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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Convex real projective structures on closed surfaces are closed


Authors: Suhyoung Choi and William M. Goldman
Journal: Proc. Amer. Math. Soc. 118 (1993), 657-661
MSC: Primary 57M50; Secondary 14P05, 53C15, 58D27
MathSciNet review: 1145415
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Abstract: The deformation space $ \mathfrak{C}(\Sigma )$ of convex $ \mathbb{R}{{\mathbf{P}}^2}$-structures on a closed surface $ \Sigma $ with $ \chi (\Sigma ) < 0$ is closed in the space $ \operatorname{Hom} (\pi ,\operatorname{SL} (3,\mathbb{R}))/\operatorname{SL} (3,\mathbb{R})$ of equivalence classes of representations $ {\pi _1}(\Sigma ) \to \operatorname{SL} (3,\mathbb{R})$. Using this fact, we prove Hitchin's conjecture that the contractible "Teichmüller component" (Lie groups and Teichmüller space, preprint) of $ \operatorname{Hom} (\pi ,\operatorname{SL} (3,\mathbb{R}))/\operatorname{SL} (3,\mathbb{R})$ precisely equals $ \mathfrak{C}(\Sigma )$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1145415-8
PII: S 0002-9939(1993)1145415-8
Article copyright: © Copyright 1993 American Mathematical Society