The symplectic nature of fundamental groups of surfaces
Open accessThe claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The symplectic structure obtained in this way is closely related to the symplectic structure on the abstract space of loops on Σ first discovered by Goldman.
“It is also interesting that the symplectic structure obtained in this way is closely related to the symplectic structure on the abstract space of loops on Σ first discovered by Goldman [145, 146].”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [145]
The opening discussion describes Goldman's construction of a symplectic structure on spaces of surface-group representations. It mentions a related Lie algebra on spaces based on closed curves, but explicitly assigns that work to a forthcoming paper. This copy therefore supports a connection to loops, not the claim that this paper discovered a symplectic structure on the abstract space of loops. (LLM, conf=0.84)
Experimental — accuracy not yet measuredBy a theorem of Goldman [21], the pairing determines a symplectic structure on T*Teich(S), and this is the standard symplectic structure on a cotangent bundle rather than the symplectic structure induced from the symplectic structure of Teichmüller space.
“By a theorem of Goldman [21], the pairing determines a symplectic structure on T ∗ Teich ( S ), and this is the standard symplectic structure on a cotangent bundle (rather than the symplectic structure induced from the symplectic structure of Teichm¨ uller space).”
From Lorentz spacetimes of constant curvature (2007) · cites this paper as [21]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
On Deformations of Complex Analytic Structures, I
Kodaira, K., & Spencer, D. C. (1958) · The Annals of Mathematics · doi:10.2307/1970009 · Cited by 347 · see citing works
Flagged first, then by citations
Research works printed but not linked: 22
- L. AHLFORS, Some remarks on Teichmiiller's space of Riemann surfaces, Ann. of Marh. 74 (1961), 171-191.
- M. ATIYAH AND R. Bon, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Sot. London A 308 (1982), 523-615.
- PI K. BROWN, "Cohomology of groups," Graduate Texts in Mathematics No. 87, Springer-Verlag, New York, 1982.
- L. BERS, Uniformization, moduli and Kleinian groups, Bull. London Muth. Sot. 4 (1972), 257-300.
- El C. EARLE, Teichmiiller theory, in "Discrete Groups and Automorphism Functions" (W. Harvey, Ed.), pp. 143-162, Academic Press, New York, 1977.
- R. Fox, The free differential calculus. I. Derivations in the group ring, Ann. ofMath. 57 (1953), 547-560.
- W. GOLDMAN, "Discontinuous Groups and the Euler Class," Thesis, University of California, Berkeley, 1980.
- W. GOLDMAN, Characteristic classes and representations of discrete subgroups of Lie groups, Bull. Amer. Math. Sot. (TV. S) 6 (1982), 91-94.
- W. GOLDMAN, Representations of fundamental groups of surfaces, in "Proceedings of the Special Year in Topology, Maryland 1983-1984," Lecture Notes in Mathematics, Springer-Verlag, New York, to appear.
- R. C. GUNNING, "Lectures on Vector Bundles over Riemann Surfaces," Princeton Univ. Press, Princeton, N.J., 1967.
- D. HEJHAL, Monodromy groups and Poincare series, Bull. Amer. Math. Sot. 84 (1978), 339-376.
- J. MARSDEN AND A. WEINSTEIN, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121-130.
- J. MILNOR, Construction of universal bundles, I, Ann. of Math. 63 (1958), 272-284.
- M. S. NARASIMHAN, Geometry of moduli of vector bundles over an algebraic curve, in "Actes Int. Congress de Math., Nice 1970," Tome 2, pp. 199-201.
- M. S. NARASIMHAN AND C. S. SESHADRI, Holomorphic vector bundles on a compact Riemann surface, Math. Ann. 155 (1964), 69-80.
- M. S. NARASIMHAN AND C. S. SESHADRI, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540-567.
- M. S. RAGHUNATHAN, "Discrete Subgroups of Lie Groups," Springer-Verlag, New York, 1972.
- G. SHIMURA, Sur les integrales attachees aux formes automorphes, J. Math. Sot. Japan 11 (1959), 291-311.
- N. STEENROD, "The Topology of Fiber Bundles," Princeton Univ. Press, Princeton, N.J., 1951.
- S. WOLPERT, An elementary formula for the Fenchel-Nielsen twist, Comm. Math. Helu. 56 (1981), 132-135.
- S. WOLPERT, The Fenchel-Nielsen deformation, Ann. of Math. 115 (1982), 501-528.
- S. WOLPERT, On the symplectic geometry of deformations of a hyperbolic surface, Ann of Math. 117 (1983), 207-234.
Cited by
Sorted by citations
Lorentz spacetimes of constant curvature
Mess, G. (2007) · Geometriae Dedicata · doi:10.1007/s10711-007-9155-7 · Cited by 278 · see citing works
Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe
Carlip, S. (2005) · Living Reviews in Relativity · doi:10.12942/lrr-2005-1 · Cited by 172 · see citing works
Lorentz spacetimes of constant curvature
Mess, G. (2007) · arXiv · Cited by 5 · see citing works
- Authors
- Goldman, W. M.
- Affiliations
- Massachusetts Institute of Technology (US)
- Corresponding author
- Goldman, William M
- ORCID
- Goldman, William M
- Venue
- Advances in Mathematics
- Publisher
- Elsevier BV
- Published
- 1984-11
- Volume
- 54
- Issue
- 2
- Pages
- 200-225
- ISSN
- 0001-8708 (ISSN-L)
- OpenAlex
- W2053441708
- Cited by
- 685
- References
- 1
- Citations by year
- 2026: 17 · 2025: 35 · 2024: 31 · 2023: 21 · 2022: 29 · 2021: 41 · 2020: 26 · 2019: 18 · 2018: 38 · 2017: 21 · 2016: 30 · 2015: 29 · 2014: 17 · 2013: 21 · 2012: 21
- FWCI
- 6.36
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Advances in Mathematics (publishedVersion)
- Funders
- National Science Foundation (8202082)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Algebra and Geometry
- Keywords
- symplectic structure, fundamental group of surfaces, moduli spaces, representation varieties, character varieties, intersection pairing, Kähler structures, Teichmüller space, Jacobi variety, reductive Lie groups, inner automorphisms, Riemann surfaces
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
