Combinatorial and group-theoretic compactifications of buildings

Caprace, Pierre-Emmanuel; Lécureux, Jean

doi:10.5802/aif.2624

Combinatorial and group-theoretic compactifications of buildings
[Compactifications combinatoires et de Chabauty des immeubles]

Caprace, Pierre-Emmanuel ¹ ; Lécureux, Jean ²

¹ UCLouvain, Département de Mathématiques, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve (Belgium)
² Université de Lyon; Université Lyon 1; INSA de Lyon; Ecole Centrale de Lyon; CNRS, UMR5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, F-69622 Villeurbanne-Cedex (France)

Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 619-672.

Résumé
Abstract

Soit $X$ un immeuble de type arbitraire. Nous introduisons une compactification de l’ensemble des résidus sphériques ${Res}_{sph} (X)$ de $X$ . Nous démontrons que celle-ci coïncide avec la compactification de Busemann de ${Res}_{sph} (X)$ , lorsqu’on munit celui-ci d’une distance combinatoire naturelle apellée la distance radicielle. Les stabilisateurs de points du bord sont moyennables ; réciproquement, tout groupe moyennable d’automorphismes de $X$ fixe un point du compactifié. De plus, nous démontrons que, sous certaines conditions de transitivité de $Aut (X)$ , cette compactification coïncide avec la compactification par la topologie de Chabauty sur les sous-groupes de $Aut (X)$ . Ceci généralise aux immeubles arbitraires des résultats de Y. Guivarc’h et B. Rémy sur le cas d’immeubles de Bruhat-Tits.

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 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. Rémy [20] in the Bruhat–Tits case.

MR Zbl

DOI : 10.5802/aif.2624

Classification : 20E42, 20G25, 22E20, 22F50, 51E24
Keywords: Compactification, building, Chabauty topology, amenable group
Mot clés : compactification, immeuble, topologie de Chabauty, groupe moyennable

Affiliations des auteurs :

Caprace, Pierre-Emmanuel ¹ ; Lécureux, Jean ²

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     title = {Combinatorial and group-theoretic compactifications of buildings},
     journal = {Annales de l'Institut Fourier},
     pages = {619--672},
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Caprace, Pierre-Emmanuel; Lécureux, Jean. Combinatorial and group-theoretic compactifications of buildings. Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 619-672. doi : 10.5802/aif.2624. https://aif.centre-mersenne.org/articles/10.5802/aif.2624/

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