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. Pick some point of a hyperbolic metric space to be the origin. A geodesic ray is a path given by an isometry such that each segment is a path of shortest length from to. Two geodesics are defined to be equivalent if there is a constant such that for all. The equivalence class of is denoted. The Gromov boundary of a geodesic and proper hyperbolic metric space is the set is a geodesic ray in.
Topology
It is useful to use the Gromov product of three points. The Gromov product of three points in a metric space is . In a tree, this measures how long the paths from to and stay together before diverging. Since hyperbolic spaces are tree-like, the Gromov product measures how long geodesics from to and stay close before diverging. Given a point in the Gromov boundary, we define the sets there are geodesic rays with and. 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 before diverging. This topology makes the Gromov boundary into a compactmetrizable space. The number of ends of a hyperbolic group is the number of components of the Gromov boundary.
Properties of the Gromov boundary
The Gromov boundary has several important properties. One of the most frequently used properties in group theory is the following: if a group acts geometrically on a δ-hyperbolic space, then is hyperbolic group and and have homeomorphic Gromov boundaries. One of the most important properties is that it is a quasi-isometry invariant; that is, if two hyperbolic metric spaces are quasi-isometric, then the quasi-isometry between them gives a homeomorphism between their boundaries. This is important because homeomorphisms of compact spaces are much easier to understand than quasi-isometries of spaces.
For a complete CAT 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 issuing from o, and positive numberst > 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 and proper geodesic δ-hyperbolic space, the cone topology coincides with the topology of Gromov boundary.
Cannon's Conjecture
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. The analog to this conjecture is known to be true for 1-spheres and false for spheres of all dimension greater than 2.
