Gromov boundary

It is useful to use the Gromov product of three points. The Gromov product of three points ${\displaystyle x,y,z}$ in a metric space is ${\displaystyle (x,y)_{z}=1/2(d(x,z)+d(y,z)-d(x,y))}$. In a tree (graph theory), this measures how long the paths from ${\displaystyle z}$ to ${\displaystyle x}$ and ${\displaystyle y}$ stay together before diverging. Since hyperbolic spaces are tree-like, the Gromov product measures how long geodesics from ${\displaystyle z}$ to ${\displaystyle x}$ and ${\displaystyle y}$ stay close before diverging.

Given a point ${\displaystyle p}$ in the Gromov boundary, we define the sets ${\displaystyle V(p,r)=\{q\in \partial X|}$ there are geodesic rays ${\displaystyle \gamma _{1},\gamma _{2}}$ with ${\displaystyle [\gamma _{1}]=p,[\gamma _{2}]=q}$ and ${\displaystyle \lim \inf _{s,t\rightarrow \infty }(\gamma _{1}(s),\gamma _{2}(t))_{O}\geq r\}}$. These open sets form a basis for the topology of the Gromov boundary.

These open sets are just the set of geodesic rays which follow one fixed geodesic ray up to a distance ${\displaystyle r}$ before diverging.

This topology makes the Gromov boundary into a compact metrizable space.

The number of ends of a hyperbolic group is the number of components of the Gromov boundary.

For a complete CAT(0) space X, the visual boundary of X, like the Gromov boundary of δ-hyperbolic space, consists of equivalence class of asymptotic geodesic rays. However, the Gromov product cannot be used to define a topology on it. For example, in the case of a flat plane, any two geodesic rays issuing from a point not heading in opposite directions will have infinite Gromov product with respect to that point. The visual boundary is instead endowed with the cone topology. Fix a point o in X. Any boundary point can be represented by a unique geodesic ray issuing from o. Given a ray ${\displaystyle \gamma }$ issuing from o, and positive numbers t > 0 and r > 0, a neighborhood basis at the boundary point ${\displaystyle [\gamma ]}$ is given by sets of the form

${\displaystyle U(\gamma ,t,r)=\{[\gamma _{1}]\in \partial X|\gamma _{1}(0)=o,d(\gamma _{1}(t),\gamma (t))<r\}.}$

The cone topology as defined above is independent of the choice of o.

If X is proper, then the visual boundary with the cone topology is compact. When X is both CAT(0) and proper geodesic δ-hyperbolic space, the cone topology coincides with the topology of Gromov boundary.[6]