Compact Projective Planes Research Articles

Automorphism groups of compact projective planes

Compact connected projective planes have been investigated extensively in the last 30 years, mostly by studying their automorphism groups. It is our aim here to remove the connectedness assumption in some general results of Salzmann [31] and Hahl [14] on automorphism groups of compact projective planes. We show that the continuous collineations of every compact projective plane form a locally compact transformation group (Theorem 1), and that the continuous collineations fixing a quadrangle in a compact translation plane form a compact group (Corollary to Theorem 3). Furthermore we construct a metric for the topology of a quasifield belonging to a compact projective translation plane, using the modular function of its additive group (Theorem 2).

Actions of SO(3) on 4-dimensional stable planes

In a previous paper [13] on 4-dimensional compact projective planes admitting a group ~ ~ SO(3) = SO3 of automorphisms, we proved that either the plane is the Desarguesian plane P2 C or • has index at most 2 in the full automorphism group. Here we treat a similar question more generally for locally compact stable planes, where lines do not always meet (see [5] for a definition). We show that either the plane is one of three open subptanes of P2 C, or its full automorphism group has very small dimension (at most 4). The proof in the projective case rests on the fact that an involutory automorphism tr contained in a connected group of automorphisms has a centre and an axis. Removing, e.g., the centre and the points of the axis, one gets a stable plane with a free involution, i.e. one without fixed points. However, in section 1 we prove for SO3-actions on stable planes of dimension 4 that every point lies on the axis of some involution. This allows us to determine the possible isotropy groups on points and to some extent also on lines. According to [14], the isotropy groups determine the topological type of the action. In section 2, these general facts about SO3-actions are used to prove our main result, viz.

Compactness in Topological Hjelmslev Planes

AbstractIn the theory of ordinary topological affine and projective planes it is known that (1) An affine plane is never compact (2) a locally compact ordered projective plane is compact and archimedean (3) a locally compact connected projective plane is compact and (4) a locally compact projective plane over a coordinate ring with bi-associative multiplication is compact. In this paper we re-examine these results within the theory of topological Hjelmslev Planes and observe that while (1) remains valid (2), (3) and (4) are false. At first glance these negative results seem to suggest we are working in too general a setting. However a closer examination reveals that the absence of compactness in our setting is a natural and expected feature which in no way precludes the possibility of obtaining significant results.

Compact 16-dimensional projective planes with large collineation groups. III

Compact 16-dimensional projective planes with large collineation groups. III

Compact projective planes with homogeneous ovals

Closed ovals exist only in 2-or 4-dimensional compact projective planes. We show that a plane of dimension 2 or 4 contains ahomogeneous closed oval iff the automorphism group contains SO2 or SO3, respectively. In 4-dimensional planes, existence of a homogeneous oval and existence of a homogeneous Baer subplane are equivalent. We determine the possible full automorphism groups of planes containing homogeneous ovals except for two possibilities in dimension 4 whose existence remains uncertain.

Compact 16-dimensional projective planes with large collineation groups. II

If a locally compact topological projective planeP of finite topological dimension satisfies dim AutP > 46, thenP is isomorphic to the Moufang plane over the alternative division ring $$\mathbb{O}$$ of the real octonions.

Compact 16-dimensional projective planes with large collineation groups

Compact 16-dimensional projective planes with large collineation groups

Compact 8-dimensional projective planes with large collineation groups

Compact 8-dimensional projective planes with large collineation groups