Stability phenomena for Martin boundaries of relatively hyperbolic groups

2019/08/28 by Matthieu Dussaule, Dussaule, Matthieu, Ilya Gekhtman +1 · 2 citations
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Probability (math.PR) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1909.01577

openalex publication_date 2019/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \Γ be a relatively hyperbolic group and let \μ be an admissible\nsymmetric finitely supported probability measure on \Γ. We extend\nFloyd-Ancona type inequalities up to the spectral radius of \μ. We then show\nthat when the parabolic subgroups are virtually abelian, the Martin boundary of\nthe induced random walk on \Γ is stable in the sense of Picardello and\nWoess. We also define a notion of spectral degenerescence along parabolic\nsubgroups and give a criterion for strong stability of the Martin boundary in\nterms of spectral degenerescence. We prove that this criterion is always\nsatisfied in small rank. so that in particular, the Martin boundary of an\nadmissible symmetric finitely supported probability measure on a geometrically\nfinite Kleinian group of dimension at most 5 is always strongly stable.\n

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• Local limit theorems on relatively hyperbolic groups with respect to virtually nilpotent subgroups
• Exotic local limit theorems at the phase transition in free products

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