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Prescribed holonomy for projective structures on compact surfaces
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Every homomorphism of the fundamental group of a closed surface of genus g to SL2C with image not conjugate into SU(2) nor into the stabilizer of a point or a pair of points in CP1 is the holonomy of a projective structure.
“It is known that every homomorphism of the fundamental group of a closed surface of genus g to SL 2 C with image which is not conjugate into SU (2) nor into the stabilizer of a point or a pair of points in CP 1 is the holonomy of a projective structure [42]”
From Lorentz spacetimes of constant curvature (2007) · cites this paper as [42]
The Introduction states Theorem 1 for a compact oriented surface of genus g > 2: a homomorphism to PSL(2,C) that lifts to SL(2,C) and has nonelementary image is the holonomy of a projective structure. The cited claim closely paraphrases this after projectivizing the SL(2,C) representation, but says only “genus g” and omits the paper’s g > 2 restriction. (LLM, conf=0.88)
Experimental — accuracy not yet measuredGallo shows that given a homomorphism from the fundamental group of a surface to SL2C with image not conjugate into SU(2) nor into the stabilizer of a point or pair of points in CP1, there is a decomposition of the surface into pairs of pants such that the holonomy of any pair of pants is quasifuchsian.
“Gallo [42] actually shows that given a homomorphism from the fundamental group of a surface to SL 2 C with image not conjugate into SU (2) nor into the stabilizer of a point or pair of points in CP 1 , there is a decomposition of the surface into pairs of pants, such that the holonomy of any pair of pants is quasifuchsian.”
From Lorentz spacetimes of constant curvature (2007) · cites this paper as [42]
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Research works printed but not linked: 7
- Gallo, D. M., 1990, Extended monodromy for bordered surfaces, J. Analyse Math., 54, 1, 10.1007/BF02796140 · DOI
- 2. W. Goldman, Discontinuous groups and the Euler class, Doctoral Thesis, Univ. of California, Berkeley, 1980.
- Goldman, William M., 1987, Projective structures with Fuchsian holonomy, J. Differential Geom., 25, 297
- Gunning, R. C., 1967, Lectures on vector bundles over Riemann surfaces
- Hejhal, Dennis A., 1975, Monodromy groups and linearly polymorphic functions, Acta Math., 135, 1, 10.1007/BF02392015 · DOI · Paper page
- Steenrod, Norman, 1951, The Topology of Fibre Bundles, 10.1515/9781400883875 · DOI
- Thurston, William P., 1982, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.), 6, 357, 10.1090/S0273-0979-1982-15003-0 · DOI
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
Lorentz spacetimes of constant curvature
Mess, G. (2007) · arXiv · Cited by 5
Metadata
- Authors
- Gallo, D. M.
- Corresponding author
- Gallo, Daniel M.
- Venue
- Bulletin of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Published
- 1989
- Volume
- 20
- Issue
- 1
- Pages
- 31-34
- ISSN
- 0002-9904 (ISSN-L)
- 1088-9485
- 0273-0979
- OpenAlex
- W2011297942
- Cited by
- 8
- References
- 0
- Citations by year
- 2022: 1
- FWCI
- 0.00
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Locations
- Bulletin of the American Mathematical Society (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Geometry Research
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Keywords
- projective structures, holonomy, compact surfaces, holonomy representations, holonomy groups
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