Foliations of 3-manifolds with solvable fundamental group

Pointwise partial hyperbolicity in three-dimensional nilmanifolds

“…If π is non-zero, its level sets define a codimension 1 foliationF π on H which quotients down to a foliation F π on the nilmanifold H/Γ. Plante showed that any Reebless C 2 foliation of H/Γ must be almost aligned to F π for some π [24]. We need a similar result for foliations which are C 1,0+ , that is, C 0 with C 1 leaves tangent to a C 0 distribution.…”

“…For C 2 -foliations, Plante (see [24]) gave a classification of foliations without torus leaves in three-dimensional manifolds with almost solvable fundamental group. His proof relies on the application of a result from [30] which uses the C 2 -hypothesis in an important way (other results which used the C 2 -hypothesis such as Novikov's theorem are now well known to work for C 0 -foliations thanks to [31]).…”

“…The same classification can be done for one-dimensional foliations of T 2 for which the proof is easier (see, for example, Section 4.A of [26] Proof. The proof can be done directly (see [24,30] for the C 2 -case). We will use Theorem B.3 instead.…”

3-manifolds supporting Anosov flows and Abelian normal subgroups

“…Theorem 3 [18,Corollary 3.3]. If M is a closed 3-manifold such that π 1 M is an almost solvable group with exponential growth and H 1 (M, Z) = 0 then every codimension one foliation of M has a compact leaf.…”

Regulating flows, topology of foliations and rigidity

“…In this case π 1 (M ) is solvable and there is important work of Plante [Pl2] completely classifying foliations in such manifolds. However Plante assumes that F is at least C 2 and he makes use of Kopell's lemma [Ko].…”