Buekenhout Tits Geometry

The designation Buekenhout-Tits geometry (also called Buekenhout geometry or diagram geometry ) stands in geometry for a common generalization of the terms projective geometry , affine geometry , block plan , linear space and many other related terms. The concept was largely developed in the years after 1956 by Jacques Tits and later his student Francis Buekenhout , after whom it is now named. The basic idea of ​​this concept is to largely refrain from details of the geometric structure and to investigate the properties of classic structures and their generalizations that are associated with the classic geometric term "flag" .

The diagram geometry was applied by Tits with some success to ( non-commutative ) finite simple groups and their classification. These groups can hardly be further broken down in a meaningful way with purely group-theoretical methods: Their normal divisional lattice is trivial and their subgroup lattice is too large for the smallest representatives and too little characteristic in its structure to offer a starting point for investigations, let alone a classification could. On the other hand, it has long been known that many of the simple groups operate on classical geometric structures or their generalizations as full automorphism groups or as one of their subgroups or factor groups (see Wittscher block diagram as an example ); these "geometric" structures are often projective planes or (more generally) Block plans. Tits initially consisted of assigning a suitable “composite, geometric” structure to a group that operates on geometric structures of different types as a group of automorphisms, which reflects as much information as possible from the various initial structures.

A Buekenhout diagram for a rank 4 geometry. The residual according to one of the types 1 to 3 is a projective space (rank 3), the residual according to two of these types, e.g. B. after is each a projective plane. The arrows indicate a (conceivable) triality (comparable to geometric duality ) - they are not themselves part of the Buekenhout diagram. Models of such self-trialling geometries can be constructed from suitable quadratic sets , for example from Klein's quadric in a seven-dimensional, finite projective space. This is no longer a basic diagram, since the axiom (TP) is only fulfilled for the residuals according to one of the types 1 to 3. ${\displaystyle \{1,2\}}$

Another important application is the investigation of induced geometries that result, for example, from quadratic sets on finite projective spaces, compare the figure at the end of the introduction. It is historically noteworthy that as early as 1896 Eliakim Hastings Moore proposed a concept for an abstract geometry that essentially corresponds to the geometry of the diagram. In Moore's time this was not pursued any further.

Guiding principles

The diagram geometry generalizes concepts that have arisen on the basis of questions from very different areas of mathematics. Therefore, many geometric terms are filled with a new, more general content that often does not generalize the otherwise usual terms in a formal sense. An incidence structure is , for example, no geometry in terms of the graph geometry, but any incidence structure can also naturally as a diagram geometry (and of rank 2) interpreted . The concept of (finite-dimensional) projective geometries can be used as the main idea here, which is why the corresponding term for projective geometries is often contrasted in this article with the term from diagram geometry. In fact, projective planes together with two types of trivial rank-2 geometries form the most important basic building blocks of the diagram geometries investigated to date.

Basic concepts from projective geometry

A Buekenhout diagram for a three-dimensional projective space. The residue after a point ( point ) or a plane ( plane ) is in each case a projective plane. In contrast, the residue is a straight line ( line ) a generalized Zweieck

A finite-dimensional projective space determines the set of its real projective subspaces (including the set of its points, but here without the empty set and the total space). Each element of can be assigned a “type” from an index set (here, for example, its respective projective dimension) using a “typing function” . The set of these types can be described by a finite set, for example . The number of types actually occurring is then the projective dimension (or the “rank”) of the total space. On the set of "elements" (real subspaces) an incidence relation is given by the symmetrized subspace relation . ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle t:S\rightarrow \Delta }$ ${\displaystyle \Delta }$ ${\displaystyle \Delta =\{{\text{Punkt, Gerade, Ebene,}}\ldots \}}$ ${\displaystyle |t(S)|}$ ${\displaystyle S}$ ${\displaystyle UIV\leftrightarrow \left(U\leq V\lor V\leq U\right)}$

A flag in such a projective geometry is a subset of totally ordered by the antisymmetric incidence (here: the unsymmetrical subspace relation) . Such a flag is also finite for an infinite projective space, provided that the dimension of this space is finite and the length of the flag is not greater than this projective dimension of the space. ${\displaystyle S}$

Definitions

geometry

Be a set, its elements and subsets are called types . A (diagram) geometry about consists of a triplet , with an amount a symmetrical and reflexive relation to , the incidence ratio is a surjective function , the typing function when the following axiom (TP = transversality property ) is satisfied: ${\displaystyle \Delta }$ ${\displaystyle \Gamma }$ ${\displaystyle \Delta }$ ${\displaystyle \Gamma =(S,I,t)}$ ${\displaystyle S}$ ${\displaystyle I}$ ${\displaystyle S}$ ${\displaystyle t}$ ${\displaystyle t:S\rightarrow \Delta }$

(TP) If there is a maximum set of pairwise incident elements, then the restriction of the typing function is to a bijection . ${\displaystyle M\subseteq S}$ ${\displaystyle t}$ ${\displaystyle M}$ ${\displaystyle t|_{M}:\;M\leftrightarrow \Delta }$

Flag, room, residual, rank

Be a geometry about . ${\displaystyle \Gamma =(S,I,t)}$ ${\displaystyle \Delta }$

• A flag of is a (possibly empty ) set of pairwise incidental elements of . ${\displaystyle \Gamma }$ ${\displaystyle F}$ ${\displaystyle S}$
• Two flags are called incident if there is also one flag. ${\displaystyle F_{1},F_{2}}$ ${\displaystyle F_{1}\cup F_{2}}$
• Maximum flags hot room (Engl. Chambers ).
• The set of all rooms of is noted as. ${\displaystyle \Gamma }$ ${\displaystyle \mathop {\mathrm {Cham} } (\Gamma )}$
• The type of a flag is the amount . ${\displaystyle F}$ ${\displaystyle t(F)}$
• For a subset of the type set , each flag of the type is also referred to as a flag. ${\displaystyle A\subseteq \Delta }$ ${\displaystyle A}$ ${\displaystyle A}$
• The kotype of a flag is the amount . ${\displaystyle F}$ ${\displaystyle \Delta \setminus t(F)}$
• The residual of a plume in is the geometry over that passes through ${\displaystyle F}$ ${\displaystyle \Gamma }$ ${\displaystyle \Gamma _{F}=(S_{F},I_{F},t_{F})}$ ${\displaystyle \Delta _{F}=t(S_{F})=\Delta \setminus t(F)}$

${\displaystyle S_{F}=\{x\in S\setminus F|\forall y\in F:xIy\}}$and the limitation of the incidence relation to and the typing function is given. ${\displaystyle I_{F}}$ ${\displaystyle I}$ ${\displaystyle S_{F}}$ ${\displaystyle t_{F}=t|_{F}}$

• The rank of is the power of , i.e. the number of types that are represented in. ${\displaystyle r(\Gamma )}$ ${\displaystyle \Gamma }$ ${\displaystyle \Delta }$ ${\displaystyle |t(s)|}$ ${\displaystyle \Gamma }$
• The rank of a flag is the number of elements of , its Quran is the rank of . ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle \Gamma _{F}}$

Basic diagram of a geometry

Be a geometry about . A pair of different elements (“types”) are called connected, that is, they form an edge of the basic diagram if there is at least one flag with the cotype , the residue of which is not a generalized two-triangle. ${\displaystyle \Gamma =(S,I,t)}$ ${\displaystyle \Delta }$ ${\displaystyle (i,j)\in \Delta ^{2}}$ ${\displaystyle \{i,j\}}$

Examples

Buekenhout diagram of a three-dimensional affine space: The residue after a "point" ( point ) is a projective space, therefore, the remaining at Residuenbildung edge between is "straight line" ( line () and "plane" plane ) are not explicitly indicated, on the other hand is the residual (for a "plane" plane ) has an affinity level that is therefore characterized. The labeling can vary from author to author.

• An incidence structure with is a rank geometry. It is called a generalized two-triangle . A generalized two-cornered triangle is usually not drawn in the diagram. ${\displaystyle ({\mathfrak {p}},{\mathfrak {G}},{\mathfrak {p}}\times {\mathfrak {G}})}$ ${\displaystyle |{\mathfrak {p}}|,|{\mathfrak {G}}|\geq 2}$ ${\displaystyle 2}$
• A projective plane is a rank geometry. It is shown in the diagram by a line without a marker. ${\displaystyle 2}$
• A three-dimensional affine space is a rank geometry with the type set , the possible dimensions of the real subspaces. If there is a point in space, then a flag is of the type , i.e. of rank and Quran . The residual of consists of all straight lines and planes that contain. This is a projective plane! In contrast, the residual of a plane is in an affine plane , so the line connecting the points and straight lines in the diagram is marked as "affine", compare the illustration on the right. The residual of a straight line consists of all points on the straight line and all planes that contain the straight line. This rank geometry is a generalized two-triangle, so no direct connection from the points to the levels is drawn in the diagram. ${\displaystyle A}$ ${\displaystyle 3}$ ${\displaystyle \{0,1,2\}}$ ${\displaystyle p}$ ${\displaystyle \{p\}}$ ${\displaystyle \{0\}}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle \{p\}}$ ${\displaystyle p}$ ${\displaystyle A}$ ${\displaystyle 2}$

literature

• Albrecht Beutelspacher , Ute Rosenbaum: Projective geometry . From the basics to the applications (=  Vieweg Studium: advanced course in mathematics ). 2nd, revised and expanded edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X ( table of contents [accessed on August 11, 2012]).
• Francis Buekenhout: Diagrams for geometries and groups . In: J. Comb. Th. (A) . tape 27 , 1979, pp. 121–151 (First version of the basic terms of diagram geometry, some terms have since had a modified (generalized) definition).
• Francis Buekenhout: The basic diagram of a geometry . In: Geometry and Groups . Springer, Berlin / Heidelberg / New York 1981, pp. 1–29 (basic terms and overview of elementary results).
• Francis Buekenhout: The geometry of diagrams . In: Geo. Ded. tape 8 , 1979, pp. 253-257 .
• BA Cooperstein: A characterization of some Lie incidence structures . In: Geo. Ded. tape 6 , 1971, p. 232-246 .
• Eliakim Hastings Moore: Tactical Memoranda . In: Amer. J. Math . tape 18 . Berlin / Heidelberg / New York 1896, p. 264-303 .
• A. Pasini: Diagrams and incidence structers . Preprint. In: Rapporto matematico . tape 21 . Istituto di Mat., Univ. Siena, Berlin / Heidelberg / New York 1980, p. 264-303 .
• J. Tits: Les groupes de Lie exceptionnels et leur interprétation géométrique . In: Bull. Soc. Math. Belg. tape 8 , 1956, pp. 48-81 .
• J. Tits: Buildings and Buekenhout Geometries . In: M. Collins (Ed.): Finite Simple Groups II . Academic Press, New York 1981, pp. 309-320 .

References and comments

1. ↑ ^{a b} Beutelspacher and Rosenbaum 2004
2. ↑ Tits (1981)
3. ↑ Buekenhout (1981)
4. ↑ ^{a b} Tits (1956)
5. ↑ Beutelspacher and Rosenbaum (2004) 4.7: The Kleinsche quadratische set
6. ^ Moore (1896)
7. ↑ Buekenhout (1981), p. 2: "Strangely enough the concept of geometry as presented here appears very clearly in a paper of EH Moore as early as 1896!" - What is meant is the article Moore (1896)
8. ↑ Buekenhout (1981), the present definition with only one axiom (TP) represents a weakening of the original definition from Buekenhout (1979). The validity of the two additional axioms (SC) and (GL) is given by Buekenhout (1981) Expressions “ strongly connected geometry” (SC) or “fulfills the linearity condition ” (LC).