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. 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. 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 {displaystyle O} of a hyperbolic metric space X {displaystyle X} to be the origin. A geodesic ray is a path given by an isometry γ : [ 0 , ∞ ) → X {displaystyle gamma :[0,infty ) ightarrow X} such that each segment γ ( [ 0 , t ] ) {displaystyle gamma ()} is a path of shortest length from O {displaystyle O} to γ ( t ) {displaystyle gamma (t)} . Two geodesics γ 1 , γ 2 {displaystyle gamma _{1},gamma _{2}} are defined to be equivalent if there is a constant K {displaystyle K} such that d ( γ 1 ( t ) , γ 2 ( t ) ) ≤ K {displaystyle d(gamma _{1}(t),gamma _{2}(t))leq K} for all t {displaystyle t} . The equivalence class of γ {displaystyle gamma } is denoted [ γ ] {displaystyle } . The Gromov boundary of a geodesic and proper hyperbolic metric space X {displaystyle X} is the set ∂ X = { [ γ ] | γ {displaystyle partial X={|gamma } is a geodesic ray in X } {displaystyle X}} . It is useful to use the Gromov product of three points. The Gromov product of three points x , y , z {displaystyle x,y,z} in a metric space is ( x , y ) z = 1 / 2 ( d ( x , z ) + d ( y , z ) − d ( x , y ) ) {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 z {displaystyle z} to x {displaystyle x} and y {displaystyle y} stay together before diverging. Since hyperbolic spaces are tree-like, the Gromov product measures how long geodesics from z {displaystyle z} to x {displaystyle x} and y {displaystyle y} stay close before diverging. Given a point p {displaystyle p} in the Gromov boundary, we define the sets V ( p , r ) = { q ∈ ∂ X | {displaystyle V(p,r)={qin partial X|} there are geodesic rays γ 1 , γ 2 {displaystyle gamma _{1},gamma _{2}} with [ γ 1 ] = p , [ γ 2 ] = q {displaystyle =p,=q} and lim inf s , t → ∞ ( γ 1 ( s ) , γ 2 ( t ) ) O ≥ r } {displaystyle lim inf _{s,t ightarrow 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 {displaystyle r} before diverging. This topology makes the Gromov boundary into a compact metrizable space.