Compact 16-dimensional planes. An update

TL;DR

This paper updates the classification of 16-dimensional compact projective planes, focusing on those with large automorphism groups, and establishes criteria for their automorphism groups to be Lie groups.

Contribution

It provides a classification of 16-dimensional planes with automorphism groups of dimension at least 35, excluding those fixing exactly one flag, and introduces criteria for automorphism groups to be Lie groups.

Findings

Classified 16-dimensional planes with large automorphism groups.

Established criteria for automorphism groups to be Lie groups.

Extended previous classifications with new theorems.

Abstract

This paper is an addition to the book [54] on Compact projective planes. Such planes, if connected and finite-dimensional, have a point space of topological dimension 2, 4, 8, or 16, the classical example in the last case being the projective closure of the affine plane over the octonion algebra. The final result in the book (which was published 20 years ago) is a complete description of all planes admitting an automorphism group of dimension at least 40. Newer results on 8-dimensional planes have been collected in [52]. Here, we present a classification of 16-dimensional planes with a group of dimension at least 35, provided the group does not fix exactly one flag, and we prove several further theorems, among them criteria for a connected group of automorphisms to be a Lie group. My sincere thanks are due to Hermann H\"ahl for many fruitful discussions.