Quadrangles of Type F4

This chapter deals with the case that the building at infinity Λ‎ of the Bruhat-Tits building Ξ‎ is a Moufang quadrangle of type F₄. It begins with the hypothesis stating that Λ‎ = (K, L, q) is a quadratic space of type F₄, K is complete with respect to a discrete valuation ν‎ and F is closed with respect to ν‎, Λ‎ is the Moufang quadrangle corresponding to a root group sequence, and R₀ and R₁ as the two residues of Ξ‎. The chapter also considers the theorem supposing that Λ‎ is of type F₄ and that R₀ and R₁ are not both indifferent, and claims that both cases really occur. Finally, it presents the proposition that R₀ and R₁ are both indifferent if and only if q is totally wild.

Residues

This chapter deals with the residues of a Bruhat-Tits building whose building at infinity is an exceptional quadrangle. It begins with the remark that if Λ‎ is an arbitrary quadratic space of type Eℓ for ℓ = 6, 7 or 8 or of typeF₄ over a field K that is complete with respect to a discrete valuation, and if in the F4-case the subfield F is closed with respect to this valuation and if Δ‎ is the corresponding Moufang quadrangle of type Eℓ or F₄, then there always exists a unique affine building Ξ‎ such that Δ‎ is the building at infinity of Ξ‎ with respect to its complete system of apartments. The chapter also considers the standard embedding of the apartment A in the Euclidean plane which takes the intersection of A and R to the set of eight triangles containing the origin. Finally, it describes a Moufang polygon with two root group sequences.

Unramified Quadrangles of Type E6, E7 and E8

This chapter deals with the case that the building at infinity of the Bruhat-Tits building Ξ‎ is a Moufang quadrangle of type E⁶, E₇, and E₈. It begins with a hypothesis that takes into account a quadratic space of type Eℓ for ℓ = 6, 7 or 8, K which is complete with respect to a discrete valuation, the two residues of Ξ‎, and the two root group sequences of a Moufang polygon. It then considers the case that Ξ‎ is an unramified quadrangle if the proposition δ‎Ψ‎ = 2 holds. It also explains two other propositions: Ξ‎ is a semi-ramified quadrangle if δ‎Λ‎ = 1 and δ‎Ψ‎ = 2 holds, and a ramified quadrangle if δ‎Λ‎ = δ‎Ψ‎ = 1 holds.

Moufang Quadrangles

This chapter proves various results about Moufang quadrangles. It first considers the notions of a proper involutory set, a proper indifferent set, and a proper anisotropic pseudo-quadratic space. It then shows that the root group sequence Ω‎ is isomorphic to a root group sequence of exactly one of six types relating to some proper involutory set, some non-trivial anisotropic quadratic space, some proper indifferent set, some proper anisotropic pseudo-quadratic space, and some quadratic space. It also describes the degree of a finite purely inseparable field extension as a power of the characteristic, an isomorphism from a root group sequence of Δ‎ to the Moufang quadrangle, and abelian and non-abelian groups.