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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Baker and Riley gave an inclusion of a free group of rank 3 in a hyperbolic group for which the Cannon-Thurston map is not well-defined. By using their result, we show that every non-elementary hyperbolic group can be included in some hyperbolic group in such a way that the Cannon-Thurston map is not well-defined. In fact we generalize their result to every non-elementary relatively hyperbolic group. 42 Y. Matsuda and S. Oguni

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We define relatively quasiconvex subgroups of relatively hyperbolic groups in the sense of Osin and show that such subgroups have expected properties. Also we state several definitions equivalent to the definition of relatively hyperbolic groups in the sense of Osin.