In $\mathbb{H}^n$, equipped with its hyperbolic metric of constant curvature $-1$, if we have two points $p,q$ on a common horosphere $\partial S$, then $$d\_{\mathbb{H}}(p,q) = 2\sinh^{-1} (d\_{\partial S} (p,q)/2)$$ where $d\_{\partial S}(p,q)$ is the shortest path from $p$ to $q$ that lies on the horosphere.


Is there any version of this identity in a general $CAT(-1)$ metric space? I must admit my hyperbolic trigonometry is very weak.



Horospherical distance in CAT($-1$) space
I suppose I'll answer my own question.


According to this paper: <https://projecteuclid.org/download/pdf_1/euclid.jdg/1214434219>
if $M$ is a Hadamard manifold with pinched sectional curvature $-b^2\leq K \leq -a^2$, and if $H$ is a horosphere and $h$ represents distance in $H$, then $$\frac{2}{a}\sinh (a/2) d(p,q)\leq h(p,q) \leq \frac{2}{b}\sinh (b/2) d(p,q)$$ with the obvious modifications if $a=0$, $b=-\infty$. As mentioned in the comments, a general inequality for a $CAT(-1)$ space seems to be asking for too much.


