Double ergodicity of the Poisson boundary and applications to bounded cohomology

We prove that the Poisson boundary of any spread out non-degenerate symmetric randomwalk on an arbitrary locally compact second countable group G is doubly $\mathcal{M}$^sep-ergodic with respect to the class $\mathcal{M}$^sep of separable coefficient Banach G-modules. The proof is direct and based on an analogous property of the bilateral Bernoulli shift in the space of increments of the random walk. As a corollary we obtain that any locally compact s-compact group G admits a measure class preserving action which is both amenable and doubly $\mathcal{M}$^sep-ergodic. This generalizes an earlier result of Burger and Monod obtained under the assumption that G is compactly generated and allows one to dispose of this assumption in numerous applications to the theory of bounded cohomology.     

Sharp phase transition theorems for hyperbolicity of random groups

We prove that in various natural models of a random quotient of a group, depending on a density parameter, for each hyperbolic group there is some critical density under which a random quotient is still hyperbolic with high probability, whereas above this critical value a random quotient is very probably trivial. We give explicit characterizations of these critical densities for the various models.     

Real entire functions of infinite order and a conjecture of Wiman

We prove that if f is a real entire function of infinite order, then ff has infinitely many non-real zeros. In conjunction with the result of Sheil-Small for functions of finite order this implies that if f is a real entire function such that ff has only real zeros, then f is in the Laguerre-Pólya class, the closure of the set of real polynomials with real zeros. This result completes a long line of development originating from a conjecture of Wiman of 1911.     

Geometric inequalities via a general comparison principle for interacting gases

The article builds on several recent advances in the Monge- Kantorovich theory of mass transport which have, among other things, led to new and quite natural proofs for a wide range of geometric inequalities such as the ones formulated by Brunn-Minkowski, Sobolev, Gagliardo- Nirenberg, Beckner, Gross, Talagrand, Otto-Villani and their extensions by many others. While this paper continues in this spirit, we however propose here a basic framework to which all of these inequalities belong, and a general unifying principle from which many of them follow. This basic inequality relates the relative total energy - internal, potential and interactive - of two arbitrary probability densities, their Wasserstein distance, their barycentres and their entropy production functional. The framework is remarkably encompassing as it implies many old geometric - Gaussian and Euclidean - inequalities as well as new ones, while allowing a direct and unified way for computing best constants and extremals. As expected, such inequalities also lead to exponential rates of convergence to equilibria for solutions of Fokker-Planck and McKean-Vlasov type equations. The principle also leads to a remarkable correspondence between ground state solutions of certain quasilinear - or semilinear - equations and stationary solutions of nonlinear Fokker-Planck type equations.     

Volume,diameter and the minimal mass of a stationary 1-cycle

In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemannian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a closed n-dimensional Riemannian manifold Mⁿ is bounded from above by (n + 2)!d/4, where d is the diameter of a manifold Mⁿ. The second result is that the minimal mass of a stationary 1-cycle on a closed Riemannian manifold Mⁿ is bounded from above by where where is the filling radius of a manifold, and where is its volume.     

Noncommutative potential theory and the sign of the curvature operator in Riemannian geometry

The aim of this work is to show that in any complete Riemannian manifold M, without boundary, the curvature operator is nonnegative if and only if the Dirac Laplacian D² generates a C*-Markovian semigroup (i.e. a strongly continuous, completely positive, contraction semigroup) on the Cliord C*-algebra of Mor, equivalently, if and only if the quadratic form $\mathcal{E}$_D of D² is a C*-Dirichlet form.     

Kazhdan's Property (T) for the unitary group of aseparable Hilbert space

We show that the unitary group of a separable Hilbert space has Kazhdan's Property (T), when it is equipped with the strong operator topology. More precisely, for every integer m 2, we give an explicit Kazhdan set consisting of m unitary operators and determine an optimal Kazhdan constant for this set. Moreover, we show that a locally compact group with Kazhdan's Property (T) has a finite Kazhdan set if and only if its Bohr compactification has a finite Kazhdan set. As a consequence, if a locally compact group with Property (T) is minimally almost periodic, then it has a finite Kazhdan set.     

Finite jet determination of local analytic CR automorphisms and their parametrization by 2-jets in the finite type case

We show that germs of local real-analytic CR automorphisms of a real-analytic hypersurface M in $\mathbb{C}$² at a point p M are uniquelydetermined by their jets of some finite order at p if and only if M is not Levi-flat near p. This seems to be the first necessary and sufficient result on finite jet determination and the first result of this kind in the infinite type case.If M is of finite type at p, we prove a stronger assertion: the local real-analytic CR automorphisms of M fixing p are analytically parametrized (and hence uniquely determined) by their 2-jets at p. This result is optimal since the automorphisms of the unit sphere are not determined by their 1-jets at a point of the sphere. The finite type condition is necessary since otherwise the needed jet order can be arbitrarily high [Kow1,2], [Z2]. Moreover, we show, by an example, that determination by 2-jets fails for finite type hypersurfaces already in $\mathbb{C)$³.We also give an application to the dynamics of germs of local biholomorphisms of $\mathbb{C)$².     

<Emphasis Type="Italic">L</Emphasis>-convex-concave body in ${\boldmath{$\mathbb{R}P^3$}}$ contains a line

We define a class of L-convex-concave subsets of ${\boldmath{$\mathbb{R}P^3$}}$, where L is a projective line in ${\boldmath{$\mathbb{R}P^3$}}$. These are sets whose sections by any plane containing L are convex and concavely depend on this plane. We prove a version of Arnolds conjecture for these sets, namely we prove that each such set contains a line.     

Representations of algebraic groups over a 2-dimensional local field

We introduce a categorical framework for the study of representations of G(F), where G is a reductive group, and F is a 2-dimensional local field, i.e. F = K((t)), where K is a local field. Our main result says that the space of functions on G(F), which is an object of a suitable category of representations of G(F) with the respect to the action of G on itself by left translations, becomes a representation of a certain central extension of G(F), when we consider the action by right translations.     

Low regularity solutions for the Kadomtsev-Petviashvili I equation

In this paper we obtain local in time existence and (suitable) uniqueness and continuous dependence for the KP-I equation for small data in the intersection of the energy space and a natural weighted L ² space. An erratum to this article is available at .     

Growth and percolation on the uniform infinite planar triangulation

A construction as a growth process for sampling of the uniform in- finite planar triangulation (UIPT), defined in [AnS], is given. The construction is algorithmic in nature, and is an efficient method of sampling a portion of the UIPT.By analyzing the progress rate of the growth process we show that a.s. the UIPT has growth rate r ⁴ up to polylogarithmic factors, in accordance with heuristic results from the physics literature. Additionally, the boundary component of the ball of radius r separating it from infinity a.s. has growth rate r ² up to polylogarithmic factors. It is also shown that the properly scaled size of a variant of the free triangulation of an m-gon (also defined in [AnS]) converges in distribution to an asymmetric stable random variable of type 1/2.By combining Bernoulli site percolation with the growth process for the UIPT, it is shown that a.s. the critical probability p _c = 1/2 and that at p _c percolation does not occur.     

On the structure of level sets of uniform and Lipschitz quotient mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{R} $$

We study two questions posed by Johnson, Lindenstrauss, Preiss, and Schechtman, concerning the structure of level sets of uniform and Lipschitz quotient mappings from . We show that if , is a uniform quotient mapping then for every has a bounded number of components, each component of separates and the upper bound of the number of components depends only on and the moduli of co-uniform and uniform continuity of .Next we prove that all level sets of any co-Lipschitz uniformly continuous mapping from to are locally connected, and we show that for every pair of a constant and a function with , there exists a natural number , so that for every co-Lipschitz uniformly continuous map with a co-Lipschitz constant and a modulus of uniform continuity , there exists a natural number and a finite set with card so that for all has exactly components, has exactly components and each component of is homeomorphic with the real line and separates the plane into exactly 2 components. The number and form of components of for are also described - they have a finite tree structure.     

Cyclicity of zeros of families of analytic functions

Let be a family of holomorphic functions in the unit disk , which are also holomorphic in a parameter . We express cyclicity (=generalized multiplicity) of a zero of at via some algebraic characteristics of the ideal generated by the Taylor coefficients of . As an example we estimate the cyclicity of the family of generalized exponential polynomials.     

Hausdorff dimension of limit sets of discrete subgroups of higher rank Lie groups

Let X be a globally symmetric space of noncompact type, and a discrete subgroup. Introducing an appropriate notion of Hausdorff measure on the geometric boundary of , we prove that for regular boundary points , the Hausdorff dimension of the radial limit set in is bounded above by the exponential growth rate of the number of orbit points close in direction to . Furthermore, for Zariski dense discrete groups we construct -invariant densities with support in every G-invariant subset of the limit set and study their properties. For a class of groups which generalises convex cocompact groups in the rank one setting, these densities allow to give a sharp estimate on the Hausdorff dimension of the radial limit set in each subset .     

Equidistribution of Kronecker sequences along closed horocycles

It is well known that (i) for every irrational number the Kronecker sequence m (m = 1,...,M) is equidistributed modulo one in the limit , and (ii) closed horocycles of length become equidistributed in the unit tangent bundle of a hyperbolic surface of finite area, as . In the present paper both equidistribution problems are studied simultaneously: we prove that for any constant the Kronecker sequence embedded in along a long closed horocycle becomes equidistributed in for almost all , provided that . This equidistribution result holds in fact under explicit diophantine conditions on (e.g. for = 2) provided that , with additional assumptions on the Fourier coefficients of certain automorphic forms. Finally, we show that for , our equidistribution theorem implies a recent result of Rudnick and Sarnak on the uniformity of the pair correlation density of the sequence n² modulo one.     

Transience and capacity of minimal submanifolds

We prove explicit lower bounds for the capacity of annular domains of minimal submanifolds P ^m in ambient Riemannian spaces N ⁿ with sectional curvatures bounded from above. We characterize the situations in which the lower bounds for the capacity are actually attained. Furthermore we apply these bounds to prove that Brownian motion defined on a complete minimal submanifold is transient when the ambient space is a negatively curved Hadamard-Cartan manifold. The proof stems directly from the capacity bounds and also covers the case of minimal submanifolds of dimension m > 2 in Euclidean spaces.