Actions of almost simple groups on eight-dimensional stable planes

Abstract

1. Generalities, and known results 1.1 Definitions. A stable plane is a linear space M = (M , M), where the point space M and the line space M are endowed with locally compact topologies such that joining of points and intersection of lines are continuous operations, and that the set of pairs of intersecting lines is open in M × M (axiom of stability). Moreover, we require that the point space M has positive and finite (topological) dimension. Whenever this is convenient, we shall tacitly identify a line of a stable plane with the set of points that are incident with it. The line that joins p, q ∈ M is written pq, and the pencil of all lines that are incident with a point p will be denoted by M p . General information about stable planes can be found in the work of R. Löwen; in particular, see [3] and [7]. Most important is the deep result [7, Theorem 1] that dim M = dim M = 2 dim M p = 2 dim(pq) ∈ {2, 4, 8, 16}. For recent developments of the theory, compare also [23]. Endowed with the compact-open topology derived from the action on M (or on M), the group Aut(M) of all continuous collineations of M is a locally compact transformation group both on M and M.

Compact planes, mostly 8-dimensional. A retrospect

“…If ∆ is compact, connected and almost simple, then dim ∆ ≤ 10 or ∆ ∼ = E 3 ∼ = PU 3 H. If ∆ is semi-simple and if dim ∆ > 16, then ∆ is a Lie group of dimension 18, 21, or 35, and ∆ is equivalent to a subgroup of PSL 3 H; see [88] (4.3) and [86] 16.1, cf. also [90] and Theorem 2.2 above.…”

The skew-hyperbolic motion group of the quaternion plane

“…Therefore, we will use the acquaintance we have got by now in order to prove that no group with center factor group PSp6[~ acts non-trivially on any 8-dimensional stable plane. This contributes to the project of determining all actions of almost simple groups on stable planes [30], compare [28,Section 9] and [20,Kap. 9].…”