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We study the dynamical properties of the laminated horocycle flow on the unit tangent bundles of 2-dimensional smooth solenoidal manifolds of finite type. These laminations are the analog of complete hyperbolic surfaces of finite area., Comment: 32 pages, 2 figures

Starting with a smooth $n$-dimensional knot $K\subset\bS^{n+2}$, and a beaded $n$-dimensional necklace subordinated to $K$, we construct a wild knot with a Cantor set of wild points (\ie the knot is not locally flat in these points). The construction uses the conformal Schottky group acting on $\bS^{n+2}$, generated by inversions on the spheres which are the boundary of the``beads''. We show that if $K$ is a fibered knot then the wild knot is also fibered. We also study cyclic branched coverings along the wild knots., Comment: 17 pages, 8 figures

We prove, using the celebrated result by Spitzer about winding of planar Brownian motion, and the existence of harmonic morphisms $f:M\to{\mathbb S}^1$ representing cohomology classes in $\text{H}^1(M,\mathbb Z)$, that there is a stochastic process $H_t:{\mathcal C}(M)\to{\text{Hom}(\text{H}^1(M;\mathbb R), \mathbb R)}\simeq{\text{H}_1(M;\mathbb R)}$ ($t\in[0,\infty)$), where ${\mathcal C}(M)= \{ \alpha:[0, \infty) \to M :\alpha \,\, \text{is continuous} \}$, which has a multivariate Cauchy distribution i.e. such that for each nontrivial cohomology class $[\omega]\in{\text{H}^1(M;\mathbb R), \mathbb R)}$, represented by a closed 1-form $\omega$, in the de Rham cohomology, the process $A^\omega_t:{\mathcal C}(M)\to\mathbb R\,$ ($t\in[0,\infty)$) with $A^\omega_t(B)=H_t(B)([\omega]),\, B\in{\mathcal C}(M)$ converges in distribution, with respect to Wiener measure on ${\mathcal C}(M)$, to a Cauchy's distribution, with parameter 1. The process describes the ``homological winding" of the Brownian paths in $M$, thus it can be regarded as a generalization of Spitzer result. The last section discusses the asymptotic behavior of holonomy along Brownian paths., Comment: The principal result is incorrect. The distributions proposed are properly normalized Normal distributions and it leaves open the fascinating question of the asymptotic behavior of homology classes of Brownian paths

In a series of papers, published in Mathematische Annalen, Bianchi and Blumenthal introduced the notions of Bianchi orbifolds and Hilbert-Blumnethal surfaces as generalizations of modular curves associated to quadratic fields. In this paper, in the same spirit, and following a similar line of reasoning, we introduce the concept of Bianchi and Hilbert-Blumenthal quaternionic orbifolds as generalizations of the Lipschitz and Hurwitz quaternionic modular orbifolds defined recently by D\'iaz, Vlacci and the first author. In particular, we describe the cusp cross-sections of the Hilbert-Blumenthal quaternionic orbifolds in terms of fundamental units of real quadratic fields. These are 7-dimensional solvmanifolds which are virtual ${\mathbb T}^6$ bundles over the circle with monodromy a linear Anosov diffeomorphism of the 6-torus., Comment: 16 pages

In this paper we survey $n$-dimensional solenoidal manifolds for $n=1,2$ and 3, and present new results about them. Solenoidal manifolds of dimension $n$ are metric spaces locally modeled on the product of a Cantor set and an open $n$-dimensional disk. Therefore, they can be "laminated" (or "foliated") by $n$-dimensional leaves. By a theorem of A. Clark and S. Hurder, topologically homogeneous, compact solenoidal manifolds are McCord solenoids i.e. are obtained as the inverse limit of an increasing tower of finite, regular covers of a compact manifold with an infinite and residually finite fundamental group. In this case their structure is very rich since they are principal Cantor-group bundles over a compact manifold and they behave like "laminated" versions of compact manifolds, thus they share many of their properties. These objects codify the commensurability properties of manifolds., Comment: This version contains some corrections and suggestions by an anonymous referee

In this paper we present a series of seemingly unrelated results of Complex Analysis which are in fact connected via a different approach to their proofs using the results of Errett Bishop of volumes and limits of analytic varieties. We start by proving Chow's theorem by a technique suggested long time ago in the beautiful book by Gabriel Stolzenberg. We think this approach is very attractive and easier for students and newcomers to understand; also the theory presented here is linked to areas of mathematics that are not usually associated with Chow's result. In addition, Bishop's results imply both Chow's and Remmert-Stein's theorems directly, meaning that this view is simpler and just as profound as Remmert-Stein's proof. After that, we give a comparison table that explains how Bishop's theorems generalize to several complex variables classical results of one complex variable and prove Montel's compactness theorem using the techniques presented here. Finally we give an alternative proof of a theorem of Edwards, Millet and Sullivan of foliations with compact leaves for the case of complex foliations in K\"ahler manifolds., Comment: arXiv admin note: substantial text overlap with arXiv:2101.09872

We give a new proof of the uniformization theorem of the leaves of a lamination by surfaces of hyperbolic conformal type. We use a laminated version of the Ricci flow to prove the existence of a laminated Riemannian metric (smooth on the leaves, transversaly continuous) with leaves of constant Gaussian curvature equal to -1, which is conformally equivalent to the original metric., Comment: 15 pages; Section 5 was rewritten to make more comprehensible the proof of Theorem 5.1.;submitted to Proc. Am. Math. Soc

We present a proof of Chow's theorem using two results of Errett Bishop retated to volumes and limits of analytic varieties. We think this approach suggested a long time ago in the beautiful book by Gabriel Stolzenberg, is very attractive and easier for students and newcomers to understand, also the theory presented here is linked to areas of mathematics that are not usually associated with Chow's theorem. Furthermore, Bishop's results imply both Chow's and Remmert-Stein's theorems directly, meaning that this approach is more economic and just as profound as Remmert-Stein's proof. At the end of the paper there is a comparison table that explains how Bishop's theorems generalize to several complex variables classical results of one complex variable., Comment: The paper is superceeded by arXiv:2111.04846 math.CV It containes new material and has some minor corrections

In this paper we present a generalization of Poincar\'e's Rotation Theory of homeomorphisms of the circle to the case of one-dimensional compact abelian groups which are solenoidal groups, {\it i.e.}, groups which fiber over the circle with fiber a Cantor abelian group. We define rotation elements, \emph{\`a la} Poincar\'e and discuss the dynamical properties of translations on these solenoidal groups. We also study the semiconjugation problem when the rotation element generates a dense subgroup of the solenoidal group. Finally, we comment on the relation between Rotation Theory and entropy for these homeomorphisms, since unlike the case of the circle, for the solenoids considered here there are homeomorphisms (not homotopic to the identity) with positive entropy.

In this paper, following Grothendieck {\it Esquisse d'un programme}, which was motivated by Belyi's work, we study some properties of surfaces $X$ which are triangulated by (possibly ideal) isometric equilateral triangles of one of the spherical, euclidean or hyperbolic geometries. These surfaces have a natural Riemannian metric with conic singularities. In the euclidean case we analyze the closed geodesics and their lengths. Such surfaces can be given the structure of a Riemann surface which, considered as algebraic curves, are defined over $\bar{\mathbb{Q}}$ by a theorem of Belyi. They have been studied by many authors of course. Here we define the notion of connected sum of two Belyi functions and give some concrete examples. In the particular case when $X$ is a torus, the triangulation leads to an elliptic curve and we define the notion of a "peel" obtained from the triangulation (which is a metaphor of an orange peel) and relate this peel with the modulus $\tau$ of the elliptic curve. Many fascinating questions arise regarding the modularity of the elliptic curve and the geometric aspects of the Taniyama-Shimura-Weil theory.

In this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering quantum toric varieties as (non-algebraic) stacks, we define their category and show that it is equivalent to a category of quantum fans. We develop a Quantum Geometric Invariant Theory (QGIT) type construction of Quantum Toric Varieties. Unlike classical toric varieties, quantum toric varieties admit moduli and we define their moduli spaces, prove that these spaces are orbifolds and, in favorable cases, up to homotopy, they admit a complex structure., Comment: 93 pages, 4 figures

We study the proalgebraic space which is the inverse limit of all finite branched covers over a normal toric variety with branching set the invariant divisor under the algebraic torus action. These are completions (compactifications) of the adelic abelian algebraic group which is the profinite completion of the algebraic torus. We prove that the vector bundle category of the proalgebraic toric completion of a toric variety is the direct limit of the respective categories of the finite toric varieties coverings defining the completion. In the case of the complex projective line we obtain as proalgebraic completion the adelic projective line P. We define holomorphic vector bundles over P. We also introduce the smooth, Sobolev and Wiener adelic loop groups and the corresponding Grassmannans; we describe their properties and prove Birkhoff's factorization for these groups. We prove that the adelic Picard group of holomorphic line bundles is isomorphic to the rationals and prove the Birkhoff-Grothendieck splitting theorem for holomorphic bundles of higher rank over P.

We generalize the Ahlfors-Bers theory to the adelic Riemann sphere. In particular, after defining the appropriate notion of a Beltrami differential in the solenoidal context, we give a sufficient condition on it such that the corresponding Beltrami equation has a quasiconformal homeomorphism solution; i.e. The Ahlfors-Bers Theorem in the solenoidal case. This additional condition on the solenoidal Beltrami differentials can be written as a Banach norm in a subspace of solenoidal differentials. Moreover, this subspace is the completion under this norm of those solenoidal differentials locally constant at the fiber. As a toy example, we show how this technique works on a linear problem: We generalize the diophantine equation complex analytic extension problem to the respective solenoidal space., Comment: arXiv admin note: text overlap with arXiv:1603.05676

A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This in particular proves that this flow is minimal, which establishes a conjecture proposed by Matsumoto. The main tool is a theorem due to Coud\`ene, which he presented as an alternative proof for the surface case. It applies to two continuous flows defining a measure-preserving action of the affine group of the line on a compact metric space, precisely matching the foliated setting. In addition, we briefly discuss the application of Coud\`ene's theorem to other kinds of foliations., Comment: 18 pages, 1 figure, to appear in Ergodic Theory and Dynamical Systems

These are notes for the CIME school on Complex non-K\"ahler geometry from July 9th to July 13th of 2018 in Cetraro, Italy. It is an overview of different properties of a class of non-K\"ahler compact complex manifolds called LVMB manifolds, obtained as the Hausdorff space of leaves of systems of commuting complex linear equations in an open set in complex projective space ${{mathbb P}_{\mathbb C}}^{n-1}$

We construct an Ahlfors-Bers complex analytic model for the Teichm\"uller space of the universal hyperbolic lamination (also known as Sullivan's Teichm\"uller space) and the renormalized Weil-Petersson metric on it as an extension of the usual one. In this setting, we prove that Sullivan's Teichm\"uller space is K\"ahler isometric biholomorphic to the space of continuous functions from the profinite completion of the fundamental group of a compact Riemann surface of genus greater than or equal to two to the Teichm\"uller space of this surface; i.e. We find natural K\"ahler coordinates for the Sullivan's Teichm\"uller space. This is the main result. As a corollary, we show the expected fact that the Nag-Verjovsky embedding is transversal to the Sullivan's Teichm\"uller space contained in the universal one., Comment: Accepted in Annales Academiae Scientiarum Fennicae Mathematica

Inspired by the work of Zagier, we study geometrically the probability measures $m_y$ with support on the closed horocycles of the unit tangent bundle $M=\text{PSL}(2,\mathbb{R})/\text{PSL}(2,\mathbb{Z})$ of the modular orbifold $\text{PSL}(2,\mathbb Z)$. In fact, the canonical projection $\mathfrak{p}:M\to\mathbb{H}/\text{PSL}(2,\mathbb Z)$ it is actually a Seifert fibration over the orbifold with two especial circle fibers corresponding to the two conical points of the modular orbifold. Zagier proved that $m_y$ converges to normalized Haar measure $m_o$ of $M$ as $y\to0$: for every smooth function $f:M\to \mathbb R$ with compact support $m_y(f)=m_0(f)+o(y^\frac12)$ as $y\to0$. He also shows that $m_y(f)=m_0(f)+o(y^{\frac34-\epsilon})$ for all $\epsilon>0$ and smooth function $f$ with compact support in $M$ if and only if the Riemann hypothesis is true. In this paper we show that the exponent $\frac12$ is optimal if $f$ is the characteristic function of certain open sets in $M$. This of course does not imply that the Riemann hypothesis is false. It is required the differentiability of the functions in the theorem., Comment: This is an updated version of the paper of the same title that appeared in: Verjovsky, A. Arithmetic geometry and dynamics in the unit tangent bundle of the modular orbifold. Dynamical systems (Santiago, 1990), 263{298, Pitman Res. Notes Math. Ser., 285, Longman Sci. Tech., Harlow, 1993

We introduce a new fundamental domain for the cusp stabilizer of a Hilbert modular group over a real quadratic field K=Q(sqrt n). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of the biplane. The region is the Cartesian product of the positive reals with a 3-dimensional tower formed by deformations of lattices in the ring of integers of K, and makes explicit the cusp cross section's Sol 3-manifold structure and Anosov diffeomorphism. We include computer generated images and data illustrating various examples., Comment: 17 pages, 4 figures, 2 tables

Inspired by the works of Zagier, we study the probability measures $\nu(t)$ with support on the flat tori which are the compact orbits of the maximal unipotent subgroup acting holomorphically on the positive orthonormal frame bundle $\mathcal{F}({M}_D)$ of 3-dimensional hyperbolic Bianchi orbifolds ${M}_D=\mathbb{H}^3/\widetilde{\Gamma}_D$, of finite volume and with only one cusp. Here $\Gamma_D=PSL(2, \mathcal{O})$, where $\mathcal{O}$ is the ring of integers of an imaginary quadratic field of class number one., Comment: We corrected a few typos and included a few details

In this paper, we prove that given two cubical links of dimension two in ${\mathbb R}^4$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister and Roseman moves for classical tame knots of dimension one and two, respectively., Comment: 9 figures. arXiv admin note: text overlap with arXiv:1302.2133

In the context of orientable circuits and subcomplexes of these as representing certain singular spaces, we consider characteristic class formulas generalizing those classical results as seen for the Riemann-Hurwitz formula for regulating the topology of branched covering maps and that for monoidal transformations which include the standard blowing-up process. Here the results are presented as cap product pairings, which will be elements of a suitable homology theory, rather than characteristic numbers as would be the case when taking Kronecker products once Poincar\'e duality is defined. We further consider possible applications and examples including branched covering maps, singular varieties involving virtual tangent bundles, the Chern-Schwartz-MacPherson class, the homology L-class, generalized signature, and the cohomology signature class.

In this paper we study topological surfaces as gridded surfaces in the 2-dimensional scaffolding of cubic honeycombs in Euclidean and hyperbolic spaces., Comment: It is indeed possible to embed the tree of life in the 2-skeleton. Every (conected) orientable surface(with an arbitrary space of ends)can be embedded into the 2-skeleton of the cubulation of ${\mathbb R}^3$

We say that a topological $n$-manifold $N$ is a cubical $n$-manifold if it is contained in the $n$-skeleton of the canonical cubulation $\mathcal{C}$ of ${\mathbb{R}}^{n+k}$ ($k\geq1$). In this paper, we prove that any closed, oriented cubical $2$-manifold has a transverse field of 2-planes in the sense of Whitehead and therefore it is smoothable by a small ambient isotopy., Comment: 8 figures

We describe several methods to construct minimal foliations by hyperbolic surfaces on closed 3-manifolds, and discuss the properties of the examples thus obtained., Comment: Some statements and proofs have been revised. To appear in Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM

Topologically the adelic Riemann sphere is the suspension of the adelic solenoid and because of this relation, here we study the adelic solenoid by studying the adelic Riemann sphere topology. The main result is the Birkhoff-Grothendieck Theorem: A holomorphic vector bundle splits as a sum of holomorphic line bundles whose Chern character is now a rational number. As a consequence, the Picard group is isomorphic to the additive group of rational numbers and the $K$--ring has new elements that factor the tautological class.

We construct open book structures on all moment-angle manifolds and describe the topology of their leaves and bindings under certain restrictions. II. We also show, using a recent deep result about contact forms due to Borman, Eliashberg and Murphy [6], that every odd-dimensional moment-angle manifold admits a contact structure. This contrasts with the fact that, except for a few, well-determined cases, even-dimensional ones do not admit symplectic structures. We obtain the same results for large families of more general intersections of quadrics., Comment: arXiv admin note: text overlap with arXiv:1303.2671

Using the rings of Lipschitz and Hurwitz integers $\mathbb{H}(\mathbb{Z})$ and $\mathbb{H}ur(\mathbb{Z})$ in the quaternion division algebra $\mathbb{H}$, we define several Kleinian discrete subgroups of $PSL(2,\mathbb{H})$, Comment: 43 pages, 15 figures

We consider a generalized Riemann-Hurwitz formula as it may be applied to rational maps between projective varieties having an indeterminacy set and fold-like singularities. The case of a holomorphic branched covering map is recalled. Then we see how the formula can be applied to iterated maps having branch-like singularities. Separately, we consider a further application involving the Chern classes of determinantal varieties when the latter are realized as the degeneracy loci of certain vector bundle morphisms., Comment: Some corections and improvements were made in this new version

We consider a minimal compact lamination by hyperbolic surfaces. We prove that if it admits a leaf whose holonomy covering is not topologically trivial, then the horocycle flow on its unitary tangent bundle is minimal., Comment: This result was presented by Fernando Alcalde at the conference Foliations 2014, that took place in Madrid, Spain, in September 2014

In this note we introduce a new family of non-commutative spaces that we call non-commutative toric varieties and we describe some of their main properties. The main technical tool in this investigation is a natural extension of LVM-theory for the irrational case. In order to introduce the moduli space of (non-commutative) toric varieties we use variations on the notion of diffeology as models for non-commutative spaces., Comment: 25 pages, 5 figures, dedicated to the 60th birthdays of Ralph L. Cohen and Gunnar Carlsson and the 70th birthday of Ib Madsen

This article presents a generalization of the notion of \emph{Poincar\'e rotation set} to homeomorphisms of the ad\`ele class group $\mathbb{A}/\mathbb{Q}$ of the rational numbers $\mathbb{Q}$, which is a connected compact abelian group which can be identified with the one-dimensional universal solenoid $\mathbf{S}$, \ie the algebraic universal covering of the circle. The definition is first introduced in general for homeomorphisms of $\mathbf{S}$ which are isotopic to a translation, and then specializing in homeomorphisms of $\mathbf{S}$ isotopic to the identity, in which case the rotation set is a closed interval contained in the base leaf (the connected component of the identity). If in the latter case the rotation interval reduces to a single element $\rho$ and $\rho$ is irrational (\ie it is a monothetic generator of $\mathbf{S}$), we show that the homeomorphism is semiconjugate to the translation $z\mapsto\rho{z}$, like in the classical theory of Poincar\'e. This theory is valid for any general compact abelian one dimensional solenoidal group $\mathbf{S}_G$, which are Pontryagin duals of dense subgroups $G$ of the rational numbers with the discrete topology. These solenoidal groups are one-dimensional laminations which are locally homeomorphic to the product of a Cantor set by an interval so they behave very much like a ``diffuse'' version of the circle. Our approach differs from others because we use Pontryagin duality of compact abelian groups to define the rotation sets. \end{abstract}, Comment: 15 pages

We construct new examples of contact manifolds in arbitrarily large dimensions. These manifolds which we call quasi moment-angle manifolds, are closely related to the classical moment-angle manifolds.

In this paper we consider the Kleinian groups acting conformally on the sphere $\mathbb{S}^{n+2}$ $(1\leq{n}\leq5)$ which have as limit sets wild spheres $K^n$ which were constructed in \cite{BHV} and prove that $K^n$ is ambient homogeneous. In other words, given two points $p,\,\,q\in{K}$ there exists a homeomorphism $\psi:\mathbb{S}^{n+2}\rightarrow\mathbb{S}^{n+2} $ such that $\psi(K)=K$ and $\psi(p)=q$., Comment: The proof contains a mistake

We construct open book structures on moment-angle manifolds and give a new construction of examples of contact manifolds in arbitrarily large dimensions., Comment: 7 figures

The Jones-Witten invariants can be generalized for non-singular smooth vector fields with invariant probability measure on 3-manifolds, giving rise to new invariants of dynamical systems [22]. After a short survey of cohomological field theory for Yang-Mills fields, Donaldson-Witten invariants are generalized to four-dimensional manifolds with non-singular smooth flows generated by homologically non-trivial p-vector fields. These invariants have the information of the flows and they are interpreted as the intersection number of these flow orbits and constitute invariants of smooth four-manifolds admitting global flows. We study the case of Kahler manifolds by using the Witten's consideration of the strong coupling dynamics of N=1 supersymmetric Yang-Mills theories. The whole construction is performed by implementing the notion of higher dimensional asymptotic cycles a la Schwartzman [18]. In the process Seiberg-Witten invariants are also described within this context. Finally, we give an interpretation of our asymptotic observables of 4-manifolds in the context of string theory with flows., Comment: 34 pages, no figures

For $n \geq 1$, the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ of the conformal $2n$-sphere is biholomorphic to the Zariski closure, taken in the complex Grassmannian manifold $\mathbf{G}(n+1, 2n+2)$, of the set of graphs of skew-symmetric linear endomorphism of $\mathbb{C}^{n+1}$. We use this fact to describe a natural stratification of the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ with $n \geq 3$, in terms of what we have called {\it generalised complex orthogonal Stiefel manifolds} of $\mathbb{C}^{n+1}$. In particular, the twistor space $\mathfrak{Z}(\mathbb{S}^{6})$ is biholomorphic to a non-singular complex quadric hypersurface in $\mathbb{P}^{7}$. We explicitly construct a real-analytic foliation, by linear 3-folds, of this quadric hypersurface such that the quotient space is isomorphic to the 6-sphere with its standard metric. This foliation is Riemannian with respect to the Fubini-Study metric and isometrically equivalent to the twistor fibration over the 6-sphere., Comment: This paper has been withdrawn by the authors in order to make several corrections and write a new version

We explain an error in our paper "A smooth foliation of the 5-sphere by complex surfaces", Ann. Math 156 (2002), p.915-930.

We construct a smooth codimension-one foliation on the five-sphere in which every leaf is a symplectic four-manifold and such that the symplectic structure varies smoothly. Our construction implies the existence of a complete regular Poisson structure on the five-sphere., Comment: We have withdrawn our paper due to a mistake in the main reference paper and errors in our construction. More over our result has been superseded by Y. Mitsumatsu (arXiv:1101.2319 Leafwise Symplectic Structures on Lawson's Foliation)

In this paper we construct infinitely many wild knots, $\mathbb{S}^{n}\hookrightarrow\mathbb{S}^{n+2}$, for $n=2,3,4$ and 5, each of which is a limit set of a geometrically finite Kleinian group. We also describe some of their properties, Comment: Accepted in Conformal Theory and Dynamical Systems (AMS)

The $n$-skeleton of the canonical cubulation $\cal C$ of $\mathbb{R}^{n+2}$ into unit cubes is called the {\it canonical scaffolding} ${\cal{S}}$. In this paper, we prove that any smooth, compact, closed, $n$-dimensional submanifold of $\mathbb{R}^{n+2}$ with trivial normal bundle can be continuously isotoped by an ambient isotopy to a cubic submanifold contained in ${\cal{S}}$. In particular, any smooth knot $\mathbb{S}^{n}\hookrightarrow\mathbb{R}^{n+2}$ can be continuously isotoped to a knot contained in ${\cal{S}}$., Comment: In this revised version we include more detailed, complete and self-contained proofs of our theorems

In this article, we focus on a very special class of foliations with complex leaves whose diffeomorphism type is fixed. They have a unique compact leaf and the noncompact leaves all accumulate onto it. We show that the complex structure along the non-compact leaves is fixed by the complex structure of the compact leaf. Reciprocally, we prove that the complex structure along a non-compact leaf determines the complex structure along the other leaves. We apply these results to the study of foliated deformations of Hopf manifolds, a foliated analogue to the notion of deformation in the large.

We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(\hat M, T^1\mathcal{F})$ of a compact minimal lamination $(M,\mathcal F)$ by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal, and examples where this action is not minimal. In the first case, we prove that if $\mathcal F$ has a leaf which is not simply connected, the horocyle flow is topologically transitive., Comment: Corrected version of previous paper "Hedlund's theorem for compact minimal laminations"