Francis Buekenhout

Diagrams for geometries and groups

J. Tits has achieved a far-reaching generalization of projective geometry, including a geometric interpretation of all simple groups of Lie type (see [35] and the references therein). One of the most fascinating features of this theory is that each of its geometries is essentially determined by a diagram (the Coxeter diagram). These nice and simple pictures with an enormous potential of information might very well appear as pieces of that universal language that some people want to elaborate in order to communicate with hypothetical extraterrestrial beings. In this geometric paradise created for us by Tits, geometers do not enter easily and this might be due to the fact that relatively few classical geometries are covered by the theory. The purpose of the present paper is to open the paradise to more geometers, i.e., to get more space(s) in it. This will be achieved by a generalization of the notion of diagram inspired by earlier work of Tits, especially [30,31, 33,351. However, the main motivation for this work and its most interesting aspect is based on the observation that more than half of the known sporadic groups (and why not all of them?) are related to a diagram appearing as an extension of a spherical diagram by one or more strokes of the same type

On the foundations of polar geometry, II

One of the two Buekenhout-Shult theorems for polar spaces required a finite rank assumption. Here we get rid of that restriction. Similarly, the polar spaces of possibly infinite rank having some line of two points are classified.

Semi — Quadratic sets in projective spaces

The purpose of this paper is to characterize semi-quadrics in projective spacesP of finite dimension 2 at least. A concept of semi-quadratic set inP is introduced: a semi-quadratic setQ inP is essentially a set of points ofP such that the union of all tangent lines at each pointp ofQ is either a hyperplane ofP orP itself. (A tangent line ofQ atp is a line contained inQ or meetingQ exactly inp). The main result is that a semi-quadratic set which is invariant under “many” perspectivities is a semi-quadric.

The number of nets of the regular convex polytopes in dimension ≤4

Abstract Classifying the nets (also called unfoldings or developments or patterns) of the regular convex polytopes under the isometry group of the polytope is equivalent to classifying the spanning trees of the facet-adjacency graph under its automorphism group. This is done for all such polytopes of dimension at most 4.

Finite distance-transitive generalized polygons

Using the classification of the finite simple groups, we classify all finite generalized polygons having an automorphism group acting distance-transitively on the set of points. This proves an old conjecture of J. Tits saying that every group with an irreducible rank 2 BN-pair arises from a group of Lie type.

Polar Spaces Having Some Line of Cardinality Two

Abstract The nonthick geometries of type C n and D n or equivalently all polar spaces having at least one line of cardinality 2 are classified. It turns out that there are two classes of such polar spaces. On the one hand, decomposable polar spaces or polar spaces which are direct sums of two or more polar spaces are obtained. On the other hand, polar spaces arising from the interval lattice of an irreducible projective geometry which can also be seen as being partitioned by a pair of disjoint maximal singular subspaces can be gotten.

An axiomatic of inversive spaces

Abstract One shows essentially that a geometry consisting of points and circles with any 3 distinct points on just one circle and such that any 4 non-concyclic points generate an inversive plane can only be the geometry induced on an ovoid of a projective space or an inversive plane; if the number of points is finite one can have only inversive planes.

All Geometries of the Mathieu Group Mll Based on Maximal Subgroups

Using a Cayley program, we get all firm, residually connected geometries whose rank-two residues satisfy the intersection. property, on which M ll acts flag-transitively, and in which the stabilizer of each element is a maximal subgroup of M ll.

Geometries of the Group PSL(2,11)

We determine all residually weakly primitive flag-transitive geometries for the groups PSL(2,11) and PGL(2,11). For the first of these we prove the existence by simple constructions while uniqueness, namely the fact that the lists are complete, relies on MAGMA programs. A central role is played by the subgroups Alt(5) in PSL(2,11). The highest rank of a geometry in our lists is four. Our work is related to various ‘atlases’ of coset geometries.

On Flag-transitive Incidence Geometries of Rank 6 for the Mathieu Group M12

We show that the Mathieu group M12 does not have geometries of rank greater or equal to 6, satisfying the RWPRI and (IP)2 conditions. Our proof of this result is based on classifications of geometries of some subgroups of M12 which have been obtained using Magma programs.