Sixteen-dimensional locally compact translation planes with large automorphism groups having no fixed points
The author classifies those 16-dimensional compact projective planes whose automorphism group contains a closed connected subgroup \(\Gamma\) which fixes a line, but no points and has dimension at least \(35\). A recent result of \textit{H. Salzmann} [Near-homogeneous 16-dimensional planes, to appear in Adv. Geom.] shows that such a plane \(\mathbb P\) is a translation plane. Therefore, \(\Gamma\) contains a closed connected subgroup \(\Lambda\) of the reduced stabilizer such that \(\Lambda\) is at least \(18\)-dimensional and fixes no points of the translation axis. According to results of the reviewer [J. Lie Theory 10, No. 1, 127-146 (2000; Zbl 0951.51007) and `Noncompact subgroups of the reduced stabilizer of a locally compact, connected translation plane', preprint], it follows that \(\Lambda\) is compact or is an almost direct product \(\Lambda = \Psi \cdot K\) of a \(2\)-fold covering group \(\Psi\) of \(\text{PSO}^+_{k}(\mathbb R,1)\), \(3\leq k \leq 10\), and a compact group \(K\). If \(\mathbb P\) is not the octonion plane, then the author proves that \(\Lambda\) is isomorphic \(\text{SU}_4 \mathbb C \cdot \text{SU}_2 \mathbb C\), \(\text{SU}_4 \mathbb C \cdot \text{SL}_2 \mathbb R\), or \(\text{SL}_2 \mathbb H \cdot \text{SU}_2 \mathbb C\). The translation planes admitting actions of these groups have been determined explicitly by the author [Abh. Math. Sem. Univ. Hamburg 70, 137-163 (2000)] and the reviewer [`Sixteen-dimensional locally compact translation planes admitting \(\text{SL}_2 \mathbb H\) as a group of collineations', preprint]. The octonion plane belongs to each of the three families, but there are a vast number of other planes of this kind. The dimension of the full group of automorphisms of the nonclassical examples equals \(35\), whence one obtains the following corollary: If the dimension of the group \(\Gamma\) above is at least \(36\), then \(\mathbb P\) is isomorphic to the octonion plane. Moreover, the paper under review contains a classification of those translation planes whose automorphism group contains a compact subgroup \(\Delta\) with \(\dim \Delta \geq 17\). Using results of \textit{H. Biller} [`Actions of compact groups on spheres and on generalized quadrangles', Thesis, Stuttgart (1999)], it is shown that the plane is isomorphic to the octonion plane, or that \(\Delta\) equals \(\text{SU}_4\mathbb C \cdot \text{SU}_2\mathbb C\) (see above) or \(\text{Spin}_7 \mathbb R\). Planes admitting an action of the latter group have been determined explicitly by the author [see Arch. Math. 48, 267-276 (1987; Zbl 0639.51017)].
collineation groups without fixed points
scientific article; zbMATH DE number 1546664