Throughout this paper, all 3-manifolds are closed and orientable.
Our aim is to
give some new examples of 3-manifolds which are virtual bundles, that is,
have finite
sheeted covers which are surface bundles over the circle. Thurston has
raised the
question as to whether all irreducible atoroidal 3-manifolds might have
this property.
The geometrisation conjecture would imply that all such 3-manifolds have
hyperbolic
metrics, and the virtual bundle question is then equivalent to showing
the existence
of geometrically infinite incompressible surfaces (compare [14]).
Not much progress has been made on this problem. Recently, Reid [9]
has given
an explicit example using an arithmetic hyperbolic 3-manifold, and Cooper,
Long
and Reid [3] have given methods of constructing immersed
incompressible surface in
surfaces bundles over the circle, which are sometimes geometrically infinite.
In [10],
examples are given of horizontal immersed incompressible surfaces in graph
manifolds which do not lift to embeddings in any finite sheeted covering
spaces.
Similar examples are given in toroidal manifolds with cubings of non-positive
curvatures in [9]. So the assumption that the manifold
has the atoroidal property is
essential in Thurston's question. Our aim is to give several simple
constructions of
large classes of examples using different polyhedral metrics of non-positive
curvature
on 3-manifolds (compare [2]).
We follow closely the construction of Thurston (compare [13]),
where he observes
that the right-angled regular hyperbolic dodecahedron can be viewed as
a cube with
arcs drawn on each face through midpoints of a pair of opposite sides so
that no two
such arcs share a common vertex. Then the foliation of the cube by planes
orthogonal
to a diagonal gives an induced foliation of any 3-manifold arising from
a torsion-free
finite index subgroup of the symmetry group of the tessellation of hyperbolic
3-space
by the right-angled dodecahedron. Clearly, any leaf of this foliation is
then an
immersed geometrically infinite surface. One explicit example is then the
4-fold cyclic
branched cover of the Borromean rings, since the Borromean rings has a
flat 2-fold
branched cover which is obtained by gluing two cubes together. This example
is
formed by a subgroup of index 4 in the above symmetry group.
Our examples come directly from this observation of Thurston, using
different
ways of dividing a 3-manifold into simple polyhedra so that there is an
overall metric
of non-positive curvature. We use some well-known tessellations of Euclidean
3-dimensional
space, described in Coxeter [4], by truncated octahedra
and tetrahedra,
plus flying saucers (compare [2]), as well as the easy
case of cubes.
In the final section we show that our classes of cubed examples and
flying saucers
admit geometric decompositions in the sense of Thurston, since they are
virtual
bundles. We would like to thank the referee for a number of helpful comments
which
improved the exposition.