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Quasi-Fuchsian group

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Record originEnglish Wikipedia
Text licenseCC BY-SA 4.0
Source revisionApr 11, 2022
Entity authorityQ7269431 ↗
Source-derived summary

In the mathematical theory of Kleinian groups, a quasi-Fuchsian group is a Kleinian group whose limit set is contained in an invariant Jordan curve. If the limit set is equal to the Jordan curve the quasi-Fuchsian group is said to be of type one, and otherwise it is said to be of type two. Some authors use "quasi-Fuchsian group" to mean "quasi-Fuchsian group of type 1", in other words the limit set is the whole Jordan curve. This terminology is incompatible with the use of the terms "type 1" and "type 2" for Kleinian groups: all quasi-Fuchsian groups are Kleinian groups of type 2 (even if they are quasi-Fuchsian groups of type 1), as their limit sets are proper subsets of the Riemann sphere. The special case when the Jordan curve is a circle or line is called a Fuchsian group, named after Lazarus Fuchs by Henri Poincaré.

Finitely generated quasi-Fuchsian groups are conjugate to Fuchsian groups under quasi-conformal transformations.

The space of quasi-Fuchsian groups of the first kind is described by the simultaneous uniformization theorem of Bers.

References

Fricke, Robert; Klein, Felix (1897), Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen. (in German), Leipzig: B. G. Teubner, ISBN 978-1-4297-0551-6, JFM 28.0334.01 {{citation}}: ISBN / Date incompatibility (help)

Fricke, Robert; Klein, Felix (1912), Vorlesungen über die Theorie der automorphen Functionen.

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This entry incorporates text from “Quasi-Fuchsian group” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.