Stability phenomena for Martin boundaries of relatively hyperbolic groups

TL;DR

This paper investigates the stability of Martin boundaries for relatively hyperbolic groups, extending inequalities and establishing criteria for stability, especially in cases with virtually abelian parabolic subgroups and small rank.

Contribution

It extends Floyd-Ancona inequalities to the spectral radius, introduces spectral degenerescence, and provides conditions for Martin boundary stability in relatively hyperbolic groups.

Findings

Martin boundary is stable for virtually abelian parabolic subgroups

Spectral degenerescence criterion ensures strong stability

Martin boundary of certain Kleinian groups is always strongly stable

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.