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The theory of buildings, created by J. Tits three decads ago, has ooered a uniied geometric treatment of nite simple groups of Lie type, both of classical and of exceptional type. (See Tits 19] and 20] for an exposition of that theory; also Ronan 15] and Brown 1].) Diagram geometry (see 13] for an exposition) is a generalization of the theory of buildings.(More)
Let 2 be a finite thick dual polar space of rank 3. We say that a hyperplane H of 2 is locally singular (respectively, quadrangular or ovoidal) if H & Q is the perp of a point (resp. a subquadrangle or an ovoid) of Q for every quad Q of 2. If H is locally singular, quadrangular, or ovoidal, then we say that H is uniform. It is known that if H is locally(More)
Let ∆ be a dual polar space of rank n ≥ 4, H be a hyperplane of ∆ and Γ := ∆ \ H be the complement of H in ∆. We shall prove that, if all lines of ∆ have more than 3 points, then Γ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar(More)
In this paper we consider partial linear spaces containing a set of subspaces isomorphic to affine planes, such that the lines and these afline planes on a fixed point form a non-degenerate polar spaces of rank at least 2. We obtain a complete classification, provided that the rank is at least 3. The study of geometries on the absolute points of polarities(More)
We study (i-)locally singular hyperplanes in a thick dual polar space of rank n. If is not of type DQ(2n, K), then we will show that every locally singular hyperplane of is singular. We will describe a new type of hyperplane in DQ(8, K) and show that every locally singular hyperplane of DQ(8, K) is either singular, the extension of a hexagonal hyperplane in(More)