Triple

T3072662
Position Surface form Disambiguated ID Type / Status
Subject Annals of Mathematics Studies E64059 entity
Predicate hasNotableWork P4 FINISHED
Object 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.
E325284 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Hyperbolic Manifolds and Discrete Groups | Statement: [Annals of Mathematics Studies, hasNotableWork, Hyperbolic Manifolds and Discrete Groups]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Hyperbolic Manifolds and Discrete Groups
Context triple: [Annals of Mathematics Studies, hasNotableWork, Hyperbolic Manifolds and Discrete Groups]
  • A. Milnor–Wood inequality
    The Milnor–Wood inequality is a result in differential geometry and topology that bounds the Euler class of flat circle bundles over surfaces, with important implications for foliations and group actions on the circle.
  • B. 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.
  • C. Dehn’s decision problems in group theory
    Dehn’s decision problems in group theory are foundational early 20th-century problems that introduced algorithmic questions about the solvability of word, conjugacy, and isomorphism problems in finitely presented groups, helping launch the field of algorithmic group theory.
  • D. Dehn complex
    The Dehn complex is a topological construction introduced by Max Dehn in the study of group presentations and decision problems, encoding relations of a group as a 2-dimensional cell complex.
  • E. Teichmüller theory
    Teichmüller theory is a branch of complex analysis and geometry that studies the deformation spaces of Riemann surfaces and their moduli, often via quasiconformal mappings.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Hyperbolic Manifolds and Discrete Groups
Triple: [Annals of Mathematics Studies, hasNotableWork, Hyperbolic Manifolds and Discrete Groups]
Generated description
"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.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Hyperbolic Manifolds and Discrete Groups
Target entity description: "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.
  • A. Milnor–Wood inequality
    The Milnor–Wood inequality is a result in differential geometry and topology that bounds the Euler class of flat circle bundles over surfaces, with important implications for foliations and group actions on the circle.
  • B. 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.
  • C. Dehn’s decision problems in group theory
    Dehn’s decision problems in group theory are foundational early 20th-century problems that introduced algorithmic questions about the solvability of word, conjugacy, and isomorphism problems in finitely presented groups, helping launch the field of algorithmic group theory.
  • D. Dehn complex
    The Dehn complex is a topological construction introduced by Max Dehn in the study of group presentations and decision problems, encoding relations of a group as a 2-dimensional cell complex.
  • E. Teichmüller theory
    Teichmüller theory is a branch of complex analysis and geometry that studies the deformation spaces of Riemann surfaces and their moduli, often via quasiconformal mappings.
  • F. None of above. chosen

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69ad857a8aec8190bfdfd9c14554ac5a completed March 8, 2026, 2:19 p.m.
NER Named-entity recognition batch_69ada14e372c81908c25c7f3e7e0c864 completed March 8, 2026, 4:18 p.m.
NED1 Entity disambiguation (via context triple) batch_69b1f8828c488190877902a6c2dfcb5e completed March 11, 2026, 11:19 p.m.
NEDg Description generation batch_69b1f930321081908f98aa5f9b8ba611 completed March 11, 2026, 11:22 p.m.
NED2 Entity disambiguation (via description) batch_69b1f9ad0b148190b98f93699598dd1e completed March 11, 2026, 11:24 p.m.
Created at: March 8, 2026, 3:02 p.m.