Tits polygons

We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a “rank 2 2 ” presentation for the group of F F -rational points of an arbitrary exceptional simple group of F F -rank at least 4 4 and to determine defining relations for the group of F F -rational points of an an arbitrary group of F F -rank 1 1 and absolute type D 4 D_4 , E 6 E_6 , E 7 E_7 or E 8 E_8 associated to the unique vertex of the Dynkin diagram that is not orthogonal to the highest root. All of these results are over a field of arbitrary characteristic.

Galois Involutions

This chapter focuses on the fixed points of a strictly semi-linear automorphism of order 2 of a spherical building which satisfies the conditions laid out in Hypothesis 30.1. It begins with the fhe definition of a spherical building satisfying the Moufang condition and a Galois involution of Δ‎, described as an automorphism of Δ‎ of order 2 that is strictly semi-linear. It can be recalled that Δ‎ can have a non-type-preserving semi-linear automorphism only if its Coxeter diagram is simply laced. The chapter assumes that the building Δ‎ being discussed is as in 30.1 and that τ‎ is a Galois involution of Δ‎. It also considers the notation stating that the polar region of a root α‎ of Δ‎ is the unique residue of Δ‎ containing the arctic region of α‎.

Pseudo-Split Buildings

This chapter introduces a class of Moufang spherical buildings known as pseudo-split buildings and considers the notion of the field of definition of a spherical building satisfying the Moufang condition. It begins with the notation: Let Δ‎ be an irreducible spherical building satisfying the Moufang condition, and let ℓ denote its rank (so ℓ is greater than or equal to 2 by definition). It then characterizes pseudo-split buildings as the spherical buildings which can be embedded in a split building of the same type. It also presents the proposition stating that every pseudo-split building is a subbuilding of a split building.

Buildings

This chapter assembles a few standard definitions, fixes some notation, and reviews a few of the results about buildings and Moufang polygons. It also summarizes the basic facts about Coxeter groups and buildings, including the fundamental properties of roots, residues, apartments, and projection maps. The chapter defines a Moufang building as spherical, thick, irreducible and of rank at least 2, and a Bruhat-Tits building as a thick irreducible affine building whose building at infinity is Moufang. Furthermore, it presents a fundamental result of Tits: that an irreducible thick spherical building of rank at least 3 satisfies the Moufang condition as do all the irreducible residues of rank at least 2 of such a building. Finally, it considers a simplicial complex, the dimension of which is its cardinality minus one.

Moufang Structures

This chapter uses the notion of a Moufang structure to show that if Δ‎ is a spherical building satisfying the Moufang condition and Γ‎ is a descent group of Δ‎, then the fixed point building Δ‎Γ‎ also satisfies the Moufang condition. The discussion begins with the notation: Let (W, S) denote the type of Δ‎, let G = Aut(Δ‎) and let G° denote the group of type-preserving elements of G. The chapter then presents the conditions for an element g of G to be unipotent and for a subgroup U of G to be unipotent. It also describes a unipotent group U stabilizing a residue R and a unipotent element fixing two chambers x and y. Finally, it considers the set of extensions that forms a group acting faithfully on R.

Moufang affine buildings have Moufang spherical buildings at infinity

We show in a direct and elementary way that the spherical building at infinity of every rank 3 affine building which satisfies Tits' Moufang condition, is itself a Moufang building. This result is also true for higher rank affine buildings by Tits' classification [4].