Reduction

Abstract

In this section 4, Δ and Δ′ denote two buildings of rank at least 3, whose Weyl complexes are finite and irreducible. For A ∈ Δ or Δ′, and i ∈ $\mathop N\limits_ =$ such that i ≥ codim A, we denote by E_i(A) the set of all chambers having a face of codimension i in common with A. The purpose of this section is to establish the following two theorems:

• 4.1.1. THEOREM. Let Σ ⊂ Δ be an apartment and C ∈ Σ be a chamber. Then, any isomorphism ϕ : Δ → Δ′ is entirely determined by its restriction to E₁(C) ∪ Σ.

• 4.1.2. THEOREM. Let C ∈ Δ and C′ ∈ Δ′ be chambers. Then, every adjacence-preserving bijection ϕ : E₂(C) → E₂(C′) extends to an isomorphism of Δ onto Δ′.