Summary: FOLIATIONS OF SOME 3­MANIFOLDS WHICH FIBER OVER
THE CIRCLE.
D. COOPER AND D.D. LONG
(Communicated by R. Fintushel)
Abstract. We show that a hyperbolic punctured torus bundle admits a foli­
ation by lines which is covered by a product foliation. Thus its fundamental
group acts freely on the plane.
1. Introduction
This paper discusses one dimensional foliations of closed three­manifolds. Every
closed three­manifold admits a one dimensional foliation, for example the three­
sphere admits a foliation by round circles (Hopf) and by smooth lines [4]. Epstein,
[3], showed that every foliation by circles is a Seifert fibration, and this class of
manifolds has been extensively studied. A manifold which fibers over the circle
admits a one dimensional foliation such that each leaf maps monotonely under the
map to the circle defining the fibration. If a closed three­manifold admits one of
the eight geometric structures described by Thurston, [5], then it is either Seifert
fibered, a torus bundle over the circle, or hyperbolic. It has been conjectured that
every hyperbolic three­manifold is finitely covered by a three­manifold which fibers
over the circle. A foliation of M induces a foliation on any cover of M by pull­back.
In the case that M fibers over the circle, or is Seifert fibered, either the universal

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics