The valuations of the near octagon

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 G n with a large automorphism group. In this paper, we classify the valuations of the near octagon G 4 . We show that each such valuation is either classical, the extension of a non-classical valuation of a G 3 -hex or is associated with a valuation of Fano-type of an H 3 -hex. In order to describe the latter type of valuation we must study the structure of G 4 with respect to an H 3 -hex. This study also allows us to construct new hyperplanes of G 4 . We also show that each valuation of G 4 is induced by a (classical) valuation of the dual polar space D H ( 7 , 4 ) .