The visual boundary is a useful topological space that can be defined for CAT(0) spaces and any (reasonably geodesic) hyperbolic space. For hyperbolic spaces, this boundary is invariant under quasi-isometry, which allows us to define it as a topological invariant of hyperbolic groups. This procedure fails for CAT(0) spaces as, even in the presence of a geometric action, the visual boundary is not invariant under quasi-isometries. Morse boundaries have been introduced as a remedy to this phenomenon. They are an asymptotic structure, which can be equipped with a topology to make it a quasi-isometry-invariant. Returning with this new tool to CAT(0) spaces, it turns out that the Morse boundary can be seen as a subspace of the visual boundary. Unfortunately, the topology it inherits as a subspace still is not invariant under quasi-isometry.

However, it turns out that this issue disappears if one assumes the existence of a suitable group action. In this talk, we will introduce a notion of hyperbolic projections and show how one can use this notion to study the asymptotic structure of a group. As a consequence, we show how that the Morse boundary, equipped with the visual topology, is a well-defined topological invariant for a large class of cubulable groups. In addition, we use this tool to introduce a new topology on Morse boundaries of mapping class groups that has a geometric interpretation in terms of medians.