The Godbillon–Vey class is the most significant secondary characteristic class for foliations. We will throw a glance at deformations and rigidity of the Godbillon–Vey class especially from a viewpoint of transverse structures of foliations.

We summarize some of the recent works, devoted to the study of one-dimensional (pseudo)group actions and codimension one foliations. We state a conjectural alternative for such actions (generalizing the already obtained results) and describe the properties in both alternative cases. We also discuss the generalizations for holomorphic one-dimensional actions. Finally, we state some open questions that seem to be already within the reach.

A matchbox manifold is a connected, compact foliated space with totally disconnected transversals; or in other notation, a generalized lamination. It is said to be Lipschitz if there exists a metric on its transversals for which the holonomy maps are Lipschitz. Examples of Lipschitz matchbox manifolds include the exceptional minimal sets for $C^{1}$-foliations of compact manifolds, tiling spaces, the classical solenoids, and the weak solenoids of McCord and Schori, among others. We address the question: When does a Lipschitz matchbox manifold admit an embedding as a minimal set for a smooth dynamical system, or more generally as an exceptional minimal set for a $C^{1}$-foliation of a smooth manifold? We give examples which do embed, and develop criteria for showing when they do not embed, and give examples. We also discuss the classification theory for Lipschitz weak solenoids.

We review various aspects of representations of the holonomy Lie algebras and the associated monodromy representations of the fundamental groups for the complement of hyperplane arrangements. In particular, we describe the relation between monodromy representations of the KZ equation and homological representations of the braid groups by means of hypergeometric integrals.

The little Teichmüller space of symmetric homeomorphisms of the circle defines a Banach foliated structure of the universal Teichmüller space. First we consider rigidity of Möbius representations given by symmetric conjugation and failure of the fixed point property for isometric group action on the little Teichmüller space. This space includes the Teichmüller space of circle diffeomorphisms with Hölder continuous derivatives. Then we characterize these diffeomorphisms by Beltrami coefficients of quasiconformal extensions and Schwarzian derivatives of their Bers embeddings. This is used for proving certain rigidity of representations by symmetric conjugation in the group of circle diffeomorphisms. We also consider Teichmüller spaces of integrable symmetric homeomorphisms, which induce another Banach foliated structure and the generalized Weil–Petersson metric on the universal Teichmüller space. As an application, we investigate the fixed point property for isometric group action on these spaces and give a condition for a group of circle diffeomorphisms with Hölder continuous derivatives to be conjugate to a Möbius group in the same class.

We recreate an unpublished proof of William Thurston from the early 1970's that any smooth 2-plane field on a manifold of dimension at least 4 is homotopic to the tangent plane field of a foliation.

In this paper, we consider a few major open problems in the theory of characteristic classes of foliations. After recalling some history and background of each of these problems, we propose possible approaches to challenge them and present several “sub-problems”.

This is a survey on several problems related to the study of groups of diffeomorphisms.

In this article, we provide a short review of general results on harmonic measures on foliated manifolds and discuss briefly about uniquely ergodic foliations.

In this paper, we review a recent progress of the study of the Diederich–Fornaess index of complex domains with emphasis on the case of domains with Levi-flat boundary. It is exhibited that for any compact Levi-flat real hypersurface, the norm of its infinitesimal holonomy must exceed the curvature of its normal bundle at a point.

An analog of the Baumslag–Solitar group $\mathit{BS}(1, k)$ acts on the torus naturally. The action is not locally rigid in higher dimension, but any perturbation of the action should be homogeneous.

We study geometric properties of characteristic classes of surfaces bundles. In particular, we show that oriented surface bundles over bases with amenable fundamental groups and dimension at least 2 have trivial simplicial volume. We show furthermore that all MMM-classes are hyperbolic in the sense of Gromov, verifying a weakened version of a conjecture due to Morita. Finally we consider surface bundles over products and restrictions on their characteristic classes

On sutured 3-manifolds $M$, we classify taut foliations almost without holonomy up to isotopy. We assume that the compact leaves lie in $\partial M$. The classification is given by finitely many convex, polyhedral cones in $H^{1}(M;\mathbb{R})$ which have disjoint interiors. The classes in the interiors of these cones determine the isotopy classes. This work relies heavily on the Handel–Miller classification of the isotopy classes of endperiodic surface automorphisms. While the Handel–Miller theory was not published by the originators, the authors have given a complete account elsewhere.

For a contact manifold $(M^{2m+1}, A)$ and an $m+1$-dimensional $dA$-isotropic $C^{2}$ foliation, we define Godbillon–Vey invariants $\{\mathit{GV}_{i}\}_{i = 0}^{m+1}$ inspired by the Godbillon–Vey invariant of a codimension-one foliation, and we demonstrate the potential of this family as a tool in geometric rigidity theory. One ingredient for the latter is the Mitsumatsu formula for geodesic flows on (Finsler) surfaces.

A hyperbolic 2-sphere is made from the double of an $n$-gon in Poincaré disc. Its geodesic flow is a transitive Anosov flow. We construct genus one Birkhoff sections for the geodesic flows of hyperbolic 2-spheres with $n$ ($\ge 3$) singularities.

Let $M$ be a closed manifold. Polterovich constructed a linear map from the vector space of quasi-morphisms on the fundamental group $\pi_{1}(M)$ of $M$ to the space of quasi-morphisms on the identity component $\mathrm{Diff}_{\Omega}^{\infty} (M)_{0}$ of the group of volume-preserving diffeomorphisms of $M$. In this paper, the restriction $H^{1}(\pi_{1}(M); \mathbb{R}) \to H^{1}(\mathrm{Diff}_{\Omega}^{\infty} (M)_{0}; \mathbb{R})$ of the linear map is studied and its relationship with the flux homomorphism is described.

Throughout the last 2–3 decades, there has been great interest in the extrinsic geometry of foliated Riemannian manifolds (see [2], [4] and [22]).

One approach is to build examples of foliations with reasonably simple singularities with leaves admitting some very restrictive geometric condition. For example (see [22], [23] and [17]), consider in particular foliations of $\mathbb{S}^{3}$ by totally geodesic or totally umbilical leaves with isolated singularities.

The article [14] provides families of foliations of $\mathbb{S}^{3}$ by Dupin cyclides with only one smooth curve of singularities. Quadrics and other families of cyclides like Darboux cyclides provide other examples. These foliations are built on solutions of a three contacts problem: we show that the surfaces of the considered family satisfying three imposed contact conditions, if they exist, form a one parameter family of surfaces which will be used to construct a foliation.

Finally we will study the four contact condition problem in the realm of Darboux–d'Alembert cyclides.

We characterize the semi-conjugacy class of a Fuchsian action of the modular group on the circle in terms of rotation numbers of two standard generators and that of their product. We also show that among lifts of a Fuchsian action of the modular group, only 5-fold lift admits a similar characterization. These results indicate similarity and difference between rotation number and linear character.

Denote by $\mathrm{Diff}_{c}^{r}(M)_{0}$ the identity component of the group of the compactly supported $C^{r}$ diffeomorphisms of a connected $C^{\infty}$ manifold $M$. We show that if $\dim(M)\geq2$ and $r\neq \dim(M)+1$, then any homomorphism from $\mathrm{Diff}_{c}^{r}(M)_{0}$ to $\mathrm{Diff}^{1}(\mathbb{R})$ or $\mathrm{Diff}^{1}(S^{1})$ is trivial.