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Sixteen-dimensional locally compact translation planes with collineation groups of dimension at least 38. (English) Zbl 1218.51009
The present paper is the last step of a large classification program to determine all 16-dimensional locally compact translation planes with collineation groups of dimension at least 38. The first two steps have already been done by the author in [Arch. Math. (Basel) 48, 267–276 (1987; Zbl 0639.51017)] and [Geom. Dedicata 36, 181–197 (1990; Zbl 0719.51011)]. The result of the classification says:
The 16-dimensional locally compact translation planes with collineation groups of dimension at least 38 are precisely6.5mm
(1)
the planes over the quasifields $$\mathbb O^{(\rho)}$$ constructed via perturbations of the octonion algebra $$\mathbb O$$ (cf. chapter 82.4 of [H. Salzmann, D. Betten, T. Grundhöfer, H. Hähl, R. Löwen and M. Stroppel, Compact projective planes. With an introduction to octonion geometry. De Gruyter Expositions in Mathematics. 21. Berlin: de Gruyter (1996; Zbl 0851.51003)]) and
(2)
the planes over the quasifields $$\mathbb O_{\alpha,\,\rho}$$ constructed via generalized mutations of $$\mathbb O$$ (cf. chapter 82.21 of ibidem).
The 16-dimensional locally compact translation planes admitting the universal covering group $$\text{Spin}_7{\mathbb R}$$ of $$SO_7{\mathbb R}$$ as collineation group are precisely the planes from (1); if $$\rho=\text{id}$$, then we have the classical octonion plane whose full automorphism group is the exceptional Lie group $$E_6(-26)$$ which is $$78$$-dimensional, otherwise the dimension of the full automorphism group is 39 or 38, depending on the homeomorphism $$\rho:{\mathbb R}\to{\mathbb R}$$.
The 16-dimensional locally compact translation planes which admit the exceptional compact Lie group $$G_2$$ as group of collineations, but not the group $$\text{Spin}_7{\mathbb R}$$ and whose full automorphism groups are of dimension at least 38, are precisely the planes from (2); the dimension of the full automorphism group is 40 (for $$\rho=\text{id}$$) or 39 or 38.
MSC:
 51H10 Topological linear incidence structures 51A40 Translation planes and spreads in linear incidence geometry
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References:
 [1] Hähl H., Geom. Dedicata 4 pp 305– (1975) [2] DOI: 10.1007/BF01214575 · Zbl 0381.51009 · doi:10.1007/BF01214575 [3] DOI: 10.1016/0166-8641(81)90029-8 · Zbl 0446.51010 · doi:10.1016/0166-8641(81)90029-8 [4] DOI: 10.1007/BF00181316 · Zbl 0622.51008 · doi:10.1007/BF00181316 [5] Hähl H., Math. (Basel) 48 pp 267– (1987) [6] DOI: 10.1007/BF01295287 · Zbl 0691.51007 · doi:10.1007/BF01295287 [7] DOI: 10.1007/BF00150787 · Zbl 0719.51011 · doi:10.1007/BF00150787 [8] DOI: 10.1007/BF01265640 · Zbl 0845.51011 · doi:10.1007/BF01265640 [9] Kramer L., Math. 563 pp 83– (2003) [10] DOI: 10.2140/pjm.2003.209.325 · Zbl 1063.51008 · doi:10.2140/pjm.2003.209.325
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