The geometry of genericity in mapping class groups and Teichmüller 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 this 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 Teichmüller space. We completely characterize sublinear Morseness in terms of the hierarchical structures of these spaces, and use this to prove that their sublinearly Morse boundaries admit continuous equivariant injections into the boundary of the curve graph. It was already known that the Gromov boundary of the curve graph is a Poisson model for sufficiently nice random walks of the mapping class group on itself and on Teichmüller space. As corollary, we prove that the corresponding hitting measure is fully supported on the image of the sublinearly Morse boundary, which was previously unknown. 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.