For finitely supported random walks on finitely generated groups G we prove that the identity map on G extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This yields new results for relatively hyperbolic groups. Our key estimate relates the Green and Floyd metrics, generalizing results of Ancona for random walks on hyperbolic groups and of Karlsson for quasigeodesics. We then apply these techniques to obtain some results concerning the harmonic measure on the limit sets of geometrically finite isometry groups of Gromov hyperbolic spaces. .
Let G X be a non-elementary action by isometries of a hyperbolic group G on a (not necessarily proper) hyperbolic metric space X. We show that the set of elements of G which act as loxodromic isometries of X has density one in the word metric on G. That is, for any finite generating set of G, the proportion of elements in G of word length at most n, which are X-loxodromics, approaches 1 as n → ∞. We also establish several results about the behavior in X of the images of typical geodesic rays in G; for example, we prove that they make linear progress in X and converge to the Gromov boundary ∂X. We discuss various applications, in particular to mapping class groups, Out(FN ), and right-angled Artin groups. and its limit in a suitable compactification (when it exists) is usually called a Patterson-Sullivan (PS) measure. In the second case, for each n one has the convolution measure μ n := μ · · · μ n times , which is precisely the distribution of the nth step of the random walk. In this case, the limit measure in a suitable compactification is the hitting measure for the random walk.The two types of counting, however, need not yield the same notion of genericity; indeed, an underlying theme of much research has been the question: Are Patterson-Sullivan measures also hitting measures for a certain random walk?This is for instance the main question of [64]. As Furstenberg showed [28], for a semisimple Lie group G, the appropriate boundary is B = G/P , where P is a minimal parabolic subgroup. Furthermore, the analogue of the Patterson-Sullivan measure is the unique measure ν on B which is invariant by a maximal compact subgroup. In fact, he proved that for any lattice Λ in G there exists a measure on Λ whose hitting measure is ν. Moreover, for groups acting properly and cocompactly on δ-hyperbolic manifolds, Connell and Muchnik [21] showed that there exists a certain random walk which produces the Patterson-Sullivan measure on the boundary. They also extended this result to arbitrary Gibbs measures in the case the space is CAT(−1) [20].However, for actions of hyperbolic groups G on spaces X, if the harmonic measure and PS measure are in the same measure class, then the metrics on G and on X are quasi-isometric [10]. Interestingly, for hyperbolic groups Gouëzel, Mathéus, and Maucourant [30] recently proved that, when considering a word metric on G, harmonic measures for symmetric random walks of finite support are never in the same measure class as Patterson-Sullivan measures, unless the group is virtually free. One should compare this to Ledrappier's result that for compact hyperbolic surfaces the hitting measure for Brownian motion coincides with the PS measure if and only if the surface has constant curvature [49]. There are many examples of harmonic measures which are singular with respect to the PS-type measure, see for example, [39].In this paper, we will use random walk methods to prove results about genericity with respect to counting in balls in the word metric on G. In particular, we consider the case of G a w...
Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green's function of the corresponding random walk. We give a complete topological characterization of the Martin boundary of finitely supported random walks on relatively hyperbolic groups with virtually abelian parabolic subgroups. In particular, in the case of nonuniform lattices in the real hyperbolic space H n , we show that the Martin boundary coincides with the CAT (0) boundary of the truncated space, and thus when n = 3, is homeomorphic to the Sierpinski carpet.
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