Let $X$ be a complete simply connected Riemannian manifold of dimension $\ge2$ and pinched negative curvature, $ X(\infty)$ its boundary at infinity, $\Gamma$ a non-elemantary discrete group of isometries of $X$.

We consider the Gromov product of $a,b\in X$ relative to $x\in X$, \[ (a,b)_x :=\frac12\big(d(x,a)+d(x,b)-d(a,b)\big). \] Based on the Busemann function, consider the horospherical distance (relative to $\xi\in X(\infty)$), given a geodesic ray $\gamma$ with $\gamma(0)=y$ and $\gamma(\infty)=\xi$, \[ \beta_\xi(x,y) := b_y(x,\xi) = \lim_{t\to\infty}d(x,\gamma(t))-d(y,\gamma(t)). \]

Given $\xi,\eta\in X(\infty)$, $\xi\ne \eta$, and $a_n\to \xi, b_n\to \eta$, then \[ (\xi,\eta)_x := \lim_{n\to\infty}(a_n,b_n)_x \] (this limit exists and is independent on the chosen sequences, see above). If $\xi=\gamma(\infty)$ and $\eta=\gamma(-\infty)$ for a geodesic $\gamma$, given $z=\gamma(0)$ (or any other point in $\gamma$), \[\begin{split} (\xi,\eta)_x &= \lim_n\,(\gamma(n),\gamma(-n))_x\\ &= \lim_n\frac12\big(d(x,\gamma(n))+d(x,\gamma(-n))-d(\gamma(n),\gamma(-n)\big)\\ &= \lim_n\frac12\big(d(x,\gamma(n))+d(x,\gamma(-n))-\Big(d(\gamma(n),z)+d(z,\gamma(-n)\Big)\big)\\ &= \lim_n\frac12\big(d(x,\gamma(n))-d(z,\gamma(n))\big)+ \lim_n\frac12\big(d(x,\gamma(-n))-d(z,\gamma(-n)\big)\\ &= \frac12(b_\xi(x,z)+b_\eta(x,z)). \end{split}\]

For the following see [Bridson, Haefliger, Chapter III.H Propositions 3.7 and 3.21] or [Bourdon 1995].

Lemma. For any $x\in X$, the function \[ d_x(\xi,\eta) :=\begin{cases} e^{-(\xi,\eta)_x}&\text{ if }\xi\ne\eta,\\ 0&\text{ otherwise}, \end{cases} \] defines a metric on $ X(\infty)$ which induces the same topology as the cone topology. Moreover, for any $x,y\in X$ \[ d_y(\xi,\eta) = e^{\frac12(b_\xi(x,y)+b_\eta(x,y))}d_x(\xi,\eta). \]

The above result will be used to state some regularity properties of Gibbs cocycles.