Gromov boundary plays a central role in many aspects of geometric group theory. In this study, we develop a theory of boundary when the condition on hyperbolicity is removed: For a given proper, geodesic metric space X and a given sublinear function κ, we define the κ-boundary, as the space of all κ-Morse quasi-geodesics rays. The sublinearly Morse boundary is QI-invariant and thus can be associated with the group that acts geometrically on X. For a large class of groups, we show that sublinearly Morse boundaries are large: they provide topological models for the Poisson boundaries of the group. This talk is mainly based on several joint projects with Ilya Gekhtman, Kasra Rafi and Giulio Tiozzo.

海报