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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A generalization of a theorem of Poincaré

Author: Irwin Kra
Journal: Proc. Amer. Math. Soc. 27 (1971), 299-302
MSC: Primary 30A58; Secondary 20H10
MathSciNet review: 0301189
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Abstract: Let $ G$ be a finitely generated Fuchsian group of the first kind. Let $ \varphi $ be a cusp form, and $ f$ a solution to $ {\theta _2}f = \varphi $, where $ {\theta _2}$ is the Schwarzian derivative. Then for every $ A \in G$ there is a Möbius transformation $ \chi (A)$ such that $ f \circ A = \chi (A) \circ f$. We show that the homomorphism $ \chi $ from $ G$ to Möbius transformations determines $ \varphi $. The theorem for the special case where $ G$ is the covering group of a compact surface was first proved by Poincaré.

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Keywords: Fuchsian group, deformation, cusp form
Article copyright: © Copyright 1971 American Mathematical Society

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