Gromov boundary extended


Gromov boundary provides a useful compactification for all infinite-diameter Gromov hyperbolic spaces. It consists of all geodesic rays starting at a given base-point and it has been an essential tool in the study of the coarse geometry of hyperbolic groups. In this study we introduce two topological spaces that are natural analogs of the Gromov boundary for a larger class of metric spaces. First we construct the sublinearly Morse boundaries and show that it is a QI-invariant topological space that can be associated to all finitely generated groups. Furthermore, for many groups, the sublinear boundary can be identified with the Poisson boundaries of the associated group, thus providing a QI-invariant model for Poisson boundaries. This result answers the open problems regarding QI-invariant models of CAT(0) groups and the mapping class group. Lastly, for a subset of the metric spaces we define a compactification of the sublinearly Morse boundary and show that in these cases they are naturally identified with the Bowditch boundary. This talk is based on a series of joint work with Kasra Rafi and Giulio Tiozzo.