Let be a building of arbitrary type. A compactification of the set of spherical residues of is introduced. We prove that it coincides with the horofunction compactification of endowed with a natural combinatorial distance which we call the root-distance. Points of admit amenable stabilisers in and conversely, any amenable subgroup virtually fixes a point in . In addition, it is shown that, provided is transitive enough, this compactification also coincides with the group-theoretic compactification constructed using the Chabauty topology on closed subgroups of . This generalises to arbitrary buildings results established by Y. Guivarc’h and B. Rémy [20] in the Bruhat–Tits case.
Soit un immeuble de type arbitraire. Nous introduisons une compactification de l’ensemble des résidus sphériques de . Nous démontrons que celle-ci coïncide avec la compactification de Busemann de , 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 fixe un point du compactifié. De plus, nous démontrons que, sous certaines conditions de transitivité de , cette compactification coïncide avec la compactification par la topologie de Chabauty sur les sous-groupes de . Ceci généralise aux immeubles arbitraires des résultats de Y. Guivarc’h et B. Rémy sur le cas d’immeubles de Bruhat-Tits.
Keywords: Compactification, building, Chabauty topology, amenable group
Mot clés : compactification, immeuble, topologie de Chabauty, groupe moyennable
Caprace, Pierre-Emmanuel 1; Lécureux, Jean 2
@article{AIF_2011__61_2_619_0, author = {Caprace, Pierre-Emmanuel and L\'ecureux, Jean}, title = {Combinatorial and group-theoretic compactifications of buildings}, journal = {Annales de l'Institut Fourier}, pages = {619--672}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {61}, number = {2}, year = {2011}, doi = {10.5802/aif.2624}, mrnumber = {2895068}, zbl = {1266.51016}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2624/} }
TY - JOUR AU - Caprace, Pierre-Emmanuel AU - Lécureux, Jean TI - Combinatorial and group-theoretic compactifications of buildings JO - Annales de l'Institut Fourier PY - 2011 SP - 619 EP - 672 VL - 61 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2624/ DO - 10.5802/aif.2624 LA - en ID - AIF_2011__61_2_619_0 ER -
%0 Journal Article %A Caprace, Pierre-Emmanuel %A Lécureux, Jean %T Combinatorial and group-theoretic compactifications of buildings %J Annales de l'Institut Fourier %D 2011 %P 619-672 %V 61 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2624/ %R 10.5802/aif.2624 %G en %F AIF_2011__61_2_619_0
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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