Group-theoretic compactification of Bruhat–Tits buildings

Yves Guivarc'h; Bertrand Rémy

Annales scientifiques de l'École Normale Supérieure (2006)

  • Volume: 39, Issue: 6, page 871-920
  • ISSN: 0012-9593

How to cite

top

Guivarc'h, Yves, and Rémy, Bertrand. "Group-theoretic compactification of Bruhat–Tits buildings." Annales scientifiques de l'École Normale Supérieure 39.6 (2006): 871-920. <http://eudml.org/doc/82703>.

@article{Guivarch2006,
author = {Guivarc'h, Yves, Rémy, Bertrand},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {semisimple groups; non-Archimedean local fields; Bruhat-Tits buildings; Chabauty topology; polyhedral compactifications; amenable subgroups; distal subgroups; geometric parametrizations of subgroups; convergence theorems; parabolic subgroups; maximal compact subgroups; Euclidean buildings; semi-homogeneous trees},
language = {eng},
number = {6},
pages = {871-920},
publisher = {Elsevier},
title = {Group-theoretic compactification of Bruhat–Tits buildings},
url = {http://eudml.org/doc/82703},
volume = {39},
year = {2006},
}

TY - JOUR
AU - Guivarc'h, Yves
AU - Rémy, Bertrand
TI - Group-theoretic compactification of Bruhat–Tits buildings
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2006
PB - Elsevier
VL - 39
IS - 6
SP - 871
EP - 920
LA - eng
KW - semisimple groups; non-Archimedean local fields; Bruhat-Tits buildings; Chabauty topology; polyhedral compactifications; amenable subgroups; distal subgroups; geometric parametrizations of subgroups; convergence theorems; parabolic subgroups; maximal compact subgroups; Euclidean buildings; semi-homogeneous trees
UR - http://eudml.org/doc/82703
ER -

References

top
  1. [1] Adams S., Ballmann W., Amenable isometry groups of Hadamard spaces, Math. Ann.312 (1998) 183-195. Zbl0913.53012MR1645958
  2. [2] Abels H., Distal affine transformation groups, J. reine angew. Math.299 (1978) 294-300. Zbl0367.20047MR470944
  3. [3] Abels H., Distal automorphism groups of Lie groups, J. reine angew. Math.329 (1981) 82-87. Zbl0463.22006MR636446
  4. [4] Allcock D., Carlson J.A., Toledo D., The complex hyperbolic geometry of the moduli space of cubic surfaces, J. Algebraic Geometry11 (2002) 659-724. Zbl1080.14532MR1910264
  5. [5] Ash A., Mumford D., Rapoport M., Tai Y.S., Smooth Compactification of Locally Symmetric Spaces, Lie Groups: History, Frontiers and Applications, vol. IV, Math. Sci. Press, Brookline, MA, 1975. Zbl0334.14007MR457437
  6. [6] Baily W.L., Borel A., Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math.84 (1966) 442-528. Zbl0154.08602MR216035
  7. [7] Behr H., Higher finiteness properties of S-arithmetic groups in the function field case I, in: Müller T.W. (Ed.), Groups: Topological, Combinatorial and Arithmetic Aspects, London Math. Soc. Lecture Notes Series, vol. 311, Cambridge University Press, 2004, pp. 27-42. Zbl1099.20022MR2073345
  8. [8] Bridson M., Haefliger A., Metric Spaces of Non-Positive Curvature, Grund. Math. Wiss., vol. 319, Springer, 1999. Zbl0988.53001MR1744486
  9. [9] Borel A., Ji L., Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications, Birkhäuser, 2006. Zbl1100.22001MR2189882
  10. [10] Burger M., Mozes S., CAT ( - 1 ) -spaces, divergence groups and their commensurators, J. Amer. Math. Soc.9 (1996) 57-93. Zbl0847.22004MR1325797
  11. [11] Burger M., Mozes S., Groups acting on trees: from local to global structure, Publ. Math. IHÉS92 (2000) 113-150. Zbl1007.22012MR1839488
  12. [12] Borel A., Linear Algebraic Groups, Graduate Texts in Math., vol. 126, Springer, 1991. Zbl0726.20030MR1102012
  13. [13] Bourbaki N., Intégration VII–VIII, Éléments de Mathématique, Hermann, 1963. MR453824
  14. [14] Bourbaki N., Topologie générale I–IV, Éléments de Mathématique, Hermann, 1971. Zbl0249.54001MR358652
  15. [15] Bourbaki N., Algèbre VIII, Éléments de Mathématique, Hermann, 1973. MR417224
  16. [16] Bourbaki N., Groupes et algèbres de Lie IV–VI, Éléments de Mathématique, Masson, 1981. Zbl0483.22001MR647314
  17. [17] Brown K.S., Buildings, Springer, 1989. Zbl0715.20017MR969123
  18. [18] Borel A., Serre J.-P., Corners and arithmetic groups, Comm. Math. Helv.48 (1973) 463-491. Zbl0274.22011MR387495
  19. [19] Borel A., Serre J.-P., Cohomologie d'immeubles et de groupes S-arithmétiques, Topology15 (1976) 211-232. Zbl0338.20055MR447474
  20. [20] Borel A., Tits J., Groupes réductifs, Publ. Math. IHÉS27 (1965) 55-150. Zbl0145.17402MR207712
  21. [21] Borel A., Tits J., Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I, Invent. Math.12 (1971) 95-104. Zbl0238.20055MR294349
  22. [22] Bruhat F., Tits J., Groupes réductifs sur un corps local, I. Données radicielles valuées, Publ. Math. IHÉS41 (1972) 5-251. Zbl0254.14017MR327923
  23. [23] Borel A., Tits J., Homomorphismes « abstraits » de groupes algébriques simples, Ann. of Math.97 (1973) 499-571. Zbl0272.14013MR316587
  24. [24] Bruhat F., Tits J., Groupes réductifs sur un corps local, II. Schémas en groupes. Existence d'une donnée radicielle valuée, Publ. Math. IHÉS60 (1984) 197-376. Zbl0597.14041MR756316
  25. [25] Bruhat F., Tits J., Schémas en groupes et immeubles des groupes classiques sur un corps local, Bull. Soc. Math. France112 (1984) 259-301. Zbl0565.14028MR788969
  26. [26] Canary R.D., Epstein D.B.A., Green P., Notes on notes of Thurston, in: Epstein D.B.A. (Ed.), Analytical and Geometrical Aspects of Hyperbolic Spaces, London Math. Soc. Lecture Notes Series, vol. 111, Cambridge University Press, 1987. Zbl0612.57009MR903850
  27. [27] Conze J.-P., Guivarc'h Y., Remarques sur la distalité dans les espaces vectoriels, C. R. Acad. Sci. Paris278 (1974) 1083-1086. Zbl0275.54028MR339108
  28. [28] de Cornulier Y., Invariant means on projective spaces, Preprint, 2004. 
  29. [29] Figá-Talamanca A., Nebbia C., Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees, London Math. Soc. Lecture Notes Series, vol. 162, Cambridge University Press, 1991. Zbl1154.22301MR1152801
  30. [30] Furstenberg H., The structure of distal flows, Amer. J. Math.85 (1963) 477-515. Zbl0199.27202MR157368
  31. [31] Furstenberg H., Boundary theory and stochastic processes on homogeneous spaces, in: Moore C.C. (Ed.), Harmonic Analysis on Homogeneous Spaces, Proc. Symp. Pure Math., vol. XXVI, AMS, Providence, RI, 1972, pp. 193-229. Zbl0289.22011MR352328
  32. [32] Ghys É., de la Harpe P., Sur les groupes hyperboliques d'après Mikhael Gromov, Progr. Math., vol. 83, Birkhäuser, 1990. Zbl0731.20025MR1086648
  33. [33] Goldman O., Iwahori N., The space of p-adic norms, Acta Math.109 (1963) 137-177. Zbl0133.29402MR144889
  34. [34] Gille P., Unipotent subgroups of reductive groups in characteristic p g t ; 0 , Duke Math. J.114 (2002) 307-328. Zbl1013.20040MR1920191
  35. [35] Guivarc'h Y., Compactifications of symmetric spaces and positive eigenfunctions of the Laplacian, in: Taylor J.Ch. (Ed.), Topics in Probability and Lie Groups; Boundary Theory, CRM Proc. Lect. Notes, vol. 28, AMS, 2001, pp. 69-116. Zbl1160.58306MR1832435
  36. [36] Guivarc'h Y., Ji L., Taylor J.Ch., Compactifications of Symmetric Spaces, Progr. Math., vol. 156, Birkhäuser, 1998. Zbl1053.31006MR1633171
  37. [37] Landvogt E., A Compactification of the Bruhat–Tits Building, Lecture Notes in Math., vol. 1619, Springer, 1996. Zbl0935.20034MR1441308
  38. [38] Lubotzky A., Mozes S., Asymptotic properties of unitary representations of tree automorphisms, in: Piccardello M. (Ed.), Harmonic Analysis and Discrete Potential Theory (Frascati, 1991), Plenum Press, New York, 1992, pp. 289-298. MR1222467
  39. [39] Lubotzky A., Mozes S., Zimmer R.J., Superrigidity for the commensurability group of tree lattices, Comm. Math. Helv.69 (1994) 523-548. Zbl0839.22011MR1303226
  40. [40] Mackey G.W., Induced representations of locally compact groups I, Ann. of Math.55 (1952) 101-139. Zbl0046.11601MR44536
  41. [41] Margulis G.A., Discrete Subgroups of Semisimple Lie Groups, Ergeb. Math. Grenzgeb. (3), vol. 17, Springer, 1991. Zbl0732.22008MR1090825
  42. [42] Moore C.C., Compactifications of symmetric spaces, Amer. J. Math.86 (1964) 201-218. Zbl0156.03202MR161942
  43. [43] Moore C.C., Amenable subgroups of semisimple groups and proximal flows, Israel J. Math.34 (1979) 121-138. Zbl0431.22014MR571400
  44. [44] Platonov V., Rapinchuk A., Algebraic Groups and Number Theory, Pure Appl. Math., vol. 139, Academic Press, 1994. Zbl0841.20046MR1278263
  45. [45] Prasad G., Elementary proof of a theorem of Bruhat–Tits–Rousseau and of a theorem of Tits, Bull. Soc. Math. France110 (1982) 197-202. Zbl0492.20029MR667750
  46. [46] Raghunathan M.S., Discrete Subgroups of Lie Groups, Ergeb. Math. Grenzgeb., vol. 68, Springer, 1972. Zbl0254.22005MR507234
  47. [47] Ronan M.A., Lectures on Buildings, Perspectives in Mathematics, vol. 7, Academic Press, 1989. Zbl0694.51001MR1005533
  48. [48] Rousseau G., Immeubles des groupes réductifs sur les corps locaux, thèse d'État, Université de Paris-Sud (Orsay), 1977. Zbl0412.22006MR491992
  49. [49] Satake I., On compactifications of the quotient spaces for arithmetically defined discontinuous groups, Ann. of Math.72 (1960) 555-580. Zbl0146.04701MR170356
  50. [50] Satake I., On representations and compactifications of symmetric Riemannian spaces, Ann. of Math.71 (1960) 77-110. Zbl0094.34603MR118775
  51. [51] Springer T.A., Linear Algebraic Groups, Progr. Math., vol. 9, Birkhäuser, 1998. Zbl0453.14022MR1642713
  52. [52] Tits J., Free subgroups in linear groups, J. Algebra20 (1972) 250-270. Zbl0236.20032MR286898
  53. [53] Tits J., Reductive groups over local fields, in: Borel A., Casselman W.A. (Eds.), Automorphic Forms, Representations and L-Functions, Proc. Symp. Pure Math. (Oregon State Univ., Corvallis, 1977), part 1, vol. XXXIII, AMS, Providence, RI, 1979, pp. 29-69. Zbl0415.20035MR546588
  54. [54] Werner A., Compactification of the Bruhat–Tits building of PGL by lattices of smaller rank, Doc. Math.6 (2001) 315-341. Zbl1048.20014MR1871666
  55. [55] Werner A., Compactification of the Bruhat–Tits building of PGL by seminorms, Math. Z.248 (2004) 511-526. Zbl1121.20024MR2097372
  56. [56] Zimmer R.J., Ergodic Theory and Semisimple Groups, Monographs Math., vol. 81, Birkhäuser, 1984. Zbl0571.58015MR776417

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.