** Vectorial metric compactification of symmetric spaces and affine buildings **

**1. The vectorial metric **

Let be a symmetric space or a building. Flats (apartments) are modelled on a Euclidean space equipped with the action of a finite Euclidean Coxeter group , the Weyl group. Weyl chambers are sectors forming fundamental domains for in . Pick one of them, .

Let be the folding projection .

Given two points , , there is a flat (isometric copy of ) containing and . Put at the origin. Then does not depend on choices, let us denote it by . This refines the canonical metric, given by , where is a -invariant Euclidean norm on .

satisfies a triangle inequality with respect to the following partial order:

This appears only in a hidden manner in Bruhat-Tits or Kostant, and in the special case of , in a paper by Ky Fan (1950).

The inequality is an equality iff belongs to the diamond of and , i.e. the intersection . In other words, iff belongs to a geodesic between and for any Finsler metric whose restriction to is linear.

Theorem 1The restriction of to pairs of geodesics is convex

This appears hidden in a paper by Kostant.

**2. Vectorial horo-compactification **

The idea is to modify the classical horo-bordification of metric spaces. Use to embed into the space of functions from to , equipped with the topology of uniform convergence on bounded subsets. Divide by the -action by translation.

Theorem 2If is locally compact, this gives an embedding whose closure is compact. It is equivariant under isometries of . This compactification dominates all linear Finsler horocompactifications (Kapovich-Leeb-Porti).

The boundary consists of functions called vectorial horofunctions.

Beware that does not dominate the compactification.

**3. Facets and transverse spaces at infinity **

Facets in are images of walls of in flats. Say two facets are *asymptotic* if they stay at bounded distance from each other. They are *strictly asymptotic* if their distance tends to 0 far from walls. Facets at infinity are strict asymptoticity equivalence classes of facets, viewed as subset of the visual compactification. They form the simplices of a simplicial complex structure on the visual compactification (Tits boundary).

Let be a facet at infinity. The tube at infinity in direction is the set of facets converging to . It is the product of a Euclidean space (the linear span of a facet) and a subsymmetric space.

Theorem 3There is a 1-1 correspondence between strong asymptoticity classes of facets and points of the boundary . Thus is stratified by subsymmetric spaces. In other words, coincides with the polyhedral or Satake compactification of .

For instance, the closure of a flat in has one point for each Weyl chamber, and a whole vectorsubspace for each wall.

Vector horofunctions can be described explicitly.