Compact 8-dimensional Projective Planes

Salzmann, Helmut

doi:10.1515/form.1990.2.15

Cited by 14 publications

(23 citation statements)

References 26 publications

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“…a) For the Spin planes, we have two conjugacy classes of polarities, both with unitals of spherical type (5,1). The group equals R/Z for the first class, and Spin 3 for the second one.…”

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confidence: 99%

“…b) Planes over mutations admit two classes of polarities, with unitals of spherical type (7,3) and (5,1), respectively. In the first case, we have = SO 3 R, while = R/Z in the second case.…”

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confidence: 99%

“…None of these unitals admits reconstruction from the group. c) Each plane over a Rees algebra has a single conjugacy class of polarities, with a unital of spherical type (5,1). Reconstruction from the group is not possible.…”

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Polar unitals in compact eight-dimensional planes

Stroppel

2004

Arch. Math.

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We determine centralizers and unitals for the polarities of eight-dimensional compact planes with at least 17-dimensional group of automorphisms, and discuss transitivity properties. Introduction.Let P = (P , L, ∈) be a compact projective plane (cf. [6]): point set P and line set L are compact spaces, and intersection and joining are continuous operations. We say that P is d-dimensional if P has topological (covering) dimension d.A polarity of P = (P , L, ∈) is a continuous involution π from P ∪ L onto L ∪ P such that p ∈ ⇐⇒ π ∈ p π holds for all p ∈ P and all ∈ L. The absolute points of the polarity π are those points p with p ∈ p π . The polar unital U π = (U π , B π ) for π is the set U π of absolute points, together with the set B π of traces (also called blocks) of lines that meet U π in more than one point. In many of the cases that we have to deal with, there is a distinguished line (at infinity) fixed by all automorphisms of the plane, and the polar unital contains exactly one point p ∞ at infinity. In these cases, we consider the affine part A π := U π \{p ∞ }.A unital (U, B) is called of spherical type (n, s) if U is homeomorphic to an n-sphere and each block b ∈ B is homeomorphic to an s-sphere. It has been conjectured that each nonempty polar unital of any 8-dimensional compact projective plane is of spherical type (7, 3) or (5, 1). Under additional hypotheses, this has been proved, cf. [3, 3.2, 3.3, 4.5].All 8-dimensional compact projective planes with at least 17-dimensional group of automorphisms have been determined, see [5], cf. [6, 84.28]. Apart from the Hughes planes discussed in Section 6, every such plane is a translation plane. A translation plane can only be self-dual if it has Lenz type V. There are three (closely related) families of compact eight-dimensional planes of Lenz type V with 18-dimensional group of automorphisms, but only one of these families contains self-dual planes (cf. [11, Thm. 6.1]): these are the Spin planes discussed in Section 3. Every eight-dimensional plane of Lenz type V with 17-dimensional group of automorphisms is coordinatized by a mutation of H, or by a Rees

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“…a) For the Spin planes, we have two conjugacy classes of polarities, both with unitals of spherical type (5,1). The group equals R/Z for the first class, and Spin 3 for the second one.…”

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Polar unitals in compact eight-dimensional planes

Stroppel

2004

Arch. Math.

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“…All such planes with automorphism groups of dimension at least 17 have been determined by Salzmann [9]. For semisimple groups, this bound lowers to 16 (see [9,(1)]).…”

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confidence: 99%

A characterization of quaternion planes

Stroppel

1990

Geom Dedicata

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ABSTRACT. The eight-dimensional planes admitting SLzl/q as a group of automorphisms are determined.Every open subset of the projective plane over I~, C, H (Hamilton quaternions) REMARKS. (a)The subplane above is the geometry induced on the set of points moved by the central involution ~ of A. Since ff cannot be planar, one can show that ~ is embedded into the projective plane over ~, and that the action of A extends to the natural one.(b) A special case of the stable planes considered here are compact eightdimensional projective planes. All such planes with automorphism groups of dimension at least 17 have been determined by Salzmann [9]. For semisimple groups, this bound lowers to 16 (see [9,(1)]). Our result extends this classification to the case of the 15-dimensional groups locally isomorphic to SL2H. (Since SOs~ cannot act on eight-dimensional projective planes, we can exclude PSLzH. ) NOTATION. Let A = SL2I~ , and let be the stabilizer of the point (1,0) in the natural linear action on H 2 and Geometriae Dedicata 36: [405][406][407][408][409][410] 1990.

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“…In the classification of compact projective planes with large automorphism groups, as accomplished by Salzmann in [22], [24], [29] and [28], it turned out that the hardest case is the one where the automorphism group E is neither semi-simple nor reductive, but has a normal subgroup 0 ~-~t. This subgroup 0 is then identified as a group of elations, and elaborate arguments show that E contains a transitive elation group, if dim E is sufficiently large.…”

Section: Introductionmentioning

confidence: 99%