Geometry, Topology and Dynamics Seminar
Olga Lukina
UIC
Molino theory for laminations
Abstract: A foliation of a compact manifold can be considered as a
generalized dynamical system, in the sense of Smale. The study of the
dynamical properties of foliations has been an active area of research for
the past 40 years. A smooth foliation is Riemannian, if the normal bundle
of the foliation admits a Riemannian metric invariant under the action of
the holonomy pseudogroup of the foliation. Riemannian foliations are very
rigid geometric structures, and they are completely classified by Molino
theory.
Ghys asked in 1988 whether Molino theory can be generalized to a
topological setting. In this setting, one considers foliations of compact
topological spaces, which do not admit normal bundles, and where the
transversals need not be locally connected. The condition analogous to the
existence and invariance of a Riemannian metric in this non-differentiable
setting, is the assumption of equicontinuity of the holonomy pseudogroup
of the foliation. Alvarez Lopez, Candel, and Moreira Galicia gave a
version of a Molino-like theory for foliated spaces under the additional
assumption that the closure of the holonomy pseudogroup is strongly
quasi-analytic, that is, it satisfies the condition of local generation.
In this talk, we consider foliated spaces with totally disconnected
transversals, which we call matchbox manifolds, and use the methods of
topological dynamics and continuum theory to develop a Molino-like
classification of all such spaces. We show that for matchbox manifolds,
the Molino sequence need not be well-defined, and specify the conditions
under which it is well-defined. We outline the classes of matchbox
manifolds, for which the local generation condition holds or does not
hold, and study other properties of these spaces. Inspired by the result
of Lubotzky about the existence of torsion in profinite completions of
torsion-free groups, we construct a class of examples with well-defined
non-trivial Molino sequences, where the non-triviality of the Molino
sequence cannot be explained by the holonomy properties of leaves in the
matchbox manifold. The examples that we construct and study show that this
class of dynamical systems is far from being completely classified.
Monday October 31, 2016 at 3:00 PM in SEO 636