### $({S}_{3},{S}_{6})$-Amalgams IV

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We relate some features of Bruhat-Tits buildings and their compactifications to tropical geometry. If G is a semisimple group over a suitable non-Archimedean field, the stabilizers of points in the Bruhat-Tits building of G and in some of its compactifications are described by tropical linear algebra. The compactifications we consider arise from algebraic representations of G. We show that the fan which is used to compactify an apartment in this theory is given by the weight polytope of the representation...

We show that the group of type-preserving automorphisms of any irreducible semiregular thick right-angled building is abstractly simple. When the building is locally finite, this gives a large family of compactly generated abstractly simple locally compact groups. Specialising to appropriate cases, we obtain examples of such simple groups that are locally indecomposable, but have locally normal subgroups decomposing non-trivially as direct products, all of whose factors are locally normal.

We prove that the boundary of a right-angled hyperbolic building is a universal Menger space. As a consequence, the 3-dimensional universal Menger space is the boundary of some Gromov-hyperbolic group.

We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group $\mathrm{G}$ over a suitable non-Archimedean field $k$ we define a map from the Bruhat-Tits building $\mathcal{B}(\mathrm{G},k)$ to the Berkovich analytic space ${\mathrm{G}}^{\mathrm{an}}$ associated with $\mathrm{G}$. Composing this map with the projection of ${\mathrm{G}}^{\mathrm{an}}$ to its flag varieties, we define a family of compactifications of $\mathcal{B}(\mathrm{G},k)$. This generalizes results by Berkovich in the case of split groups. Moreover,...

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 ${\mathcal{C}}_{\mathrm{sph}}\left(X\right)$ 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 ${\mathcal{C}}_{\mathrm{sph}}\left(X\right)$ admit amenable stabilisers in $\text{Aut}\left(X\right)$ and conversely, any amenable subgroup virtually fixes a point in ${\mathcal{C}}_{\mathrm{sph}}\left(X\right)$. 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 $\Gamma \to {\mathrm{\mathbf{G}\mathbf{L}}}_{n}$ peut se définir en termes de l’action de $\Gamma $ sur l’immeuble de Tits de ${\mathrm{\mathbf{G}\mathbf{L}}}_{n}$. 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.

The question of embedding fields into central simple algebras $B$ over a number field $K$ was the realm of class field theory. The subject of embedding orders contained in the ring of integers of maximal subfields $L$ of such an algebra into orders in that algebra is more nuanced. The first such result along those lines is an elegant result of Chevalley [6] which says that with $B={M}_{n}\left(K\right)$ the ratio of the number of isomorphism classes of maximal orders in $B$ into which the ring of integers of $L$ can be embedded...

We give a new proof of a useful result of Guy Rousseau on Galois-fixed points in the Bruhat-Tits building of a reductive group.