Combinatorial and group-theoretic compactifications of buildings

[Compactifications combinatoires et de Chabauty des immeubles]

Caprace, Pierre-Emmanuel ; Lécureux, Jean

Annales de l'Institut Fourier, Tome 61 (2011), p. 619-672 / Harvested from Numdam

Résumé

Soit $X$ un immeuble de type arbitraire. Nous introduisons une compactification de l’ensemble des résidus sphériques $\text{Res}_{\mathrm{sph}}(X)$ de $X$. Nous démontrons que celle-ci coïncide avec la compactification de Busemann de $\text{Res}_{\mathrm{sph}}(X)$, 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 $\text{Aut}(X)$, cette compactification coïncide avec la compactification par la topologie de Chabauty sur les sous-groupes de $\text{Aut}(X)$. Ceci généralise aux immeubles arbitraires des résultats de Y. Guivarc’h et B. Rémy sur le cas d’immeubles de Bruhat-Tits.

Abstract

Let $X$ be a building of arbitrary type. A compactification $\mathscr{C}_{\mathrm{sph}}(X)$ of the set $\text{Res}_{\mathrm{sph}}(X)$ of spherical residues of $X$ is introduced. We prove that it coincides with the horofunction compactification of $\text{Res}_{\mathrm{sph}}(X)$ endowed with a natural combinatorial distance which we call the root-distance. Points of $\mathscr{C}_{\mathrm{sph}}(X)$ admit amenable stabilisers in $\text{Aut}(X)$ and conversely, any amenable subgroup virtually fixes a point in $\mathscr{C}_{\mathrm{sph}}(X)$. In addition, it is shown that, provided $\text{Aut}(X)$ is transitive enough, this compactification also coincides with the group-theoretic compactification constructed using the Chabauty topology on closed subgroups of $\text{Aut}(X)$. This generalises to arbitrary buildings results established by Y. Guivarc’h and B. Rémy  [20] in the Bruhat–Tits case.

Détail

Publié le : 2011-01-01

MR 2895068 | Zbl 1266.51016

DOI : https://doi.org/10.5802/aif.2624
Classification:  20E42,  20G25,  22E20,  22F50,  51E24
Mots clés: compactification, immeuble, topologie de Chabauty, groupe moyennable

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