Group-theoretic compactification of Bruhat-Tits buildings

Guivarc'h, Yves; Rémy, Bertrand

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 39 (2006) no. 6, pp. 871-920.

[1] Adams S., Ballmann W., Amenable isometry groups of Hadamard spaces, Math. Ann. 312 (1998) 183-195. | MR | Zbl

[2] Abels H., Distal affine transformation groups, J. reine angew. Math. 299 (1978) 294-300. | MR | Zbl

[3] Abels H., Distal automorphism groups of Lie groups, J. reine angew. Math. 329 (1981) 82-87. | MR | Zbl

[4] Allcock D., Carlson J.A., Toledo D., The complex hyperbolic geometry of the moduli space of cubic surfaces, J. Algebraic Geometry 11 (2002) 659-724. | MR | Zbl

[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. | MR | Zbl

[6] Baily W.L., Borel A., Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. 84 (1966) 442-528. | MR | Zbl

[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. | MR | Zbl

[8] Bridson M., Haefliger A., Metric Spaces of Non-Positive Curvature, Grund. Math. Wiss., vol. 319, Springer, 1999. | MR | Zbl

[9] Borel A., Ji L., Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications, Birkhäuser, 2006. | MR | Zbl

[10] Burger M., Mozes S., $\text{CAT}(-1)$-spaces, divergence groups and their commensurators, J. Amer. Math. Soc. 9 (1996) 57-93. | MR | Zbl

[11] Burger M., Mozes S., Groups acting on trees: from local to global structure, Publ. Math. IHÉS 92 (2000) 113-150. | Numdam | MR | Zbl

[12] Borel A., Linear Algebraic Groups, Graduate Texts in Math., vol. 126, Springer, 1991. | MR | Zbl

[13] Bourbaki N., Intégration VII-VIII, Éléments de Mathématique, Hermann, 1963. | MR

[14] Bourbaki N., Topologie générale I-IV, Éléments de Mathématique, Hermann, 1971. | MR | Zbl

[15] Bourbaki N., Algèbre VIII, Éléments de Mathématique, Hermann, 1973. | MR

[16] Bourbaki N., Groupes et algèbres de Lie IV-VI, Éléments de Mathématique, Masson, 1981. | MR | Zbl

[17] Brown K.S., Buildings, Springer, 1989. | MR | Zbl

[18] Borel A., Serre J.-P., Corners and arithmetic groups, Comm. Math. Helv. 48 (1973) 463-491. | MR | Zbl

[19] Borel A., Serre J.-P., Cohomologie d'immeubles et de groupes S-arithmétiques, Topology 15 (1976) 211-232. | MR | Zbl

[20] Borel A., Tits J., Groupes réductifs, Publ. Math. IHÉS 27 (1965) 55-150. | Numdam | MR | Zbl

[21] Borel A., Tits J., Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I, Invent. Math. 12 (1971) 95-104. | MR | Zbl

[22] Bruhat F., Tits J., Groupes réductifs sur un corps local, I. Données radicielles valuées, Publ. Math. IHÉS 41 (1972) 5-251. | Numdam | MR | Zbl

[23] Borel A., Tits J., Homomorphismes « abstraits » de groupes algébriques simples, Ann. of Math. 97 (1973) 499-571. | MR | Zbl

[24] Bruhat F., Tits J., Groupes réductifs sur un corps local, II. Schémas en groupes. Existence d'une donnée radicielle valuée, Publ. Math. IHÉS 60 (1984) 197-376. | Numdam | MR | Zbl

[25] Bruhat F., Tits J., Schémas en groupes et immeubles des groupes classiques sur un corps local, Bull. Soc. Math. France 112 (1984) 259-301. | Numdam | MR | Zbl

[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. | MR | Zbl

[27] Conze J.-P., Guivarc'H Y., Remarques sur la distalité dans les espaces vectoriels, C. R. Acad. Sci. Paris 278 (1974) 1083-1086. | MR | Zbl

[28] de Cornulier Y., Invariant means on projective spaces, Preprint, 2004.

[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. | MR | Zbl

[30] Furstenberg H., The structure of distal flows, Amer. J. Math. 85 (1963) 477-515. | MR | Zbl

[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. | MR | Zbl

[32] Ghys É., De La Harpe P., Sur les groupes hyperboliques d'après Mikhael Gromov, Progr. Math., vol. 83, Birkhäuser, 1990. | MR | Zbl

[33] Goldman O., Iwahori N., The space of p-adic norms, Acta Math. 109 (1963) 137-177. | MR | Zbl

[34] Gille P., Unipotent subgroups of reductive groups in characteristic $p>0$, Duke Math. J. 114 (2002) 307-328. | MR | Zbl

[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. | MR | Zbl

[36] Guivarc'H Y., Ji L., Taylor J.Ch., Compactifications of Symmetric Spaces, Progr. Math., vol. 156, Birkhäuser, 1998. | MR | Zbl

[37] Landvogt E., A Compactification of the Bruhat-Tits Building, Lecture Notes in Math., vol. 1619, Springer, 1996. | MR | Zbl

[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. | MR

[39] Lubotzky A., Mozes S., Zimmer R.J., Superrigidity for the commensurability group of tree lattices, Comm. Math. Helv. 69 (1994) 523-548. | MR | Zbl

[40] Mackey G.W., Induced representations of locally compact groups I, Ann. of Math. 55 (1952) 101-139. | MR | Zbl

[41] Margulis G.A., Discrete Subgroups of Semisimple Lie Groups, Ergeb. Math. Grenzgeb. (3), vol. 17, Springer, 1991. | MR | Zbl

[42] Moore C.C., Compactifications of symmetric spaces, Amer. J. Math. 86 (1964) 201-218. | MR | Zbl

[43] Moore C.C., Amenable subgroups of semisimple groups and proximal flows, Israel J. Math. 34 (1979) 121-138. | MR | Zbl

[44] Platonov V., Rapinchuk A., Algebraic Groups and Number Theory, Pure Appl. Math., vol. 139, Academic Press, 1994. | MR | Zbl

[45] Prasad G., Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits, Bull. Soc. Math. France 110 (1982) 197-202. | Numdam | MR | Zbl

[46] Raghunathan M.S., Discrete Subgroups of Lie Groups, Ergeb. Math. Grenzgeb., vol. 68, Springer, 1972. | MR | Zbl

[47] Ronan M.A., Lectures on Buildings, Perspectives in Mathematics, vol. 7, Academic Press, 1989. | MR | Zbl

[48] Rousseau G., Immeubles des groupes réductifs sur les corps locaux, thèse d'État, Université de Paris-Sud (Orsay), 1977. | MR | Zbl

[49] Satake I., On compactifications of the quotient spaces for arithmetically defined discontinuous groups, Ann. of Math. 72 (1960) 555-580. | MR | Zbl

[50] Satake I., On representations and compactifications of symmetric Riemannian spaces, Ann. of Math. 71 (1960) 77-110. | MR | Zbl

[51] Springer T.A., Linear Algebraic Groups, Progr. Math., vol. 9, Birkhäuser, 1998. | MR | Zbl

[52] Tits J., Free subgroups in linear groups, J. Algebra 20 (1972) 250-270. | MR | Zbl

[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. | MR | Zbl

[54] Werner A., Compactification of the Bruhat-Tits building of PGL by lattices of smaller rank, Doc. Math. 6 (2001) 315-341. | MR | Zbl

[55] Werner A., Compactification of the Bruhat-Tits building of PGL by seminorms, Math. Z. 248 (2004) 511-526. | MR | Zbl

[56] Zimmer R.J., Ergodic Theory and Semisimple Groups, Monographs Math., vol. 81, Birkhäuser, 1984. | MR | Zbl