Abstract. We prove the following theorem: Let 7 ~ be an 8-dimensional compact topological projective plane. If the connected component A of its automorphism group has dimension at least 12, then A is a Lie group.Mathematics Subject Classification (1991): 51H 10.For any compact connected topological projective plane 7', whose point space P has dimension at most 4, we know that P is a manifold (in fact P ~ P2~ or P ~ P2C respectively) and that the automorphism group of 7" is a Lie group (see [13]). If the point space has dimension greater than 4, it is not known whether or not it is a manifold. The automorphism group of the projective plane is always a locally compact transformation group of P, but it is still an open question if it is always a Lie group. For the most interesting cases, however, i.e. for automorphism groups of sufficiently large dimension, we can show that they are Lie groups. Thus, the purpose of this paper is to prove the following theorem: THEOREM. The connected component A of the automorphism group of an 8-dimensional compact connected projective plane is a Lie group if dim A > 12.Remark. An analogous result for 16-dimensional planes was published by LOwen and Salzmann [8]. They proved that an automorphism group is a Lie group if its dimension is at least 36. Using methods of the present paper, this bound can be improved considerably: 28 is already sufficient. PreliminariesThe notation in this paper is as follows: 79 will always denote a compact connected 8-dimensional projective plane with point space P and line space/2. The connected component of its automorphism group is called A, and A _< A is the stabilizer of a nondegenerate quadrangle. For X C P we denote by Ax the stabilizer of X, and by A[x ] its pointwise stabilizer. The set of fixed points of an element E A is denoted by Fs. For the dimension of a coset space we write dim A/F = dimA-dimP = A.F.

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The paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E6(-26). A 16-dimensional, compact projective plane ~ admitting an automorphism group of dimension 41 or more is classical, [18] 87.5 and 87.7. For the special case of a semisimple group d acting on ~ the same result can be obtained if dimd _-> 37, see [i6]. Our aim is to lower this bound. We show: if d is semisimple and dim d _-> 29, then ~ is either classical or a Moufang-Hughes plane or A is isomorphic to Spin 9 (~., r), r E {0, 1}.The underlying paper contains the first part of the proof showing that A is in fact almost simple.O. Introduction. Let ~ be a 16-dimensional compact projective plane with point space P and line space ~. An example of such a plane is the Moufang plane ~2 9 over the octonions, the so-called classical plane of dimension 16. In order to classify such planes we consider a subgroup d of the automorphism group Aut(~) of ~'. In the compact-open topology, Aut (~) is a locally compact transformation group, [18] 44.3. The group A shall be closed and connected. The larger the topological dimension of A, the better are the chances to obtain a complete classification of planes ~ with d < Aut (~).We also take into account the structure of A. Semisimple and non-semisimple groups behave in a completely different way. For projective planes the main difference between these two types is based on the fact that semisimple groups (with one exception) always contain involutions. Here, we will restrict the discussion to semisimple groups. Now we can formulate the problem: classify all 16-dimensional compact projective planes ~' admitting a semisimple group A < Aut (~') with dimA => d for some suitably chosen d. In [16] this problem was solved for d = 37. In this case ~ is always classical. In the present paper and in [9] we will lower this bound to d = 29. Our aim is to prove the following Theorem. Let ~ be a 16-dimensional compact projective plane and .4 a connected and closed subgroup of the automorphism group of ~. If .4 is semisimple and at least 29-dimensional, then one of the following statements holds:Mathematics Subject Classification (1991): 51H10.

Let ~P = (P,/2) be a compact projective plane with 0 < dim P < oo and let • be a compact connected subgroup of Aut(P). If dim ~5 _> dim E -dim P, where E is the elliptic motion group of the corresponding classical plane, then ff ~ E or ff is isomorphic to a point stabilizer E0 in E, cf.[31]. Here we consider the case • -----E0. It is shown that the action of ~ on the point space P is equivalent to the classical action of E0. For dimP E {8, 16} the plane P is uniquely determined by a 2-dimensional subplane g with SO2]R < Aut(g).Mathematics Subject Classification (1991): 51H 10.

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