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

@article{Fukaya2018ACC,
  title={A coarse Cartan–Hadamard theorem with application to the coarse Baum–Connes conjecture},
  author={Tomohiro Fukaya and Shin-ichi Oguni},
  journal={Journal of Topology and Analysis},
  year={2018}
}
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. 

Figures from this paper

Open Cones and $K$-theory for $\ell^p$ Roe Algebras
In this paper, we verify the l coarse Baum-Connes conjecture for open cones and show that the K-theory for l Roe algebras of open cones are independent of p ∈ [1,∞). Combined with the result of T.Expand
Expanders are counterexamples to the $\ell^p$ coarse Baum-Connes conjecture
We consider an l coarse Baum-Connes assembly map for 1< p <∞, and show that it is not surjective for expanders arising from residually finite hyperbolic groups.
The coarse Baum-Connes conjecture for certain extensions and relative expanders
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 fromExpand
Visual maps between coarsely convex spaces
The class of coarsely convex spaces is a coarse geometric analogue of the class of nonpositively curved Riemannian manifolds. It includes Gromov hyperbolic spaces, CAT(0) spaces, proper injectiveExpand
Coronas for properly combable spaces
This paper is a systematic approach to the construction of coronas (i.e. Higson dominated boundaries at infinity) of combable spaces. We introduce three additional properties for combings:Expand
Locally elliptic actions, torsion groups, and nonpositively curved spaces
Extending and unifying a number of well-known conjectures and open questions, we conjecture that locally elliptic actions (that is, every element has a bounded orbit) of finitely generated groups onExpand
Helly meets Garside and Artin
A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is,Expand
Coarse assembly maps
A coarse assembly map relates the coarsification of a generalized homology theory with a coarse version of that homology theory. In the present paper we provide a motivic approach to coarse assemblyExpand
Expanders are counterexamples to the coarse $p$-Baum-Connes conjecture
We show that certain expanders are counterexamples to the coarse $p$-Baum-Connes conjecture.
Coarse injectivity, hierarchical hyperbolicity, and semihyperbolicity
We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely injective spaces, and strongly shortcut spaces. We show that every hierarchically hyperbolicExpand
...
1
2
...

References

SHOWING 1-10 OF 45 REFERENCES
Coronae of product spaces and the coarse Baum–Connes conjecture☆
Abstract 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 satisfyExpand
THE COARSE BAUM–CONNES CONJECTURE FOR RELATIVELY HYPERBOLIC GROUPS
We study a group which is hyperbolic relative to a finite family of infinite subgroups. We show that the group satisfies the coarse Baum–Connes conjecture if each subgroup belonging to the familyExpand
The coarse Baum–Connes conjecture for Busemann nonpositively curved spaces
We prove that the coarse assembly maps for proper metric spaces which are non-positively curved in the sense of Busemann are isomorphisms, where we do not assume that the spaces are with boundedExpand
The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space
Corollary 1.2. Let Γ be a finitely generated group. If Γ, as a metric space with a word-length metric, admits a uniform embedding into Hilbert space, and its classifying space BΓ has the homotopyExpand
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 K-homology of the corona with the K-theory of the Roe algebra, viaExpand
Large‐type Artin groups are systolic
We prove that Artin groups from a class containing all large-type Artin groups are systolic. This provides a concise yet precise description of their geometry. Immediate consequences are new resultsExpand
Coronas for properly combable spaces
This paper is a systematic approach to the construction of coronas (i.e. Higson dominated boundaries at infinity) of combable spaces. We introduce three additional properties for combings:Expand
Dismantlability of weakly systolic complexes and applications
In this paper, we investigate the structural properties of weakly systolic complexes introduced recently by the second author and of their 1-skeletons, the weakly bridged graphs. We present severalExpand
Boundaries and JSJ Decompositions of CAT(0)-Groups
Let G be a one-ended group acting discretely and co-compactly on a CAT(0) space X. We show that ∂X has no cut points and that one can detect splittings of G over two-ended groups and recover its JSJExpand
Simplicial nonpositive curvature
We introduce a family of conditions on a simplicial complex that we call local k-largeness (k≥6 is an integer). They are simply stated, combinatorial and easily checkable. One of our themes is thatExpand
...
1
2
3
4
5
...