Compact groups on compact projective planes
A compact, connected projective plane \({\mathcal P}\) has a point set \(P\) of covering dimension \(\dim P = 2 \ell |16\) (if \(\dim P < \infty\) [cf. the ref. et al., Compact projective planes, Berlin: de Gruyter (1995) (abbreviated by CPP for further citation in this review), Section 54.11]. The automorphism group \(\Sigma\) of \({\mathcal P}\) is a locally compact transformation group of \(P\); if \(\ell \leq 2\) or if \(\dim \Sigma \geq 4 (\ell - 1)\), then \(\Sigma\) is even a Lie group, see CPP \S 87. Let \(E\) denote the (compact) elliptic motion group of the classical \(2 \ell\)-dimensional Moufang plane, and put \(k_\ell = \dim E - 2 \ell\). If \(\Phi\) is a compact subgroup of \(\Sigma\), and if \(\dim \Phi > k_\ell\), then \(\Phi \cong E\) and \({\mathcal P}\) is classical; this is true even for the more general class of stable planes [cp. \textit{M. Stroppel}, Forum Math. 6, No. 3, 339-359 (1994; Zbl 0799.51008)]. The author investigates the case \(\dim \Phi = k_\ell\). As a transformation group of the point set, \(\Phi\) is then isomorphic to a point stabilizer \(E_0\). For \(\ell < 8\) other proofs of this fact are given in CPP, see 71.10 and 84.9. Consider a flag \((p,L)\) not incident with the two fixed elements of \(\Phi\). If \(\ell = 4\) or \(\ell = 8\), then \(\Lambda = \Phi_{p,L}\) is isomorphic to \(SO_3 \mathbb{R}\) or to \(G_2\) respectively, and the fixed elements of \(\Lambda\) form a two-dimensional subplane \({\mathcal E}\) admitting a group \(SO_2 \mathbb{R}\). Up to isomorphy, the plane \({\mathcal P}\) is uniquely determined by \({\mathcal E}\). It is difficult to decide which planes \({\mathcal E}\) actually lead to a projective plane \({\mathcal P}\), cp. a similar construction of four-dimensional planes with a group \(C^\times \cdot SU_2 \mathbb{C}\) by \textit{P. Sperner} [Geom. Dedicata 34, No. 3, 301-312 (1990; Zbl 0702.51011)].
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