Combinatorial and group-theoretic compactifications of buildings
Annales de l'Institut Fourier, Volume 61 (2011) no. 2, pp. 619-672.

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.

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.

DOI: 10.5802/aif.2624
Classification: 20E42,  20G25,  22E20,  22F50,  51E24
Keywords: Compactification, building, Chabauty topology, amenable group
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)
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Caprace, Pierre-Emmanuel; Lécureux, Jean. Combinatorial and group-theoretic compactifications of buildings. Annales de l'Institut Fourier, Volume 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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