Maps between buildings that preserve a given Weyl distance

Let Δ and Δ′ be two buildings of the same type (W, S), viewed as sets of chambers endowed with“distance” functions δ and δ′, respectively, admitting values in the common Weyl group W, which is a Coxeter group with standard generating set S. For a given element ω ε W, we study surjective maps ? : Δ → Δ′ with the property that δ(C, D) = ω if and only if Δ′ (?(C), ?(D)) = ω. The result is that the restrictions of ? to all residues of certain spherical types—determined by ω—are isomorphisms. We show with counterexamples that this result is optimal. We also demonstrate that, in many cases, this is enough to conclude that ? is an isomorphism. In particular, ? is an isomorphism if Δ and Δ′ are 2-spherical and every reduced expression of ω involves all elements of S.