Wrapping polygons in polygons

This paper is developed toI2(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. TheI2(2g).c-geometries obtained in this way may be regarded as the “standard” ones. We characterize them in this paper. For everyI2(2g).c-geometry Γ, we define a numberw(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 thatw(Γ)=1 if and only if Γ is “standard” and we apply that result to a number of special cases.