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Anosov Flows, Surface Groups and Curves in Projective Space

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Francois Labourie

Differential Geometry · Submitted 19 Jan 2004

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In the preprint "Anosov flows, Surface groups and Curves in Projective Space", we extend this interpretation to higher dimension and show every representation in Hitchin's component is attached to a (special) curve in projective space, thus giving a geometric interpretation of these representations.

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arXiv IDmath/0401230 (version 2)
Pith IDCXKAVYNK, Pith's permanent record of this paper and its review
CitationFrancois Labourie. "Anosov Flows, Surface Groups and Curves in Projective Space." arXiv:math/0401230 (2004). https://pith.science/paper/CXKAVYNK
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,706 input tokens.

What the abstract itself states, as probabilities: claims first 0.11 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.87 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

In 1990, Hitchin's proved a component of the space of representations of a surface group in SL(n,R) is homeomorphic to a ball. For n=2,3 this component has been identified with the holonomies of geometric structures (hyperbolic for n=2, or real projective for n=3). In the preprint "Anosov flows, Surface groups and Curves in Projective Space", we extend this interpretation to higher dimension and show every representation in Hitchin's component is attached to a (special) curve in projective space, thus giving a geometric interpretation of these representations. We also prove these representations are faithful, discrete and purely loxodromic (or hyperbolic)

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In 1990, Hitchin's proved a component of the space of representations of a surface group in SL(n,R) is homeomorphic to a ball. background 1.00 0.03 0.03 asserted 1.00
s1 For n=2,3 this component has been identified with the holonomies of geometric structures (hyperbolic for n=2, or real projective for n=3). background 1.00 0.06 0.03 asserted 1.00
s2 In the preprint "Anosov flows, Surface groups and Curves in Projective Space", we extend this interpretation to higher dimension and show every representation in Hitchin's component is attached to a (special) curve in projective space, thus giving a geometric interpretation of these representations. contribution 0.88 0.96 0.88 asserted 1.00
s3 We also prove these representations are faithful, discrete and purely loxodromic (or hyperbolic) result 0.99 0.94 0.20 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

In 1990, Hitchin's proved a component of the space of representations of a surface group in SL(n,R) is homeomorphic to a ball. For n=2,3 this component has been identified with the holonomies of geometric structures (hyperbolic for n=2, or real projective for n=3). In the preprint "Anosov flows, Surface groups and Curves in Projective Space", we extend this interpretation to higher dimension and show every representation in Hitchin's component is attached to a (special) curve in projective space, thus giving a geometric interpretation of these representations. We also prove these representations are faithful, discrete and purely loxodromic (or hyperbolic)

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References
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2 arXiv papers cite this in 6 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 3 sentences without a confident label

Compares with it 1 paper
  • It would be interesting to compare the rigid geometric structures constructed by Goldman [ref] , Guichard–Wienhard [ref] and [ref] to the flexible higher complex structure.

    Higher Complex Structures and Higher Teichm\"uller Theory · Geometric approaches to higher Teichmüller theory · 0.81

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