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.

[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 $\mathrm{CAT}\left(0\right)$ 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 $\mathrm{CAT}\left(0\right)$, 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 ${ℝ}^n$ 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