TITLE: What spaces have well known horofunctions?
QUESTION [8 upvotes]: Following Gromov, take a metric space $(X,d)$ and consider $C(X)/\mathbb{R}$ the set of continuous functions to $\mathbb{R}$ with the topology of uniform convergence on compact sets after taking the quotient by constant functions (i.e. two functions are equivalent if the difference is a constant).   Embed $X$ into this space by means of the map:
$x \mapsto f(x) = [d(x,\cdot)]$
A horofunction is an element of the closure of $f(X)$ that is not in $f(X)$.
In $\mathbb{R}^d$ all horofunctions are given by inner product with a unit vector.
In the upper-half plane model of the hyperbolic disc the function $h(z) = \log(\text{Im}(z))$ is a horofunction and all others can be obtained by composing with hyperbolic isometries.
What other spaces have well known horofunctions?
For example, are horofunctions of the model geometries $\text{Nil}$, $\text{Sol}$, and $\widetilde{\text{Sl}(2,R)}$ in dimension $3$ known relatively explicitly?
I know there is a general relationship between horofunctions and geodesics (Busseman functions) in non-positively curved spaces.  However I'm interested in spaces where one can compute relatively explicitly (e.g. spaces where one knows what the distance function looks like).

REPLY [2 votes]: One possible place to look (other than symmetric spaces) is the warped product of hyperbolic metrics, a metric of the form $dr^2+e^{C_1r} dx^2 + e^{C_2r}dy^2$, $C_1, C_2 >0$. Then $r$ is a horofunction. If one could compute the horofunction for another point in the Gromov boundary, then since the isometry group acts transitively on these other points of the Gromov boundary, one would have a computation for all points. However, I imagine computing the other horofunctions could be tricky, since I don't know how to compute the geodesics explicitly.