Stability phenomena for Martin boundaries of relatively hyperbolic groups

Let Gamma be a relatively hyperbolic group and let mu be an admissible symmetric finitely supported probability measure on Gamma. We extend Floyd-Ancona type inequalities from Gekhtman et al. (Martin boundary covers Floyd boundary, 2017. arXiv:1708.02133) up to the spectral radius R of mu. We use them to find the precise homeomorphism type of the r-Martin boundary, which describes r-harmonic functions, for every r <= R. We also define a notion of spectral degeneracy along parabolic subgroups which is crucial to describe the homeomorphism type of the R-Martin boundary. Finally, we give a criterion for (strong) stability of the Martin boundary in the sense of Picardello andWoess (in: Potential theory, de Gruyter, 1992) in terms of spectral degeneracy. We then prove that this criterion is always satisfied in small rank, so that in particular, the Martin boundary of an admissible symmetric finitely supported probability measure on a geometrically finite Kleinian group of dimension at most 5 is always strongly stable.