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16-dimensional compact projective planes with a large group fixing two points and only one line. (English) Zbl 1263.51009
Following the long-standing program to classify all finite-dimensional connected compact topological projective planes whose automorphism groups are sufficiently large, the authors consider a connected closed subgroup \(\Delta\) of the automorphism group \(\Sigma\) of a topological projective plane \(\mathcal{P}\) with a compact point set of (covering) dimension 16. B. Priwitzer and the second author [J. Lie Theory 8, No. 1, 83–93 (1998; Zbl 0902.51012)] have shown that \(\Delta\) is a Lie group if its dimension is at least 27. In case \(\Delta\) fixes at least 3 points and \(\dim\Delta\geq 32\), the second author [Adv. Geom. 2003, Spec. Issue, 153–S157 (2003; Zbl 1043.51011)] has proved that \(\mathcal{P}\) is a translation plane. The present authors [Arch. Math. 85, No. 1, 89–100 (2005; Zbl 1077.51003)] determined those planes where \(\Delta\) is of dimension at least 33 and fixes exactly two points and two lines. There are also partial results when \(\Delta\) has only one fixed point.
In the paper under review the authors deal with the case that \(\Delta\) is of dimension at least 34 and fixes exactly two points \(u\) and \(v\) and one line. They show that under these assumptions the translation group of \(\mathcal{P}\) is at least 15-dimensional. Furthermore, either \(\Delta\) has a subgroup isomorphic to \(\mathrm{Spin}_7\mathbb R\) and \(\dim\Delta\geq 36\), or \(\mathcal{P}\) is a translation plane, \(\dim\Delta=34\) and a maximal semi-simple subgroup of \(\Delta\) is isomorphic to \(\mathrm{SU}_4\mathbb C\). The statement on the dimension of the translation group is obtained by a detailed analysis of how various groups can act, repeatedly using R. Bödi’s result [Geom. Dedicata 53, No. 2, 201–216 (1994; Zbl 0829.51007)] on the automorphism groups of 8-dimensional ternary fields, and showing that one of the translation groups \(T_{[u]}\) and \(T_{[v]}\) must be linearly transitive.
Those planes where \(\dim\Delta\geq 35\) are described explicitly as precisely the planes that can be coordinatized by the following topological Cartesian fields, which are a modification of the octonion algebra \((\mathbb O,+,.)\). Let \((\mathbb R,+,\ast,1)\) be a topological Cartesian field with unit element 1 such that \((-r)\ast s=-(r\ast s)\) for all \(r,s\in\mathbb R\). Let \(\rho\) be a homeomorphism of \([0,\infty)\) to itself that fixes 1. Define a new multiplication \(\circ\) on \(\mathbb O\) by \(s\circ x=| s|^{-1}s(| s|\ast\mathrm{Re}\,x+\rho(| s|)\cdot\mathrm{Pu}\,x) \) where \(s\neq 0\), \(\mathrm{Re}\,x=\frac{1}{2}(x+\bar x)\) and \(\mathrm{Pu}\,x=\frac{1}{2}(x-\bar x)\) is the real and pure part of \(x\), respectively, and \(0\circ x=0\) for all \(x\).
If \(\mathcal{P}\) is not the classical Moufang plane, then \(\dim\Delta<40\) and \(\Sigma\) fixes each of the two fixed points of \(\Delta\). Furthermore, the cases where \(\Sigma\) has dimension 38 or 39 are characterized by the form of \(\ast\) and \(\rho\).
51H10 Topological linear incidence structures
51A35 Non-Desarguesian affine and projective planes
51A40 Translation planes and spreads in linear incidence geometry
22E99 Lie groups