Some Exceptional 2-Adic Buildings
نویسنده
چکیده
Tits [ 191 has initiated the study of geometries having properties strongly resembling those of buildings (compare [6]). His main result is that, in general, such a geometry is the image of a building under a suitable type of morphism. These geometries that are almost buildings were called GABS in [lo], and finite examples were described in [2, 8, 10, 15, 161 that are not buildings but have highly transitive groups. In this paper we will proceed in the opposite direction. An unexpected type of description is given for the afftne buildings associated to some low-dimensional 2-adic orthogonal groups. This description makes it easy to produce infinitely many finite GABS having large groups. Specifically, we construct subgroups of 0’(8, a,), O(7, a,), LV(6, Q,) and G2(U4J that are flag-transitive on the corresponding affine buildings and can be written using matrices with entries in the subring Z [4] of the rationals. (These flag-transitive groups are just Q(Z[+], f,) and the automorphism group of the non-split Cayley algebra over Z [f ], where k = 8, 7 or 6 and fk is the quadratic form C: xi .) If m is any odd integer > 1 and these matrices are viewed modulo m then a finite GAB is obtained having a flag-transitive group and one of the following diagrams.
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