Entropy and drift for word metric on relatively hyperbolic groups

Abstract

We are interested in the Guivarc'h inequality for admissible random walks on finitely generated relatively hyperbolic groups, endowed with a word metric. We show that for random walks with finite super-exponential moment, if this inequality is an equality, then the Green distance is roughly similar to the word distance, generalizing results of Blachère, Haïssinsky and Mathieu for hyperbolic groups [4]. Our main applications are for relatively hyperbolic groups with some virtually abelian parabolic subgroup of rank at least 2, relatively hyperbolic groups with spherical Bowditch boundary, and free products with at least one virtually nilpotent factor. We show that for such groups, the Guivarc'h inequality with respect to a word distance and a finitely supported random walk is always strict.

The Hausdorff dimension of the harmonic measure for relatively hyperbolic groups

“…The archetypal groups with infinitely many ends are free products of the form A * B, where A = Z/2Z or B = Z/2Z. Using again the strictness of the fundamental inequality established in [16,Theorem 1.5] for certain free products, we can state the following corollary.…”

Entropy and drift for Gibbs measures on geometrically finite manifolds

“…In this context, Gouezel, Matheus and Maucourant proved that the inequality (1) is strict for any superexponential moment generating random walk on a word-hyperbolic group which is not virtually free. Dussaule-Gekhtman [15] extended this result to large classes of relatively hyperbolic groups, including finite covolume isometry groups of pinched negatively curved manifolds and geometrically finite Kleinian groups.…”

Stability phenomena for Martin boundaries of relatively hyperbolic groups

“…Since Ω is (λ, c)-starlike around e, if w ∈ B η0 (e) ∩ Ω, there is a path from w to e that stays inside Ω and whose length is bounded by λη 0 + c. In particular, we have G(x, w; Ω) ≤ C η0 G(x, e; Ω) and G(w, y; Ω) ≤ C η0 G(e, y; Ω). Summing over all possible w ∈ B η0 , we obtain (9) G(x, y; reg(η 0 , δ) ∩ Ω) ≤ C ′ η0 G(x, e; Ω)G(e, y; Ω). We now find an upper bound of G(x, y; reg(η 0 , δ) c ∩ Ω).…”