Sublinearly Morse boundary, II: Proper geodesic spaces

We build an analogue of the Gromov boundary for any proper geodesic metric space, hence for any finitely generated group. More precisely, for any proper geodesic metric space X and any sublinear function, we construct a boundary for X, denoted by @ X, that is quasi-isometrically invariant and metrizable. As an application, we show that when G is the mapping class group of a finite type surface or a relatively hyperbolic group, with minimal assumptions, the Poisson boundary of G can be realized on the –Morse boundary of G equipped with the word metric associated to any finite generating set.