Point-line characterizations of Lie geometries (Q2783473)

In the study of Lie incidence geometries, \textit{A. M. Cohen} and \textit{B. N. Cooperstein}'s early results [Geom. Dedicata 15, 73-105 (1983; Zbl 0541.51010)] have been central. In their characterisation of certain parapolar spaces with thick lines they made three basic assumptions of (1) constant finite symplectic rank at least 3; (2) every singular subspace having finite projective rank; (3) a gap occurs in the spectrum of projective ranks of \(x^\perp\cap S\) where \(x\) is a point not incident with the symplecton \(S\). (\(x^\perp\cap S\) has projective rank at most 1 where either rank 1 or rank 0 does not occur, or is a maximal singular subspace of \(S\).) NEWLINENEWLINENEWLINEIn the paper under review the authors concentrate on the second theorem of Cohen and Cooperstein, which characterises buildings of type \(C_{n,1},D_{n,1},D_{5,5}, E_{6,4},E_{6,1},E_{7,7},E_{8,1}\). They dispense with the first assumption, greatly weaken the second one and replace assumption (3) with something weaker and quite different. The authors obtain two main theorems. The first one deals with parapolar spaces \(\Gamma\) of symplectic rank at least 3 such that \(x^\perp\cap S\) is never just a point where \(x\) is a point not incident with the symplecton \(S\); such that some mild local conditions are satisfied; and such that every maximal singular subspace has finite projective rank if all symplecta have rank at least 4. Then \(\Gamma\) is either \(E_{6,4}\), \(E_{7,7}\) or \(E_{8,1}\); a metasymplectic space; or a polar Grassmannian of lines of a non-degenerate polar space of rank at least 4. In case of finite polar rank all these geometries are truncations of buildings and one of \(B_{n,2}\) or \(D_{n,2}\) for \(n\geq 4\) in the last alternative. NEWLINENEWLINENEWLINEIn the second theorem strong parapolar spaces \(\Gamma\) are considered such that \(x^\perp\cap S\) is nonempty for each point \(x\) and symplecton \(S\); such that the set of points at distance at most 2 from a point forms a geometric hyperplane; and such that if there is no symplecton of rank 2 then every maximal singular subspace has finite projective rank. It is shown that \(\Gamma\) is either \(D_{6,6}\), \(A_{5,3}\) or \(E_{7,1}\); a classical dual polar space of rank 3; or a product geometry formed from a line and a non-degenerate polar space of rank at least 3. To prove their second result the authors show that \(\Gamma\) possesses the weak hexagon property. Such spaces all whose symplecta have rank at least 3 have been classified by \textit{M. El-Atrash} and \textit{E. Shult} [Geom. Dedicata 71, 221-235 (1998; Zbl 0912.51002)] and lead to the first 3 geometries. If \(\Gamma\) contains a symplecton which is a grid and one which is not, then one obtains a product geometry. For their first theorem the authors show that the point-collinearity graph is simply connected and that point-residue geometries are uniformly isomorphic. In case of finite singular rank \(\Gamma\) belongs to certain locally truncated diagrams. From this the authors obtain a graph morphism between associated point-collinearity graphs which is an isomorphism in their situation. In the infinite singular rank case point-residue geometries must be one of the geometries occuring in their second theorem.