Flag-transitive \(C_2. L_n\) geometries. (English) Zbl 0897.51005
M. Ronan gave in Proc. Lond. Math. Soc., III. Ser. 53, 385-406 (1986; Zbl 0643.51010) an example of a thick flag-transitive \(C_2.L\)-geometry which is not a truncation of a polar space. He also determined sufficient conditions for a \(C_2.L\)-geometry to be a truncation of a polar space. These conditions were weakened by B. Baumeister and A. Pasini [Geom. Dedicata 71, No. 1, 33-59 (1998)].
In this paper the authors show that every locally finite, thick flag-transitive \(C_n.L\)-geometry with \(n \geq 3\) is a truncation of a polar space. Moreover, they show that there are no flag-transitive thick \(C_2.Af.A_{n-2}.L\)-geometries with classical generalized quadrangles as lower residues of elements of type \(2\), except possibly when \(q = 3\) or \(4\). They mention some known examples whose lower residues are non-classical generalized quadrangles.
In this paper the authors show that every locally finite, thick flag-transitive \(C_n.L\)-geometry with \(n \geq 3\) is a truncation of a polar space. Moreover, they show that there are no flag-transitive thick \(C_2.Af.A_{n-2}.L\)-geometries with classical generalized quadrangles as lower residues of elements of type \(2\), except possibly when \(q = 3\) or \(4\). They mention some known examples whose lower residues are non-classical generalized quadrangles.
Reviewer: B.Baumeister (Halle)
MSC:
| 51E24 | Buildings and the geometry of diagrams |
| 51A50 | Polar geometry, symplectic spaces, orthogonal spaces |
| 05E20 | Group actions on designs, etc. (MSC2000) |
Citations:
Zbl 0643.51010References:
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