Geometrical Constructions of Compactifications

Abstract

In this chapter several geometrical compactifications are described that are relevant to the rest of this book. The first one is the conic compactification (see §3.1). When X is identified with p, it amounts to adjoining a sphere of codimension 1 at infinity to a Euclidean space in the usual way. It turns out that this sphere X(∞) at infinity may be given the structure of a simplicial complex Δ(X) (see Theorem 3.15 and Proposition 3.18) with respect to which it is a spherical Tits building (Definition 3.14). This is accomplished by identifying the sets of points in X(∞) stabilized by the various parabolic subgroups (see Proposition 3.9).