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It is a well-known and elementary fact that a holomorphic function on a compact complex manifold without boundary is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property holds in the setting of holomorphically foliated spaces.

This paper is devoted to a systematic study of the geometry of nondegenerate ${\mathbb R}^n$-actions on $n$-manifolds. The motivations for this study come from both dynamics, where these actions form a special class of integrable dynamical systems and the understanding of their nature is important for the study of other Hamiltonian and non-Hamiltonian integrable systems, and geometry, where these actions are related to a lot of other geometric objects, including reflection groups, singular affine structures, toric and quasi-toric manifolds, monodromy phenomena, topological invariants, etc. We construct a geometric theory of these actions, and obtain a series of results, including: local and semi-local normal forms, automorphism and twisting groups, the reflection principle, the toric degree, the monodromy, complete fans associated to hyperbolic domains, quotient spaces, elbolic actions and toric manifolds, existence and classification theorems.

International audience; We study the dynamics of the automorphisms group of K3 surfaces. Assuming that the surface contains two elliptic fibrations that are invariant by non-periodic automorphisms, we give the classification of invariant probability measures. We also describe the closure of orbits and then give applications to the repartition of rational points on K3 surfaces.

Given a compact manifold M , we denote by Diffr0.M / the identity component of the group of C –diffeomorphisms of M . A way to understand this group is to try to describe its subgroups. In other words, given a group G , does there exist an injective group morphism from the group G to the group Diffr0.M /? In case the answer is positive, one can try to describe the group morphisms from the group G to the group Diffr0.M / (ideally up to conjugacy, but this is often an unattainable goal). The concept of distortion element allows one to obtain rigidity results on group morphisms from G to Diffr0.M /. It will provide some very partial answers to these questions. Here is the definition. Remember that a group G is finitely generated if there exists a finite generating set G : any element g in this group is a product of elements of G and their inverses, g D s 1 1 s 2 2 s n , where the si are elements of G and the i are equal to C1 or 1. The smallest integer n in such a decomposition is denoted by lG.g/. The map lG is invariant under inverses and satisfies the triangle inequality lG.gh/ lG.g/C lG.h/. Therefore, for any element g in the group G , the sequence .lG.g //n 0 is sub-additive, so the sequence .lG.g/=n/n converges. When the limit of this sequence is zero, the element g is said to be distorted or a distortion element in the group G . Notice that this notion does not depend on the generating set G . In

We propose a program to study groups acting faithfully on S^1 in terms of number of pairwise transverse dense invariant laminations. We give some examples of groups which admit a small number of invariant laminations as an introduction to such groups. Main focus of the present paper is to characterize Fuchsian groups in this scheme. We prove a group acting on S^1 is conjugate to a Fuchsian group if and only if it admits three very-full laminations with a variation of the transversality condition. Some partial results toward a similar characterization of hyperbolic 3-manifold groups which fiber over the circle have been obtained. This work was motivated by the universal circle theory for tautly foliated 3-manifolds developed by Thurston and Calegari-Dunfield., Comment: 27 pages, 10 figures

An analog of the Baumslag-Solitar group BS(1,k) naturally acts on the sphere by conformal transformations. The action is not locally rigid in higher dimension, but exhibits a weak form of local rigidity. More precisely, any perturbation preserves a smooth conformal structure., 21 pages, no figures

Given a general pseudo-Anosov flow in a three manifold, the orbit space of the lifted flow to the universal cover is homeomorphic to an open disk. We compactify this orbit space with an ideal circle boundary. If there are no perfect fits between stable and unstable leaves and the flow is not topologically conjugate to a suspension Anosov flow, we then show: The ideal circle of the orbit space has a natural quotient space which is a sphere and is a dynamical systems ideal boundary for a compactification of the universal cover of the manifold. The main result is that the fundamental group acts on the flow ideal boundary as a uniform convergence group. Using a theorem of Bowditch, this yields a proof that the fundamental group of the manifold is Gromov hyperbolic and it shows that the action of the fundamental group on the flow ideal boundary is conjugate to the action on the Gromov ideal boundary. This implies that pseudo-Anosov flows without perfect fits are quasigeodesic flows and we show that the stable/unstable foliations of these flows are quasi-isometric. Finally we apply these results to foliations: if a foliation is R-covered or with one sided branching in an atoroidal three manifold then the results above imply that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity., 69 pages. Major revision, more explanations and simplified some simplified proofs. Detailed explanations of scalloped regions, parabolic points and perfect fit horoballs. 22 figures (3 new figures)

We extend the one dimensional Denjoy-Sacksteder theorems to some diffeomorphism groups of smooth compact closed n-dimensional manifolds. More precisely, we show the existence of a non trivial stabilizer for actions of “quasi-conformal groups” admitting an “exceptional” minimal set which is of “strongly decreasing type”; our results include the classical Denjoy-Sacksteder theorems.

In this paper we study the action of non abelian subgroup G generated by affine homotheties on C^n. We prove that there exist a subgroup H of C\{0}, a G-invariant affine subspace E of C^n and b in E such that the closure of any orbit G(z) is equal to H(z-a)+E, z in C^n. In particular, every orbit in E is dense in it. Moreover, if the complementary U=C^n \E is non empty, every orbit of U is minimal in it.

Let $BS(1, n)=< a, b | aba^{-1} = b^n >$ be the solvable Baumslag-Solitar group, where $ n\geq 2$. It is known that B(1, n) is isomorphic to the group generated by the two affine maps of the line : $f_0(x) = x + 1$ and $h_0(x) = nx $. The action on $S^1 = \RR \cup {\infty}$ generated by these two affine maps $f_0$ and $h_0 $ is called the standard affine one. We prove that any representation of BS(1,n) into $Diff^1(S^1)$ is (up to a finite index subgroup) semiconjugated to the standard affine action.

For [math] and [math] , consider the matrix [math] and the rotation matrix [math] . Let [math] denote some product of [math] instances of [math] and [math] of [math] , with the condition [math] ( [math] ). We analyze the measure of the set of [math] for which [math] ( [math] ). This can be regarded as a model problem for the Bochi–Fayad conjecture.

We show that if $r\geq 3$ and $\alpha$ is a faithful $Z^r$-Cartan action on a torus $T^d$ by automorphisms, then any closed subset of $(T^d)^2$ which is invariant and topologically transitive under the diagonal $\bZ^r$-action by $\alpha$ is homogeneous, in the sense that it is either the full torus $(T^d)^2$, or a finite set of rational points, or a finite disjoint union of parallel translates of some d-dimensional invariant subtorus. A counterexample is constructed for the rank 2 case., Comment: 40 pages

In this paper, we give a characterization of the action of any abelian subgroup G of GL(n, C) on C^n. We prove that any orbit of G is regular with order m

International audience; In this paper, we consider codimension one Anosov actions of IR^k, k ≥ 1, on closed connected orientable manifolds of dimension n+k with n≥ 3. We show that the fundamental group of the ambient manifold is solvable if and only if the weak foliation of codimension one is transversely affine. We also study the situation where one 1-parameter subgroup of IR^k admits a cross-section, and compare this to the case where the whole action is transverse to a fibration over a manifold of dimension n. As a byproduct, generalizing a Theorem by Ghys in the case k=1, we show that, under some assumptions about the smoothness of the sub-bundle E^ss ⊕ E^uu, and in the case where the action preserves the volume, it is topologically equivalent to a suspension of a linear Anosov action of Z^k on T^n.

We prove that a transversely equicontinuous minimal lamination on a locally compact metric space $Z$ has a transversely invariant Radon measure. Moreover if the space $Z$ is compact, then the tranversely invariant Radon measure is shown to be unique up to a scaling., 10 pages

We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky's property ($\tau$) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to answering the question of Lubotzky and Zuk, whether for families of subgroups, property ($\tau$) is inherited to the lattice of subgroups generated by the family. On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicite estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expander covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance., Comment: Corrections made based on the referee's comments

We show that if a connected closed orientable manifold $M$ admits a codimension one locally free smooth action $\phi$ of a connected nilpotent Lie group such that any orbit of $\phi$ is non-compact, then $M$ is homeomorphic to a nilmanifold. And as an example of such an action, we study also a homogeneous action.

Given a harmonic measure of a hyperbolic lamination on a compact metric space, a positive harmonic function is defined on the universal cover of a typical leaves. We discuss some properties of this function. Especially if all the leaves are hyperbolic, ergodic harmonic measures are divided into two classes., 22 pages

In this paper we formulate and prove a structure theorem for area preserving diffeomorphisms of genus zero surfaces with zero entropy. As an application we relate the existence of faithful actions of a finite index subgroup of the mapping class group of a closed surface $\Sigma_g$ on $S^2$ by area preserving diffeomorphisms to the existence of finite index subgroups of bounded mapping class groups $MCG(S, \partial S)$ with non-trivial first cohomology., Comment: 88 pages, 4 figures

We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky's property ($��$) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to answering the question of Lubotzky and Zuk, whether for families of subgroups, property ($��$) is inherited to the lattice of subgroups generated by the family. On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicite estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expander covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance., Corrections made based on the referee's comments

According to Thurston's stability theorem, every group of C^1 diffeomorphisms of the closed interval is locally indicable (.e., every finitely generated subgroup factors through Z). We show that, even for finitely generated groups, the converse of this statement is not true. More precisely, we show that the semi-direct product between F_2 an Z^2, although locally indicable, does not embed into Diff_+^1 (]0,1[). (Here F_2 is any free subgroup of SL(2,Z), and its action on Z^2 is the projective one.) Moreover, we show that for every non-solvable subgroup G of SL(2,Z), the semi-direct product between G and Z^2 does not embed into Diff^1_+(S^1)., Comment: 8 pages, no figure

In this article we collect a series of observations that constrain actions of many groups on compact manifolds. In particular, we show that "generic" finitely generated groups have no smooth volume preserving actions on compact manifolds while also producing many finitely presented, torsion free groups with the same property., References added, minor changes

By the Thurston stability theorem, a group of C^1 orientation-preserving diffeomorphisms of the closed unit interval is locally indicable. We show that the local order structure of orbits gives a stronger criterion for nonsmoothability that can be used to produce new examples of locally indicable groups of homeomorphisms of the interval that are not conjugate to groups of C^1 diffeomorphisms., 4 pages, 1 figure

Using methods and results of decomposition theory we construct minimal actions of groups of homeomorphisms of some classical fractals (the Sierpi\'nski carpet and its generalizations, and the Menger curve) with positive entropy. Suspending these group actions we get minimal foliated spaces which have positive geometric entropy and are modelled on these fractals.

Let $R_1, R_2,\ldots,R_m$ be rotations generating ${\mathbb{SO}}_{d+1}$ , $d\geq 2$ , and let $f_1, f_2,\ldots,f_m$ be small smooth perturbations of them. We show that $\{f_\alpha\}$ can be linearized simultaneously if and only if the associated random walk has zero Lyapunov exponents. As a consequence, we obtain stable ergodicity of actions of random rotations in even dimensions

We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens’ theorem for the full G G -shift for a finitely-generated torsion-free nilpotent group G G . Using bounds for the Möbius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape \[ ∑ | τ | ≤ N 1 e h | τ | ∼ C N α ( log ⁡ N ) β \sum _{\vert \tau \vert \le N}\frac {1}{e^{h\vert \tau \vert }}\sim CN^{\alpha }(\log N)^{\beta } \] where | τ | \vert \tau \vert is the cardinality of the finite orbit τ \tau and h h denotes the topological entropy. For the usual orbit-counting function we find upper and lower bounds, together with numerical evidence to suggest that for actions of non-cyclic groups there is no single asymptotic in terms of elementary functions.

Let Gamma be a finitely generated, amenable group. Using an idea of E Ghys, we prove that if Gamma has a nontrivial, orientation-preserving action on the real line, then Gamma has an infinite, cyclic quotient. (The converse is obvious.) This implies that if Gamma has a faithful action on the circle, then some finite-index subgroup of Gamma has the property that all of its nontrivial, finitely generated subgroups have infinite, cyclic quotients. It also means that every left-orderable, amenable group is locally indicable. This answers a question of P Linnell., This is the version published by Algebraic & Geometric Topology on 15 December 2006

If $\cal {G}$ is a finitely generated group with generators $\{g_1,\dots,g_j\}$ , then an infinite-order element $f \in \cal {G}$ is a distortion element of $\cal {G}$ provided that ${\lim \, \inf_{n \to \infty} |f^n|/n = 0,}$ where $|f^n|$ is the word length of $f^n$ in the generators. Let $S$ be a closed orientable surface, and let $\rm{Diff}(S)_0$ denote the identity component of the group of $C^1$ -diffeomorphisms of $S$ . Our main result shows that if $S$ has genus at least two and that if $f$ is a distortion element in some finitely generated subgroup of $\rm{Diff}(S)_0$ , then $\rm {supp}(\mu) \subset \rm {Fix}(f)$ for every $f$ -invariant Borel probability measure $\mu$ . Related results are proved for $S = T^2$ or $S^2$ . For $\mu$ a Borel probability measure on $S$ , denote the group of $C^1$ -diffeomorphisms that preserve $\mu$ by $\rm{Diff}_{\mu}(S)$ . We give several applications of our main result, showing that certain groups, including a large class of higher-rank lattices, admit no homomorphisms to $\rm{Diff}_{\mu}(S)$ with infinite image

We prove global rigidity results for some linear abelian actions on tori. The type of actions we deal with includes in particular maximal rank semisimple actions on $\T^N$., Comment: 17 pages

Let $Y=(f,g,h):\mathbb{R}^{3} \to \mathbb{R}^{3}$ be a $C^{2}$ map and let $\spec(Y)$ denote the set of eigenvalues of the derivative $DY_p$, when $p$ varies in $\mathbb{R}^3$. We begin proving that if, for some $\epsilon>0,$ $\spec(Y)\cap (-\epsilon,\epsilon)=\emptyset,$ then the foliation $\mathcal{F}(k),$ with $k\in \{f,g,h\},$ made up by the level surfaces $\{k={\rm constant}\},$ consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek's Jacobian Conjecture for polynomial maps of $\mathbb{R}^n.$, Comment: 13 pages and 3 figures

We study the topological dynamics of the action of the diagonal subgroup on quotients Gamma\PSL(2,R)*PSL(2,R), where Gamma is an irreducible lattice. Closed orbits are described and a set of points of dense orbit is explicitly given. Such properties are expressed using the Furstenberg boundary of the symmetric space H*H., Comment: 18 pages, 2 figures

Let $G$ be a simply connected Lie group and consider a Lie $G$ foliation $\mathscr F$ on a closed manifold $M$ whose leaves are all dense in $M$ . Then the space of ends ${\mathscr E}(F)$ of a leaf $F$ of $\mathscr F$ is shown to be either a singleton, a two points set, or a Cantor set. Further if $G$ is solvable, or if $G$ has no cocompact discrete normal subgroup and $\mathscr F$ admits a transverse Riemannian foliation of the complementary dimension, then ${\mathscr E}(F)$ consists of one or two points. On the contrary there exists a Lie $\widetilde{SL}(2,\bm{R})$ foliation on a closed 5-manifold whose leaf is diffeomorphic to a 2-sphere minus a Cantor set.

We introduce the so-called BT-category of borelian-topological spaces: it will be a natural frame for a measurable classification of usual foliations and laminations. We focus on the two-dimensional case: borelian laminations by surfaces. We prove two main results: (1) Any borelian lamination by planes is the suspension of a $\Z^2$-action on a Borel space iff this lamination is hyperfinite. (2) Any borelian lamination by surfaces is amenable if parabolic, i.e. if it admits a complex structure parabolic on each leaf. The third result is an improvement of (2) in case of laminations endowed with a transverse quasi-invariant measure $\mu$. The statement is the following: (3) Any borelian lamination by planes, cylinders and tori is $\mu$-amenable if and only if it admits a metric which is flat on $\mu$-almost all leafs., Comment: 32 pages, 1 figure

We exhibit rigid rotations of spheres as distortion elements in groups of diffeomorphisms, thereby answering a question of J Franks and M Handel. We also show that every homeomorphism of a sphere is, in a suitable sense, as distorted as possible in the group Homeo(S^n), thought of as a discrete group. An appendix by Y de Cornulier shows that Homeo(S^n) has the strong boundedness property, recently introduced by G Bergman. This means that every action of the discrete group Homeo(S^n) on a metric space by isometries has bounded orbits., Comment: This is the version published by Geometry & Topology on 26 March 2006 (V7: typesetting corrections)

In this paper, we characterize the dynamic of every abelian subgroups $\mathcal{G}$ of GL($n$, $\mathbb{K}$), $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. We show that there exists a $\mathcal{G}$-invariant, dense open set $U$ in $\mathbb{K}^{n}$ saturated by minimal orbits with $\mathbb{K}^{n}- U$ a union of at most $n$ $\mathcal{G}$-invariant vectorial subspaces of $\mathbb{K}^{n}$ of dimension $n-1$ or $n-2$ on $\mathbb{K}$. As a consequence, $\mathcal{G}$ has height at most $n$ and in particular it admits a minimal set in $\mathbb{K}^{n}-\{0\}$., Comment: 16 pages

Let G be the Heisenberg group of real lower triangular 3x3 matrices with unit diagonal. A locally free smooth action of G on a manifold M^4 is given by linearly independent vector fields X_1, X_2, X_3 such that X_3 = [X_1,X_2] and [X_1,X_3] = [X_2, X_3] = 0. The C^1 topology for vector fields induces a topology in the space of actions of G on M^4. An action is compact if all orbits are compact. Given a compact action $\theta$, we investigate under which conditions its C^1 perturbations $\tilde\theta$ are guaranteed to be compact. There is more than one interesting definition of stability, and we show that in the case of the Heisenberg group, unlike for actions of R^n, the definitions do not turn out to be equivalent., Comment: 17 pages, no figures (corrected typos and added references)

In this paper the author determines necessary and sufficient conditions for existence of the Ehresmann connection on a manifold foliated by locally free action of the commutative Lie group. Also here we describe structure of $C_0(M)|_L$ for a leaf $L \subset M$ in case such a connection exists. Finally we give some results on structure of the spectrum of the family of Schr\"odinger operators related to the leaves of the foliation., Comment: is not well written and completely unreadable without a detailed knowledge of the older literature

We prove that certain volume preserving actions of Lie groups and their lattices do not preserve rigid geometric structures in the sense of Gromov. The actions considered are the "exotic" examples obtained by Katok and Lewis and the first author, by blowing up closed orbits in the well known actions on homogeneous spaces. The actions on homogeneous spaces all preserve affine connections, whereas the action along the exceptional divisor preserves a projective structure. The fact that these structures cannot in some way be "glued together" to give a rigid structure on the entire space is not obvious. We also define the notion of an almost rigid structure. The paradigmatic example of a rigid structure is a global framing and the paradigmatic example of an almost rigid structure is a framing that is degenerate along some exceptional divisor. We show that the actions discussed above do possess an invariant almost rigid structure. Gromov has shown that a manifold with rigid geometric structure invariant under a topologically transitive group action is homogeneous on an open dense set. How generally this open dense set can be taken to be the entire manifold is an important question with many dynamical applications. Our results indicate one way in which the geometric structure cannot degenerate off the open dense set.

In this work, we prove the so-called "Shadow lemma" on geometrically finite manifolds of variable negative curvature. We deduce a result of non divergence of the horospheres of such manifolds., Comment: 33 pages, 10 figures, in french

In this paper we consider the ergodic properties of linear actions on the plane of (2×2)-matrices with entries in the complex numbers, quaternions, or Clifford numbers.

Let $\Gamma$ be a discrete group with property $(T)$ of Kazhdan. We prove that any Riemannian isometric action of $\Gamma$ on a compact manifold $X$ is locally rigid. We also prove a more general foliated version of this result. The foliated result is used in our proof of local rigidity for standard actions of higher rank semisimple Lie groups and their lattices in \cite{FM2}. One definition of property $(T)$ is that a group $\Gamma$ has property $(T)$ if every isometric $\Gamma$ action on a Hilbert space has a fixed point. We prove a variety of strengthenings of this fixed point properties for groups with property $(T)$. Some of these are used in the proofs of our local rigidity theorems., Comment: 69 pages. See also http://comet.lehman.cuny.edu/fisher/ Substantial revision including reordering of many arguments and streamlining of some statements

If $M$ is a compact or convex-cocompact negatively curved manifold, we associate to any Gibbs measure on $\tm$ a quasi-invariant transverse measure for the horospherical foliation, and prove that this measure is uniquely determined by its Radon-Nikodym cocycle. (This extends the Bowen-Marcus unique ergodicity result for this foliation.) We shall also prove equidistribution properties for the leaves of the foliation w.r.t. these Gibbs measures. We use these results in the study of invaiant meausres for horospherical foliations on regular covers of $M$., 28 pages, 6 figures

An action of a topological group $G$ on a topological space $X$ is orbit nonproper if, for some $x\in X$, the map $g\mapsto gx:G\to X$ is nonproper. We describe the collection of connected, simply connected Lie groups admitting a locally faithful, orbit nonproper action by isometries of a connected Lorentz manifold.

Let $G$ be a non-unimodular solvable Lie group which is a semidirect product of $R^m$ and $R^n$. We consider a codimension one locally free volume preserving action of $G$ on a closed manifold. It is shown that, under some conditions on the group $G$, such an action is homogeneous. It is also shown that such a group $G$ has a homogeneous action if and only if the structure constants of $G$ satisfy certain algebraic conditions.

We consider endomorphism actions of arbitrary discrete semigroups on a connected metrizable topological group G. We give necessary and sufficient conditions for expansiveness of such actions when G is a Lie group or a compact finite-dimensional group., Comment: 21 pages, no figures

We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $\Gamma $ by automorphisms of compact abelian groups in terms of Fuglede-Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $\Gamma $-action by means of a `fundamental homoclinic point', and the description of entropy in terms of the renormalized logarithmic growth-rate of the set of $\Gamma_n$-fixed points, where $(\Gamma_n, n\ge1)$ is a decreasing sequence of finite index normal subgroups of $\Gamma $ with trivial intersection., Comment: Using a recent result of Y. Choi the discussion of expansiveness was strengthened and shortened. The paper will appear in: Ergodic Theory and Dynamical Systems