Compact Projective Planes Research Articles

Polarities of Schellhammer planes

Abstract We construct polarities for projective planes admitting a group of automorphisms that acts regularly on the complements of an anti-flag in the point and line set. In particular, we determine all polarities of the compact connected planes studied by I. Schellhammer and by P. Sperner. For both classes of planes we solve the conjugacy problem, and determine the set of absolute points for each one of the polarities. Among these polarities we find the first examples of elliptic polarities in non-Moufang compact projective planes.

16-dimensional compact projective planes with a large group fixing two points and only one line

We complete the determination of all pairs ( P , Δ ) , where P is a compact projective plane with a 1 6 -dimensional point set, Δ is an automorphism group of P of dimension at least 3 5 , and Δ does not fix exactly one point and one line. If Δ fixes two points and only one line, then Δ contains a 1 5 -dimensional translation group and a compact subgroup S p i n 7 ℝ ; hence dim Δ ≥ 3 6 . The planes are described by their coordinatizing Cartesian fields, more explicitly for dim Δ > 3 6 .

16-dimensional compact projective planes with a collineation group of dimension ≥ 35

A compact projective plane is isomorphic to the classical Moufang plane if it admits an automorphism group of dimension at least 38 which fixes a non-incident point-line pair.

Affine parts of topological unitals

Abstract We study different, non-compact groups acting on unitals in compact projective planes in such a way that the geometry can be reconstructed from the group action. These unitals have many ambient automorphisms, but are not induced by polarities. In passing, we construct polarities for a class of eight-dimensional translation planes, and study the corresponding unitals.

16-dimensional compact projective planes with a large group fixing two points and two lines

We determine all planes having the properties of the title with a group of dimension at least 33.

Topological affine planes with affine connections

Abstract We consider the question whether the system of lines of a two-dimensional topological plane can be described as the system of geodesics of a Riemannian metric or an affine connection. In [4] we have shown that non-classical compact projective planes do not admit Riemannian metrics. Here we use different methods that are suited to two-dimensional planes in general. We apply them to three families of topological affine planes with large collineation groups. It turns out that for the standard models of the skew-parabola planes and the planes over cartesian fields no affine connection, and hence no Riemannian metric exist. However, for the Moulton planes affine connections do exist and we determine all of them. Among them is at least one Riemannian connection. We also give an example where no Riemannian metric exists. Moreover, we derive a characterization of the classical affine plane in the case that ℝ2 acts by vector space translations as a subgroup of the collineation group.

16-dimensional compact projective planes with 3 fixed points

Let 9 = (P, £) be a topological projective plane with a compact point set P of finite (covering) dimension d = dim P > 0. A systematic treatment of such planes can be found in the book Compact Projective Planes [15]. Each line L e £ is homotopy equivalent to a sphere §/ with / 1 8, and d = 2/, see [15] (54.1 1). In all known examples, L is in fact homeomorphic to §/. Taken with the compact-open topology, the automorphism group Σ = Aut^ (of all continuous collineations) is a locally compact transformation group of Ρ with a countable basis, the dimension dim Σ is finite [15] (44.3 and 83.2). The classical examples are the planes κ over the three locally compact, connected fields IK with ( — dim IK and the 16-dimensional Moufang plane G = έΡ® over the octonion algebra Θ. If 9 is a classical plane, then Aut^ is an almost simple Lie group of dimension Q, where C = 8, €2 = 16, €4 = 35, and Cg = 78. In all other cases, dim Σ < ^ Q + 1 < 5Λ Planes with a group of dimension sufficiently close to C( can be described explicitly. More precisely, the classification program seeks to determine all pair s (&, Δ), where Δ is a connected closed subgroup of Aut & and bf ^ dim Δ 5^ for a suitable bound be 4/ — 1 . This has been accomplished for t ^ 2 and also for 64 = 17. Here, the case ( = 8 will be studied; the value of bt varies with the configuration of the fixed elements ofA. Most theorems that have been obtained so far require additional assumptions on the structure of Δ. If dim Δ 27, then Δ is always a Lie group [12]. By the structure theory of Lie groups, there are 3 possibilities: (i) Δ is semi-simple, or (ii) Δ contains a central torus subgroup, or (iii) Δ has a minimal normal vector subgroup, cf. [15] (94.26). The first two cases are understood fairly well:

Absolute Points of Continuous and Smooth Polarities

This paper deals with sets of absolute points of continuous or smooth polarities in compact, connected or smooth projective planes, called topological polar unitals or smooth polar unitals, respectively. We will show that topological polar unitals are Z2-homology spheres. In the four-dimensional case, a topological polar unital U is either a topological oval, or any line which intersects U in more than one point intersects in a set homeomorphic to S1. Smooth polar unitals turn out to be smoothly embedded submanifolds of the point space. Moreover, secants intersect such unitals transversally. For these unitals, we will obtain full information on the existence of secants, tangents and exterior lines through given points according to their position with respect to the unital. The main result of this paper states that the possible dimensions of smooth polar unitals coincide with those of sets of absolute points of continuous polarities in the classical projective planes P2F, Fϵ{R,C,H,O}. Finally, we will prove that smooth polar unitals in four-dimensional smooth projective planes are topological ovals or are homeomorphic to S3.

Four-dimensional compact projective planes admitting an affine Hughes group

By constructing all 4-dimensional affine Hughes planes that are not translation planes, we contribute the last missing part to the classification of all compact 4-dimensional projective planes having a non-solvable automorphism group of dimension at least 6. We use the information on the possible actions of the automorphism group obtained by the third author [5] and the construction principle employing derived planes that was given by the second author [4]. The unique solvabilty of certain nonlinear systems of equations has to be shown; this is done using topological degree techniques and delicate analytic arguments. Also the proof of continuity of the geometric operations is quite subtle.

Sixteen-Dimensional Locally Compact Translation Planes with Large Automorphism Groups having no Fixed Points

The 16-dimensional compact projective planes whose automorphism group contains a closed connected subgroup fixing a line, but no point and having dimension at least 35 are determined. It is shown that these planes all belong to three families of planes determined by H. Lowe and the author, and hence are explicitly known. A major stepping stone to this goal is a result by H. Salzmann according to which every such plane is a translation plane.

Convergence of automorphisms of compact projective planes

We show that a pointwise convergent sequence (σn)n∈N of continuous collineations of a compact projective plane converges uniformly if and only if the pointwise limitα of (σn)n∈N has a quadrangle in its image. Moreoverα is then a continuous collineation. Furthermore, we derive that a homomorphism between topological projective planes is continuous if and only if it is continuous at some point.

Smooth Hughes planes are classical

We prove that the only compact projective Hughes planes which are smooth projective planes are the classical planes over the complex numbers $\Bbb C $ , the quaternions $\Bbb H $ , and the Caley numbers $\Bbb O $ . As a by-product this shows that an 8-dimensional smooth projective plane which admits a collineation group of dimension $d \geq 17$ is isomorphic to the quaternion projective plane ${\cal P _2\Bbb H }$ . For topological compact projective planes this is true if $d \geq 19$ , and this bound is sharp.

Compact 16-dimensional projective planes

Compact 16-dimensional projective planes

Large Almost Simple Groups Acting on 16-Dimensional Compact Projective Planes

This 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 clasical, [23] 87.5 and 87.7. For the special case of a semisimple group Δ acting on ? the same result can be obtained if dim \(\), see [22]. Our aim is to lower this bound. We show: if Δ is semisimple and dim \(\), then ? is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin9 (ℝ, r), r∈{0, 1}. The proof consists of two parts. In [16] it has been shown that Δ is in fact almost simple or isomorphic to SL3?ċSpin3ℝ. In the underlying paper we can therefore restrict our considerations to the case that Δ is almost simple, and the corresponding planes are classified.

Four-Dimensional Compact Projective Planes with a Solvable Collineation Group

We consider a four-dimensional compact projective plane \(\pi = (\mathfrak{B},\mathfrak{L})\) whose collineation group Σ is six-dimensional and solvable with a nilradical N isomorphic to Nil×R, where Nil denotes the three-dimensional, simply connected, non-Abelian, nilpotent Lie group. We assume that Σ fixes a flag p ∈ W, acts transitively on \(\mathfrak{L}_p \backslash \{ W\}\) and fixes no point in the set W\p. Under these conditions, we will prove that either Σ contains a three-dimensional group of elations or Σ acts doubly transitively on \(\mathfrak{L}_p \backslash \{ W\}\).

Characterization of 16-dimensional Hughes planes

The well-known finite Hughes planes have compact analoga with 16-dimensional point space. The automorphism group of such a plane is a 36-dimensional Lie group. Theorem: Assume that the compact projective plane $\cal P $ is not isomorphic to the classical Moufang plane over the octonions. Let $\Delta $ be a closed subgroup of $\hbox {Aut} \,\cal P $. If $\dim \Delta \ge 31$ and if $\Delta $ has a normal torus subgroup, then $\cal P $ is a Hughes plane, $\Delta = \hbox {Aut} \,\cal P $, and $\Delta ^{\prime } \cong \hbox {PSL}_3 \Bbb H $.

Affine Hughes groups acting on 4-dimensional compact projective planes

Suppose that the real projective plane is contained as a Baer subplane in a compact 4-dimensional plane . Salzmann [15] has shown that if the group of projective collineations of extends to , then is isomorphic to the complex plane. This means that there are no compact 4-dimensional Hughes planes. If the group AGL2ℝ of offine collineations of extends to , then we call the extended group an offine Hughes group. Here, non-classical examples exist. We show that, up to duality, every action of AGL2ℝ on a compact 4-dimensional plane is an offine Hughes group, and we describe explicitly the possible actions on the point space of as transformation groups. In subsequent work (jointly with N. Knarr and H. Klein), this will be used to determine the possible planes , as well. This will complete the classification of 4-dimensional planes admitting a non-solvable group of dimension at least 6.

16-dimensional smooth projective planes with large collineation groups

Smooth projective planes are projective planes defined on smooth manifolds (i.e. the set of points and the set of lines are smooth manifolds) such that the geometric operations of join and intersection are smooth. A systematic study of such planes and of their collineation groups can be found in previous works of the author. We prove in this paper that a 16-dimensional smooth projective plane which admits a collineation group of dimension d ≥ 39 is isomorphic to the octonion projective plane P2 O. For topological compact projective planes this is true if d ≥ 41. Note that there are nonclassical topological planes with a collineation group of dimension 40.

Large semisimple groups on 16-dimensional compact projective planes are almost simple

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 P 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 Δ acting on P the same result can be obtained if dim δ ≧ 37, see [16]. Our aim is to lower this bound. We show: if Δ is semisimple and dim δ ≧ 29, then P is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin9 (ℝ, r), r ∈ {0, 1 }. The underlying paper contains the first part of the proof showing that Δ is in fact almost simple.

Compact projective planes: with an introduction to octonion geometry

Compact projective planes: with an introduction to octonion geometry