Gromov boundary
In mathematics, the Gromov boundary of a δ-hyperbolic space (especially a hyperbolic group) is an abstract concept generalizing the boundary sphere of hyperbolic space.
Conceptually, the Gromov boundary is the set of all points at infinity.
For instance, the Gromov boundary of the real line is two points, corresponding to positive and negative infinity.
There are several equivalent definitions of the Gromov boundary of a geodesic and proper δ-hyperbolic space.
One of the most common uses equivalence classes of geodesic rays.
A geodesic ray is a path given by an isometry
The Gromov boundary of a geodesic and proper hyperbolic metric space
In a tree (graph theory), this measures how long the paths from
Since hyperbolic spaces are tree-like, the Gromov product measures how long geodesics from
in the Gromov boundary, we define the sets
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
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.
The Gromov boundary is a quasi-isometry invariant; that is, if two Gromov-hyperbolic metric spaces are quasi-isometric, then the quasi-isometry between them induces a homeomorphism between their boundaries.
[2][3] This is important because homeomorphisms of compact spaces are much easier to understand than quasi-isometries of spaces.
This invariance allows to define the Gromov boundary of a Gromov-hyperbolic group: if
acts properly discontinuously and cocompactly (for instance its Cayley graph).
This is well-defined as a topological space by the invariance under quasi-isometry and the Milnor-Schwarz lemma.
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.
Any boundary point can be represented by a unique geodesic ray issuing from o.
issuing from o, and positive numbers t > 0 and r > 0, a neighborhood basis at the boundary point
is given by sets of the form 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.
[5] Cannon's conjecture concerns the classification of groups with a 2-sphere at infinity: Cannon's conjecture: Every Gromov hyperbolic group with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.
[6] The analog to this conjecture is known to be true for 1-spheres and false for spheres of all dimension greater than 2.
