Annals of Combinatorics 2(1998) 325-349

Wrapping Polygons in Polygons

Antonio Pasini and Giustina Pica

Department of Mathematics, University of Siena, Via del Capitano 15, I--53100 Siena, Italy

Department of Mathematics, University of Naples, Via Claudio 21, I--80125 Napoli, Italy

Received July 22, 1998

AMS Subject Classification: 51E24, 51D15, 05B25

Abstract. This paper is devoted to I2(2g).c-geometries, namely, point-line-plane structures where planes are generalized 2g-gons with exactly two lines on every point and any two intersecting lines belong to a unique plane. I2(2g).c-geometries appear in several contexts, sometimes in connection with sporadic simple groups. Many of them are homomorphic images of truncations of geometries belonging to Coxeter diagrams. The I2(2g).c-geometries obtained in this way may be regarded as the ``standard'' ones. We characterize them in this paper. For every I2(2g).c-geometry Γ, we define a number w(Γ), which counts the number of times we need to walk around a 2g-gon contained in a plane of Γ, building up a wall of planes around it, before closing the wall. We prove that w(Γ) = 1 if and only if Γ is ``standard'' and we apply that result to a number of special cases.

Keywords: diagram geometry, near polygons, semi-biplanes, Coxeter complexes

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