Abstract
This chapter describes finite diagram geometries extending buildings. Buildings of spherical type (in particular finite) are the natural geometric counterpart of simple groups of Lie–Chevalley type (in particular finite). In this context, sporadic groups appear as orphans and call for an extension of the theory of buildings. This situation reproduces to some extent (not entirely) the conditions under which Tits started research on geometries that would explain the five exceptional simple complex Lie groups E6, E7, E8, F4 and G2. Among the sporadic groups, the five Mathieu groups were definitely loaded with geometric meaning and some inclusions or ‘towers' of simple groups went in the same direction. The three Fischer groups Fi22, Fi23, Fi24 extend geometrically the Mathieu groups M22, M23, M24. The latter appear as three consecutive ‘extensions' of the projective plane of order 4 and the former as extensions of the rank 3 polar space corresponding to U(6, 2), whose planes are precisely projective planes of order 4. It also became clear that it was possible to work with such structure in some generality.
Published Version
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