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

 

A uniqueness theorem of reflectable deformations of a Fuchsian group


Author: Jharna Sengupta
Journal: Proc. Amer. Math. Soc. 104 (1988), 1148-1152
MSC: Primary 30F30; Secondary 20H10, 30F35, 32G15
MathSciNet review: 969051
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Abstract: Let $ \Gamma $ be a Fuchsian group of signature $ (p,n,m;{\nu _1},{\nu _2}, \ldots ,{\nu _n})$; $ 2p - 2 + m + \sum\nolimits_{j = 1}^n {(1 - 1/{\nu _j}) > 0} $. Let $ {I_1},{I_2}, \ldots ,{I_m}$ be a maximal set of inequivalent components of $ \Omega \cap {\mathbf{\hat R}}$; $ \Omega $ is the region of discontinuity and $ {\mathbf{\hat R}}$ is the extended real line. Let $ \phi $ be a quadratic differential for $ \Gamma $. Let $ f$ be a solution of the Schwarzian differential equation $ Sf = \phi $. If $ \phi $ is reflectable, $ f$ maps each $ {I_j}$ into a circle $ {C_j}$. For each $ \gamma \in \Gamma $ there is a Moebius transformation $ \mathcal{X}(\gamma )$ such that $ f \circ \gamma = \mathcal{X}(\gamma ) \circ f$. We prove that $ \phi $ is determined by the homomorphism $ \mathcal{X}$ and the circles $ {C_1},{C_2}, \ldots ,{C_m}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0969051-7
PII: S 0002-9939(1988)0969051-7
Article copyright: © Copyright 1988 American Mathematical Society