Kleinian Groups:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Berlin, Heidelberg
Springer Berlin Heidelberg
1988
|
| Schriftenreihe: | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
287 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | The modern theory of Kleinian groups starts with the work of Lars Ahlfors and Lipman Bers; specifically with Ahlfors' finiteness theorem, and Bers' observation that their joint work on the Beltrami equation has deep implications for the theory of Kleinian groups and their deformations. From the point of view of uniformizations of Riemann surfaces, Bers' observation has the consequence that the question of understanding the different uniformizations of a finite Riemann surface poses a purely topological problem; it is independent of the conformal structure on the surface. The last two chapters here give a topological description of the set of all (geometrically finite) uniformizations of finite Riemann surfaces. We carefully skirt Ahlfors' finiteness theorem. For groups which uniformize a finite Riemann surface; that is, groups with an invariant component, one can either start with the assumption that the group is finitely generated, and then use the finiteness theorem to conclude that the group represents only finitely many finite Riemann surfaces, or, as we do here, one can start with the assumption that, in the invariant component, the group represents a finite Riemann surface, and then, using essentially topological techniques, reach the same conclusion. More recently, Bill Thurston wrought a revolution in the field by showing that one could analyze Kleinian groups using 3-dimensional hyperbolic geome try, and there is now an active school of research using these methods |
| Beschreibung: | 1 Online-Ressource (XIII, 328 p) |
| ISBN: | 9783642615900 9783642648786 |
| ISSN: | 0072-7830 |
| DOI: | 10.1007/978-3-642-61590-0 |
Internformat
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| 500 | |a The modern theory of Kleinian groups starts with the work of Lars Ahlfors and Lipman Bers; specifically with Ahlfors' finiteness theorem, and Bers' observation that their joint work on the Beltrami equation has deep implications for the theory of Kleinian groups and their deformations. From the point of view of uniformizations of Riemann surfaces, Bers' observation has the consequence that the question of understanding the different uniformizations of a finite Riemann surface poses a purely topological problem; it is independent of the conformal structure on the surface. The last two chapters here give a topological description of the set of all (geometrically finite) uniformizations of finite Riemann surfaces. We carefully skirt Ahlfors' finiteness theorem. For groups which uniformize a finite Riemann surface; that is, groups with an invariant component, one can either start with the assumption that the group is finitely generated, and then use the finiteness theorem to conclude that the group represents only finitely many finite Riemann surfaces, or, as we do here, one can start with the assumption that, in the invariant component, the group represents a finite Riemann surface, and then, using essentially topological techniques, reach the same conclusion. More recently, Bill Thurston wrought a revolution in the field by showing that one could analyze Kleinian groups using 3-dimensional hyperbolic geome try, and there is now an active school of research using these methods | ||
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Datensatz im Suchindex
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| dewey-full | 512.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
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| dewey-search | 512.2 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-3-642-61590-0 |
| format | Electronic eBook |
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| isbn | 9783642615900 9783642648786 |
| issn | 0072-7830 |
| language | English |
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| spelling | Maskit, Bernard Verfasser aut Kleinian Groups by Bernard Maskit Berlin, Heidelberg Springer Berlin Heidelberg 1988 1 Online-Ressource (XIII, 328 p) txt rdacontent c rdamedia cr rdacarrier Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 287 0072-7830 The modern theory of Kleinian groups starts with the work of Lars Ahlfors and Lipman Bers; specifically with Ahlfors' finiteness theorem, and Bers' observation that their joint work on the Beltrami equation has deep implications for the theory of Kleinian groups and their deformations. From the point of view of uniformizations of Riemann surfaces, Bers' observation has the consequence that the question of understanding the different uniformizations of a finite Riemann surface poses a purely topological problem; it is independent of the conformal structure on the surface. The last two chapters here give a topological description of the set of all (geometrically finite) uniformizations of finite Riemann surfaces. We carefully skirt Ahlfors' finiteness theorem. For groups which uniformize a finite Riemann surface; that is, groups with an invariant component, one can either start with the assumption that the group is finitely generated, and then use the finiteness theorem to conclude that the group represents only finitely many finite Riemann surfaces, or, as we do here, one can start with the assumption that, in the invariant component, the group represents a finite Riemann surface, and then, using essentially topological techniques, reach the same conclusion. More recently, Bill Thurston wrought a revolution in the field by showing that one could analyze Kleinian groups using 3-dimensional hyperbolic geome try, and there is now an active school of research using these methods Mathematics Geometry, algebraic Group theory Algebraic topology Group Theory and Generalizations Algebraic Topology Algebraic Geometry Mathematik Kleinsche Gruppe (DE-588)4164159-0 gnd rswk-swf Riemannsche Fläche (DE-588)4049991-1 gnd rswk-swf Kleinsche Gruppe (DE-588)4164159-0 s 1\p DE-604 Riemannsche Fläche (DE-588)4049991-1 s 2\p DE-604 https://doi.org/10.1007/978-3-642-61590-0 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Maskit, Bernard Kleinian Groups Mathematics Geometry, algebraic Group theory Algebraic topology Group Theory and Generalizations Algebraic Topology Algebraic Geometry Mathematik Kleinsche Gruppe (DE-588)4164159-0 gnd Riemannsche Fläche (DE-588)4049991-1 gnd |
| subject_GND | (DE-588)4164159-0 (DE-588)4049991-1 |
| title | Kleinian Groups |
| title_auth | Kleinian Groups |
| title_exact_search | Kleinian Groups |
| title_full | Kleinian Groups by Bernard Maskit |
| title_fullStr | Kleinian Groups by Bernard Maskit |
| title_full_unstemmed | Kleinian Groups by Bernard Maskit |
| title_short | Kleinian Groups |
| title_sort | kleinian groups |
| topic | Mathematics Geometry, algebraic Group theory Algebraic topology Group Theory and Generalizations Algebraic Topology Algebraic Geometry Mathematik Kleinsche Gruppe (DE-588)4164159-0 gnd Riemannsche Fläche (DE-588)4049991-1 gnd |
| topic_facet | Mathematics Geometry, algebraic Group theory Algebraic topology Group Theory and Generalizations Algebraic Topology Algebraic Geometry Mathematik Kleinsche Gruppe Riemannsche Fläche |
| url | https://doi.org/10.1007/978-3-642-61590-0 |
| work_keys_str_mv | AT maskitbernard kleiniangroups |