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The only compact 8-dimensional projective planes admitting an automorphism group of dimension at least 17 are 5 one-parameter families of division ring planes, the Hughes planes, the nearfield planes, a two-parameter family of proper translation planes and their duals. 1980 Mathematics Subject Classification (1985 Revision): 51H10.= (p, fi) is a topological projective plane with a compact connected point set P of finite (covering) dimension, then each line L e fi is homotopy-equivalent to a sphere §' with t = 2 m < 8, and dim P = 2/, see [21] . Moreover, L is homeomorphic to §' if L is a manifold. In the compact-open topology, the group Σ = Aut & of continuous collineations is a locally compact transformation group [9] . If Σ is transitive on P, then & is one of the 4 classical planes over the real or complex numbers or the quaternions or octonions [20], and Σ is a simple Lie group of dimension g = 8, 16, 35, or 78 respectively. For dim P < 4, all planes with dim Σ > g/2 are known, cp. [29; 32]. In the present paper, the classification of all 8-dimensional planes satisfying dim Σ> 17 = [g/2] will be completed. In particular, dim Σ < 18 for non-arguesian planes. This had already been proved in [37] except in the case where Σ fixes exactly one flag.From now on let dim P = 8 and assume first dim Σ > 14. Then (0) Σ is a Lie group.The proof is quite similar to that given for 16-dimensional planes in [23]. The arguments will not be repeated here.The author expresses his thanks for the hospitality of the University of Toronto where an essential part of this paper was written.Brought to you by | University of Queensland -UQ Library Authenticated Download Date | 7/16/15 11:01 PM

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This paper is an addition to the book [54] on Compact projective planes. Such planes, if connected and finite-dimensional, have a point space of topological dimension 2, 4, 8, or 16, the classical example in the last case being the projective closure of the affine plane over the octonion algebra. The final result in the book (which was published 20 years ago) is a complete description of all planes admitting an automorphism group of dimension at least 40. Newer results on 8-dimensional planes have been collected in [52]. Here, we present a classification of 16-dimensional planes with a group of dimension ≥35, provided the group does not fix exactly one flag, and we prove several further theorems, among them criteria for a connected group of automorphisms to be a Lie group. My sincere thanks are due to Hermann Hähl for many fruitful discussions. closure of an affine plane coordinatized by a mutation O (t) = (O, +, •) of the octonions, where c•z = t·cz + (1−t)·zc with t > 1 2 ; either t = 1, O (t) = O, and P ∼ = O , or dim Σ = 40 and Σ fixes the line at infinity and some point on this line ([54] 87.7).The ultimate goal is to describe all pairs (P, ∆), where ∆ is a connected closed subgroup of Aut P and dim ∆ ≥ b for a suitable bound b in the range 27 ≤ b < 40. This bound varies with the structure of ∆ and the configuration F ∆ of the fixed elements (points and lines) of ∆. In some cases, only ∆ can be determined, but there seems to be no way to find an explicit description of all the corresponding planes.

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