# Measure on the Boundary of a Hyperbolic Group

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?

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.
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.