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

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
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)
@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, Tome 61 (2011) no. 2, pp. 619-672. doi : 10.5802/aif.2624. https://aif.centre-mersenne.org/articles/10.5802/aif.2624/

[1] Abramenko, Peter; Brown, Kenneth S. Buildings, Graduate Texts in Mathematics, 248, Springer, New York, 2008 (Theory and applications) | MR | Zbl

[2] Anantharaman-Delaroche, C.; Renault, J. Amenable groupoids, Monographies de L’Enseignement Mathématique, 36, L’Enseignement Mathématique, Geneva, 2000 | MR | Zbl

[3] Ballmann, Werner; Gromov, Mikhaïl; Schroeder, Viktor Manifolds of nonpositive curvature, Progress in Mathematics, 61, Birkhäuser Boston Inc., Boston, MA, 1985 | MR | Zbl

[4] Balser, Andreas; Lytchak, Alexander Centers of convex subsets of buildings, Ann. Global Anal. Geom., Volume 28 (2005) no. 2, pp. 201-209 | DOI | MR | Zbl

[5] Borel, Armand; Ji, Lizhen Compactifications of symmetric and locally symmetric spaces, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 2006 | MR | Zbl

[6] Bourbaki, Nicolas Groupes et Algèbres de Lie, Chapitre IV-VI, Springer-Verlag, 2007

[7] Bourbaki, Nicolas Intégration, Chapitre VIII, Springer-Verlag, 2007

[8] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer-Verlag, Berlin, 1999 | MR | Zbl

[9] Brodzki, J.; Campbell, S. J.; Guentner, E.; Niblo, G.; Wright, N. J. Property A and CAT(0) Cube Complexes Journal of Functional Analysis (to appear)

[10] Brown, Kenneth S. Buildings, Springer-Verlag, New York, 1989 | MR | Zbl

[11] Bruhat, F.; Tits, Jacques Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math., Volume 41 (1972), pp. 5-251 | DOI | Numdam | MR | Zbl

[12] Caprace, Pierre-Emmanuel Amenable groups and Hadamard spaces with a totally disconnected isometry group, Comment. Math. Helv., Volume 84 (2009), pp. 437-455 | DOI | MR

[13] Caprace, Pierre-Emmanuel; Haglund, Frederic On geometric flats in the CAT (0) realization of Coxeter groups and Tits buildings, Canad. J. Math., Volume 61 (2009) no. 4, pp. 740-761 | DOI | MR

[14] Caprace, Pierre-Emmanuel; Lytchak, Alexander At infinity of finite-dimensional CAT(0) spaces, Math. Ann., Volume 346 (2010) no. 1, pp. 1-21 | DOI | MR | Zbl

[15] Davis, Michael W. Buildings are CAT (0), Geometry and cohomology in group theory (Durham, 1994) (London Math. Soc. Lecture Note Ser.), Volume 252, Cambridge Univ. Press, Cambridge, 1998, pp. 108-123 | MR | Zbl

[16] Deodhar, Vinay V. A note on subgroups generated by reflections in Coxeter groups, Arch. Math. (Basel), Volume 53 (1989) no. 6, pp. 543-546 | MR | Zbl

[17] Farb, B. Relatively hyperbolic groups, Geom. Funct. Anal., Volume 8 (1998) no. 5, pp. 810-840 | DOI | MR | Zbl

[18] Ghys, Étienne; de la Harpe, Pierre L’action au bord des isométries, Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988) (Progr. Math.), Volume 83, Birkhäuser Boston, Boston, MA, 1990, pp. 135-163 | Zbl

[19] Guivarc’h, Yves; Ji, Lizhen; Taylor, J. C. Compactifications of symmetric spaces, Progress in Mathematics, 156, Birkhäuser Boston Inc., Boston, MA, 1998 | MR | Zbl

[20] Guivarc’h, Yves; Rémy, Bertrand Group-theoretic compactification of Bruhat-Tits buildings, Annales scientifiques de l’ENS, Volume 39 (2006), pp. 871-920 | MR | Zbl

[21] Guralnick, Dan Coarse decompositions for boundaries of CAT(0) groups (2008) (Preprint)

[22] Hruska, G. Christopher; Kleiner, Bruce Hadamard spaces with isolated flats, Geom. Topol., Volume 9 (2005), p. 1501-1538 (electronic) (with an appendix by the authors and Mohamad Hindawi) | DOI | MR | Zbl

[23] Katok, Svetlana Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992 | MR | Zbl

[24] Kloeckner, Benoît The space of closed subgroups of n is stratified and simply connected, J. Topol. , Volume 2 (2009) no. 3, pp. 570-588 | DOI | MR | Zbl

[25] Landvogt, Erasmus A compactification of the Bruhat-Tits building, Lecture Notes in Mathematics, 1619, Springer-Verlag, Berlin, 1996 | MR | Zbl

[26] Leeb, Bernhard A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry, Bonner Mathematische Schriften [Bonn Mathematical Publications], 326, Universität Bonn Mathematisches Institut, Bonn, 2000 | MR | Zbl

[27] Ronan, Mark Lectures on buildings, Perspectives in Mathematics, 7, Academic Press Inc., Boston, MA, 1989 | MR | Zbl

[28] Rousseau, Guy Immeubles des groupes réducitifs sur les corps locaux, U.E.R. Mathématique, Université Paris XI, Orsay, 1977 (Thèse de doctorat, Publications Mathématiques d’Orsay, No. 221-77.68) | MR | Zbl

[29] Siebenmann, L. C. Deformation of homeomorphisms on stratified sets. I, II, Comment. Math. Helv., Volume 47 (1972), p. 123-136; ibid. 47 (1972), 137–163 | DOI | MR | Zbl

[30] Tits, Jacques Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin, 1974 | MR | Zbl

[31] Tits, Jacques A local approach to buildings, The geometric vein, Springer, New York, 1981, pp. 519-547 | MR | Zbl

[32] Willis, George Totally disconnected groups and proofs of conjectures of Hofmann and Mukherjea, Bull. Austral. Math. Soc., Volume 51 (1995) no. 3, pp. 489-494 | DOI | MR | Zbl

Cité par Sources :