Gromov boundary

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 O of a hyperbolic metric space X to be the origin. A geodesic ray is a path given by an isometry γ : [ 0 , ∞ ) → X such that each segment γ ( [ 0 , t ] ) is a path of shortest length from O to γ ( t ) .

We say that two geodesics γ 1 , γ 2 are equivalent if there is a constant K such that d ( γ 1 ( t ) , γ 2 ( t ) ) ≤ K for all t . The equivalence class of γ is denoted [ γ ] .

The Gromov boundary of a geodesic and proper hyperbolic metric space X is the set ∂ X = { [ γ ] | γ is a geodesic ray in X } .

Topology

It is useful to use the Gromov product of three points. The Gromov product of three points x , y , z in a metric space is ( 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 z to x and y stay together before diverging. Since hyperbolic spaces are tree-like, the Gromov product measures how long geodesics from z to x and y stay close before diverging.

Given a point p in the Gromov boundary, we define the sets V ( p , r ) = { q ∈ ∂ X | there are geodesic rays γ 1 , γ 2 with [ γ 1 ] = p , [ γ 2 ] = q and lim inf s , t → ∞ ( γ 1 ( s ) , γ 2 ( t ) ) O ≥ 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 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.

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 G acts geometrically on a δ-hyperbolic space, then G is hyperbolic group and G and X 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.

Examples

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.