16-dimensional smooth projective planes with large collineation groups (Q1275312)

Smooth projective planes are projective planes with an additional differentiable structure. That is, point set and line set are smooth manifolds and the geometric operations joining two different lines and intersecting two different lines are smooth. As for compact topological projective planes one is interested in the classification of sufficiently homogeneous smooth projective planes. The paper under review considers 16-dimensional smooth planes, that is, smooth planes where the point space is a 16-dimensional manifold. The group of smooth automorphisms of such a plane is a Lie group, hence it has a dimension. The main result states that if the dimension of the automorphism group of a 16-dimensional smooth plane is at least 39, then the plane is classical, that is, it is isomorphic to the octonion projective plane. Compare this with the corresponding result in the topological case [cf. \textit{H. Salzmann} et al., Compact projective planes, De Gruyter, Berlin (1995; Zbl 0851.51003)]: If the dimension of the automorphism group of a 16-dimensional compact projective plane is at least 41, then the plane is classical and there are nonclassical 16-dimensional compact projective planes with a 40-dimensional group of automorphisms. To obtain the result, the author examines the stabilizer of two lines and the stabilizer of a flag to reduce the possibilities for large groups to act as a automorphism group. The perhaps most powerful tool obtained is the stabilizer theorem which in the nonclassical case provides bounds for the dimension of the stabilizer of two lines and some information on the Levi subgroup of this stabilizer.