Buekenhout geometry

The term Buekenhout - Tits geometry (also Buekenhout - geometry or geometry chart called ) is in the geometry for a common generalization of the concepts of projective geometry, affine geometry, block diagram, linear space and many other related terms. The concept was significantly later his pupil Francis Buekenhout developed in the years after 1956 by Jacques Tits and according to which it is now named. The basic idea of ​​this concept is largely refrain from details of the geometrical structure and for that to investigate the properties of classical structures and their generalizations, which are connected with the classical geometric term " Flag".

The diagram geometry has been used by tits with some success on ( noncommutative ) finite simple groups and their classification. These groups can be personalized with purely group theoretical methods up to now hardly dismantle further useful: Your normal subgroup association is trivial and too big and in its structure too little characteristic than that he give her sub-group association even at the smallest representatives of a starting point for investigation, let alone a classification could. On the other hand, has long been known that many of the simple groups of classical geometric structures or their generalizations as full automorphism or as a operate their sub - or factor groups (see as an example Witt Cutter Plan), often these " geometric " structures projective planes (or in general ) block plans. The approach consists first of tits, a group that operates on geometric structures of different species as a group of automorphisms of a suitable " composite geometric " assign structure that reflects as much essential information of the different output structures.

Another important application is the investigation of induced geometries, arising for example from quadratic quantities on finite projective spaces, see the illustration at the end of the introduction. Historically noteworthy that already in 1896, Eliakim Hastings Moore has developed a concept for an abstract geometry, which essentially corresponds to the diagram geometry proposed. At Moores times but this was not pursued further.

• 2.1 geometry
• 2.2 flag, rooms, residuum, rank
• 2.3 Basic diagram of a geometry

Guiding principles

The diagram - geometry generalized concepts that have arisen because of issues from very different areas of mathematics. Therefore, many geometric terms to be filled with a new general content that is not generalized often the usual criteria in a formal sense. An incidence structure is for example no geometry in the sense of diagram geometry, but any incidence structure can also be in a natural way as a diagram geometry ( and indeed of rank 2 ) understand. As a guiding principle can apply the concept of projective geometries ( finite ) Here, therefore, the corresponding term is used frequently in this article compared to the term from the diagram geometry for projective geometries. In fact, projective planes form along with two kinds of trivial rank 2 geometries, the most important building blocks of the studied to date diagram geometries.

Fundamental concepts of projective geometry

A finite projective space determines the set S of its real projective subspaces ( including the amount of his points, but here without the empty set and the whole space ). Each element of S can be achieved by " typing " feature " type" of a stock index (here about its respective projective dimension ) are assigned. The amount of these types can be described by a finite set, for example. The number of types actually occurring is then the projective dimension ( or " rank" ) of the entire room. On the set S of " elements " ( real subspaces ) is given by the symmetrized subspace relation an incidence relation.

A flag in such a projective geometry is represented by the antisymmetric incidence ( here: the non- symmetrized subspace relation) totally ordered subset of S. Such a flag is also for an infinite projective space finite, provided only that the dimension of this space is finite and the length the flag is not greater than this dimension of the projective space.

Definitions

Geometry

Be a lot, its elements and subsets are referred to as types. A ( chart ) geometry is on a triple, where S is a set, I is a symmetric and reflexive relation on S, is the incidence relation, t is a surjective function, the typing function if the following axiom (TP = transversality property ) is satisfied:

Flag room, residuum, rank

Be a geometry over.

• A flag of a ( possibly empty) set F pairs incidental elements of S.
• Two flags are called incident, albeit a flag is.
• Maximum flags hot room (german chambers ).
• The set of all room is as quoted.
• The type of a flag F is the set.
• For a subset of the type set, each flag is called the type A as an A - flag.
• The Kotyp a flag F is the set.
• The residue of a flag F is the geometry by

• The rank of is the cardinality of, ie the number of types that are represented in.
• The rank of a flag F is the number of elements of F, its corank is the rank of.

Basic diagram of a geometry

Be a geometry over. A pair of different elements ( "types" ) are called connected ie they form an edge of the base diagram, if at least one flag with the Kotyp exists, the residual is not a generalized -gon.

Examples

• An incidence structure is a rank -2 geometry. It is called a generalized -gon. In the diagram, a generalized Zweieck usually is not drawn.
• A projective plane is a rank -2 geometry. It is represented by a line with no marker on the chart.
• A three-dimensional affine space A is a rank -3 geometry with the type size, the possible dimensions of the real subspaces. If p is a point in space, then a flag type, ie of rank 1 and corank 2, the remainder is made up of all straight lines and planes that contain p. This is a projective plane! In contrast, the residue of a level in A is an affine plane, therefore, the line connecting points and lines in the diagram as " affine " marks, see the figure on the right. The residue of a straight line consists of all points on the line and all levels, the line included. This rank -2 geometry is a generalized -gon, so the graph is no direct connection of the dots will be drawn to the planes.