Ratio-limit boundaries of random walks on relatively hyperbolic groups

Adam Dor-On, Matthieu Dussaule, Ilya Gekhtman

Published 2023-03-19Version 1

We study boundaries arising from limits of ratios of transition probabilities for random walks on relatively hyperbolic groups. We determine significant limitations of a strategy employed by Woess for computing the ratio-limit boundary in the hyperbolic case. 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 \emph{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.