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Locally compact Hughes planes

Markus Stroppel

1994
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Canadian mathematical bulletin
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Among the eight-dimensional stable planes. the compact connected generalized Hughes planes and the geometries induced on the outer points are characterized by the property that these planes admit an effective action of the group SL, C. I. Introduction. In [15 \, H. Salzmann describes the compact connected (generalized) Hughes planes (cl [2]), and characterizes these planes among the eight-dimensional compact projecti ve planes: These are the only planes of this type that admit a semi -simple
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... e) group i1 of automorphisms with dimi1 ::::> 16. This semi-simple group is isomorphic with the almost simple group SL, C, except in the desarguesian case (i.e., the plane over Hamilton's quaternions~. This plane admits-in addition to SL3 C-the thirty-fivedimensional simple group PSL3 ~, the twenty-one-dimensional simple groups PSUj ~(O) and PSU3 WI), and the semi-simple group Aut(~) ex: SL2 ~ of dimension 18, cl ! 17]). IflP';: is a non-desarguesian compact connected Hughes plane then there is an invariant Baer subplane which is desarguesian (i.e., isomorphic with the projective plane over C). The points and lines of this subplane are called inner, while all other elements of IP' ;: are termed outer. The geometry induced on the set of outer points forms an eight-dimensional stable plane M" (in the sense of R. Lbwen [12J), on which SL3 C acts point-transitively with two line orbits (namely, inner and outer lines, respectively). The planes IP' " and M" will be defined precisely in the following chapter. The present paper is devoted to the proof of the following result: Each eightdimensional stable plane M = (M, 'fo{) that admits SLj C as a group of automorphisms is isomorphic with IP' y or M;: for some cpo This result contributes to a determination of the "most homogeneous" eight-dimensional stable planes, as begun in ! 17]. Of particular importance for our proof is the fact that, in contrast to the situation of the (finite) planes originally described by D. R. Hughes, the planes IP'; and M; admit the group SL3 C rather than the simple group PSL3 C: the set of fixed lines of the center Z = {d I [ ' = I} shall give us a frame for the reconstruction of the geometry. 2. The planes. Let IP' = (P, P) be a compact connected generalized Hughes plane. According to 12J there is a ternary operation T defined on some affine line K of IP' such that the operations x + y = T( I, x, y) and x 0 y = T(x, y, 0) make (K, +, 0) a nearfield. Note, however, that T is linear if, and only if, the plane IP' is desarguesian. Since T can be described by geometric operations, the nearfield (K, +, 0) is a locally compact connected Received by the editors June 2, 1992.

doi:10.4153/cmb-1994-017-x
fatcat:pamewbtxmvbj5buotj5ohwk6ai