Compact Tits quadrangles as Lie geometries of topological Laguerre spaces. (English) Zbl 1135.51012
The authors develop a theory of topological Laguerre and Lie geometries of higher rank and show that the Lie geometry associated with a locally compact connected Laguerre space is a compact connected generalized quadrangle. If the geometric dimension of the Laguerre space is at least three, the corresponding generalized quadrangle is of Tits type. Conversely, every compact connected generalized quadrangle of Tits type arises from a Laguerre space. Geometrically, this is not surprising since every Laguerre space of rank at least three is ovoidal. The main part of the paper consists in showing that the corresponding topologies are well behaved.
The rank two case has already been handled by A. E. Schroth [“Topological circle planes and topological quadrangles”, Harlow Essex: Longman (1995; Zbl 0839.51013)].
The rank two case has already been handled by A. E. Schroth [“Topological circle planes and topological quadrangles”, Harlow Essex: Longman (1995; Zbl 0839.51013)].
Reviewer: Norbert Knarr (Gießen)
MSC:
| 51H15 | Topological nonlinear incidence structures |
| 51H10 | Topological linear incidence structures |
| 51E12 | Generalized quadrangles and generalized polygons in finite geometry |
| 51B15 | Laguerre geometries |
| 51B25 | Lie geometries in nonlinear incidence geometry |
Keywords:
topological Laguerre geometry; topological Lie geometry; compact generalized quadrangle; Tits quadrangleCitations:
Zbl 0839.51013References:
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