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.^[1]

Pick some point ${\displaystyle O}$  of a hyperbolic metric space ${\displaystyle X}$  to be the origin. A geodesic ray is a path given by an isometry ${\displaystyle \gamma :[0,\infty )\rightarrow X}$  such that each segment ${\displaystyle \gamma ([0,t])}$  is a path of shortest length from ${\displaystyle O}$  to ${\displaystyle \gamma (t)}$ .

Two geodesics ${\displaystyle \gamma _{1},\gamma _{2}}$  are defined to be equivalent if there is a constant ${\displaystyle K}$  such that ${\displaystyle d(\gamma _{1}(t),\gamma _{2}(t))\leq K}$  for all ${\displaystyle t}$ . The equivalence class of ${\displaystyle \gamma }$  is denoted ${\displaystyle [\gamma ]}$ .

The Gromov boundary of a geodesic and proper hyperbolic metric space ${\displaystyle X}$  is the set ${\displaystyle \partial X=\{[\gamma ]|\gamma }$  is a geodesic ray in ${\displaystyle X\}}$ .

Topology edit

It is useful to use the Gromov product of three points. The Gromov product of three points ${\displaystyle x,y,z}$  in a metric space is ${\displaystyle (x,y)_{z}=1/2(d(x,z)+d(y,z)-d(x,y))}$ . In a tree (graph theory), this measures how long the paths from ${\displaystyle z}$  to ${\displaystyle x}$  and ${\displaystyle y}$  stay together before diverging. Since hyperbolic spaces are tree-like, the Gromov product measures how long geodesics from ${\displaystyle z}$  to ${\displaystyle x}$  and ${\displaystyle y}$  stay close before diverging.

Given a point ${\displaystyle p}$  in the Gromov boundary, we define the sets ${\displaystyle V(p,r)=\{q\in \partial X|}$  there are geodesic rays ${\displaystyle \gamma _{1},\gamma _{2}}$  with ${\displaystyle [\gamma _{1}]=p,[\gamma _{2}]=q}$  and ${\displaystyle \lim \inf _{s,t\rightarrow \infty }(\gamma _{1}(s),\gamma _{2}(t))_{O}\geq r\}}$ . 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 ${\displaystyle r}$  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.

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

• The Gromov boundary of a tree 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.^[4]

Visual boundary of CAT(0) space edit

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

${\displaystyle U(\gamma ,t,r)=\{[\gamma _{1}]\in \partial X|\gamma _{1}(0)=o,d(\gamma _{1}(t),\gamma (t))<r\}.}$ 

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.^[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.

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