Stability phenomena for Martin boundaries of relatively hyperbolic groups

Abstract

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 up to the spectral radius of $\mu$. We then show that when the parabolic subgroups are virtually abelian, the Martin boundary of the induced random walk on $\Gamma$ is stable in the sense of Picardello and Woess. We also define a notion of spectral degenerescence along parabolic subgroups and give a criterion for strong stability of the Martin boundary in terms of spectral degenerescence. We 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.