This chapter considers the notion of parallel residues in a building. It begins with the assumption that Δ is a building of type Π, which is arbitrary except in a few places where it is explicitly assumed to be spherical. Δ is not assumed to be thick. The chapter then elaborates on a hypothesis which states that S is the vertex set of Π, (W, S) is the corresponding Coxeter system, d is the W-distance function on the set of ordered pairs of chambers of Δ, and ℓ is the length function on (W, S). It also presents a notation in which the type of a residue R is denoted by Typ(R) and concludes with the condition that residues R and T of a building will be called parallel if R = projR(T) and T = projT(R).
The Lipschitz continuity of the distance function to the cut locus
