CAT(0) groups and Coxeter groups whose boundaries are scrambled sets

In this paper, we study CAT(0) groups and Coxeter groups whose boundaries are scrambled sets. Suppose that a group G acts geometrically (i.e. properly and cocompactly by isometries) on a proper CAT(0) space X. (Such a group G is called a CAT(0) group.) Then the group G acts by homeomorphisms on the boundary ∂X of X and we can define a metric d_∂X on the boundary ∂X. The boundary ∂X is called a scrambled set if, for any α,β∈∂X with α≠β, (1) lim sup{d_∂X(gα,gβ)∣g∈G}>0 and (2) lim inf{d_∂X(gα,gβ)∣g∈G}=0. We investigate when boundaries of CAT(0) groups (and Coxeter groups) are scrambled sets.     

The local structure of length spaces with curvature bounded above

We show that a number of different notions of dimension coincide for length spaces with curvature bounded above. We then apply this result, showing that if X is a locally compact CAT(0) space with cocompact isometry group, then the dimension of the Tits boundary and the asymptotic cone(s) of X are determined by the maximal dimension of a flat in X. Received April 21, 1998     

Transfinite Hausdorff dimension

Making extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorphisms), in fact like Hausdorff dimension, it does not increase under Lipschitz maps, and it also satisfies the intermediate dimension property (Theorem 2.7). The primary goal of transfinite Hausdorff dimension is to classify metric spaces with infinite Hausdorff dimension. Indeed, if tHD(X)?ω₀, then HD(X)=+∞. We prove that tHD(X)?ω₁ for every separable metric space X, and, as our main theorem, we show that for every ordinal number α<ω₁ there exists a compact metric space Xα (a subspace of the Hilbert space l₂) with tHD(Xα)=α and which is a topological Cantor set, thus of topological dimension 0. In our proof we develop a metric version of Smirnov topological spaces and we establish several properties of transfinite Hausdorff dimension, including its relations with classical Hausdorff dimension.     

Subsets of a Hilbert space,having finite hausdorff dimension

Let X₂, X₂ be Hilbert spaces, X₂ X₁, X₂ is dense in X₁, the imbedding is compact,m X₂, dim_H ⁱ m and h⁽ⁱ⁾(m) are the Hausdorff dimension and the limit capacity (information dimension) of the setm with respect to the metrics of the spaces X_i (i=1, 2). Two examples are constructed. 1) An example of a setm bounded in X₂, such that: a) h⁽¹⁾(m) < (and, consequently, dim_H ¹ m); b)m cannot be covered by a countable collection of sets, compact in X₂ (and, consequently, dim_H ² m=). 2) an Example of a setm, compact in X₂, such that h⁽¹⁾(m) < and h⁽²⁾(m)=.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 154–165, 1987.     

Flows, fixed points and rigidity for Kleinian groups

The main application of the techniques developed in this paper is to prove a relative version of Mostow rigidity, called pattern rigidity. For a cocompact group G, by a G-invariant pattern we mean a G-invariant collection of closed proper subsets of the boundary of hyperbolic space which is discrete in the space of compact subsets minus singletons. Such a pattern arises for example as the collection of translates of limit sets of finitely many infinite index quasiconvex subgroups of G. We prove that (in dimension at least three) for G ₁, G ₂ cocompact Kleinian groups, any quasiconformal map pairing a G ₁-invariant pattern to a G ₂-invariant pattern must be conformal. This generalizes a previous result of Schwartz who proved rigidity in the case of limit sets of cyclic subgroups, and Biswas and Mj (Pattern rigidity in hyperbolic spaces: duality and pd subgroups, arxiv:math.GT/08094449, 2008) who proved rigidity for Poincare Duality subgroups. Pattern rigidity is a consequence of the study conducted in this paper of the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group G ₁ and a quasiconformal conjugate h ^?1 G ₂ h of a cocompact group G ₂. We show that if the conjugacy h is not conformal then this group contains a flow, i.e. a non-trivial one parameter subgroup. Mostow rigidity is an immediate consequence.     

The wave equation on asymptotically de Sitter-like spaces

In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds (X^○,g) which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y_±, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to +∞, and to the other manifold as the parameter goes to −∞, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y₊ to Y₋.     

On the existence of continuous (approximate) roots of algebraic equations

The present paper considers the existence of continuous roots of algebraic equations with coefficients being continuous functions defined on compact Hausdorff spaces. For a compact Hausdorff space X, C(X) denotes the Banach algebra of all continuous complex-valued functions on X with the sup norm ∥⋅_∥∞. The algebra C(X) is said to be algebraically closed if each monic algebraic equation with C(X) coefficients has a root in C(X). First we study a topological characterization of a first-countable compact (connected) Hausdorff space X such that C(X) is algebraically closed. The result has been obtained by Countryman Jr, Hatori-Miura and Miura-Niijima and we provide a simple proof for metrizable spaces.Also we consider continuous approximate roots of the equation zn−f=0 with respect to z, where f∈C(X), and provide a topological characterization of compact Hausdorff space X with dimX?1 such that the above equation has an approximate root in C(X) for each f∈C(X), in terms of the first ?ech cohomology of X.     

At infinity of finite-dimensional CAT(0) spaces

We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space X has a non-empty intersection in the visual bordification ${ \overline{X} = X \cup \partial X}We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space X has a non-empty intersection in the visual bordification [`(X)] = X è?X{ \overline{X} = X \cup \partial X} . Using this fact, several results known for proper CAT(0) spaces may be extended to finite-dimensional spaces, including the existence of canonical fixed points at infinity for parabolic isometries, algebraic and geometric restrictions on amenable group actions, and geometric superrigidity for non-elementary actions of irreducible uniform lattices in products of locally compact groups.     

Topological conformal field theories and Calabi-Yau categories

This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov-Witten invariants (at all genera). These Gromov-Witten type invariants depend on a Calabi-Yau A_∞ category, which plays the role of the target in ordinary Gromov-Witten theory. When we use an appropriate A_∞ version of the derived category of coherent sheaves on a Calabi-Yau variety, this constructs the B model at all genera. When the Fukaya category of a compact symplectic manifold X is used, it is shown, under certain assumptions, that the usual Gromov-Witten invariants are recovered. The assumptions are that open-closed Gromov-Witten theory can be constructed for X, and that the natural map from the Hochschild homology of the Fukaya category of X to the ordinary homology of X is an isomorphism.     

Some properties of the injective tensor product of L[0,1] and a Banach space

Let X be a (real or complex) Banach space and 1<p,p′<∞ such that 1/p+1/p′=1. Then , the injective tensor product of L^p[0,1] and X, has the Radon-Nikodym property (resp. the analytic Radon-Nikodym property, the near Radon-Nikodym property, contains no copy of c₀, is weakly sequentially complete) if and only if X has the same property and each continuous linear operator from L^p′[0,1] to X is compact.     

A generalization of the Strassen converse

This note is devoted to a generalization of the Strassen converse. Let gn:R^∞→[0,∞], n?1 be a sequence of measurable functions such that, for every n?1, and for all x,y∈R^∞, where 0<C<∞ is a constant which is independent of n. Let be a sequence of i.i.d. random variables. Assume that there exist r?1 and a function ?:[0,∞)→[0,∞) with limt→∞?(t)=∞, depending only on the sequence such that lim supn→∞gn(X₁,X₂,…)=?(Er|X|) a.s. whenever Er|X|<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X₁,X₂,…)<∞ a.s. implies Er|X|<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,…)=∞ for all c≠0). Some applications are provided to illustrate this result.     

Geometric quantization for proper actions

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an analog of the Guillemin-Sternberg geometric quantization conjecture holds if M is symplectic with a Hamiltonian action of G that is proper and cocompact. This essentially solves a conjecture of Hochs and Landsman.     

On splitting theorems for CAT(0) spaces and compact geodesic spaces of non-positive curvature

In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group Γ is said to be rigid, if Γ determines its boundary up to homeomorphisms of a CAT(0) space on which Γ acts geometrically. C. Croke and B. Kleiner have constructed a non-rigid CAT(0) group. As an application of the splitting theorems for CAT(0) spaces, we obtain that if Γ₁ and Γ₂ are rigid CAT(0) groups then so is Γ₁ × Γ₂.     

On the dimension of global sections of adjoint bundles for polarized 3-folds and 4-folds

Let (X,L) be a polarized manifold of dimension n defined over the field of complex numbers. In this paper, we treat the case where n=3 and 4. First we study the case of n=3 and we give an explicit lower bound for h⁰(K_X+L) if κ(X)≥0. Moreover, we show the following: if κ(K_X+L)≥0, then h⁰(K_X+L)>0 unless κ(X)=−∞ and h¹(O_X)=0. This gives us a partial answer of Effective Non-vanishing Conjecture for polarized 3-folds. Next for n=4 we investigate the dimension of H⁰(K_X+mL) for m≥2. If n=4 and κ(X)≥0, then a lower bound for h⁰(K_X+mL) is obtained. We also consider a conjecture of Beltrametti-Sommese for 4-folds and we can prove that this conjecture is true unless κ(X)=−∞ and h¹(O_X)=0. Furthermore we prove the following: if (X,L) is a polarized 4-fold with κ(X)≥0 and h¹(O_X)>0, then h⁰(K_X+L)>0.     

Inverse problems and index formulae for Dirac operators

We consider a Dirac-type operator DP on a vector bundle V over a compact Riemannian manifold (M,g) with a non-empty boundary. The operator DP is specified by a boundary condition P(u_|∂M)=0 where P is a projector which may be a non-local, i.e., a pseudodifferential operator. We assume the existence of a chirality operator which decomposes L²(M,V) into two orthogonal subspaces X₊⊕X₋. Under certain conditions, the operator DP restricted to X₊ and X₋ defines a pair of Fredholm operators which maps X₊→X₋ and X₋→X₊ correspondingly, giving rise to a superstructure on V. In this paper we consider the questions of determining the index of DP and the reconstruction of and DP from the boundary data on ∂M. The data used is either the Cauchy data, i.e., the restrictions to ∂M×R₊ of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e., the set of the eigenvalues and the boundary values of the eigenfunctions of DP. We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in M×C⁴, M⊂R³.     

Continuous fields of C*-algebras over finite dimensional spaces

Let X be a finite dimensional compact metrizable space. We study a technique which employs semiprojectivity as a tool to produce approximations of C(X)-algebras by C(X)-subalgebras with controlled complexity. The following applications are given. All unital separable continuous fields of C*-algebras over X with fibers isomorphic to a fixed Cuntz algebra On, n∈{2,3,…,∞}, are locally trivial. They are trivial if n=2 or n=∞. For n?3 finite, such a field is trivial if and only if (n−1)[A₁]=0 in K₀(A), where A is the C*-algebra of continuous sections of the field. We give a complete list of the Kirchberg algebras D satisfying the UCT and having finitely generated K-theory groups for which every unital separable continuous field over X with fibers isomorphic to D is automatically locally trivial or trivial. In a more general context, we show that a separable unital continuous field over X with fibers isomorphic to a KK-semiprojective Kirchberg C*-algebra is trivial if and only if it satisfies a K-theoretical Fell type condition.     

On the base dimension I and the property of universality

In Iliadis (2005) [13] for an ordinal α the notion of the so-called (bn-Ind?α)-dimensional normal base C for the closed subsets of a space X was introduced. This notion is defined similarly to the classical large inductive dimension Ind. In this case we shall write here I(X,C)?α and say that the base dimension I of the space X by the normal base C is less than or equal to α. The classical large inductive dimension Ind of a normal space X, the large inductive dimension Ind₀ of a Tychonoff space X defined independently by Charalambous and Filippov, as well as, the relative inductive dimension defined by Chigogidze for a subspace X of a Tychonoff space Y may be considered as the base dimension I of X by normal bases Z(X) (all closed subsets of X), Z(X) (all functionally closed subsets of X), and , respectively.In the present paper, we shall consider normal bases of spaces consisting of functionally closed subsets. In particular, we introduce new dimension invariant : for a space X, is the minimal element α of the class O∪{−1,∞}, where O is the class of all ordinals, for which there exists a normal base C on X consisting of functionally closed subsets such that I(X,C)?α. We prove that in the class of all completely regular spaces X of weight less than or equal to a given infinite cardinal τ such that there exist universal spaces. However, the following questions are open.(1) Are there universal elements in the class of all normal (respectively, of all compact) spaces X of weight ?τ with ?(2) Are there universal elements in the class of all Tychonoff (respectively, of all normal) spaces X of weight ?τ with Ind₀(X)?n∈ω? (Note that for a compact space X.)     

Minimal sets in compact connected subspaces

Let X be a topological space, f:X→X be a continuous map, and Y be a compact, connected and closed subset of X. In this paper we show that, if the boundary X_∂Y contains exactly one point v and f(v)∈Y, then Y contains a minimal set of f.     

Criteria for embedding of spaces constructed by the method of means with arbitrary quasi-concave functional parameters

We consider a generalization ?(X₀,X₁)_p0,p1 of the method of means to arbitrary non-degenerate functional parameter. In this case non-trivial embedding ?(X₀,X₁)_p0,p1⊂ψ(X₀,X₁)_q0,q1 take place. We find necessary and sufficient condition for such embedding if 1?q₀?p₀?∞ and 1?q₁?p₁?∞ or 1?p₀?q₀?∞ and 1?p₁?q₁?∞.     

The strongest intrinsic meaning of sequential-evaluation convergence

For classical Banach sequence spaces c₀(X), l_∞(X) and lp(X) (0<p<+∞) we have found the strongest intrinsical meanings of their β-duals, and two basic convergence results are established in the β-duals.