Gromov hyperbolic group
E911231
A Gromov hyperbolic group is a finitely generated group whose Cayley graph exhibits negative curvature–like properties, leading to rich geometric, dynamical, and algorithmic behavior.
All labels observed (4)
| Label | Occurrences |
|---|---|
| CAT(-1) group | 1 |
| Gromov hyperbolic group canonical | 1 |
| Gromov hyperbolic groups | 1 |
| Gromov's paper "Hyperbolic groups" in Essays in Group Theory (1987) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11215229 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Gromov hyperbolic group Context triple: [Dehn function, connectedTo, Gromov hyperbolic group]
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A.
geometric group theory
Geometric group theory is a branch of mathematics that studies groups by interpreting them as geometric objects and analyzing their actions on spaces using tools from geometry and topology.
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B.
Hyperbolic Manifolds and Discrete Groups
"Hyperbolic Manifolds and Discrete Groups" is a foundational mathematical monograph that develops the theory of hyperbolic geometry and its deep connections with discrete group actions and low-dimensional topology.
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C.
Kleinian group
A Kleinian group is a discrete subgroup of Möbius transformations acting on hyperbolic 3-space, central to the study of Riemann surfaces, complex dynamics, and low-dimensional topology.
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D.
Dehn function
The Dehn function is a mathematical tool in geometric group theory that measures the complexity of filling loops with discs in a space or group, quantifying the difficulty of solving the word problem.
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E.
Culler–Vogtmann Outer space
Culler–Vogtmann Outer space is a topological space that parametrizes marked metric graphs, serving as an analogue of Teichmüller space for studying the outer automorphism group of a free group.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Gromov hyperbolic group Target entity description: A Gromov hyperbolic group is a finitely generated group whose Cayley graph exhibits negative curvature–like properties, leading to rich geometric, dynamical, and algorithmic behavior.
-
A.
geometric group theory
Geometric group theory is a branch of mathematics that studies groups by interpreting them as geometric objects and analyzing their actions on spaces using tools from geometry and topology.
-
B.
Hyperbolic Manifolds and Discrete Groups
"Hyperbolic Manifolds and Discrete Groups" is a foundational mathematical monograph that develops the theory of hyperbolic geometry and its deep connections with discrete group actions and low-dimensional topology.
-
C.
Kleinian group
A Kleinian group is a discrete subgroup of Möbius transformations acting on hyperbolic 3-space, central to the study of Riemann surfaces, complex dynamics, and low-dimensional topology.
-
D.
Dehn function
The Dehn function is a mathematical tool in geometric group theory that measures the complexity of filling loops with discs in a space or group, quantifying the difficulty of solving the word problem.
-
E.
Culler–Vogtmann Outer space
Culler–Vogtmann Outer space is a topological space that parametrizes marked metric graphs, serving as an analogue of Teichmüller space for studying the outer automorphism group of a free group.
- F. None of above. chosen
Statements (84)
| Predicate | Object |
|---|---|
| instanceOf |
geometric group theory concept
ⓘ
group theory concept ⓘ mathematical concept ⓘ |
| appearsInWork |
Gromov's paper "Hyperbolic groups" in Essays in Group Theory (1987)
ⓘ
linked to:
Gromov hyperbolic group
|
| characterizedBy |
Cayley graph is Gromov hyperbolic for one (equivalently any) finite generating set
ⓘ
existence of δ such that all geodesic triangles in Cayley graph are δ-thin ⓘ |
| field |
dynamical systems
ⓘ
geometric group theory ⓘ geometry ⓘ group theory ⓘ theoretical computer science ⓘ |
| generalizationOf | fundamental groups of closed negatively curved Riemannian manifolds ⓘ |
| hasDefinition |
a finitely generated group whose Cayley graph is Gromov hyperbolic
ⓘ
a finitely generated group whose Cayley graph is a δ-hyperbolic metric space for some δ ≥ 0 ⓘ |
| hasExample |
cocompact lattice in a rank-one simple Lie group
ⓘ
free group of rank at least 2 ⓘ fundamental group of a closed hyperbolic n-manifold ⓘ fundamental group of a closed hyperbolic surface ⓘ fundamental groups of many graphs of groups with hyperbolic vertex groups and quasi-convex edge groups ⓘ small cancellation groups satisfying C′(1/6) ⓘ word-hyperbolic Coxeter groups ⓘ |
| hasProperty |
Bowditch boundary coincides with Gromov boundary
ⓘ
Cannon–Thurston maps exist in many natural extensions ⓘ Cayley graph has thin triangles ⓘ Cayley graph is a geodesic metric space ⓘ Cayley graph is roughly isometric to a proper hyperbolic metric space ⓘ Cayley graph is δ-hyperbolic ⓘ Cayley graph satisfies Rips condition ⓘ Cayley graph satisfies slim triangle condition ⓘ Morse geodesics ⓘ Poisson boundary often identifies with Gromov boundary ⓘ
linked to:
Gromov boundary
acts as a convergence group on its boundary ⓘ automatic group ⓘ biautomatic group ⓘ boundary at infinity is compact ⓘ boundary at infinity is locally connected in many cases ⓘ boundary at infinity is metrizable ⓘ boundary at infinity is perfect if group is non-elementary ⓘ boundary dynamics are topologically mixing for many actions ⓘ cohomological dimension equals topological dimension of boundary plus one (under mild hypotheses) ⓘ every abelian subgroup is virtually cyclic ⓘ every amenable hyperbolic group is virtually cyclic ⓘ exponential growth (if infinite and non-virtually cyclic) ⓘ finite asymptotic dimension ⓘ finite intersection of quasi-convex subgroups is quasi-convex ⓘ finite number of conjugacy classes of finite subgroups ⓘ finite presentation ⓘ finitely generated ⓘ growth series is rational for many hyperbolic groups ⓘ is relatively hyperbolic only to finite subgroups ⓘ linear Dehn function ⓘ linear isoperimetric inequality ⓘ negatively curved in the large-scale sense ⓘ no subgroups isomorphic to ℤ² unless virtually cyclic ⓘ non-elementary hyperbolic groups are not amenable ⓘ non-elementary hyperbolic groups have exponential growth of conjugacy classes ⓘ non-elementary hyperbolic groups have uncountably many pairwise non-conjugate subgroups ⓘ quasi-convex subgroups are finitely generated ⓘ quasi-convex subgroups are hyperbolic ⓘ quasi-isometry invariant ⓘ random walks have positive drift ⓘ satisfies Tits alternative ⓘ satisfies linear isodiametric inequality ⓘ satisfies linear-time solution to word problem with respect to automatic structure ⓘ satisfies property of stability of quasi-geodesics ⓘ solvable conjugacy problem ⓘ solvable word problem ⓘ strongly geodesically automatic ⓘ virtually torsion-free ⓘ visual metrics can be defined on the boundary ⓘ word-hyperbolic ⓘ |
| introducedBy | Mikhail Gromov ⓘ |
| introducedIn | 1980s ⓘ |
| namedAfter | Mikhail Gromov ⓘ |
| relatedConcept |
Bowditch boundary
ⓘ
CAT(-1) group ⓘ
linked to:
Gromov hyperbolic group
Cannon conjecture ⓘ Gromov boundary ⓘ Gromov hyperbolic space ⓘ automatic group ⓘ convergence group ⓘ quasi-convex subgroup ⓘ relatively hyperbolic group ⓘ small cancellation theory ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Gromov hyperbolic group Description of subject: A Gromov hyperbolic group is a finitely generated group whose Cayley graph exhibits negative curvature–like properties, leading to rich geometric, dynamical, and algorithmic behavior.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Gromov hyperbolic group
Gromov hyperbolic group
→
appearsInWork
→
Gromov's paper "Hyperbolic groups" in Essays in Group Theory (1987)
ⓘ
linked to: Gromov hyperbolic group
linked to: Gromov hyperbolic group