Coronae of relatively hyperbolic groups and coarse cohomologies

Fukaya, Tomohiro; Oguni, Shin Ichi

doi:10.1142/s1793525316500151

Cited by 4 publications

(4 citation statements)

References 28 publications

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“…By Proposition 2.2, ∂ ∂X is an isomorphism. Combining Propositions 3.4 and 4.3, we see that c(X) is also an isomorphism (see [5,Proof of Theorem 4.8] and also [6,Section 3.2]). By tracing the diagram, we have the conclusion.…”

Section: Coronamentioning

confidence: 78%

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On coarse geometric aspects of Hilbert geometry

Mineyama

Oguni

2018

Monatsh Math

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We begin a coarse geometric study of Hilbert geometry. Actually we give a necessary and sufficient condition for the natural boundary of a Hilbert geometry to be a corona, which is a nice boundary in coarse geometry. In addition, we show that any Hilbert geometry is uniformly contractible and with coarse bounded geometry. As a consequence of these we see that the coarse Novikov conjecture holds for a Hilbert geometry with a mild condition. Also we show that the asymptotic dimension of any two-dimensional Hilbert geometry is just two. This implies that the coarse Baum-Connes conjecture holds for any two-dimensional Hilbert geometry via Yu's theorem.

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Section: Coronamentioning

confidence: 78%

“…Since the following argument is standard in coarse geometry, we write briefly (for the precise definition, see [9,6]).…”

Section: Coronamentioning

confidence: 99%

On coarse geometric aspects of Hilbert geometry

Mineyama

Oguni

2018

Monatsh Math

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“…Here K * (X), c r X, K * (c r X), KX * (X), K * (W ), A(X), µ(X), c(X), ∂ W , b W and T W are the K-theory of X, the reduced stable Higson corona of X, the K-theory of c r X, the coarse K-theory of X, the reduced K-theory of W , the co-assembly map of X, the coarse co-assembly map of X, the character map of X, the boundary map of the cohomological long exact sequence of (X ∪ W, W ), the map induced by the inclusion of C(W ) into c r X and the transgression map, respectively (see [1,Section 4], [9,Chapter 4], [2,Sections 3,4] for details).…”

Section: Coarse K-theories and Coarse Cohomologies Of Proper Busemannmentioning

confidence: 99%

“…There are some classes whose element X has a coarse compactification X ∪ W satisfying that T W and b W are isomorphisms. Such a typical class consists of unbounded proper geodesic hyperbolic spaces in the sense of Gromov (see [2,Corollary 5.3]). Indeed the Gromov completions are desired coarse compactifications.…”

Section: Introductionmentioning

confidence: 99%

The coarse Baum–Connes conjecture for Busemann nonpositively curved spaces

Fukaya¹,

Oguni²

2016

Kyoto J. Math.

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A coarse Cartan–Hadamard theorem with application to the coarse Baum–Connes conjecture

Fukaya

Oguni

2018

J. Topol. Anal.

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We establish a coarse version of the Cartan-Hadamard theorem, which states that proper coarsely convex spaces are coarsely homotopy equivalent to the open cones of their ideal boundaries. As an application, we show that such spaces satisfy the coarse Baum-Connes conjecture. Combined with the result of Osajda-Przytycki, it implies that systolic groups and locally finite systolic complexes satisfy the coarse Baum-Connes conjecture.The class of geodesic coarsely convex spaces includes geodesic Gromov hyperbolic spaces [14, §2, Proposition 25] and CAT(0)-spaces, more generally, Busemann non-positively curved spaces [5] [24]. We remark that this class is closed under direct product, therefore, it includes products of these spaces. An important subclass of geodesic coarsely convex spaces is a class of systolic complexes.Systolic complexes are connected, simply connected simplicial complexes with combinatorial conditions on links. They satisfy one of the basic feature of CAT(0)-spaces, that is, the balls around convex sets are convex. This class of simplicial complexes was introduced by Chepoi [6] (under the name of bridged complexes), and independently, by Januszkiewich-Świątkowski [19] and Haglund [16]. Osajda-Przytycki [23] introduced Euclidean geodesics, which behave like CAT(0) geodesics, to construct boundaries of systolic complexes. Their result implies the following. Theorem 1.2 ([23, Corollary 3.3, 3.4]). The 1-skeleton of systolic complexes are geodesic coarsely convex spaces.A group is systolic if it acts geometrically by simplicial automorphisms on a systolic complex. Osajda-Przytycki used their result to show that systolic groups admit EZstructures. This implies the Novikov conjecture for torsion-free systolic groups. Now it is natural to ask whether systolic groups satisfy the the coarse Baum-Connes conjecture.Let X be a proper metric space. The coarse assembly map is a homomorphism from the coarse K-homology of X to the K-theory of the Roe-algebra of X. The coarse Baum-Connes conjecture [17] states that for "nice" proper metric spaces, the coarse assembly maps are isomorphisms.

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Coronae of relatively hyperbolic groups and coarse cohomologies

Cited by 4 publications

References 28 publications

On coarse geometric aspects of Hilbert geometry

On coarse geometric aspects of Hilbert geometry

The coarse Baum–Connes conjecture for Busemann nonpositively curved spaces

A coarse Cartan–Hadamard theorem with application to the coarse Baum–Connes conjecture

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