Showing papers in "Geometric and Functional Analysis in 2005"
TL;DR: In this article, the Fraisse theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures are studied.
Abstract: (A) In this paper we study some connections between the Fraisse theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. A prime concern of topological dynamics is the study of continuous actions of (Hausdorff) topological groups G on (Hausdorff) compact spaces X.
419 citations
TL;DR: In this article, an analogue of Szemeredi's regularity lemma in the context of abelian groups is presented and used to derive some results in additive number theory.
Abstract: Szemeredi’s regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemeredi’s regularity lemma in the context of abelian groups and use it to derive some results in additive number theory. One is a structure theorem for sets which are almost sum-free. If
$$A \subseteq \{1,\ldots,N\}$$ has δ N2 triples (a1, a2, a3) for which a1 + a2 = a3 then A = B ∪ C, where B is sum-free and |C| = δ′N, and
$$\delta^{\prime} \rightarrow 0$$
as
$$\delta \rightarrow 0.$$
Another answers a question of Bergelson, Host and Kra. If
$$\alpha, \epsilon > 0,$$
if
$$N\,>\,N_{0}(\alpha, \epsilon)$$
and if
$$A \subseteq \{1,\ldots,N\}$$
has size α N, then there is some d ≠ 0 such that A contains at least
$$(\alpha^{3}-\epsilon)N$$
three-term arithmetic progressions with common difference d.
210 citations
TL;DR: In this paper, it was shown that the concentration of solutions occurs at some geodesics of ∂Ω when ǫ → 0, where ∆ ≥ 0.
Abstract: We prove new concentration phenomena for the equation −ɛ2 Δu + u = up in a smooth bounded domain \(\Omega \subseteq \mathbb{R}^3 \) and with Neumann boundary conditions. Here p > 1 and ɛ > 0 is small. We show that concentration of solutions occurs at some geodesics of ∂Ω when ɛ → 0.
94 citations
TL;DR: In this paper, it was shown that any n-point metric space (X, d) embeds in Hilbert space with distortion, where α is a geometric estimate on the decomposability of X and λ is the doubling constant of X. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O(( log n)2).
Abstract: We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Frechet embeddings for finite metrics, due to Bourgain (1985) and Rao (1999). We prove that any n-point metric space (X, d) embeds in Hilbert space with distortion
$$O{\left( {{\sqrt {\alpha _{X} \cdot \log n} }} \right)},$$
where α
X
is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an
$$O{\left( {{\sqrt {(\log \lambda _{X} )\log n} }} \right)}$$
distortion embedding, where λ
X
is the doubling constant of X. Since λ
X
≤ n, this result recovers Bourgain’s theorem, but when the metric X is, in a sense, “low-dimensional,” improved bounds are achieved. Our embeddings are volume-respecting for subsets of arbitrary size. One consequence is the existence of (k, O(log n)) volume-respecting embeddings for all 1 ≤ k ≤ n, which is the best possible, and answers positively a question posed by U. Feige. Our techniques are also used to answer positively a question of Y. Rabinovich, showing that any weighted n-point planar graph embeds in
$${\ell }^{{O(\log n)}}_{\infty } $$
with O(1) distortion. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O((log n)2).
93 citations
TL;DR: In this paper, an isoperimetric inequality of euclidean type for complete metric spaces admitting a cone-type inequality was proved for all Banach spaces and all simply-connected metric spaces of non-positive curvature.
Abstract: In this paper we prove an isoperimetric inequality of euclidean type for complete metric spaces admitting a cone-type inequality. These include all Banach spaces and all complete, simply-connected metric spaces of non-positive curvature in the sense of Alexandrov or, more generally, of Busemann. The main theorem generalizes results of Gromov and Ambrosio-Kirchheim.
92 citations
TL;DR: In this article, simple methods to establish the property of Rapid Decay for a number of groups arising geometrically were presented. But these methods are not suitable for non-cocompact lattices in rank one Lie groups.
Abstract: We explain simple methods to establish the property of Rapid Decay for a number of groups arising geometrically. Those lead to new examples of groups with the property of Rapid Decay, notably including non-cocompact lattices in rank one Lie groups.
82 citations
TL;DR: In this article, the authors proposed a method to solve the problem of the problem.No Abstracts. No Abstracts, no Abstracts No Abstract, No abstracts, No Abstract
Abstract: No Abstract..
79 citations
TL;DR: In this paper, the authors proposed a method to solve the problem of the problem.No Abstracts. No Abstracts, no Abstracts No Abstract, No abstracts, No Abstract
Abstract: No Abstract..
77 citations
TL;DR: In this paper, it was shown that hyperbolic groups admit affine affine isometric actions on $l^p$-spaces, where p is the number of vertices.
Abstract: In this paper, we show that hyperbolic groups admit proper affine isometric actions on $l^p$-spaces
72 citations
TL;DR: In this paper, it was shown that a K3 surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve.
Abstract: A K3 surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that similar statement holds for the generic couple (X, H) with X a deformation of (K3)[n] and H an ample divisor of square 2 for Beauville’s quadratic form. If n = 2 then according to the conjecture X is a double cover of a singular) sextic 4-fold in \(\mathbb{P}^{5} .\) It follows from the conjecture that a deformation of (K3)[n] carrying a divisor (not necessarily ample) of degree 2 has an anti-symplectic birational involution. We test the conjecture. In doing so we bump into some interesting geometry: examples of two antisymplectic involutions generating an interesting dynamical system, a case Strange duality and what is probably an involution on the moduli space degree-2 quasi-polarized (X, H) where X is a deformation of (K3)[2].
71 citations
TL;DR: In this paper, the authors proved that key functionals (such as the volume and the number of vertices) of a random polytope is strongly concentrated, using a martingale method.
Abstract: We prove that key functionals (such as the volume and the number of vertices) of a random polytope is strongly concentrated, using a martingale method. As applications, we derive new estimates for high moments and the speed of convergence of these functionals.
TL;DR: In this article, the relation between the existence of Kahler-Ricci solitons and a certain functional associated to some complex Monge-Ampere equation on compact complex manifolds with positive first Chern class was discussed.
Abstract: In this paper, we discuss the relation between the existence of Kahler–Ricci solitons and a certain functional associated to some complex Monge–Ampere equation on compact complex manifolds with positive first Chern class. In particular, we obtain a strong inequality of Moser–Trudinger type on a compact complex manifold admitting a Kahler–Ricci soliton.
TL;DR: In this paper, the authors introduce a notion of energy for harmonic currents of bi-degree (1, 1) on a complex Kahler manifold (M, ω) and define T ∧ T ∞ ∞ ω k−2, for positive harmonic currents.
Abstract: We introduce a notion of energy for harmonic currents of bi- degree (1, 1) on a complex Kahler manifold (M, ω). This allows us to define T ∧ T ∧ ω k−2 , for positive harmonic currents. We then show that for a lamination with singularities of a compact set in P 2 , without directed positive closed currents, there is a unique positive harmonic current which minimizes energy. If X is a compact laminated set in P 2 of class C 1 it carries a unique positive harmonic current T of mass 1. The current T can be obtained by an Ahlfors type construction starting with an arbitrary leaf of X. When X has a totally disconnected set of singularities, contained in a countable union of analytic sets, the above construction still gives positive harmonic currents.
TL;DR: In this paper, the existence of the non-uniqueness phase for the Bernoulli percolation on unimodular transitive locally finite graphs admitting nonconstant harmonic Dirichlet functions was shown.
Abstract: The main goal of this paper is to answer question~1.10 and settle conjecture~1.11 of Benjamini-Lyons-Schramm \cite{BLS99} relating harmonic Dirichlet functions on a graph to those of the infinite clusters in the uniqueness phase of Bernoulli percolation. We extend the result to more general invariant percolations, including the Random-Cluster model. We prove the existence of the nonuniqueness phase for the Bernoulli percolation (and make some progress for Random-Cluster model) on unimodular transitive locally finite graphs admitting nonconstant harmonic Dirichlet functions. This is done by using the device of $\ell^2$ Betti numbers.
TL;DR: The notion of the perimeter of a map between two finite 2-complexes is introduced in this paper, which is a criterion for positively determining the coherence of a group and is based on the notion of local quasiconvexity.
Abstract: A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2-complexes which is introduced here. In the groups to which this theory applies, a presentation for a finitely generated subgroup can be computed in quadratic time relative to the sum of the lengths of the generators. For many of these groups we can show in addition that they are locally quasiconvex. As an application of these results we prove that one-relator groups with sufficient torsion are coherent and locally quasiconvex and we give an alternative proof of the coherence and local quasiconvexity of certain 3-manifold groups. The main application is to establish the coherence and local quasiconvexity of many small cancellation groups.
TL;DR: In this article, the authors prove a rigidity and a characterization result for building and spherical joins using sets of antipodal points, and prove that these points can be used for building construction.
Abstract: We prove a rigidity and a characterization result for buildings and spherical joins using sets of antipodal points.
TL;DR: In this article, the authors prove local limit theorems for products of independent random variables on the Heisenberg group which are identically distributed with respect to an arbitrary centered and compactly supported probability measure.
Abstract: We prove local limit theorems for products of independent random variables on the Heisenberg group which are identically distributed with respect to an arbitrary centered and compactly supported probability measure μ. We also provide uniform estimates for translates of a bounded set by comparing μ
n
to the associated heat kernel. This, in turn, enables us to show the equidistribution of Heisenberg-unipotent random walks on finite volume homogeneous spaces G / Γ.
TL;DR: In this paper, the authors give explicit formulae for local normal zeta functions of torsion-free, class-2-nilpotent groups, subject to conditions on the associated Pfaffian hypersurface which are generically satisfied by groups with small center and sufficiently large abelianization.
Abstract: We give explicit formulae for the local normal zeta functions of torsion-free, class-2-nilpotent groups, subject to conditions on the associated Pfaffian hypersurface which are generically satisfied by groups with small centre and sufficiently large abelianization. We show how the functional equations of two types of zeta functions – the Weil zeta function associated to an algebraic variety and zeta functions of algebraic groups introduced by Igusa – match up to give a functional equation for local normal zeta functions of groups. We also give explicit formulae and derive functional equations for an infinite family of class-2-nilpotent groups known as Grenham groups, confirming conjectures of du Sautoy.
TL;DR: The filling area conjecture for genus 1 fillings of the circle was shown to hold in this article, extending P. Pu's result in genus 0, where the singular points are Weierstrass points.
Abstract: We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu’s result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingredients. On the one hand, we exploit integral geometric comparison with orbifold metrics of constant positive curvature on real surfaces of even positive genus. Here the singular points are Weierstrass points. On the other hand, we exploit an analysis of the combinatorics on unions of closed curves, arising as geodesics of such orbifold metrics.
TL;DR: The relation between the first eigenvalues of successive higher Laplacians of a simplicial simplicial complex of a graph G = (V, E) is studied in this article.
Abstract: The flag complex of a graph G = (V, E) is the simplicial complex X(G) on the vertex set V whose simplices are subsets of V which span complete subgraphs of G. We study relations between the first eigenvalues of successive higher Laplacians of X(G). One consequence is the following:
TL;DR: In this paper, the authors present several classes of real Banach Lie-Poisson spaces whose characteristic distributions are integrable, the integral manifolds being symplectic leaves just as in finite dimensions, and investigate when these leaves are embedded submanifolds or when they have Kahler structures.
Abstract: We present several large classes of real Banach Lie–Poisson spaces whose characteristic distributions are integrable, the integral manifolds being symplectic leaves just as in finite dimensions. We also investigate when these leaves are embedded submanifolds or when they have Kahler structures. Our results apply to the real Banach Lie–Poisson spaces provided by the self-adjoint parts of preduals of arbitrary W*-algebras, as well as of certain operator ideals.
TL;DR: In this article, the authors consider an entropy-type invariant which measures the polynomial volume growth of submanifolds under the iterates of a map, and show that this invariant is at least 1 for every diffeomorphism in the symplectic isotopy class of the Dehn-Seidel twist.
Abstract: We consider an entropy-type invariant which measures the polynomial volume growth of submanifolds under the iterates of a map, and we show that this invariant is at least 1 for every diffeomorphism in the symplectic isotopy class of the Dehn–Seidel twist.
TL;DR: In this paper, it was shown that if A is a separable, nuclear, and absorbing (or strongly purely infinite) C*-algebra which is homotopic to zero in an ideal-system preserving way, then A is the inductive limit of C *-algebras.
Abstract: We show that if A is a separable, nuclear, \(\mathcal{O}_\infty \)-absorbing (or strongly purely infinite) C*-algebra which is homotopic to zero in an ideal-system preserving way, then A is the inductive limit of C*-algebras of the form \(C_0 (\Gamma ,\upsilon ) \otimes M_k ,\) where Γ is a finite connected graph (and \(C_0 (\Gamma ,\upsilon )\) is the algebra of continuous functions on Γ that vanish at a distinguished point \(\upsilon \in \Gamma \)).
TL;DR: In this paper, the authors give a quantitative proof for a theorem of Martio, Rickman and Vaisala [MRV] on the rigidity of the local homeomorphism property of spatial quasiregular mappings with distortion close to one.
Abstract: We give a quantitative proof for a theorem of Martio, Rickman and Vaisala [MRV] on the rigidity of the local homeomorphism property of spatial quasiregular mappings with distortion close to one. The proof is based on a distortion theory established by using two main tools. First, we use a conformal invariant between sphere families and components of their preimages under quasiregular mappings. Secondly, we use Hall’s quantitative isoperimetric inequality result [H] to relate two different types of distortion.
TL;DR: In this paper, a conjecture relating GIT stability of a polarized algebraic variety to the existence of a Kahler metric of constant scalar curvature was shown to hold for reductive algebraic varieties.
Abstract: G. Tian and S.K. Donaldson formulated a conjecture relating GIT stability of a polarized algebraic variety to the existence of a Kahler metric of constant scalar curvature. In [D3] Donaldson partially confirmed it in the case of projective toric varieties. In this paper we extend Donaldson’s results and computations to a new case, that of reductive varieties.
TL;DR: In this article, the notion of security for polygons and flat surfaces was introduced and studied, and it was shown that a lattice polygon is secure iff it is arithmetic.
Abstract: We introduce and study the notion of security for polygons and flat surfaces. Let P be one. For x, y ∈ P let G(x, y) be the set of geodesics connecting x and y. We say that P is secure if for any x, y ∈ P all geodesics in G(x, y) can be blocked by a finite set B ⊂ P. We prove, in particular, that a lattice polygon is secure iff it is arithmetic.
TL;DR: In this article, a study of several notions of size of subsets of groups is presented, and connections between properties of Haar null sets and algebraic properties (amenability, FC) of the underlying group are made.
Abstract: This is a study of several notions of size of subsets of groups. The first part (sections 3–5) concerns a purely algebraic setting with the underlying group discrete. The various notions of size considered there are similar to each other in that each of them assesses the size of a set using a family of measures and translations of the set; they differ in the type of measures used and the type of translations allowed. The way these various notions relate to each other is tightly and, perhaps, unexpectedly connected with the algebraic structure of the group. An important role is played by amenable, ICC (infinite conjugacy class), and FC (finite conjugacy class) groups. The second part of the paper (section 6), which was the original motivation for the present work, deals with a well-studied notion of smallness of subsets of Polish, not necessarily locally compact, groups – Haar null sets. It contains applications of the results from the first part in solving some problems posed by Christensen and by Mycielski. These applications are the first results detecting connections between properties of Haar null sets and algebraic properties (amenability, FC) of the underlying group.
TL;DR: In this article, the authors investigated analogues for curves of the Kakeya problem for straight lines, which arise from H"ormander-type oscillatory integrals in the same way as the straight line case comes from the restriction and Bochner-Riesz problems.
Abstract: We investigate analogues for curves of the Kakeya problem for straight lines. These arise from H"ormander-type oscillatory integrals in the same way as the straight line case comes from the restriction and Bochner-Riesz problems. Using some of the geometric and arithmetic techniques developed for the straight line case by Bourgain, Wolff, Katz and Tao, we are able to prove positive results for families of parabolas whose coefficients satisfy certain algebraic conditions.
TL;DR: In this paper, a compact Kahler manifold admitting a hypercomplex structure (M, I, J, K) admits a natural HKT-metric, which is used to construct a holomorphic symplectic form on (m, I).
Abstract: Let (M, I) be a compact Kahler manifold admitting a hypercomplex structure (M, I, J, K). We show that (M, I, J, K) admits a natural HKT-metric. This is used to construct a holomorphic symplectic form on (M, I).
TL;DR: In this article, Voiculescu showed that for any discrete finitely-generated group G and any self-adjoint n-tuple X 1, X n of generators of the group algebra CG, the non-microstate free entropy dimension is exactly equal to β 1(G) − β 0(G )+1, where βi are the � 2 -Betti numbers of G.
Abstract: We show that for any discrete finitely-generated group G and any self-adjoint n-tuple X1 ,...,X n of generators of the group algebra CG, Voiculescu's non-microstates free entropy dimension δ ∗ (X1 ,...,X n) is exactly equal to β1(G) − β0(G )+1 , whereβi are the � 2 -Betti numbers of G.