Horosphere of a metric tree

Question

I have a project I have to do. In order to do it I need to investigate W.E. Grosso's translation of "The Green Book" on Hyperbolic Group Theory, as found here.

I try to understand the term horosphere as defined on page 91. I do not understand what kind of objects are in that set. Is it an ordered pair of elements of the tree? Perhaps there is a typo?

Here is the relevant excerpt:

Let $X$ be a metric tree and $a$ a point on its boundary $\partial X$. For $x,y \in X$, the distances $|x-a|$ and $|y-a|$ aren't defined; however, their difference $|y-a|-|x-a|$ is well defined. More precisely, we define $$ \beta_a (x,y) = |x-w|-|y-w| \in \mathbb{R}$$ where $w$ is a point on the intersetciob of the rays $[x,a)$ and $[y,a)$—we remark that the number $\beta_a(x,y)$ doesn't depend on the choice of $w$.

9. Definition. The horosphere centered at $a \in \partial X$ is the set $\{ y \in X " \beta_a(x,y) = 0 \}$.

Answer 1

First let me point out that the terminology seems deficient: since $x$ is a parameter in the definition of a horosphere, it should be part of the terminology. In other words, one should speak of "The horosphere through $x$ centered at $a \in \partial X$", and as stated this is by definition equal to the set $$\{y \in X \, | \, \beta_a(x,y)=0\} $$ The set theory notation is quite clear on this issue: this horosphere is a set of points in $X$; each object in the horosphere is a point $y \in X$.

For instance, the specific point $x$ itself is in the horosphere because $\beta_a(x,x)=0$.

On the other hand, referring to the picture you drew, the specific point $y$ is NOT in the horosphere, because $\beta_a(x,y) = |x-w|-|y-w|$ is a positive number.

Here's a way to think of it intuitively. Imagine that you hang the tree on a wall, by putting a nail through the point $a$ and letting the tree hang downward from that point (of course, the point $a$ is infinitely far up, but perhaps your infinite imagination can deal with this anyway). Then the horosphere through $x$ is the subset of all points in the tree that are on the same horizontal level as $x$.