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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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 bounded coarse geometry. Also it is shown that we can calculate the coarse K-homology and the K-theory of the Roe algebra by using the visual boundaries.

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 family satisfies the coarse Baum–Connes conjecture and admits a finite universal space for proper actions. If the group is torsion-free, then it satisfies the analytic Novikov conjecture.

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