Extending polar spaces of rank at least 3
An incidence system is a pair \(\Gamma = \Gamma ({\mathcal P}, {\mathcal B})\), where \({\mathcal P}\) is a set of points and \({\mathcal B}\) is a family of subsets of \({\mathcal P}\) called blocks such that each block of \({\mathcal B}\) has at least two distinct points, and the incidence between points and blocks is defined by inclusion. Two points \(p,q\) are said to be adjacent if they lie in a common block. The point graph of \(\Gamma\) is a graph with vertex set \({\mathcal P}\) and with adjacency same as in \(\Gamma\). The residue \(\Gamma_p\) of a point \(p\) is the incidence system of points adjacent to \(p\) excluding \(p\) itself together with blocks on \(p\) with the point \(p\) removed. We say that \(\Gamma\) is connected if its point graph is connected. And finally, connected incidence system \(\Gamma = \Gamma ({\mathcal P}, {\mathcal B})\) is an extended polar space or extended (projective) plane if its point residues are finite, thick, nondegenerate polar spaces, or finite nondegenerate projective planes, respectively. In this paper under review, which is a continuation of an earlier author's work [J. Comb. Theory, Ser. A 68, No. 1, 100-114 (1994; Zbl 0808.05060)], the following main theorem is proved: If \(\Gamma\) is an extended polar space admitting extended planes, then either (i) \(\Gamma\) is a standard quotient of an affine polar space over \(GF(2)\), or (ii) \(\Gamma\) is the extended polar space for the sporadic Fischer group \(Fi_{22}\), or (iii) there are exactly two extended planes on each block and \(\Gamma\) is an extended \(Q^+_5 (4)\)-polar space. From this theorem it immediately follows that the extended polar space for the sporadic Fischer group \(Fi_{22}\) is the only extended polar space which has more than two extended planes on a block and is not isomorphic to a quotient of an affine polar space over \(GF(2)\). New examples of \(EGQ(4,1)\) and \(EGQ(4,2)\) are presented, as well.
scientific article; zbMATH DE number 823494