The Hausdorff dimension of the harmonic measure for relatively hyperbolic groups

Abstract

The paper studies the Hausdorff dimension of harmonic measures on various boundaries of a relatively hyperbolic group which are associated with random walks driven by a probability measure with finite first moment. With respect to the Floyd metric and the shortcut metric, we prove that the Hausdorff dimension of the harmonic measure equals the ratio of the entropy and the drift of the random walk.If the group is infinitely-ended, the same dimension formula is obtained for the end boundary endowed with a visual metric. In addition, the Hausdorff dimension of the visual metric is identified with the growth rate of the word metric. These results are complemented by a characterization of doubling visual metrics for accessible infinitely-ended groups : the visual metrics on the end boundary is doubling if and only if the group is virtually free. Consequently, there are at least two different bi-Hölder classes (and thus quasi-symmetric classes) of visual metrics on the end boundary.

Harmonic measures on Bowditch boundaries of groups hyperbolic relative to virtually nilpotent subgroups

“…where ξ gH ∈ ∂X is the fixed point of gHg −1 . Then by comparison with (14) we get 1 r ν y (S G (ξ gH , B G (y, r))) ≈ r e kH (y)−ρG(y,Γ) , (16) as y in a horoball and r can be chosen arbitrarily. Note that (16) holds for any y ∈ H g as x may be chosen suitably so that the case in Step 2 is satisfied.…”

“…Note that (16) holds for any y ∈ H g as x may be chosen suitably so that the case in Step 2 is satisfied. We may use (16) to get a similar lower bound as follows. We may assume without loss of generality y = (e, n) for some n ∈ N, n ≥ 10 • r. Note that for some r > 0, ν (e,n) (S G (ξ H , B G ((e, n), r)) ≈ r ν (e,n) (S G (ξ H , U H (e, g H (a ρX ((e,n),e)+r ))).…”

“…The main issue in our case is to extend the shadow lemma to the shadow of balls centered at non-orbit points. Another approach from Kaimanovich [32], Le Prince [38], Tanaka [48], applicable rather generally for groups acting with exponential growth on proper hyperbolic spaces and jumps given by measures with finite first moment (see also Dussaule-Yang [16] for harmonic measures of random walks on Cayley graphs of relatively hyperbolic groups), utilizes properties of the asymptotic entropy, drift (Kaimanovich [33]; Maher-Tiozzo [40] in the non-proper case) in the hyperbolic setting and the ergodicity of the action of Γ with respect to the harmonic measure in the boundary (for the lower bound on the dimension, [48]). The dimension of µ, the D Γ -dimensional Patterson-Sullivan density of Γ for its action on X, where D Γ is the critical exponent of Γ acting on X (µ being unique up to constant multiples as it is non-atomic, doubling and ergodic for the Γ-action, see Remark 4.3 and Remark 4.12 below) is computed by an approach of Stratmann-Velani [46], using a shadow lemma to study cusp excursions into horoballs.…”

“…The question of ascertaining the dimension of the harmonic measure in related settings has been treated in Ledrappier [35,37], Kaimanovich [32], Le Prince [38], [5], Tanaka [48]. It has been treated for random walks on relatively hyperbolic groups in [16], for harmonic measures in both Floyd and Bowditch boundaries. The random walks considered in the above works are walks that jump along orbits.…”