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In this work we exhibit examples of $5$-manifolds that are not homeomorphic to any leaf of any $C^2$ codimension one foliation of any compact $6$-manifold but are homeomorphic to (proper) leaves of some $C^1$ codimension one foliations and also to (proper) leaves of some $C^\inf$ codimension $2$ foliations. The topology of these manifolds is characterized by Sturmian-like sequences., Comment: 16 pages, 2 figures

A noncompact (oriented) surface satisfies the condition $(\star)$ if their isolated ends are ''accumulated by genus''. We show that every surface satisfying this condition is homeomorfic to the leaf of a minimal codimension one foliation on a closed $3$-manifold whose generic leaf is not simply connected. Moreover, the above result is also true for any prescription of a countable family of noncompact surfaces (satisfying $(\star)$): they can coexist in the same minimal codimension one foliation as above. All the given examples are hyperbolic foliations, meaning that they admit a leafwise Riemannian metric of constant negative curvature., Comment: 14 pages, 4 figures

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed $3$-manifold. These foliations are (or are covered by) suspensions of continuous minimal actions of surface groups on the circle. Moreover, the above result is also true for any prescription of a countable family of topologies of open surfaces: they can coexist in the same minimal foliation. All the given examples are hyperbolic foliations, meaning that they admit a leafwise Riemannian metric of constant negative curvature. Many oriented Seifert manifolds with a fibered incompressible torus and whose associated orbifold is hyperbolic admit minimal foliations as above. The given examples are not transversely $C^2$-smoothable., Comment: 36 pages, 6 figures. Overall redaction improved in this version

Following the recent advances in the study of groups of circle diffeomorphisms, we describe an efficient way of classifying the topological dynamics of locally discrete, finitely generated, virtually free subgroups of the group $\mathsf{Diff}^\omega_+(\mathbb S^1)$ of orientation preserving real-analytic circle diffeomorphisms, which include all subgroups of $\mathsf{Diff}^\omega_+(\mathbb S^1)$ acting with an invariant Cantor set. An important tool that we develop, of independent interest, is the extension of classical ping-pong lemma to actions of fundamental groups of graphs of groups. Our main motivation is an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403--453] from foliation theory, which we solve in this restricted but significant setting: this and other consequences of the classification will be treated in more detail in a companion work., Comment: v4 39 pages, 12 figures; minor corrections; final version to appear in J. Comb. Algebra

We show that any simply connected topological closed $4$-manifold punctured along any compact, totally disconnected tame subset $\Lambda$ admits a continuum of smoothings which are not diffeomorphic to any leaf of a $C^{1,0}$ codimension one foliation on a compact manifold. This includes the remarkable case of $S^4$ punctured along a tame Cantor set. This is the lowest reasonable regularity for this realization problem. These results come from a new criterion for nonleaves in $C^{1,0}$ regularity. We also include a new criterion for nonleaves in the $C^2$-category. Some of our smooth nonleaves are "exotic", i.e., homeomorphic but not diffeomorphic to leaves of codimension one foliations on a compact manifold., Comment: 29 pages, 2 figures. Improving overall readability (with a more exhaustive exposition of the proof of the main Theorem 3.6)

This article takes the inspiration from two milestones in the study of non minimal actions of groups on the circle: Duminy's theorem about the number of ends of semi-exceptional leaves and Ghys' freeness result in analytic regularity. Our first result concerns groups of analytic diffeomorphisms with infinitely many ends: if the action is non expanding, then the group is virtually free. The second result is a Duminy's theorem for minimal codimension one foliations: either non expandable leaves have infinitely many ends, or the holonomy pseudogroup preserves a projective structure., Comment: We can now make a precise reference to Deroin's work arXiv:1811.10298. 54 pages, 2 figures

We study the possibility of realizing exotic smooth structures on punctured simply connected $4$-manifolds as leaves of a codimension one foliation on a compact manifold. In particular, we show the existence of uncountably many smooth open $4$-manifolds which are not diffeomorphic to any leaf of a codimension one transversely $C^{2}$ foliation on a compact manifold. These examples include some exotic ${\mathbb R}^4$'s and exotic cylinders $S^3\times{\mathbb R}$., Comment: 18 pages. New formulation of Section 1, this allows to include more examples of exotic non-leaves

Classical Ljusternik-Schnirelmann category is upper bounded by the number of critical points of any bounded from below differentiable functions of Palais-Smale type. Here we achieve an adaptation of this result for the tangential category of foliations. We introduce a weaker type of Palais-Smale function, obtaining a slight improvement in the classical theorem of critical points., Comment: 10 pages

In the author's Ph.D., a version of the tangential LS category for foliated spaces depending on a transverse invariant measure, called the measured category, was introduced. Unfortunately, the measured category vanishes easily. When it is zero, the rate of convergence to zero of the quantity involved in the definition, by taking arbitrarily large homotopies, gives a new invariant, called the secondary measured category. Several versions of classical results are proved for the secondary measured category. It is also shown that the secondary measured category is a transverse invariant related to the growth of (pseudo)groups. The equality between secondary category and the growth of a group is done in the case of free suspensions by Rohlin groups., Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:1112.5009

A version of the tangential LS category is introduced for topological laminations with a transverse invariant measure. Here, we use the transverse measure of the contraction of a tangential categorical open set instead of counting this set. This new measured category is invariant by leafwise homotopy equivalences preserving the transverse measures, and the condition of being zero or positive is a transverse invariant. The usual tangential LS category is also bounded by the number of certain critical sets. It is also proved that the measured category is semicontinuous when the foliated structure and the transverse invariant measure varies on a fixed manifold, which is a version of a result of W. Singhof and E. Vogt for the tangential category. Hopefully, this relation between LS category and critical sets will be useful to deal with foliated versions of variational problems, like the existence of closed leafwise geodesics, and even to simplify the proofs of classical results on manifolds., Comment: 22 pages

We give two new versions of the LS category for the set-up of measurable laminations defined by Berm\'udez. Both of these versions must be considered as "tangential categories". The first one, simply called (LS) category, is the direct analogue for measurable laminations of the tangential category of (topological) laminations introduced by Colman Vale and Mac\'ias Virg\'os. For the measurable lamination that underlies any lamination, our measurable tangential category is a lower bound of the tangential category. The second version, called the measured category, depends on the choice of a transverse invariant measure. We show that both of these "tangential categories" satisfy appropriate versions of some well known properties of the classical category: the homotopy invariance, a dimensional upper bound, a cohomological lower bound (cup length), and an upper bound given by the critical points of a smooth function., Comment: 22 pages

A new notion of cohomology is introduced for MT-spaces, which are measurable and topological spaces whose measurable structure may not agree with the Borel $\sigma$-algebra of their topology. The main examples of MTspaces are measurable foliations. This is a singular version of the measurable simplicial cohomology defined by Heitsch and Lazarov for foliations and extended by Bermudez for MT-spaces. Basic topics of algebraic topology are adapted, and applications to the theory of foliations are given. Moreover we introduce a new notion of singular L2-cohomology for MT-spaces., Comment: 16 pages

It is shown that any transverse invariant measure of a foliated space can be considered as a measure on the ambient space., Comment: We consider generalized measures, not only probability measures, and possibly not $\sigma$-finite. Transverse invariant measures can extend in several ways to the ambient space, we discuss some reasonable conditions where this extension is unique