Gromov boundary: Difference between revisions
→Definition: Added actual definition and topology |
|||
Line 7: | Line 7: | ||
==Definition== |
==Definition== |
||
There are several equivalent definitions of the Gromov boundary. One of the most common uses equivalence classes of [[geodesic#Metric geometry|geodesic]] rays. |
|||
{{Empty section|date=November 2013}} |
|||
Pick some point <math>O</math> of a hyperbolic metric space <math>X</math> to be the origin. A '''geodesic ray''' is a path given by an [[isometry]] <math>\gamma:[0,\infty)\rightarrow X</math> such that each segment <math>\gamma([0,t])</math> is a path of shortest length from <math>O</math> to <math>\gamma(t)</math>. |
|||
We say that two geodesics <math>\gamma_1,\gamma_2</math> are equivalent if there is a constant <math>K</math> such that <math>d(\gamma_1(t),\gamma_2(t))\leq K</math> for all <math>t</math>. The [[equivalence class]] of <math>\gamma</math> is denoted <math>[\gamma]</math>. |
|||
The '''Gromov boundary''' of a hyperbolic metric space <math>X</math> is the set <math>\partial X=\{[\gamma]|\gamma</math> is a geodesic ray in <math>X\}</math>. |
|||
===Topology=== |
|||
It is useful to use the '''Gromov product''' of three points. The Gromov product of three points <math>x,y,z</math> in a metric space is |
|||
<math>(x,y)_z=1/2(d(x,z)+d(y,z)-d(x,y))</math>. In a [[tree (graph theory)]], this measures how long the paths from <math>z</math> to <math>x</math> and <math>y</math> stay together before diverging. Since hyperbolic spaces are tree-like, the Gromov product measures how long geodesics from <math>z</math> to <math>x</math> and <math>y</math> stay close before diverging. |
|||
Given a point <math>p></math> in the Gromov boundary, we define the sets <math>V(p,r)=\{q\in \partial X|</math> there are geodesic rays <math>\gamma_1,\gamma_2</math> with <math>[\gamma_1]=p, [\gamma_2]=q</math> and <math>\lim \inf_{s,t\rightarrow \infty}(\gamma_1(s),\gamma_2(t))_O\geq r\}</math>. These open sets form a [[basis (topology)|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 <math>r</math> before diverging. |
|||
==Properties of the Gromov boundary== |
==Properties of the Gromov boundary== |
Revision as of 17:14, 28 November 2013
In mathematics, the Gromov boundary of a delta-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.
Definition
There are several equivalent definitions of the Gromov boundary. One of the most common uses equivalence classes of geodesic rays.
Pick some point of a hyperbolic metric space to be the origin. A geodesic ray is a path given by an isometry such that each segment is a path of shortest length from to .
We say that two geodesics are equivalent if there is a constant such that for all . The equivalence class of is denoted .
The Gromov boundary of a hyperbolic metric space is the set is a geodesic ray in .
Topology
It is useful to use the Gromov product of three points. The Gromov product of three points in a metric space is . In a tree (graph theory), this measures how long the paths from to and stay together before diverging. Since hyperbolic spaces are tree-like, the Gromov product measures how long geodesics from to and stay close before diverging.
Given a point in the Gromov boundary, we define the sets there are geodesic rays with and . 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 before diverging.
Properties of the Gromov boundary
The Gromov boundary has several important properties:
- The Gromov boundary of a hyperbolic group is compact.
- If a group acts geometrically on a delta-hyperbolic space, then is hyperbolic group and and have homeomorphic Gromov boundaries.
- The Gromov boundary of a tree is a Cantor set.
- The number of ends of a hyperbolic group is the number of components of the Gromov boundary.
Generalizations
Visual boundary of CAT(0) space
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.[1]
The analog to this conjecture is known to be true for 1-spheres and false for spheres of all dimension greater than 2.
References
- ^ James W. Cannon. The combinatorial Riemann mapping theorem. Acta Mathematica 173 (1994), no. 2, pp. 155–234.
- Kapovich, Ilya, and Nadia Benakli. "Boundaries of hyperbolic groups." Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001) 296 (2002): 39-93.