Relations between the half turns of the hyperbolic plane
| Autor: | Dragomir Ž. Djoković |
|---|---|
| Rok vydání: | 1982 |
| Předmět: |
Numerical Analysis
Algebra and Number Theory Group (mathematics) Hyperbolic geometry Covering group Mathematical analysis 0211 other engineering and technologies Lie group 021107 urban & regional planning 010103 numerical & computational mathematics 02 engineering and technology 16. Peace & justice PSL 01 natural sciences Combinatorics Turn (geometry) Generating set of a group Discrete Mathematics and Combinatorics Identity component Geometry and Topology 0101 mathematics Mathematics |
| Zdroj: | Linear Algebra and its Applications. 47:57-88 |
| ISSN: | 0024-3795 |
| DOI: | 10.1016/0024-3795(82)90226-9 |
| Popis: | Let R a denote the half turn about the point a of the hyperbolic plane H . If the points a , b , c , d lie on the same line and the pair ( c , d ) is obtained from the pair ( a , b ) by a translation, then we have R a R b = R c R d . We study the group G whose generating set is { R a : a ∈ H } and whose defining relations are the ones mentioned above together with the relations R 2 a = 1. We show that G can be made into a Lie group, G has two connected components, and its identity component G 0 is the universal covering group of PSL 2 ( R ). In particular, it follows that all relations between the half turns in PSL 2 ( R ) follow from the abovementioned relations and a single additional relation of length five. |
| Databáze: | OpenAIRE |
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