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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 $$\rho$$.
##### MSC:
 51H10 Topological linear incidence structures 51A35 Non-Desarguesian affine and projective planes 51A40 Translation planes and spreads in linear incidence geometry 22E99 Lie groups