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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 κ, we construct a boundary for X, denoted ∂κX, 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 κ-Morse boundary of G equipped the word metric associated to any finite generating set.
Qing et al. (Thu,) studied this question.