a powerful tool to prove strong results for the associated groups.

If the algebraic group associated to a spherical building is defined over a field with a discrete valuation, then Bruhat and Tits constructed in [@BT:GRC:72] a second building associated to this group, a so-called affine building. These buildings are simplicial complexes which are covered by tessellated Euclidean spaces.

Affine buildings were classified by Bruhat and Tits in several following papers (see [@Wei:SAB:09] for a complete account). Higher rank affine buildings (more or less) correspond bijectively to semisimple isotropic algebraic groups over fields with discrete valuations. The classification heavily uses the fact that such buildings have a spherical building ‘at infinity’ such that the classification of spherical buildings can be used as an ingredient.

It is common to endow spherical and affine buildings with metrics which come from the canonical metrics on the spheres and Euclidean spaces by which they are covered. With these metrics, spherical buildings are naturally CAT(1)-spaces whereas Euclidean buildings are naturally CAT(0)-spaces. It turns out that buildings can also be characterised by these metric properties. All buildings have a geometric realisation as CAT(0)-spaces, the Davis realisation.

The notion of a building has been generalised in various 