Thick metric spaces,relative hyperbolicity,and quasi-isometric rigidity

We study the geometry of non-relatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with non-relatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficient for a metric space to be non-hyperbolic relative to any non-trivial collection of subsets. Thick finitely generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin groups; and others. Non-uniform lattices in higher rank semisimple Lie groups are thick and hence non-relatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples of non-relatively hyperbolic groups having cut points in any asymptotic cone, resolving several questions of Drutu and Sapir about the structure of relatively hyperbolic groups. Outside of group theory, Teichmüller spaces for surfaces of sufficiently large complexity are thick with respect to the Weil–Peterson metric, in contrast with Brock–Farb’s hyperbolicity result in low complexity.     

Boundary amenability of relatively hyperbolic groups

Let K be a fine hyperbolic graph and Γ be a group acting on K with finite quotient. We prove that Γ is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups are exact. We prove this by showing that the group Γ acts amenably on a compact topological space. We include some applications to the theories of group von Neumann algebras and of measurable orbit equivalence relations.     

The isocohomological property, higher Dehn functions, and relatively hyperbolic groups

The property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the (weak) isocohomological property for the group. In the case when a group is of type HF^∞, i.e. that has a classifying space with the homotopy type of a polyhedral complex with finitely many cells in each dimension, we show that the isocohomological property is geometric and is equivalent to the property that the universal cover of the classifying space has polynomially bounded higher Dehn functions. If a group is hyperbolic relative to a collection of subgroups, each of which is polynomially combable, respectively HF^∞ and isocohomological, then we show that the group itself has these respective properties. Combining with the results of Connes-Moscovici and Dru?u-Sapir we conclude that a group satisfies the strong Novikov conjecture if it is hyperbolic relative to subgroups which are of property RD, of type HF^∞ and isocohomological.     

Existence and uniqueness for a nonlinear discrete hyperbolic system

The existence, uniqueness and asymptotic behaviour of the solutions to a nonlinear discrete hyperbolic system subject to some extreme conditions and initial data are investigated in a real Hilbert space.     

Differential operators on Hilbert modular forms

We investigate differential operators and their compatibility with subgroups of SL₂n(R). In particular, we construct Rankin-Cohen brackets for Hilbert modular forms, and more generally, multilinear differential operators on the space of Hilbert modular forms. As an application, we explicitly determine the Rankin-Cohen bracket of a Hilbert-Eisenstein series and an arbitrary Hilbert modular form. We use this result to compute the Petersson inner product of such a bracket and a Hilbert modular cusp form.     

Geometric invariants of spaces with isolated flats

We study those groups that act properly discontinuously, cocompactly, and isometrically on CAT(0) spaces with isolated flats. The groups in question include word hyperbolic CAT(0) groups as well as geometrically finite Kleinian groups and numerous two-dimensional CAT(0) groups. For such a group we show that there is an intrinsic notion of a quasiconvex subgroup which is equivalent to the subgroup being undistorted. We also show that the visual boundary of the CAT(0) space is actually an invariant of the group. More generally, we show that each quasiconvex subgroup of such a group has a canonical limit set which is independent of the choice of overgroup.The main results in this article were established by Gromov and Short in the word hyperbolic setting and do not extend to arbitrary CAT(0) groups.     

Ideal bicombings for hyperbolic groups and applications

For every hyperbolic group and more general hyperbolic graphs, we construct an equivariant ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant which implies that several Measure Equivalence and Orbit Equivalence rigidity results established in Monod and Shalom (Orbit equivalence rigidity and bounded cohomology, preprint, to appear) hold for all non-elementary hyperbolic groups and their non-elementary subgroups. We also derive superrigidity results for actions of general irreducible lattices on a large class of hyperbolic metric spaces.     

Coronae of product spaces and the coarse Baum–Connes conjecture

We study the coarse Baum–Connes conjecture for product spaces and product groups. We show that a product of CAT(0) groups, polycyclic groups and relatively hyperbolic groups which satisfy some assumptions on peripheral subgroups, satisfies the coarse Baum–Connes conjecture. For this purpose, we construct and analyze an appropriate compactification and its boundary, “corona”, of a product of proper metric spaces.     

On Fourier coefficients of Eisenstein series and Niebur Poincaré series of integral weight

We give a Katok-Sarnak type correspondence for Niebur type Poincaré series and Eisenstein series on the three-dimensional hyperbolic space.     

Representation of Probabilistic Data by Quantum-Like Hyperbolic Amplitudes

In this paper we study the problem of representation of statistical data of any origin by hyperbolic probabilistic amplitudes (normalized vectors in hyperbolic Hilbert space). It generalizes the conventional QM which is based on complex Hilbert space. We performed extended numerical simulation. Similar to the conventional quantum formalism for Bloch’s sphere, we visualize results of simulation for a special class of statistical data on so called Bloch’s hyperboloid. The notion of hyperbolic qubit is introduced.     

Expansive canonical coordinates are hyperbolic

Theorem. Let ?:X→X be an expansive homeomorphism of a compact metric space onto itself and let ? have canonical coordinates. Then there exists a metric compatible with the topology of X with respect to which the canonical coordinates are hyperbolic.     

Asymptotic behavior of semigroups of ρ-non-expansive and holomorphic mappings on the Hilbert Ball

We study generated semigroups of those self-mappings of the Hilbert ball which are non-expansive with respect to the hyperbolic metric. We find optimal convergence rates for such semigroups to interior stationary and boundary sink points. Since the hyperbolic metric is not defined on the boundary, the usual approach treats these two cases separately. In contrast with this practice, we use a special non-Euclidean “distance” (which induces the original topology) to present a unified theory. Our approach leads to new results even in the one-dimensional case. When the semigroups consist of holomorphic self-mappings, we obtain the rather unexpected phenomenon of universal rates of convergence of an exponential type. In particular, in the case of a boundary sink point we establish a continuous analog of the celebrated Julia–Wolff–Carathéodory theorem. Received: January 3, 2001; in final form: November 28, 2001?Published online: October 30, 2002     

Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls

We show that the open unit ball of the space of operators from a finite-dimensional Hilbert space into a separable Hilbert space (we call it “operator ball”) has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). We apply this result to find dual pairs of invariant subspaces in Pontryagin spaces. In Appendix A we present results of Itai Shafrir about hyperbolic metrics on the operator ball.     

Hyperbolic Hilbert space

The hyperbolic complex space is one class of non-Euclidean spaces with continuous singular points. It corresponds with Minkowski space, and it has the characteristic that the space-time direction is different in nature. Regard the hyperbolic complex space as original spaces. We can abstract a class of the hyperbolic inner product space and the hyperbolic Hilbert space.     

Higher dimensional isoperimetric functions in hyperbolic groups

We introduce the notion of an -combing and use it to show that hyperbolic groups satisfy linear isoperimetric inequalities for filling real cycles in each positive dimension. S. Gersten suggested the concept of metabolicity (over or ) for groups which implies hyperbolicity. Metabolicity admits several equivalent definitions: by vanishing of -cohomology, using combings, and others. We prove several criteria for a group to be hyperbolic, -metabolicity being among them. In particular, a finitely presented group G is hyperbolic iff for any normed vector space V and any . Received December 9, 1998     

Hyperbolic Coxeter groups of large dimension

We construct examples of Gromov hyperbolic Coxeter groups of arbitrarily large dimension. We also extend Vinbergs theorem to show that if a Gromov hyperbolic Coxeter group is a virtual Poincaré duality group of dimension n, then n 61.Coxeter groups acting on their associated complexes have been extremely useful source of examples and insight into nonpositively curved spaces over last several years. Negatively curved (or Gromov hyperbolic) Coxeter groups were much more elusive. In particular their existence in high dimensions was in doubt.In 1987 Gabor Moussong [M] conjectured that there is a universal bound on the virtual cohomological dimension of any Gromov hyperbolic Coxeter group. This question was also raised by Misha Gromov [G] (who thought that perhaps any construction of high dimensional negatively curved spaces requires nontrivial number theory in the guise of arithmetic groups in an essential way), and by Mladen Bestvina [B2].In the present paper we show that high dimensional Gromov hyperbolic Coxeter groups do exist, and we construct them by geometric or group theoretic but not arithmetic means.     

Boundary dimension in negatively curved spaces

LetX be a negatively curved (Gromov hyperbolic) space. We construct a bound on dim X when a group of isometries acts cocompactly onX. We construct an example of a negatively curved space with infinite-dimensional boundary.     

Some remarks on hilbert functions of veronese algebras

We study the Hilbert polynomials of finitely generated graded algebras R, with generators not all of degree one (i.e. non-standard). Given an expression P(R,t)=a(t)/(1-t^l ) ⁿ for the Poincare series of R as a rational function, we study for 0 ≤ i ≤ l the graded subspaces ? _kR_kl+i (which we denote R[l;i]) of R, in particular their Poincaré series and Hilbert functions. We prove, for example, that if R is Cohen-Macaulay then the Hilbert polynomials of all non-zeroR[l;i] share a common degree. Furthermore, if R is also a domain then these Hilbert polynomials have the same leading coefficient.     

Hyperbolic quantum mechanics

We start to develop the quantization formalism in a hyperbolic Hilbert space. Generalizing Born’s probability interpretation, we found that unitary transformations in such a Hilbert space represent a new class of transformations of probabilities which describe a kind of hyperbolic interference. The most interesting problem which prompted by our investigation is to find experimental evidence of hyperbolic interference. The hyperbolic quantum formalism can also be interesting as a new theory of probability waves that can be developed in parallel with the standard quantum theory. Comparative analysis of these two wave theories could be useful for understanding of the role of various structures of the standard quantum formalism. In particular, one of distinguishing feature of the hyperbolic quantum formalism is the restricted validity of the superposition principle.     

Fibred coarse embeddings,a-T-menability and the coarse analogue of the Novikov conjecture

The connection between the coarse geometry of metric spaces and analytic properties of topological groupoids is well known. One of the main results of Skandalis, Tu and Yu is that a space admits a coarse embedding into Hilbert space if and only if a certain associated topological groupoid is a-T-menable. This groupoid characterisation then reduces the proof that the coarse Baum–Connes conjecture holds for a coarsely embeddable space to known results for a-T-menable groupoids. The property of admitting a fibred coarse embedding into Hilbert space was introduced by Chen, Wang and Yu to provide a property that is sufficient for the maximal analogue to the coarse Baum–Connes conjecture and in this paper we connect this property to the traditional coarse Baum–Connes conjecture using a restriction of the coarse groupoid and homological algebra. Additionally we use this results to give a characterisation of the a-T-menability for residually finite discrete groups.