Dehn function
E265418
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Dehn function canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T2416899 — 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: Dehn function Context triple: [Max Dehn, hasEponym, Dehn function]
-
A.
Menger curvature
Menger curvature is a geometric concept that quantifies the curvature of a set or curve in metric spaces by using the reciprocal of the radius of the circle passing through three points.
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B.
Ackermann function
The Ackermann function is a classic example of a computable function that grows faster than any primitive recursive function, often used in theoretical computer science to illustrate extreme computational complexity.
-
C.
geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
-
D.
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.
-
E.
Perelman’s entropy functionals
Perelman’s entropy functionals are analytic quantities introduced by Grigori Perelman to study the behavior and singularities of the Ricci flow, playing a central role in his proof of the Poincaré and geometrization conjectures.
- 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: Dehn function Target entity description: 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.
-
A.
Menger curvature
Menger curvature is a geometric concept that quantifies the curvature of a set or curve in metric spaces by using the reciprocal of the radius of the circle passing through three points.
-
B.
Ackermann function
The Ackermann function is a classic example of a computable function that grows faster than any primitive recursive function, often used in theoretical computer science to illustrate extreme computational complexity.
-
C.
geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
-
D.
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.
-
E.
Perelman’s entropy functionals
Perelman’s entropy functionals are analytic quantities introduced by Grigori Perelman to study the behavior and singularities of the Ricci flow, playing a central role in his proof of the Poincaré and geometrization conjectures.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
invariant in geometric group theory
ⓘ
mathematical concept ⓘ |
| alsoKnownAs |
combinatorial isoperimetric function
ⓘ
isoperimetric function of a group ⓘ |
| appliesTo | Cayley 2-complex of a finitely presented group ⓘ |
| comparedBy | equivalence relation f ≃ g if each is bounded above by a linear rescaling of the other ⓘ |
| complexityClass |
at least quadratic for non-hyperbolic nilpotent groups of step ≥ 2
ⓘ
linear for hyperbolic groups ⓘ quadratic for many automatic groups ⓘ |
| connectedTo |
Gromov hyperbolic group
ⓘ
automatic group ⓘ filling radius ⓘ isodiametric function ⓘ nilpotent group ⓘ |
| dependsOn | finite presentation of a group ⓘ |
| describes |
area needed to fill a null-homotopic word
ⓘ
complexity of filling loops by discs ⓘ isoperimetric inequality for a group ⓘ |
| domain | finitely presented group ⓘ |
| exampleValue |
Heisenberg group has cubic Dehn function
ⓘ
free group has linear Dehn function ⓘ fundamental group of a closed hyperbolic surface has linear Dehn function ⓘ integer lattice Z^2 has quadratic Dehn function ⓘ |
| field |
combinatorial group theory
ⓘ
geometric group theory ⓘ |
| formalDefinition | minimal function f(n) such that every null-homotopic word of length ≤ n can be filled with at most f(n) 2-cells ⓘ |
| growthType | can realize many different asymptotic growth rates ⓘ |
| historicalOrigin | introduced in the context of Dehn’s algorithm for the word problem ⓘ |
| input | integer n representing word length ⓘ |
| invariantType | quasi-isometry invariant up to equivalence of functions ⓘ |
| mathematicalNature | asymptotic function on the natural numbers ⓘ |
| measures | minimal number of relators needed to express the identity word ⓘ |
| namedAfter | Max Dehn ⓘ |
| output | maximal area of van Kampen diagrams for null-homotopic words of length at most n ⓘ |
| property | well-defined up to equivalence of functions under change of finite presentation ⓘ |
| relatedConcept |
Dehn algorithm
ⓘ
asymptotic cone of a group ⓘ filling area ⓘ filling length ⓘ isoperimetric inequality ⓘ van Kampen diagram ⓘ word problem for groups ⓘ |
| relates | combinatorial properties of presentations to geometric properties of Cayley graphs ⓘ |
| studiedIn | geometric group theory literature ⓘ |
| usedFor |
classifying finitely presented groups up to quasi-isometry
ⓘ
measuring difficulty of the word problem in a group ⓘ studying large-scale geometry of groups ⓘ |
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: Dehn function Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.