Martin boundary covers Floyd boundary

Abstract

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. .

Entropy and drift for word metric on relatively hyperbolic groups

“…When the Bowditch boundary is a topological sphere of dimension ≥ 2 (or more generally, the conical limit set minus any two points has a finite number of connected components), we show that the stable translation length spectrum of the Green metric is not arithmetic. This uses continuity of cross-ratios for the Green metric on the conical limit set, which we prove using the relative Ancona inequalities from [19]. Since arithmeticity of the stable translation spectrum is preserved under rough similarity of metrics, we are able to conclude Theorem 1.6.…”

“…We thus get an identification of the set of conical limit points as a subset of the Martin boundary. The results of [19] in fact hold for measures with super-exponential first moment.…”

“…In general, we do not know the Martin boundary of a finitely supported admissible measure µ on a relatively hyperbolic group. However, Gekhtman-Gerasimov-Potyagailo-Yang proved in [19] that the Martin boundary always covers the Bowditch boundary (it actually covers the Floyd boundary). Moreover, the pre-image of a conical limit point consists of one point.…”

“…The arguments in this "geometric metric" case are considerably simpler because one has to deal with the geometry of the hyperbolic space X instead of the Cayley graph and the relevant results about Patterson-Sullivan measures have been proved by Coornaert [9]. For symmetric measures, the equivalence of the items analogous to (2), (3), (4) above were proved in Section 11 of [19]. A complete proof will be a special case of a result concerning Gibbs measures in [21].…”

“…We show that ν(Ω(g)) is approximately e −dG(e,g) . This second shadow lemma is based on the following relative Ancona inequalities proved in [19]. A transition point on a geodesic is a point that is not deep inside a parabolic subgroup (see the precise definition in Section 2).…”