Asymptotic Cones and Boundaries of CAT(0) Spaces

We explore the relationships between the asymptotic cones of a CAT(0) space and its boundary under both the standard visual (i.e., cone) topology and the Tits metric. We show that the set of asymptotic cones of a proper cocompact CAT(0) space admits canonical connecting maps under which the direct limit is isometric to the Euclidean cone on the Tits boundary. We also demonstrate how maps between asymptotic cones induce maps between Tits boundaries, which we use to show that virtually free Abelian groups are exactly the CAT(0) groups with compact Tits boundary.