Gromov boundary

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 \gamma:[0,\infty)\rightarrow X such that each segment \gamma([0,t]) is a path of shortest length from O to \gamma(t). Two geodesics \gamma_1,\gamma_2 are defined to be equivalent if there is a constant K such that d(\gamma_1(t),\gamma_2(t))\leq K for all t. The equivalence class of \gamma is denoted [\gamma]. The Gromov boundary of a geodesic and proper hyperbolic metric space X is the set \partial X=\{[\gamma]|\gamma 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 \gamma_1,\gamma_2 with [\gamma_1]=p, [\gamma_2]=q and \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 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. ==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 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 \gamma issuing from o, and positive numbers t > 0 and r > 0, a neighborhood basis at the boundary point [\gamma] is given by sets of the form : U(\gamma, t, r) = \{[\gamma_1]\in\partial X | \gamma_1(0)=o, d( \gamma_1(t),\gamma(t)) 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. ==Notes==