Gromov boundary
In mathematics, the Gromov boundary of a
Definition
[edit]There are several equivalent definitions of the Gromov boundary of a geodesic and proper
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 .
Two geodesics are defined to be equivalent if there is a constant such that for all . The equivalence class of is denoted .
The Gromov boundary of a geodesic and proper hyperbolic metric space is the set is a geodesic ray in .
Topology
[edit]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.
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
[edit]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 is such a group, its Gromov boundary is by definition that of any proper geodesic space space on which 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
[edit]- 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.[4]
Variations
[edit]Visual boundary of CAT(0) space
[edit]For a complete CAT(0) space X, the visual boundary of X, like the Gromov boundary of
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
Cannon's Conjecture
[edit]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.
Notes
[edit]References
[edit]- Bridson, Martin R.; Haefliger, André (1999), Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319, Springer-Verlag, ISBN 3-540-64324-9, MR 1744486
- Cannon, James W. (1994), "The combinatorial Riemann mapping theorem", Acta Mathematica, 173 (2): 155–234, doi:10.1007/bf02398434
- Champetier, C. (1995), "Propriétés statistiques des groupes de presentation finie", Advances in Mathematics, 116: 197–262, doi:10.1006/aima.1995.1067
- Coornaert, M.; Delzant, T.; Papadopoulos, A. (1990), Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov, Lecture Notes in Mathematics (in French), vol. 1441, Springer-Verlag, ISBN 3-540-52977-2
- de la Harpe, Pierre; Ghys, Etienne (1990), Sur les groupes hyperboliques d'après Mikhael Gromov (in French), Birkhäuser
- Gromov, M. (1987), "Hyperbolic groups", in S. Gersten (ed.), Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, pp. 75–263
- Kapovich, Ilya; Benakli, Nadia (2002), "Boundaries of hyperbolic groups", Combinatorial and geometric group theory, Contemporary Mathematics, vol. 296, pp. 39–93
- Roe, John (2003), Lectures on Coarse Geometry, University Lecture Series, vol. 31, American Mathematical Society, ISBN 978-0-8218-3332-2