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By Knarr N.

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2 of [60]. A natural question is whether there exist generalized n-gons Γ with an automorphism group acting transitively on the set of pairs (f, Ω) with f a flag and Ω an ordinary (n + 1)-gon containing f . Let us call this the Joswig property for (Γ, G), since it was Michael Joswig who first asked me if one could classify such pairs in the finite case (and this gave rise to the papers [42] and [58], where this classification is completed). In the infinite case, Katrin Tent [32] constructs for each n ≥ 2 infinitely many examples of pairs (Γ, G), with Γ a generalized n-gon, possessing the Joswig property.

Now consider an element y incident with x, and opposite xσ . Then again δ(y, y σ ) = 2m and the unique element z of Γm (y) ∩ Γm (y σ ) is fixed by σ. It is easy to see that z , x, y, z , xσ , y σ belong to a common ordinary (2m + 1)-gon, and so z is opposite z . By varying y, we see that we obtain different elements z opposite z . All these elements z are all either points or lines. Consequently, these elements are not mutually opposite, hence the unique paths joining them are also pointwise fixed by σ.

The elements of H are called generalized homologies. The following theorem joins results of different papers and authors [3, 38, 39, 55, 56]. 7 If a finite generalized polygon is (x, y)-transitive for all pairs of opposite elements x, y, then it is a Moufang polygon (but not all finite Moufang polygons arise, in particular, all finite planes and hexagons do arise, but no octagon arises and the generalized quadrangles obtained from Hermitian varieties in 4-dimensional projective space do not arise either).

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4-Dimensionale projektive Ebenen mit grosser abelscher Kollineationsgruppe by Knarr N.


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