Combinatorial and group-theoretic compactifications of buildings

Caprace, Pierre-Emmanuel; Lécureux, Jean

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

Soit $X$ un immeuble de type arbitraire. Nous introduisons une compactification de l’ensemble des résidus sphériques ${\text{Res}}_{\mathrm{sph}}\left(X\right)$ de $X$. Nous démontrons que celle-ci coïncide avec la compactification de Busemann de ${\text{Res}}_{\mathrm{sph}}\left(X\right)$, 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 $\text{Aut}\left(X\right)$, cette compactification coïncide avec la compactification par la topologie de Chabauty sur les sous-groupes de $\text{Aut}\left(X\right)$. 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 ${𝒞}_{\mathrm{sph}}\left(X\right)$ 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 ${𝒞}_{\mathrm{sph}}\left(X\right)$ admit amenable stabilisers in $\text{Aut}\left(X\right)$ and conversely, any amenable subgroup virtually fixes a point in ${𝒞}_{\mathrm{sph}}\left(X\right)$. In addition, it is shown that, provided $\text{Aut}\left(X\right)$ is transitive enough, this compactification also coincides with the group-theoretic compactification constructed using the Chabauty topology on closed subgroups of $\text{Aut}\left(X\right)$. This generalises to arbitrary buildings results established by Y. Guivarc’h and B. Rémy  [20] in the Bruhat–Tits case.

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

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