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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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[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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