16-dimensional compact projective planes with a large group fixing two points and two lines (Q2487022)

The program to classifiy all compact projective planes of positive finite dimension whose automorphism groups are sufficiently large has been sucessfully completed for 2-, 4- and 8-dimensional compact projective planes, cf. the book by \textit{H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen} and \textit{M. Stroppel} [Compact projective planes, de Gruyter, Expositions in Mathematics 21, Berlin (1996; Zbl 0851.51003)] for a comprehensive overview, including 16-dimensional planes, and an exposition of many of the methods involved. The case of 16-dimensional compact projective planes is harder to deal with and results so far are not as extensive as in the lower dimenional cases. The automorphism group \(\Gamma\) of such a projective plane \({\mathcal P}\) is locally compact with respect to the compact-open topology. Moreover, \textit{B. Priwitzer} and the second author [J. Lie Theory 8, 83--93 (1998; Zbl 0902.51012)] showed that the connected component of \(\Gamma\) is even a Lie group provided that \(\dim\Gamma \geq 27\). Hence the structure theory of Lie groups can be applied in the study of 16-dimensional compact projetctive planes in this case. Most results that have been obtained by various authors in this direction require additional assumptions on the dimension, structure or action of a closed connected group of automorphism or that \({\mathcal P}\) is a translation plane. In the paper under review the authors deal with the following situation. \(\Delta\) is a closed connected subgroup of \(\Gamma\) of dimension at least 33 and it is assumed that \(\Delta\) fixes exactly two points \(u\), \(v\) and two lines \(W=uv\) and \(Y=ov\), where the points \(o\), \(u\), \(v\) form a triangle. They show that the translation group \(T=\Delta_{[v,W]}\) is transitive, that the complement \(\Delta_o\) of \(T\) has compact commutator group \(\Phi\cong\text{Spin}_8{\mathbb R}\) and that in fact dim \(\Delta\geq 36\). If \(\Delta\) even has dimension at least 38, then \({\mathcal P}\) is isomorphic to the classical Moufang plane over the octonion algebra \(({\mathbb O},+,.)\). Moreover, the planes \({\mathcal P}\) satisfying the above general assumptions are precisely those planes that can be coordinatized by the following topological Cartesian fields \(({\mathbb O},+,\circ)\). Let \(({\mathbb R},+,*,1)\) be a topological Cartesian field with unit element 1 such that \((-r)*s=-r*s=r*(-s)\) for all \(r,s\in{\mathbb R}\). Define the new multiplication \(\circ\) on \({\mathbb O}\) by \(a\circ x=| a| *| x| (| a| ~| x| )^{-1}ax\) for \(a,x\neq 0\) and \(0\circ x=a\circ 0=0\).