Stability phenomena for Martin boundaries of relatively hyperbolic groups

Abstract

Let Γ be a relatively hyperbolic group and let µ be an admissible symmetric finitely supported probability measure on Γ. We extend Floyd-Ancona type inequalities from [11] up to the spectral radius of µ. We then show that when the parabolic subgroups are virtually abelian, the Martin boundary of the induced random walk on Γ is stable in the sense of Picardello and Woess [28]. 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.This can be written as P f = rf , where P is the Markov operator associated with µ. Every r-harmonic positive function can be represented as an integral over the Martin boundary. Precisely, for every such function f , there exists a probability measure ν f on ∂ rµ Γ such that for all x ∈ Γ,In general, the measure ν f is not unique. To obtain uniqueness, we restrict our attention to the minimal boundary that we now define.

Local limit theorems in relatively hyperbolic groups I: rough estimates

“…Along the way, we obtain other estimates involving the Green function of the parabolic subgroups and the Green function of the whole group. We also show that non-spectral degeneracy of the measure μ in the sense of [15] implies that μ is divergent in the sense of Definition 1.2, so that non-spectral degeneracy implies spectral positive recurrence.…”

“…As announced in the introduction, we also give more explanations of Definitions 1.1-1.3, making an analogy with similar definitions in the context of counting theorems on Kleinian groups. We recall the definition of spectral degeneracy, introduced in [15], and explain why non-spectral degeneracy implies finiteness of the Green moments.…”

“…In other words, the Green function is roughly multiplicative along relative geodesics (uniformly in r). One of the main results of [15] is that weak relative Ancona inequalities hold for any finitely supported, symmetric and admissible probability measure μ on a relatively hyperbolic group . Note that there also exist strong relative Ancona inequalities, which were also proved in [15].…”

“…One of the main results of [15] is that weak relative Ancona inequalities hold for any finitely supported, symmetric and admissible probability measure μ on a relatively hyperbolic group . Note that there also exist strong relative Ancona inequalities, which were also proved in [15]. Since strong inequalities are technical to state and since we will not need them in this paper, we do not mention them here.…”

“…The second paper will be devoted to proving an asymptotic of the form (1), with α = 3/2, for non-spectrally degenerate random walks. This property of spectral degeneracy was introduced in [15] to study stability of the Martin boundary. In this first paper, we introduce a looser condition, namely spectral positive recurrence, and prove some weaker estimates than (1) for spectrally positive-recurrent random walks.…”