Gromov boundary · Lablypedia

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 $γ : [0, \infty) \to X$ such that each segment $γ ([0, t])$ is a path of shortest length from $O$ to $γ (t)$ .

Two geodesics $γ_{1}, γ_{2}$ are defined to be equivalent if there is a constant $K$ such that $d (γ_{1} (t), γ_{2} (t)) \leq K$ 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 $\partial X = {[γ] | γ$ 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 $V (p, r) = {q \in \partial X |$ there are geodesic rays $γ_{1}, γ_{2}$ with $[γ_{1}] = p, [γ_{2}] = q$ and $lim inf_{s, t \to \infty} (γ_{1} (s), γ_{2} (t))_{O} \geq r}$ . 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 $r$ 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.

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Gromov boundary of a group

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. 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 $G$ is such a group, its Gromov boundary is by definition that of any proper geodesic space space on which $G$ 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.

Examples

The Gromov boundary of a regular tree of degree d≥3 is a Cantor space.
The Gromov boundary of hyperbolic n-space is an (n-1)-dimensional sphere.
The Gromov boundary of the fundamental group of a compact hyperbolic Riemann surface is the unit circle.
The Gromov boundary of most hyperbolic groups is a Menger sponge.

Variations

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 $γ$ issuing from o, and positive numbers t > 0 and r > 0, a neighborhood basis at the boundary point $[γ]$ is given by sets of the form

U (γ, t, r) = {[γ_{1}] \in \partial X | γ_{1} (0) = o, d (γ_{1} (t), γ (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.

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.