4-dimensional elation Laguerre planes with 9-dimensional automorphism groups.
| dc.contributor.author | Findlay, Rosemary | |
| dc.date.accessioned | 2024-11-07T22:45:26Z | |
| dc.date.available | 2024-11-07T22:45:26Z | |
| dc.date.issued | 2024 | |
| dc.description.abstract | Topological Laguerre planes are circle geometries with one parallelism relation and a topology on the point set and circle set such that the geometric operations of joining, touching, parallel projection and intersection are continuous. If the point set is finite dimensional, locally compact and connected then it is either of dimension 2 or 4 and the automorphism group of the Laguerre plane is a Lie group. The 4-dimensional elation Laguerre planes with automorphism groups that have dimension at least 10 are all known and have seen signi cant investigation since 1992. The aim of this research is to look at the 4-dimensional elation Laguerre planes with 9-dimensional automorphism groups. We prove that all 4-dimensional elation Laguerre planes with 9- dimensional automorphism groups have a parallel class which is fixed by the automorphism group. With this result we outline a method of classifi cation by drawing on the correspondence between elation Laguerre planes and dual translation planes. This reduces the classi fication to checking if the already classi fied 4-dimensional dual translation planes with group dimension greater than 7 can appear as derived planes at points of the fixed parallel class. We proceed using this method to prove that there is no 4-dimensional elation Laguerre plane with 9-dimensional automorphism group that has, as a derived plane at points of it's fixed parallel class, a dual-translation plane that has an automorphism group of group dimension 8 or more. We additionally construct a first example of a 4-dimensional elation Laguerre plane with 9-dimensional automorphism group thus proving that such planes do exist. | |
| dc.identifier.uri | https://hdl.handle.net/10092/107730 | |
| dc.language | English | |
| dc.language.iso | en | |
| dc.rights | All Right Reserved | |
| dc.rights.uri | https://canterbury.libguides.com/rights/theses | |
| dc.title | 4-dimensional elation Laguerre planes with 9-dimensional automorphism groups. | |
| dc.type | Theses / Dissertations | |
| thesis.degree.discipline | Mathematics | |
| thesis.degree.grantor | University of Canterbury | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy | |
| uc.college | Faculty of Engineering |