoto}\thanks{The 
first author is supported in part by Grant-in-Aid for Scientific 
research  (B)~09440042.} \and Takashi \sn{Tsuboi}\thanks{
We thank the second author for his help in providing this model file,
whose mathematical content has been profoundly distorted
{\it ad usum exempli}.}}

\begin{document}

\maketitle

\begin{abstract}
For two foliations of a three-manifold, we consider the question whether 
their transverse intersection is unique.
\end{abstract}

\section{Introduction}

In this paper we consider the transverse intersection of two foliations 
of a three-manifold (\cite{Haefliger}). 

\begin{theorem}\label{Main1}
Let $A$ be an element of $\SL(2;\ZZ)$ such that $\nb{\Tr A}>2$.
Then the intersection $\Ff^u\cap h^*\Ff^s$ 
is isotopic to the Anosov flow~$\Ff^u\cap\Ff^s$.
\end{theorem}
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Our Theorem \ref{Main1} is stronger than that in \cite{Ghys0}.
In \S \ref{Preliminary}, we construct several examples 
of multifoliations. 

\section{Unit tangent bundles of the hyperbolic surfaces}\label{Preliminary}

\begin{lemma}
If $\gamma$ is a compact leaf of the intersection $\Ff^u\cap \Ff^s$,
then the $\pi_1(M)$ orbit of the image  $p(\widetilde{\gamma})$ of the 
lift $\widetilde{\gamma}\subset \Mtilde$ is discrete in  $Q^u\times Q^s$.
\end{lemma}

\begin{proof}
For any compact disk $D$ of $Q^u\times Q^s$, one can take a 
lift $\Dtilde$ in~$\Mtilde$.\qed
\end{proof}

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These foliations are transverse to each other and form a multifoliation.
The intersections of several of their leaves are shown in Figure~5. 

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\caption{}
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\begin{thebibliography}{10}

\bibitem{Ghys0}
{\sc Ghys, \Emaj.} Flots d'Anosov sur les $3$-vari\'et\'es fibr\'ees en 
cercles. {\it Ergodic Theory Dynam. Systems 4, no. 1\/} (1984), 67--80.
 
\bibitem{Ghys1}
\EMdash Flots transversalement affines et tissus feuillet\'es.
Analyse globale et physique math\'ematique (Lyon, 1989). 
{\it M\'em. Soc. Math. France (N.S.) 46\/} (1991), 123--150. 

\bibitem{Haefliger}
{\sc Haefliger, A.} Groupo\"{\i}des d'holonomie et classifiants. 
{\it Structure transverse de feuilletages (Toulouse, 1982).
Ast\'erisque 116\/} (1984), 70--97.

\end{thebibliography}

\EMdate{6 f\'evrier 2001}

\begin{address}
Takashi Tsuboi \\
University of Tokyo
Tokyo 153-8914
Japan
\email{tsuboi@ms.u-tokyo.ac.jp}
\end{address}

\end{document}  
