The geometry of genericity in mapping class groups and Teichm\"uller spaces via CAT(0) cube complexes

Random walks on spaces with hyperbolic properties tend to sublinearly track geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-Tiozzo recently introduced the sublinearly Morse boundary and proved that this boundary is a quasi-isometry invariant which captures a notion of generic direction in a broad context. In this article, we develop the geometric foundations of sublinear Morseness in the mapping class group and Teichm\"uller space. We prove that their sublinearly Morse boundaries are visibility spaces and admit continuous equivariant injections into the boundary of the curve graph. Moreover, we completely characterize sublinear Morseness in terms of the hierarchical structures of these spaces. Our techniques include developing tools for modeling the hulls of median rays in hierarchically hyperbolic spaces via CAT(0) cube complexes. Part of this analysis involves establishing direct connections between the geometry of the curve graph and the combinatorics of hyperplanes in the approximating cube complexes.