To every Gromov hyperbolic space X one can associate a space at infinity called the Gromov boundary of X. This boundary is a fundamental tool for studying hyperbolic groups and hyperbolic 3-manifolds. As shown by Gromov, quasi-isometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a well-defined notion of the boundary of a hyperbolic group. Croke and Kleiner showed that visual boundary of non-positively curved (CAT(0)) groups is not well-defined, since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. For any sublinear function, we consider a subset of the visual boundary called  sublinear boundary and show that it is a QI-invariant. This is to say,  the sublinear-boundary of a CAT(0) group is well-defined. In the case of Right-angled Artin group, we show that the Poisson boundary of random walks on groups is naturally identified with the (/sqrt{t log t})-boundary. This talk is based on projects with Kasra Rafi and Giulio Tiozzo.

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