# Volume growth and visibility in hyperbolic spaces

I am interested in the following two questions for complex, quaternionic and octonionic hyperbolic spaces, equipped with their usual metrics and measures:

For brevity, I will denote the volume of the ball of radius $r$ by $V_r$, and a biLipschitz equivalence of two functions by $\asymp$.

Does there exist a constant $\lambda$ such that $V_r \asymp \lambda^r$, and, if $\lambda$ has been computed, what is it?

What is the visibilty constant of the space, i.e. for which $\varepsilon$ is it true that there is a visual metric $d$ on the boundary satisfying $$d(x,y)\asymp \exp(-\varepsilon(x.y)_e)$$ where $(x.y)_e$ is the Gromov product with respect to a fixed basepoint $e$?

• The correct thing to try to calculate is $\log_e(\lambda)/\varepsilon$ as this is invariant with respect to rescaling the metric. I have heard a suggestion that a lower bound on this value is the conformal dimension of the boundary and that an equality can actually be achieved, not necessarily by the spaces listed above but by some well-chosen quasi-isometric space. Is anyone aware if this is correct, and if so, where it might appear in the literature? – DavidHume Sep 8 '16 at 1:40
• is this useful? Section 5, especially Example 5.12, of arxiv.org/pdf/1312.4646v3.pdf – BN2 Sep 27 '16 at 16:09
• Yes thanks, it isn't a complete account of what I wanted but it is certainly a good place to start. – DavidHume Sep 29 '16 at 18:12