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.

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

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.

  • $\begingroup$ 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 Nov 22 '13 at 14:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.