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3 added more details, corrected spelling

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 isare:

• Does anybody has have study the limit of these measuresas $n\to\infty$? Is there any type of (weak) convergence?
• Are these measures in some way related with the harmonic Hausdorff measure on the boundary$\partial\Gamma$ of $\Gamma$ ? to the Patterson-Sullivan measure? to the harmonic measure?
2 edited title

1

# Meaure 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 $$\nu_n := \frac{1}{|\partial B(1,n)|}\sum_{\gamma\,\in\partial B(1,n)}{\delta_{\gamma}}.$$

My question is:

• Does anybody has study the limit of these measures as $n\to\infty$? Is there any type of (weak) convergence? Are these measures in some way related with the harmonic measure on the boundary $\partial\Gamma$ of $\Gamma$ ?