Measure on the Boundary of a Hyperbolic Group

Question

Let $\Gamma$ be a non-elementary Gromov's $\delta$-hyperbolic group. Let $B(1,n)$ be the set of elements at distance at most $n$ from the identity and let $\partial B(1,n)$ be the elements at distance $n$ (with the word metric). Consider the probability measures $\mu_n$ and $\nu_n$ defined as $$ \mu_n := \frac{1}{|B(1,n)|}\sum_{\gamma\\, \in B(1,n)}{\delta_{\gamma}} $$ and $$ \nu_n := \frac{1}{|\partial B(1,n)|}\sum_{\gamma\\,\in\partial B(1,n)}{\delta_{\gamma}}. $$ It is clear that these measures converge weakly (there is a convergent sub-sequence) on the compact space $\Gamma\cup\partial\Gamma$. Moreover, the limit measure is supported on $\partial\Gamma$.

My question are:

• Does anybody have study the limit measures?
• Are these related with the Hausdorff measure on the boundary? to the Patterson-Sullivan measure? to the harmonic measure?

Answer 1

I don't know about the specific sums you suggest, but here are some well established alternatives.

Try Kaimanovich's paper "The Poisson boundary of hyperbolic groups", which is about boundaries arising from random walks, which are shown to coincide with the Gromov boundary.

Another somewhat different approach is found in Coornaert's paper Patterson-Sullivan measures on the boundary of a hyperbolic space in the sense of Gromov. The idea, which goes back to the original Patterson-Sullivan measures on limit sets of Kleinian groups, is to construct measures on the boundary using sums over the whole group, where group elements are weighted by a function that decays exponentially in the distance; the base of the exponential is chosen delicately, based on geometric properties of the group. The resulting measures on the boundary are ``quasiconformal'' measures.

Answer 2

Letting $\mu_\infty$ be the Patterson-Sullivan measure on $\partial \Gamma$, any limit measure of the sequence $(\mu_n)$ is equivalent to $\mu_\infty$. Moreover, the Radon-Nikodym derivatives are bounded from above and below. This is given by Lemma 2.13 in Gouëzel, Mathéus and Maucourant's paper Entropy and drift in word hyperbolic groups (Invent. math. (2018) 211:1201–1255). As suggested by Lee Mosher, the arguments follow from arguments of Coornaert's paper.

However, even if the Poisson boundary coincides with the Gromov boundary as a measured space, the harmonic measure on the boundary is not in general equivalent to the Patterson-Sullivan measure, see Theorem 1.5 in the same paper of Gouëzel, Mathéus and Maucourant.