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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.

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  • $\begingroup$ Dear @Harry Baik: I believe this question should probably be tagged with 'gt.geometric-topology'. Thank you. $\endgroup$
    – Ricardo Andrade
    Commented Nov 22, 2013 at 12:44
  • $\begingroup$ Dear @RicardoAndrade I added a tag, thank you for your suggestion! $\endgroup$
    – Harry Baik
    Commented Nov 22, 2013 at 13:16

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This sort of thing was worked out by Coornaert, who constructed Patterson--Sullivan measures on the boundaries of word-hyperbolic groups:

Michel Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov [Patterson-Sullivan measures on the boundary of a hyperbolic space in the sense of Gromov], Pacific J. Math. 159 (1993), no. 2, 241–270.

A nice account was given by Danny Calegari in these notes. I think you'll find what you need in there.

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    $\begingroup$ 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. $\endgroup$
    – HJRW
    Commented Nov 22, 2013 at 14:41

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