Characterizations of the G(2)(4) and L-3(4) near octagons

A triple (S, S, Q) consisting of a near polygon S, a line spread S of S and a set Q of quads of S is called a polygonal triple if certain nice properties are satisfied, among which there is the requirement that the point-line geometry S' formed by the lines of S and the quads of Q is itself also a near polygon. This paper addresses the problem of classifying all near polygons S that admit a polygonal triple (S, S, Q) for which a given generalized polygon S' is the associated near polygon. We obtain several nonexistence results and show that the G(2)(4) and L-3(4) near octagons are the unique near octagons that admit polygonal triples whose quads are isomorphic to the generalized quadrangle W(2) and whose associated near polygons are respectively isomorphic to the dual split Cayley hexagon HD(4) and the unique generalized hexagon of order (4, 1).

Publication year:2021

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