Projections in Moduli Spaces of the Kleinian Groups
A two-generator Kleinian group f,g can be naturally associated with a discrete group f,ϕ with the generator ϕ of order two and where f,ϕfϕ−1=f,gfg−1⊂f,g,f,ϕ: f,gfg−1=2. This is useful in studying the geometry of the Kleinian groups since f,g will be discrete only if f,ϕ is, and the moduli space of g...
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Wiley
2022-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2022/6311193 |
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| author | Hala Alaqad Jianhua Gong Gaven Martin |
| author_facet | Hala Alaqad Jianhua Gong Gaven Martin |
| author_sort | Hala Alaqad |
| collection | DOAJ |
| description | A two-generator Kleinian group f,g can be naturally associated with a discrete group f,ϕ with the generator ϕ of order two and where f,ϕfϕ−1=f,gfg−1⊂f,g,f,ϕ: f,gfg−1=2. This is useful in studying the geometry of the Kleinian groups since f,g will be discrete only if f,ϕ is, and the moduli space of groups f,ϕ is one complex dimension less. This gives a necessary condition in a simpler space to determine the discreteness of f,g. The dimension reduction here is realised by a projection of principal characters of the two-generator Kleinian groups. In applications, it is important to know that the image of the moduli space of Kleinian groups under this projection is closed and, among other results, we show how this follows from Jørgensen’s results on algebraic convergence. |
| format | Article |
| id | doaj-art-2af0390396364f49802f72687a420742 |
| institution | Kabale University |
| issn | 1687-0409 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-2af0390396364f49802f72687a4207422025-02-03T05:53:49ZengWileyAbstract and Applied Analysis1687-04092022-01-01202210.1155/2022/6311193Projections in Moduli Spaces of the Kleinian GroupsHala Alaqad0Jianhua Gong1Gaven Martin2Department of Mathematical SciencesDepartment of Mathematical SciencesSchool of Natural and Computational SciencesA two-generator Kleinian group f,g can be naturally associated with a discrete group f,ϕ with the generator ϕ of order two and where f,ϕfϕ−1=f,gfg−1⊂f,g,f,ϕ: f,gfg−1=2. This is useful in studying the geometry of the Kleinian groups since f,g will be discrete only if f,ϕ is, and the moduli space of groups f,ϕ is one complex dimension less. This gives a necessary condition in a simpler space to determine the discreteness of f,g. The dimension reduction here is realised by a projection of principal characters of the two-generator Kleinian groups. In applications, it is important to know that the image of the moduli space of Kleinian groups under this projection is closed and, among other results, we show how this follows from Jørgensen’s results on algebraic convergence.http://dx.doi.org/10.1155/2022/6311193 |
| spellingShingle | Hala Alaqad Jianhua Gong Gaven Martin Projections in Moduli Spaces of the Kleinian Groups Abstract and Applied Analysis |
| title | Projections in Moduli Spaces of the Kleinian Groups |
| title_full | Projections in Moduli Spaces of the Kleinian Groups |
| title_fullStr | Projections in Moduli Spaces of the Kleinian Groups |
| title_full_unstemmed | Projections in Moduli Spaces of the Kleinian Groups |
| title_short | Projections in Moduli Spaces of the Kleinian Groups |
| title_sort | projections in moduli spaces of the kleinian groups |
| url | http://dx.doi.org/10.1155/2022/6311193 |
| work_keys_str_mv | AT halaalaqad projectionsinmodulispacesofthekleiniangroups AT jianhuagong projectionsinmodulispacesofthekleiniangroups AT gavenmartin projectionsinmodulispacesofthekleiniangroups |