Near-homogeneous 16-dimensional planes (Q2726406)

The paper under review is a contribution to the classification of 16-dimensional compact projective planes. Here, a compact projective plane is a projective plane whose point and line sets carry compact topologies such that the geometric operations of joining two distinct points and intersecting two distinct lines are continuous. It is known that the topological dimension of the point space of a compact projective plane equals the topological dimension of its line space, and we will speak of an \(n\)-dimensional compact projective plane if this common dimension is exactly \(n\).NEWLINENEWLINENEWLINEThe classification proceeds by looking at the following two parameters.NEWLINENEWLINENEWLINE\(\bullet\) The structure of the automorphism group \(\Delta\). By general results, \(\Delta\) is a locally compact group of finite dimension.NEWLINENEWLINENEWLINE\(\bullet\) The structure of the configuration of elements fixed by \(\Delta\).NEWLINENEWLINENEWLINEFor example, if \(\Delta\) fixes neither a point nor a line, and is of dimension at least 29, i.e. \(\dim\Delta\geq 29\), then all possible planes are known, explicitly. The next fixed structure occurs if \(\Delta\) fixes exactly one element, up to duality we may assume that \(\Delta\) fixes exactly one line but no point. This is exactly the situation considered by Salzmann in this paper. Salzmann proves, assuming the conditions stated above, that the projective plane is a translation plane if \(\dim\Delta\geq 35\). In fact, the line fixed by \(\Delta\) turns out to be a translation axis.NEWLINENEWLINENEWLINECombining with results due to Hähl and Löwe, all these translation planes are already classified.