Abstract
The Morse local-to-global property generalizes the local-to-global property for quasi-geodesics in a hyperbolic space. We show that graph products of infinite Morse local-to-global groups have the Morse local-to-global property. To achieve this, we generalize the maximization procedure from Abbott, Behrstock, and Durham for relatively hierarchically hyperbolic groups with clean containers. Under mild conditions satisfied by graph products, we show that stable embeddings into a relatively hierarchically hyperbolic space are exactly those which are quasi-isometrically embedded in the top level hyperbolic space by the orbit map. This shows that graph products of any infinite groups with no isolated vertices are Morse detectable. As an additional tool to understand the Morse local-to-global property in any relatively hierarchically hyperbolic group, we show that if every locally Morse local hierarchy path in a relatively hierarchically hyperbolic group is a global Morse quasi-geodesic, then the group has the Morse local-to-global property