Sixteen-dimensional Locally Compact Translation Planes Admitting Sl2 H as a Group of Collineations

In this paper, all 16-dimensional locally compact translation planes admitting the unimodular quaternion group SL2H as a group of collineations will be determined explicitly. Besides the classical plane over the octonions there are a vast number of planes having this property, cf. the Classification Theorem (2.8). Indeed, the class of these planes covers an interesting borderline case: Among all 16-dimensional locally compact translation plane, only the classical plane admits the action of a noncompact almost simple Lie group of dimension larger than dimSL2H = 15, cf. [7, Theorem A]. The connected component Ge of the automorphism group G of a nonclassical example is composed of the translation group, the group of homotheties, the group SL2H, and a compact group ∆ isomorphic to {e},SO2R, SO2R× SO2R, or SU2C, cf. Theorem 3.8. Thus, dimG is at most 35. It is worth mentioning that Γ = Ge leaves precisely one projective line (namely the translation axis) invariant, but does not fix any projective points. In general, a 16-dimensional compact projective plane whose automorphism group contains a closed connected subgroup Γ having this property and satisfying dimΓ ≥ 35 is necessarily a translation plane, thanks to a theorem of H. Salzmann [10]. Recently, H. Hähl has shown in [4] that there are precisely three families of such planes: A subfamily of the planes considered here1 , and the planes admitting SU4C ·SU2C or SU4C ·SL2R as a group of collineations, determined in Hähl [5]. In particular, dimΓ ≥ 36 implies that the plane is isomorphic to the octonion plane.