2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/211974Let G be a one-ended group acting discretely and co-compactly on a CAT(0) space X. We show that the boundary of X has no cut points and that one can detect splittings of $G$ over two-ended groups and recover its JSJ decomposition from the boundary. We show that any discrete action of a group G on a CAT(0) space X satisfies a convergence type property. This is used in the proof of the results above but it is also of independent interest. In particular, if G acts co-compactly on X, then one obtains as a Corollary that if the Tits diameter of the boundary of X is bigger than $\frac {3π} 2$ then it is infinite and G contains a free subgroup of rank 2.minor corrections,38 pages, 2 figures, to appear in GAFAGroup TheoryGeometric TopologyMetric Geometry20F67,20E06,20E34,57M07text