  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 $\kappa$, we construct a
boundary for $X$, denoted $\mathcal{\partial}_{\kappa} X$, 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, then with minimal assumptions the Poisson boundary of $G$ can
be realized on the $\kappa$-Morse boundary of $G$ equipped the word metric
associated to any finite generating set.

Geometric Topology;Dynamical Systems;Group Theory