Sixteen-dimensional compact translation planes with automorphism groups of dimension at least 35 (Q2087448)

All 16-dimensional locally compact translation planes with automorphism groups of dimension at least 35 that fix no points on the translation axis were determined by \textit{H. Hähl} [Geom. Dedicata 83, No. 1--3, 105--117 (2000; Zbl 0973.51011)]. In a series of papers, \textit{H. Hähl} [Adv. Geom. 11, No. 2, 371--380 (2011; Zbl 1218.51009)] classified all 16-dimensional locally compact translation planes with automorphism groups of dimension at least 38. In the paper under review, the author investigates 16-dimensional compact translation planes \(\mathbb{P}\) with automorphism groups of dimensions \(d\) between 35 and 37 and makes substantial progress in their classification. In particular, all planes with \(d = 37\) are completely determined. In this case, \(\mathbb{P}\) has Lenz-type V, a maximal compact subgroup \(\Delta\) of the reduced stabilizer is an almost direct product of \(\mathrm{U}_2\mathbb{H}\) and \(\mathrm{SO}_2\mathbb{R}\), and \(\mathbb{P}\) is isomorphic to one of the planes found by \textit{H. Hähl} and \textit{E. Meyer} [J. Geom. 111, No. 3, Paper No. 46, 19 p. (2020; Zbl 1464.51011)]. In the next case \(d=36\), the author achieves an almost complete classification with only one of four possible situations left open: \(\mathbb{P}\) has Lenz-type V and \(\Delta\) is locally isomorphic to \(\mathrm{U}_3\mathbb{C}\cdot\mathrm{SO}_2\mathbb{R}\) or \((\mathrm{SU}_2\mathbb{C})^3\cdot\mathrm{SO}_2\mathbb{R}\). The author announces this case will be solved in a forthcoming paper. In the last case \(d=35\), it is shown that besides the planes of Hähl [loc. cit.], two new families of planes occur with one more scenario unresolved: \(\Delta\) is locally isomorphic to \(\mathrm{U}_3\mathbb{C}\) or \((\mathrm{SU}_2\mathbb{C})^3\), the reduced stabilizer contains a compression subgroup \(\Upsilon\) and is isomorphic to a semidirect product of \(\Delta \cdot \Upsilon\) and an 8-dimensional unipotent collineation group. \par The proofs rely on the identification of subgroups of shears with axis the vertical line \(S\), eliminating certain compact groups with \(9\le \dim \Delta\le 16\) (like \(\mathrm{G}_2(-14)\)) up to local isomorphism, and a thorough analysis of the cases where \(\Delta\) contains \(\mathrm{SU}_4\mathbb{C}\) or \(\mathrm{U}_2\mathbb{H}\) as a direct factor.