Dense near octagons with four points on each line, III

This is the third paper dealing with the classification of the dense near octagons of order (3, t). Using the partial classification of the valuations of the possible hexes obtained in [12], we are able to show that almost all such near octagons admit a big hex. Combining this with the results in [11], where we classified the dense near octagons of order (3, t) with a big hex, we get an incomplete classification for the dense near octagons of order (3, t): There are 28 known examples and a few open cases. For each open case, we have a rather detailed description of the structure of the near octagons involved

Dual embeddings of dense near polygons

Let e: S -> Sigma be a full polarized projective embedding of a dense near polygon S, i.e., for every point p of S, the set H(p) of points at non-maximal distance from p is mapped by e into a hyperplane Pi(p) of Sigma. We show that if every line of S is incident with precisely three points or if S satisfies a certain property (P(de)) then the map p bar right arrow Pi p defines a full polarized embedding e* (the so-called dual embedding of e) of S into a subspace of the dual Sigma* of Sigma. This generalizes a result of [6] where it was shown that every embedding of a thick dual polar space has a dual embedding. We determine which known dense near polygons satisfy property (P(de)). This allows us to conclude that every full polarized embedding of a known dense near polygon has a dual embedding

An alternative definition of the notion valuation in the theory of near polygons.

Valuations of dense near polygons were introduced in \cite{DB-Va:1}. A valuation of a dense near polygon $\mathcal{S}=(\mathcal{P},\mathcal{L},\mathrm{I})$ is a map $from the point-set$\mathcal{P}$of$\mathcal{S}$to the set$\N$of nonnegative integers satisfying very nice properties with respect to the set of convex subspaces of$\mathcal{S}$. In the present paper, we give an alternative definition of the notion valuation and prove that both definitions are equivalent. In the case of dual polar spaces and many other known dense near polygons, this alternative definition can be significantly simplified

Glued near polygons

AbstractWe generalize a construction given in [3] to derive new near polygons from spreads of symmetry in generalized quadrangles