COMBINATORIAL AND GROUP-THEORETIC COMPACTIFICATIONS OF BUILDINGS

Let X be a building of arbitrary type. A compactification phi(sph)(X) of the set Res(sph)(X) of spherical residues of X is introduced. We prove that it coincides with the horofunction compactification of Res(sph)(X) endowed with a natural combinatorial distance which we call the root-distance. Points of phi(sph) (X) admit amenable stabilisers in Aut(X) and conversely, any amenable subgroup virtually fixes a point in phi(sph)(X). In addition, it is shown that, provided Aut(X) is transitive enough, this compactification also coincides with the group-theoretic compactification constructed using the Chabauty topology on closed subgroups of Aut(X). This generalises to arbitrary buildings results established by Y. Guivarc'h and B. Remy [20] in the Bruhat-Tits case.