# Density of ends of long words in a hyperbolic group

Let $G$ be a Gromov hyperbolic group with a generating set $S$. For each $g \in G$, let $\xi_g$ be the point in the ideal boundary corresponding to the sequence $(g^n)$. Let $l_S(g)$ be the word length of $g$ w.r.t. the generating set $S$.

I would like to know how dense the following set is on the ideal boundary of $G$ (denote by $\partial G$); $$E_L := \{ \xi_g : l_S(g) \le L \}$$ More precisely, is the set $E_L\quad$ $e^{-\alpha L}$-dense in $\partial G$ for some positive number $\alpha$? If so, for which $\alpha$? I suspect there is some relation between the optimal $\alpha$ and the Hausdorff dimension of $\partial G$.. but not sure how I should investigate further.

Thank you.

• Dear @Harry Baik: I believe this question should probably be tagged with 'gt.geometric-topology'. Thank you. – Ricardo Andrade Nov 22 '13 at 12:44
• Dear @RicardoAndrade I added a tag, thank you for your suggestion! – Harry Baik Nov 22 '13 at 13:16

• So, I'm not sure what you mean by '$e^{-\alpha L}$ dense', but Lemma 2.5.8 might be what you're looking for. – HJRW Nov 22 '13 at 14:41