Abstract: 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 is an essential tool in the study of the coarse geometry of hyperbolic groups. In this study we introduce a generalization of the Gromov boundary for all finitely generated groups. We construct the sublinearly Morse boundaries and show that it is a QI-invariant topological space and it is metrizable. We show the geometric genericity of points in this boundary using Patterson Sullivan measure on the visual boundary of CAT(0) spaces. As an application we discuss the connection between the sublinearly Morse boundary and random walk on groups. We answer open problems regarding QI-invariant models of random walk on CAT(0) groups and on mapping class groups. If time permits, we also will also look at how this boundary behaves under sublinear bilipschitz equivalences.
