# 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?
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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.