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 $e$?