2023/03/19 by Adam Dor-On, Dor-On, Adam, Matthieu Dussaule +3
Mathematics · #20F67 #37A55. Secondary: 37B05 #47L80 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Operator Algebras (math.OA) #Primary: 60J50 #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2303.10769
openalex publication_date 2023/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study boundaries arising from limits of ratios of transition probabilities for random walks on relatively hyperbolic groups. We extend, as well as determine significant limitations of, a strategy employed by Woess for computing ratio-limit boundaries for the class of hyperbolic groups. On the one hand we employ results of the second and third authors to adapt this strategy to spectrally non-degenerate random walks, and show that the closure of minimal points in R-Martin boundary is the unique smallest invariant subspace in ratio-limit boundary. On the other hand we show that the general strategy can fail when the random walk is spectrally degenerate and adapted on a free product. Using our results, we are able to extend a theorem of the first author beyond the hyperbolic case and establish the existence of a co-universal quotient for Toeplitz C*-algebras arising from random walks which are spectrally non-degenerate on relatively hyperbolic groups. Finally, we exhibit an example of a relatively hyperbolic group carrying two random walks such that the ratio limit boundaries are not equivariantly homeomorphic and no two equivariant quotients of their respective Toeplitz C*-algebras are equivariantly *-isomorphic.