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.
Definition.
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 formula_1 of a hyperbolic metric space formula_2 to be the origin. A geodesic ray is a path given by an isometry formula_3 such that each segment formula_4 is a path of shortest length from formula_1 to formula_6.
Two geodesics formula_7 are defined to be equivalent if there is a constant formula_8 such that formula_9 for all formula_10. The equivalence class of formula_11 is denoted formula_12.
The Gromov boundary of a geodesic and proper hyperbolic metric space formula_2 is the set formula_14 is a geodesic ray in formula_15.
Topology.
It is useful to use the Gromov product of three points. The Gromov product of three points formula_16 in a metric space is
formula_17. In a tree (graph theory), this measures how long the paths from formula_18 to formula_19 and formula_20 stay together before diverging. Since hyperbolic spaces are tree-like, the Gromov product measures how long geodesics from formula_18 to formula_19 and formula_20 stay close before diverging.
Given a point formula_24 in the Gromov boundary, we define the sets formula_25 there are geodesic rays formula_7 with formula_27 and formula_28. 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 formula_29 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.
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 formula_30 acts geometrically on a δ-hyperbolic space, then formula_30 is hyperbolic group and formula_30 and formula_2 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.
Generalizations.
Visual boundary of CAT(0) space.
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 formula_11 issuing from "o", and positive numbers "t" > 0 and "r" > 0, a neighborhood basis at the boundary point formula_12 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.
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.