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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...

A universal construction for projective Hjelmslev planes of level $n$

G. Hanssens, H. Van Maldeghem (1989)

Compositio Mathematica

All geometries of the Mathieu group $M_{11}$ based on maximal subgroups.

Buekenhout, Francis, Dehon, Michel, Leemans, Dimitri (1996)

Experimental Mathematics

Bruhat-Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings

Bertrand Rémy, Amaury Thuillier, Annette Werner (2010)

Annales scientifiques de l'École Normale Supérieure

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

Buildings and non-positively curved polygons of finite groups.

Brown, Paul R. (1998)

The New York Journal of Mathematics [electronic only]

Chamber systems, geometries and parabolic systems whose diagram contains only bonds of strength 1 and 2.

G. Stroth (1990)

Inventiones mathematicae

Combinatorial and group-theoretic compactifications of buildings

Pierre-Emmanuel Caprace, Jean Lécureux (2011)

Annales de l’institut Fourier

Let $X$ be a building of arbitrary type. A compactification $𝒞_{sph} (X)$ 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 $𝒞_{sph} (X)$ admit amenable stabilisers in $Aut (X)$ and conversely, any amenable subgroup virtually fixes a point in $𝒞_{sph} (X)$ . In addition, it is shown that, provided $Aut (X)$ is transitive enough, this compactification also coincides with the group-theoretic...

Combing Euclidean buildings.

Noskov, Gennady A. (2000)

Geometry & Topology

Complète réductibilité

Jean-Pierre Serre (2003/2004)

Séminaire Bourbaki

La notion de complète réductibilité d’une représentation linéaire $Γ \to {𝐆𝐋}_{n}$ peut se définir en termes de l’action de $Γ$ sur l’immeuble de Tits de ${𝐆𝐋}_{n}$ . 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.

Eigenfunctions of the Laplace operators for buildings of type ${\tilde{B}}_{2}$

A. M. Mantero, A. Zappa (2002)

Bollettino dell'Unione Matematica Italiana

We consider for an affine building of type ${\tilde{B}}_{2}$ Helgason's conjecture with respect to Laplace operators defined over different types of vertices. We prove that there are cases in which the conjecture fails, since there exist eigenfunctions which are not the Poisson transform of finitely additive measures at the maximal boundary of the building.

Extending locally truncated chamber systems by sheaves.

Pasini, A. (2003)

Advances in Geometry

Flag transitive $c \cdot A f^{*}$ geometries.

Baumeister, Barbara, Del Fra, Alberto, Meixner, Thomas, Pasini, Antonio (1996)

Beiträge zur Algebra und Geometrie

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