Higson Corona Research Articles

Ring and module structures onK-theory of leaf spaces and their application to longitudinal index theory

Pursuing conjectures of John Roe, we use the stable Higson corona of foliated cones to construct a new $K$-theory model for the leaf space of a foliation. This new $K$-theory model is -- in contrast to Alain Connes' $K$-theory model -- a ring. We show that Connes' $K$-theory model is a module over this ring and develop an interpretation of the module multiplication in terms of indices of twisted longitudinally elliptic operators.

On spaces with connected Higson coronas

In this paper, we characterise metric spaces which have topologically connected Higson coronas. The characterisation is given by a natural categorical condition applied in the coarse category. We also give a characterisation in terms of coarse cohomology, and consider the special case of finitely generated groups and the more general case of abstract coarse spaces. Along the way, we exhibit some connections to the notion of ω-excisive decomposition introduced by Higson, Roe and Yu, and give a categorical characterisation of such decompositions.

Coarse co-assembly as a ring homomorphism

The K -theory of the stable Higson corona of a coarse space carries a canonical ring structure. This ring is the domain of an unreduced version of the coarse co-assembly map of Emerson and Meyer. We show that the target also carries a ring structure and co-assembly is a ring homomorphism, provided that the given coarse space is contractible in a coarse sense.

Coronae of relatively hyperbolic groups and coarse cohomologies

We construct a corona of a relatively hyperbolic group by blowing-up all parabolic points of its Bowditch boundary. We relate the [Formula: see text]-homology of the corona with the [Formula: see text]-theory of the Roe algebra, via the coarse assembly map. We also establish a dual theory, that is, we relate the [Formula: see text]-theory of the corona with the [Formula: see text]-theory of the reduced stable Higson corona via the coarse co-assembly map. For that purpose, we formulate generalized coarse cohomology theories. As an application, we give an explicit computation of the [Formula: see text]-theory of the Roe-algebra and that of the reduced stable Higson corona of the fundamental groups of closed 3-dimensional manifolds and of pinched negatively curved complete Riemannian manifolds with finite volume.

Metric Compactifications and Coarse Structures

AbstractLet TB be the category of totally bounded, locally compact metric spaces with the C0 coarse structures. We show that if X and Y are in TB, then X and Y are coarsely equivalent if and only if their Higson coronas are homeomorphic. In fact, the Higson corona functor gives an equivalence of categories TB → K, where K is the category of compact metrizable spaces. We use this fact to show that the continuously controlled coarse structure on a locally compact space X induced by some metrizable compactification is determined only by the topology of the remainder .

Some ‘homological’ properties of the stable Higson corona

We establish certain ‘homological properties’ of the stable Higson corona used by Emerson and Meyer to study the Dirac-dual-Dirac approach to the Baum–Connes conjecture [5]. These are used to obtain explicit isomorphisms between the K-theory groups of stable Higson coronas, and the K-theory groups of certain geometrically defined boundaries. This is sufficient to give a simple proof of the strong Novikov conjecture for torsion-free hyperbolic groups and torsion-free groups acting properly and cocompactly on CAT(0) spaces, and also provides an input into an index theorem in single operator theory [15], [16].

Sublinear Higson corona of Euclidean cone

Let X be a proper metric space. The sublinear Higson compactification hLX is a variant of the Higson compactification. Its boundary hLX\X is denoted νLX, and is called the sublinear Higson corona of X. The sublinear Higson corona is a functor from the category of coarse spaces to that of compact Hausdorff spaces. Let P be a compact metric space and X be an unbounded proper metric space. We show that the sublinear Higson corona of a product space P × X equipped with a cone metric is homeomorphic to a product P × νLX. Especially, the sublinear Higson corona of the n-dimensional Euclidean space is homeomorphic to the product of an (n − 1)-dimensional sphere and the sublinear Higson corona of natural numbers.

Weak P-points in coronas of G-spaces

Let G be a group, X be a discrete G-space, X ⁎ be the remainder of the Stone–Čech compactification of X. The corona X ˇ of X is a factor-space of X ⁎ by the smallest by inclusion, closed in X ⁎ × X ⁎ equivalence on X ⁎ containing the orbit equivalence ∼ ( p ∼ q ⇔ ∃ g ∈ G : q = g p ) . For a countable group G and a countable G-space X we prove that the corona of X contains a weak P-point and a P-point provided that there exists a P-point in ω ⁎ . Then we transfer this statement to the Higson coronas of a proper metric spaces of bounded geometry.

Coarse dynamics and fixed-point theorem

AbstractWe study semigroup actions on a coarse space and the induced actions on the Higson corona from a dynamical point of view. Our main theorem states that if an action of an abelian semigroup on a proper coarse space satisfies certain conditions, the induced action has a fixed point in the Higson corona. As a corollary, we deduce a coarse version of Brouwer’s fixed-point theorem.

Coarse dynamics and fixed-point theorem

AbstractWe study semigroup actions on a coarse space and the induced actions on the Higson corona from a dynamical point of view. Our main theorem states that if an action of an abelian semigroup on a proper coarse space satisfies certain conditions, the induced action has a fixed point in the Higson corona. As a corollary, we deduce a coarse version of Brouwer’s fixed-point theorem.

On the asymptotic extension dimension

We introduce an asymptotic counterpart of the extension dimension defined by Dranishnikov. The main result establishes the relationship between the asymptotic extensional dimension of a proper metric space and the extension dimension of its Higson corona.

C*-reflexivity doesn't pass to quotients

Using a recently obtained criterion of $C^*$-reflexivity for commutative $C^*$-algebras, we show that the $C^*$-algebra of continuous functions on the Higson corona is not $C^*$-reflexive. This implies that $C^*$-reflexivity doesn't pass to quotient $C^*$-algebras.

Coarse fixed point theorem

We study group actions on a coarse space and the induced actions on the Higson corona from a dynamical point of view. Our main theorem states that if an action of an abelian group on a proper metric space satisfies certain conditions, the induced action has a fixed point in the Higson corona. As a corollary, we deduce a coarse version of Brouwer’s fixed point theorem.

On asymptotic Assouad–Nagata dimension

For a large class of metric spaces X including discrete groups we prove that the asymptotic Assouad–Nagata dimension AN-asdim X of X coincides with the covering dimension dim ( ν L X ) of the Higson corona of X with respect to the sublinear coarse structure on X. Then we apply this fact to prove the equality AN-asdim ( X × R ) = AN-asdim X + 1 . We note that the similar equality for Gromov's asymptotic dimension asdim generally fails to hold [A. Dranishnikov, Cohomological approach to asymptotic dimension, Preprint, 2006]. Additionally we construct an injective map ξ : cone ω ( X ) ∖ [ x 0 ] → ν L X from the asymptotic cone without the basepoint to the sublinear Higson corona.

DUALIZING THE COARSE ASSEMBLY MAP

We formulate and study a new coarse (co-)assembly map. It involves a modification of the Higson corona construction and produces a map dual in an appropriate sense to the standard coarse assembly map. The new assembly map is shown to be an isomorphism in many cases. For the underlying metric space of a group, the coarse co-assembly map is closely related to the existence of a dual Dirac morphism and thus to the Dirac dual Dirac method of attacking the Novikov conjecture.

Coronas of balleans

We introduce balleans as asymptotical counterparts of uniform topological spaces. Using slowly oscillating functions, for every ballean we define two compact spaces: corona and binary corona. These spaces can be considered as generalizations of the Higson's coronas of metric spaces and the spaces of ends of groups, respectively. We consider some balleans related to an infinite group and prove some results concerning their coronas. At the end we apply these results to describe the compact right-zero semigroups which are continuous homomorphic images of G * , the reminder of the Stone–Čech compactification of discrete group G.

Higson compactifications obtained by expanding and contracting the half-open interval

In this paper, we study Higson compactifications of the half-open interval obtained by expanding and contracting the base space. We show that the Higson coronas of the half-open interval obtained by these operations are indecomposable continua. Moreover, we show that the Stone-Cech compactification can be approximated by such Higson compactifications.

On the Higson corona of uniformly contractible spaces

Let X be a proper metric space and let νX be its Higson corona. We prove that the covering dimension of νX does not exceed the asymptotic dimension asdim X of X introduced by M. Gromov. In particular, it implies that dim νR n = n for euclidean and hyperbolic metrics on R n . We prove that for finitely generated groups Γ′ ⊃ Γ with word metrics the inequality dim νΓ′ ⩽ dim νΓ holds. Also we prove that a small action at infinity of a geometrically finite group Γ on some compactification X′ of the universal covering space X = EΓ enables one to map the Higson compactification onto X′. In that case the rational acyclicity of X′ implies the conjecture by S. Weinberger for X which is a form of the Novikov Conjecture for Γ.

Cohomology, dimension and large Riemannian manifolds

This paper surveys recent results on dimension and cohomology of the Higson corona of uniformly contractable manifolds.