|
|
| Line 1: |
Line 1: |
| [[File:F2 Cayley Graph.png|thumb|The [[Cayley graph]] of a [[free group]] with two generators. This is a [[hyperbolic group]] whose Gromov boundary is a [[Cantor set]]. Hyperbolic groups and their boundaries are important topics in [[geometric group theory]], as are Cayley graphs.]]
| | I am Edwina from Stoney Creek studying Educational Studies. I did my schooling, secured 94% and hope to find someone with same interests in Kayaking.<br><br>my web page ... [http://www.carrom.com/?option=com_k2&view=itemlist&task=user&id=84985 http://www.hostgator1centcoupon.info/] |
| | |
| [[File:Hyperbolic domains 642.png|thumb|150px|The (6,4,2) triangular hyperbolic tiling. The [[triangle group]] corresponding to this tiling has a circle as its Gromov boundary.]]
| |
| | |
| 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 [[Point at infinity|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#Metric geometry|geodesic]] rays. <ref>Kapovich, Ilya, and Nadia Benakli. "Boundaries of hyperbolic groups." Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001) 296 (2002): 39-93.</ref>
| |
| | |
| 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.
| |
| | |
| This topology makes the Gromov boundary into a [[compact space|compact]] [[metrization theorem|metrizable]] space.
| |
| | |
| The number of [[end (topology)|ends]] of a hyperbolic group is the number of [[connected set|components]] of the Gromov boundary.
| |
| | |
| ==Properties of the Gromov boundary==
| |
| The Gromov boundary has several important properties. One of the most frequently used properties in group theory is the following: if a group <math>G</math> [[geometric group action|acts geometrically]] on a [[delta-hyperbolic space]], then <math>G</math> is [[hyperbolic group]] and <math>G</math> and <math>X</math> have homeomorphic Gromov boundaries.<ref>Gromov, M. Hyperbolic groups. ''Essays in group theory'', 75–263, Math. Sci. Res. Inst. Publ., 8, ''Springer'', ''New York'', 1987</ref>
| |
| | |
| One of the most important properties is that it is a [[quasi-isometry]] invariant; that is, if two hyperbolic metric spaces are quasi-isometric, then the quasi-isometry between them gives a [[homeomorphism]] between their boundaries [[homeomorphic]].<ref> Michel Coornaert, Thomas Delzant and Athanase Papadopoulos, "Géométrie et théorie des groupes : les groupes hyperboliques de Gromov", Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990, x+165 pp. MR 92f:57003, ISBN 3-540-52977-2</ref><ref>* É. Ghys and P. de la Harpe (editors), ''Sur les groupes hyperboliques d'après Mikhael Gromov.'' Progress in Mathematics, 83. Birkhäuser Boston, Inc., Boston, MA, 1990. xii+285 pp. ISBN 0-8176-3508-4</ref> This is important because homeomorphisms of compact spaces are much easier to understand than quasi-isometries of spaces.
| |
| | |
| ==Examples==
| |
| *The Gromov boundary of a [[tree (graph theory)|tree]] is a [[Cantor set]].
| |
| *The Gromov boundary of [[hyperbolic space|hyperbolic n-space]] is an ''n''-dimensional [[sphere]].
| |
| *The Gromov boundary of ''most'' hyperbolic groups is a [[Menger sponge]] <ref>C. Champetier, ''Proprietes statistiques des groupes de presentation finie'', Adv. Math. 116 (1995), no. 2, 197--262. MR 96m:2005</ref>
| |
| | |
| ==Generalizations== | |
| | |
| ===Visual boundary of CAT(0) space===
| |
| {{Expand section|date=November 2013}}
| |
| | |
| ==Cannon's Conjecture==
| |
| {{Main|Cannon's conjecture}}
| |
| Cannon's conjecture concerns the classification of groups with a 2-sphere at infinity:
| |
| | |
| '''Cannon's conjecture''': Every [[Mikhail Gromov (mathematician)|Gromov]] [[hyperbolic group]] with a 2-sphere at infinity [[Geometric group action|acts geometrically]] on [[hyperbolic space|hyperbolic 3-space]].<ref name="RM">James W. Cannon. [http://www.springerlink.com/content/9w0608p039151254/ ''The combinatorial Riemann mapping theorem''.] [[Acta Mathematica]] 173 (1994), no. 2, pp. 155–234.</ref>
| |
| | |
| The analog to this conjecture is known to be true for 1-spheres and false for spheres of all dimension greater than 2.
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| | |
| [[Category:Geometric group theory]]
| |
| [[Category:Properties of groups]]
| |
I am Edwina from Stoney Creek studying Educational Studies. I did my schooling, secured 94% and hope to find someone with same interests in Kayaking.
my web page ... http://www.hostgator1centcoupon.info/