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Proceedings of the American Mathematical Society

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

 

 

Conformal inequivalence of annuli and the first-order theory of subgroups of $ {\rm PSL}(2,\,{\bf R})$


Author: Lee A. Rubel
Journal: Proc. Amer. Math. Soc. 88 (1983), 679-683
MSC: Primary 30C20; Secondary 03C60, 20G20, 30C25
DOI: https://doi.org/10.1090/S0002-9939-1983-0702298-9
MathSciNet review: 702298
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Abstract: An algebraic proof is given of the classical fact that two different concentric circular annuli $ A(r)$ and $ A(s)$ are conformally inequivalent, where $ A(r) = \{ z \in {\mathbf{C}}:1 < \left\vert z \right\vert < r\} $. Indeed, it is shown that the covering groups of these annuli are not elementarily equivalent in the context of $ {\text{PSL}}(2,{\mathbf{R}})$. Considering the universal covering surface as $ U$, the upper half-plane, the covering group of a bounded plane domain is naturally contained in $ {\text{PSL}}(2,{\mathbf{R}})$ as the group of covering transformations.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0702298-9
Article copyright: © Copyright 1983 American Mathematical Society