Let $K$ be a non-archimedean local field. This paper gives an explicit isomorphism between the dual of the special representation of $G{L}_{n+1}\left(K\right)$ and the space of harmonic cochains defined on the Bruhat-Tits building of $G{L}_{n+1}\left(K\right)$, in the sense of E. de Shalit [11]. We deduce, applying the results of a paper of P. Schneider and U. Stuhler [9], that there exists a $G{L}_{n+1}\left(K\right)$-equivariant isomorphism between the cohomology group of the Drinfeld symmetric space and the space of harmonic cochains.

Let $X$ be a building of arbitrary type. A compactification ${𝒞}_{\mathrm{sph}}\left(X\right)$ of the set ${\text{Res}}_{\mathrm{sph}}\left(X\right)$ of spherical residues of $X$ is introduced. We prove that it coincides with the horofunction compactification of ${\text{Res}}_{\mathrm{sph}}\left(X\right)$ endowed with a natural combinatorial distance which we call the root-distance. Points of ${𝒞}_{\mathrm{sph}}\left(X\right)$ admit amenable stabilisers in $\text{Aut}\left(X\right)$ and conversely, any amenable subgroup virtually fixes a point in ${𝒞}_{\mathrm{sph}}\left(X\right)$. In addition, it is shown that, provided $\text{Aut}\left(X\right)$ is transitive enough, this compactification also coincides with the group-theoretic...

In this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular, we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner property. This is motivated by the fact that the quasiconformal structure of the boundary led to many results of rigidity in hyperbolic spaces since G.D.Mostow. In the case of buildings...

La notion de complète réductibilité d’une représentation linéaire $\Gamma \to {\mathrm{𝐆𝐋}}_n$ peut se définir en termes de l’action de $\Gamma$ sur l’immeuble de Tits de ${\mathrm{𝐆𝐋}}_n$. Cela suggère une notion analogue pour tous les immeubles sphériques, et donc aussi pour tous les groupes réductifs. On verra comment cette notion se traduit en termes topologiques et quelles applications on peut en tirer.