Parapolar spaces of infinite rank (Q6549918)

In [Adv. Geom. 2, No. 2, 147--188 (2002; Zbl 0998.51002)], \textit{A. Kasikova} and \textit{E. Shult} characterized strong parapolar spaces \(\Delta = (\mathcal{P},\mathcal{L})\) that have the following three properties: \N\begin{itemize}\N\item[\((\mathrm{H}'_1)\)] every point is collinear to at least one point of each symplecton; \N\item[\((\mathrm{H}'_2\))] for each point \(p\), the set \(\{q \in \mathcal{P} \mid d_\Delta(q, p) \le 2\}\) forms a proper subspace of \(\Delta\); and \item[\((\mathrm{F}')\)] if no symplecton has rank two, all singular subspaces are finite dimensional.\N\end{itemize} \NIn the paper under review, the authors show that a strong parapolar space that satisfies \((\mathrm{H}'_1)\) and \((\mathrm{H}'_2)\) also satisfies \((\mathrm{F}')\), that is, the assumption of having only symplecta of finite rank can be removed. To obtain this result, the authors first analyse which steps in Kasikova and Shult's characterization [loc. cit.] do not require \((\mathrm{F}')\). They then prove that there must be some symplecton of finite rank, and finally use results by \textit{A. De Schepper} et al. [Forum Math. Sigma 9, Paper No. e2, 27 p. (2021; Zbl 1461.51006); Q. J. Math. 73, No. 1, 369--394 (2022; Zbl 1485.51004)] on strong parapolar spaces where every symplecton has finite rank \(\ge 3\).\N\NThe present authors furthermore show that in a strong parapolar space that satisfies \((\mathrm{H}'_1)\) and \((\mathrm{H}'_2)\) no two symplecta intersect in exactly one point. In the last section a construction is provided of a strong parapolar space of diameter two such that all symplecta are infinite dimensional, and every two symplecta intersect in an infinite-dimensional projective subspace. A counterexample is obtained from an increasing sequence of certain prepolar spaces.