Combinatorial and group-theoretic compactifications of buildings

Pierre-Emmanuel Caprace[1]; Jean Lécureux[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)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 2, page 619-672
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Caprace, Pierre-Emmanuel, and Lécureux, Jean. "Combinatorial and group-theoretic compactifications of buildings." Annales de l’institut Fourier 61.2 (2011): 619-672. <http://eudml.org/doc/219688>.

@article{Caprace2011,
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.},
affiliation = {UCLouvain, Département de Mathématiques, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve (Belgium); 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)},
author = {Caprace, Pierre-Emmanuel, Lécureux, Jean},
journal = {Annales de l’institut Fourier},
keywords = {Compactification; building; Chabauty topology; amenable group; compactification},
language = {eng},
number = {2},
pages = {619-672},
publisher = {Association des Annales de l’institut Fourier},
title = {Combinatorial and group-theoretic compactifications of buildings},
url = {http://eudml.org/doc/219688},
volume = {61},
year = {2011},
}

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
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 2
SP - 619
EP - 672
AB - 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.
LA - eng
KW - Compactification; building; Chabauty topology; amenable group; compactification
UR - http://eudml.org/doc/219688
ER -

References

top
  1. Peter Abramenko, Kenneth S. Brown, Buildings, 248 (2008), Springer, New York Zbl1214.20033MR2439729
  2. C. Anantharaman-Delaroche, J. Renault, Amenable groupoids, 36 (2000), L’Enseignement Mathématique, Geneva Zbl0960.43003MR1799683
  3. Werner Ballmann, Mikhaïl Gromov, Viktor Schroeder, Manifolds of nonpositive curvature, 61 (1985), Birkhäuser Boston Inc., Boston, MA Zbl0591.53001MR823981
  4. Andreas Balser, Alexander Lytchak, Centers of convex subsets of buildings, Ann. Global Anal. Geom. 28 (2005), 201-209 Zbl1082.53032MR2180749
  5. Armand Borel, Lizhen Ji, Compactifications of symmetric and locally symmetric spaces, (2006), Birkhäuser Boston Inc., Boston, MA Zbl1100.22001MR2189882
  6. Nicolas Bourbaki, Groupes et Algèbres de Lie, Chapitre IV-VI, (2007), Springer-Verlag 
  7. Nicolas Bourbaki, Intégration, Chapitre VIII, (2007), Springer-Verlag 
  8. Martin R. Bridson, André Haefliger, Metric spaces of non-positive curvature, 319 (1999), Springer-Verlag, Berlin Zbl0988.53001MR1744486
  9. J. Brodzki, S. J. Campbell, E. Guentner, G. Niblo, N. J. Wright, Property A and CAT(0) Cube Complexes Zbl1233.20036
  10. Kenneth S. Brown, Buildings, (1989), Springer-Verlag, New York Zbl0922.20033MR969123
  11. F. Bruhat, Jacques Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5-251 Zbl0254.14017MR327923
  12. Pierre-Emmanuel Caprace, Amenable groups and Hadamard spaces with a totally disconnected isometry group, Comment. Math. Helv. 84 (2009), 437-455 Zbl1233.20037MR2495801
  13. Pierre-Emmanuel Caprace, Frederic Haglund, On geometric flats in the CAT ( 0 ) realization of Coxeter groups and Tits buildings, Canad. J. Math. 61 (2009), 740-761 Zbl1231.20034MR2541383
  14. Pierre-Emmanuel Caprace, Alexander Lytchak, At infinity of finite-dimensional CAT(0) spaces, Math. Ann. 346 (2010), 1-21 Zbl1184.53038MR2558883
  15. Michael W. Davis, Buildings are CAT ( 0 ) , Geometry and cohomology in group theory (Durham, 1994) 252 (1998), 108-123, Cambridge Univ. Press, Cambridge Zbl0978.51005MR1709955
  16. Vinay V. Deodhar, A note on subgroups generated by reflections in Coxeter groups, Arch. Math. (Basel) 53 (1989), 543-546 Zbl0688.20028MR1023969
  17. B. Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998), 810-840 Zbl0985.20027MR1650094
  18. Étienne Ghys, Pierre de la Harpe, L’action au bord des isométries, Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988) 83 (1990), 135-163, Birkhäuser Boston, Boston, MA Zbl0731.20025
  19. Yves Guivarc’h, Lizhen Ji, J. C. Taylor, Compactifications of symmetric spaces, 156 (1998), Birkhäuser Boston Inc., Boston, MA Zbl1053.31006MR1633171
  20. Yves Guivarc’h, Bertrand Rémy, Group-theoretic compactification of Bruhat-Tits buildings, Annales scientifiques de l’ENS 39 (2006), 871-920 Zbl1126.20029MR2316977
  21. Dan Guralnick, Coarse decompositions for boundaries of CAT(0) groups, (2008) 
  22. G. Christopher Hruska, Bruce Kleiner, Hadamard spaces with isolated flats, Geom. Topol. 9 (2005), 1501-1538 (electronic) Zbl1087.20034MR2175151
  23. Svetlana Katok, Fuchsian groups, (1992), University of Chicago Press, Chicago, IL Zbl0753.30001MR1177168
  24. Benoît Kloeckner, The space of closed subgroups of ℝ n is stratified and simply connected, J. Topol. 2 (2009), 570-588 Zbl1187.22010MR2546586
  25. Erasmus Landvogt, A compactification of the Bruhat-Tits building, 1619 (1996), Springer-Verlag, Berlin Zbl0935.20034MR1441308
  26. Bernhard Leeb, A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry, (2000), Universität Bonn Mathematisches Institut, Bonn Zbl1005.53031MR1934160
  27. Mark Ronan, Lectures on buildings, 7 (1989), Academic Press Inc., Boston, MA Zbl0694.51001MR1005533
  28. Guy Rousseau, Immeubles des groupes réducitifs sur les corps locaux, (1977), U.E.R. Mathématique, Université Paris XI, Orsay Zbl0412.22006MR491992
  29. L. C. Siebenmann, Deformation of homeomorphisms on stratified sets. I, II, Comment. Math. Helv. 47 (1972), 123-136; ibid. 47 (1972), 137–163 Zbl0252.57012MR319207
  30. Jacques Tits, Buildings of spherical type and finite BN-pairs, (1974), Springer-Verlag, Berlin Zbl0295.20047MR470099
  31. Jacques Tits, A local approach to buildings, The geometric vein (1981), 519-547, Springer, New York Zbl0496.51001MR661801
  32. George Willis, Totally disconnected groups and proofs of conjectures of Hofmann and Mukherjea, Bull. Austral. Math. Soc. 51 (1995), 489-494 Zbl0832.22005MR1331442

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.