Geometry Of Rank Research Articles

Geometry of rank 2 distributions with nonzero Wilczynski invariants

In the famous 1910 “cinq variables” paper Cartan showed in particular that for maximally nonholonomic rank 2 distributions in ℝ5 with non-zero covariant binary biquadratic form the dimension of the pseudo-group of local symmetries does not exceed 7 and among such distributions he described the one-parametric family of distributions for which this pseudo-group is exactly 7-dimensional. Using the novel interpretation of the Cartan covariant binary biquadratic form via the classical Wilczynski invariant of curves in projective spaces associated with abnormal extremals of the distributions [4, 27, 28] one can generalize this Cartan result to rank 2 distributions in ℝn satisfying certain genericity assumption, called maximality of class, for arbitrary n ≥ 5.In the present paper for any rank 2 distribution of maximal class with at least one nonvanishing generalized Wilczynski invariants we construct the canonical frame on a (2n — 3)-dimensional bundle and describe explicitly the moduli spaces of the most symmetric models. The relation of our results to the divergence equivalence of Lagrangians of higher order is given as well.

Attractors of foliations with transversal parabolic geometry of rank one

Attractors of foliations with transversal parabolic geometry of rank one

A rank 3 geometry for the O’Nan group connected with the Livingstone graph

We construct a rank 3 geometry ( O 0 N) over the diagram a a a 1 10 1 c 8 5 8 whose automorphism group is the O’Nan sporadic simple group. The maximal parabolic subgroups are the Janko group J1, 2 × S5 and the Mathieu group M11. Our construction is based on a convenient amalgam of known geometries of rank 2 for J1 and M11 extracted from the subgroup lattice of O 0 N.

Uniform extensions of partial geometries

A geometry of rank 2 is an incidence system (P, \(\mathcal{B}\)), where P is a set of points and \(\mathcal{B}\) is a set of subsets from P, called blocks. Two points are called collinear if they lie in a common block. A pair (a, B) from (P, \(\mathcal{B}\)) is called a flag if its point belongs to the block, and an antiflag otherwise. A geometry is called φ-uniform (φ is a natural number) if for any antiflag (a, B) the number of points in the block B collinear to the point a equals 0 or φ, and strongly φ-uniform if this number equals φ. In this paper, we study φ-uniform extensions of partial geometries pGα(s, t) with φ = s and strongly φ-uniform geometries with φ = s − 1. In particular, the results on extensions of generalized quadrangles, obtained earlier by Cameron and Fisher, are extended to the case of partial geometries.

A rank six geometry related to the McLaughlin sporadic simple group

In the literature, there are but a few incidence geometries on which the McLaughlin sporadic group $${\mathsf{McL}}$$ acts as a flag-transitive automorphism group. Their highest rank is four. In the present paper, we construct a geometry of rank six on which $${\mathsf{McL}}$$ acts flag-transitively and which has the following diagram.

A Family of Geometries Related to the Suzuki Tower#

ABSTRACT Michio Suzuki constructed a sequence of five simple groups G i , with i = 0,…, 4, and five graphs Δ i , with i = 0,…, 4, such that Δ i appears as a subgraph of Δ i+1 for i = 0,…, 3 and G i is an automorphism group of Δ i for i = 0,…, 4. The largest group G 4 was a new sporadic group of order 448 345 497 600. It is now called the Suzuki group Suz. These groups and graphs form what Jacques Tits calls the Suzuki tower. In a previous work, we constructed a rank four geometry Γ(HJ) on which the Hall-Janko sporadic simple group acts flag-transitively and residually weakly primitively. In this article, we show that Γ(HJ) belongs to a family of five geometries in bijection with the Suzuki tower. The largest of them is a geometry of rank six, on which the Suzuki sporadic group acts flag-transitively and residually weakly primitively.

Reconstructing ternary Dowling geometries

We prove that the only geometry of rank n > 4 all of whose proper contractions are ternary Dowling geometries is the ternary Dowling geometry. We use this to prove a stronger version of a conjecture of Joseph Kung and James Oxley.

Tt* geometry of rank two

A notion of a Frobenius manifold with a nice real structure was introduced by Hertling. It is called CDV structure (Cecotti-Dubrovin-Vafa structure). In this paper, we introduce a “positivity condition” on CDV structures and show that any Frobenius manifold of rank two with real dimension can be equipped with a positive CDV structure. We extend naturally the symmetries of Frobenius structures given by Dubrovin to symmetries of CDV structures, which will play an important role.

Extending Locally Truncated Geometries

Given a set of types I, a type 0∈I, a subset J of I containing 0, and a diagram I0 over I\\{0}, a geometry Γ over the set of types J is said to be locally truncated of I0-type if the J\\{0}-residues of Γ are truncations of geometries or chamber systems belonging to I0. We give a sufficent condition for such a geometry to be the J-truncation of a chamber system over the set of types I with all I\\{0}-residues belonging to I0. Then we apply our result to some special cases. We exploit it to classify flag-transitive cn.c*-geometries of rank n+2⩾4 and cn.c*-geometries of order 2. We give a new proof of a theorem of Ronan on Cn.L-geometries and we construct chamber systems for a number of sporadic groups in which certain well known geometries are involved as truncations or residues.

On embeddings of Ahrens-Szekeres generalized quadrangles

We construct a series of embeddings of the ASGQ's into affine 3-space, parameterized by the points of an affine plane. Points and lines of the 3-space, together with the images of all embeddings, form a diagram geometry of rank 3.

On C2.L-geometries

A C2.L-geometry is a geometry of rank 3 with elements called points, lines and quads, where residues of points are linear spaces, residues of lines are generalized digons and residues of quads are generalized quadrangles. Some sufficient conditions can be found in the literature for a C2.L-geometry to be a quotient of a truncated Cn-building. We shall weaken those conditions in this paper.

GF(2)-expansions

Let Γ be a finite geometry of rank n ≥ 2 with a selected type of elements, called'points'. Let m be the number of'points' of Γ. Under some mild hypotheses on Γ we can consider an affine expansion of Γ to AG(m,2). We prove that the geometries obtained by applying this construction to matroids are simply connected. Then we exploit this result to study universal covers of certain geometries arising from hyperbolic quadrics and symplectic varieties over GF.

Bounding the size of certain rank 3 geometries with designs as rank 2 residues

We consider locally finite geometries of rank 3 belonging to certain diagrams where all strokes represent classes of non-trivial designs, at least one of them consisting of symmetric designs other than projective planes. The gonality diagram of such geometry is of spherical type, whereas the diameter diagram is of affine type. In cases like this, the criteria available from 24 cannot tell us anything about the finiteness or infiniteness of a geometry. In the main theorem of this paper we prove that, certain imply finiteness additional conditions on the parameters of the designs associated with the strokes of the diagram, a geometry is necessarily finite.

Gluing two affine spaces

A construction is described in [2] by which, given two or more geometries of the same rank n, each equipped with a suitable parallelism giving rise to the same geometry at infinity, we can glue them together along their geometries at infinity, thus obtaininig a new geometry of rank n+ k− 1, k being the number of geometries we glue. In this paper we will examine a special case of that construction, namely the gluing of two affine spaces.

Rank three geometries associated with {${\rm PSL}(3,4)$}

In [Bu86] F. Buekenhout started a research program in order to find a unifying combinatorial approach to all finite simple groups, inspired by J. Tits’ famous theory of buildings (see e.g. [Ti74]), that is to classify diagram geometries fulfilling various conditions for all of these groups. This attempt has recently led to several collections of geometries (see [BCD95a], [BCD95b], [BDL94] and [BDL95]) using Cayley (see [De94]). The present paper should be regarded as a part of this program. Although PSL(3, 4) is a group of Lie-type and therefore wellknown in this context, it is of great interest since there are many connections between this group and some sporadic groups, e.g. the large Mathieu groups because of the famous construction of the 5-(24,8,1) design for M24 from the projective plane of order four (see [Lu69]), and PSL(3, 4) is a large subgroup of M22. The geometries in this paper are also of another interest because they are examples of buildinglike geometries in the sense of [BuPa95], which are e.g. extensions of (generalized) polygons by linear spaces and thick diagram geometries. We construct three geometries of rank three admitting PSL(3, 4) as a flagtransitive automorphism group in an easy combinatorial way using objects like hyperovals, Baer subplanes and unitals in PG(2, 4). Also, we describe these geometries by giving their diagrams and we determine their full automorphism and correlation groups. Our main result is the following.

On Af.Af* geometries

This paper is intended to be a first step towards the classification of finite flag-transitive geometries of rank 3 with affine planes and dual affine point-residues. We describe those of diameter 1. In the case of diameter > 1, we describe minimal quotients, assuming that the number of lines through two points is large enough.

A property of the hyperplanes of a matroid and an extension of Dilworth's theorem

In this paper, I show that in any combinatorial geometry, that is a matroid without loops or parallel edges, there exist at least as many hyperplanes as elements. This is equivalent to there being at least as many copoints as points in a geometric lattice, and hence extends Dilworth’s theorem [4] which proves equality only for modular lattices. Since proving this result I have discovered that both Baster-field and Kelly [l] and Greene [5] have previously established it. However my proof is totally different as it is incorporated in settling the case of equality. For I shall also show that the only geometries which have precisely as many hyperplanes as elements are the union of disjoint connected matroids, each of which is either a projective geometry or is a geometry of rank two or less. This is not true of matroids in general. The equivalent result for matroids is that there are at least as many hyperplanes as closed sets of rank one. In [3], Dembowski and Wagner essentially prove that if the blocks of a symmetric block design form the hyperplanes of a matroid, then this matroid is a projective geometry. This result may be deduced as a corollary of the main result of this paper. Lastly I show that the conditions of the DembowskiWagner theorem may be relaxed, inasmuch as I prove that if the hyperplanes of a combinatorial geometry are such that any two intersect in h elements, then it is either a projective geometry or has a single base or is the union of a single element and a geometry of rank two.