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

Abstract

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.

Open Cones and $K$-theory for $\ell^p$ Roe Algebras

“…As an application of the theorem and corollary, combined with the result of T. Fukaya and S.-I. Oguni in [8], we show that the ℓ p coarse Baum-Connes conjecture holds for any proper coarsely convex space (cf. Definition 4.1) and the K-theory for ℓ p Roe algebras of such spaces are independent of p ∈ [1, ∞).…”

Locally elliptic actions, torsion groups, and nonpositively curved spaces

“…• G satisfies the Farrell-Jones conjecture (see [KR17] and [CCG + 20, Theorem 1.5]). • G satisfies the coarse Baum-Connes conjecture (see [FO20] Recall that a geodesic metric space is called injective if any family of pairwise intersecting closed balls has a non-empty global intersection. We refer the reader to [Lan13] for a presentation of injective metric space, and also the following result of Isbell.…”

Visual maps between coarsely convex spaces

“…Let X i L be the systems of good quasi-geodesics of X i . Then by [7,Proposition 3.3], all element in the system of good quasi-geodesics X 1 ×X 2 L of X 1 × X 2 is of the following form:…”

“…The class of coarsely convex spaces is introduced by the second author and Shin-ichi Oguni [7]. This class includes Gromov hyperbolic spaces, and Busemann nonpositively curved spaces, especially, CAT(0) spaces.…”

“…Let X be a proper coarsely convex space. In [7], the authors constructed the ideal boundary ∂X of X, and they showed that the open cone O∂X is coarsely homotopy equivalent to X via the "exponential map" and the "logarithmic map". As a corollary, it follows that the coarse Baum-Connes conjecture holds for coarsely convex spaces.…”