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 sublinear-Morse boundary to more broadly capture these generic directions.

In joint work with Abdul Zalloum, we develop the geometric foundations of sublinear-Morseness in the mapping class group and Teichmuller 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 structure on these spaces.

Our techniques include developing tools for modeling sublinearly-Morse rays via CAT(0) cube complexes. Part of this analysis involves establishing a direct connection between the geometry of the curve graph and the combinatorics of hyperplanes in these cubical models.