Coronae of product spaces and the coarse Baum–Connes conjecture

Fukaya, Tomohiro; Oguni, Shin Ichi

doi:10.1016/j.aim.2015.01.022

Cited by 10 publications

(7 citation statements)

References 15 publications

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“…Relatively hyperbolic groups. In [10], the authors studied the coarse Baum-Connes conjecture for relatively hyperbolic groups. Theorem 6.10 ( [10]).…”

Section: Coarse Homotopy Between Exp (O∂mentioning

confidence: 99%

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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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“…Relatively hyperbolic groups. In [10], the authors studied the coarse Baum-Connes conjecture for relatively hyperbolic groups. Theorem 6.10 ( [10]).…”

Section: Coarse Homotopy Between Exp (O∂mentioning

confidence: 99%

“…In [10], the authors studied the coarse Baum-Connes conjecture for relatively hyperbolic groups. Theorem 6.10 ( [10]). Let G be a finitely generated group and P = {P 1 , .…”

mentioning

confidence: 99%

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

Fukaya

Oguni

2018

J. Topol. Anal.

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“…In [43], there is also a counterexample to the coarse Novikov conjecture when the condition of bounded geometry is omitted. More recent positive results on the coarse Baum-Connes conjecture include [9][10][11][12] by Fukaya and Oguni.…”

Section: Introductionmentioning

confidence: 98%

Expanders are counterexamples to the $\ell^p$ coarse Baum-Connes conjecture

Chung¹,

Nowak²

2018

Preprint

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“…the fundamental group of a closed manifold) equipped with a word length metric, and the Gromov's zero-in-the-spectrum conjecture and the positive scalar curvature conjecture when X is a Riemannian manifold. See [55] for a comprehensive survey for the coarse Baum-Connes conjecture, and [5,9,18,19,20,21,22,24,41,48,49,50,51,53] for some recent developments.…”

Section: Introductionmentioning

confidence: 99%

The coarse Baum-Connes conjecture for certain relative expanders

Deng¹,

Wang²,

Yu³

2021

Preprint

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Let (1 → Nn → Gn → Qn → 1) n∈N be a sequence of extensions of finitely generated groups with uniformly finite generating subsets. We show that if the sequence (Nn)n∈N with the induced metric from the word metrics of (Gn) n∈N has property A, and the sequence (Qn) n∈N with the quotient metrics coarsely embeds into Hilbert space, then the coarse Baum-Connes conjecture holds for the sequence (Gn) n∈N , which may not admit a coarse embedding into Hilbert space. It follows that the coarse Baum-Connes conjecture holds for the relative expanders and group extensions exhibited by G. Arzhantseva and R. Tessera, and special box spaces of free groups discovered by T. Delabie and A. Khukhro, which do not coarsely embed into Hilbert space, yet do not contain a weakly embedded expander. This in particular solves an open problem raised by G. Arzhantseva and R. Tessera [3]. Contents

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Coronae of product spaces and the coarse Baum–Connes conjecture

Cited by 10 publications

References 15 publications

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

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

Expanders are counterexamples to the $\ell^p$ coarse Baum-Connes conjecture

The coarse Baum-Connes conjecture for certain relative expanders

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