The valuations of the near octagon G4

Abstract

In De Bruyn (2003) [4] it was shown that the dual polar space D H (2 n - 1, 4), n ≥ 2, has a sub-near 2 n-gon Gn with a large automorphism group. In this paper, we classify the valuations of the near octagon G4. We show that each such valuation is either classical, the extension of a non-classical valuation of a G3-hex or is associated with a valuation of Fano-type of an H3-hex. In order to describe the latter type of valuation we must study the structure of G4 with respect to an H3-hex. This study also allows us to construct new hyperplanes of G4. We also show that each valuation of G4 is induced by a (classical) valuation of the dual polar space D H (7, 4). © 2009 Elsevier B.V. All rights reserved.