Advertisement

Mathematische Zeitschrift

, Volume 272, Issue 1–2, pp 51–86 | Cite as

Compactification d’espaces de représentations de groupes de type fini

  • Anne Parreau
Article

Abstract

Let Γ be a finitely generated group and G be a noncompact semisimple connected real Lie group with finite center. We consider the space \({\fancyscript X(\Gamma, G)}\) of conjugacy classes of semisimple representations of Γ into G, which is the maximal Hausdorff quotient of \({{\rm Hom}(\Gamma, G)/G}\) . We define the translation vector of \({g \in G}\), with value in a Weyl chamber, as a natural refinement of the translation length of g in the symmetric space associated with G. We construct a compactification of \({\fancyscript X(\Gamma, G)}\) , induced by the marked translation vector spectrum, generalizing Thurston’s compactification of the Teichmüller space. We show that the boundary points are projectivized marked translation vector spectra of actions of Γ on affine buildings with no global fixed point. An analoguous result holds for any reductive group G over a local field.

Keywords

Moduli spaces of representations Higher teichmüller theory Reductive groups Symmetric spaces Euclidean buildings Asymptotic cones 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alperin R.: An elementary account of Selberg’s lemma. L’Ens. Math. 33, 269–373 (1987)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ballmann, W., Gromov, M., Schroeder, V.: Manifolds of nonpositive curvature. In: Progress in Mathematics, vol. 61. Birkhaäuser, Basel (1985)Google Scholar
  3. 3.
    Bass H.: Groups of integral representation type. Pac. J. Math. 86, 15–51 (1980)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Benoist Y.: Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal. 7, 1–47 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Benoist Y.: A survey on divisible convex sets. Geometry, Analysis and Topology of Discrete Groups, Advanced Lectures in Mathematics, vol. 6. International Press, Somerville (2008)Google Scholar
  6. 6.
    Bestvina M.: Degenerations of the hyperbolic space. Duke Math. J. 56, 143–161 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Borel A.: Linear algebraic groups. Graduate Texts in Mathematics, vol. 126. Springer, New York (1991)Google Scholar
  8. 8.
    Bourbaki N.: Topologie Générale. Hermann, Paris (1971)zbMATHGoogle Scholar
  9. 9.
    Bremigan R.J.: Quotients for algebraic group actions over non-algebraically closed fields. J. Reine Angew. Math. 453, 21–47 (1994)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bridson M.R., Haefliger A.: Metric spaces with non-positive curvature. Grund. Math. Wiss, vol. 319. Springer, Berlin (1999)Google Scholar
  11. 11.
    Bruhat F., Tits J.: Groupes réductifs sur un corps local. I. Données radicielles valuées. Publ. Math. IHES 41, 5–252 (1972)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Bruhat F., Tits J.: Groupes réductifs sur un corps local. II. Sché mas en groupes. Existence d’une donnée radicielle valuée. Pub. Math. IHES 60, 197–376 (1984)MathSciNetGoogle Scholar
  13. 13.
    Bruhat F., Tits J.: Schémas en groupes et immeubles des groupes classiques sur un corps local. Bull. Soc. Math. Fr. 112, 259–301 (1984)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Chiswell I.M.: Non standard analysis and the Morgan–Shalen compactification. Q. J. Math. Oxf. 42, 257–270 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Choi S., Goldman W.M.: Convex real projective structures on closed surfaces are closed. Proc. Am. Math. Soc. 118, 657–661 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Dal’Bo F., Kim I.: Marked length rigidity for symmetric spaces. Comment. Math. Helv. 77, 399–407 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Eberlein P.: Geometry of Non-Positively Curved Manifolds. Chicago Lectures in Mathematics, The University of Chicago Press, Chicago (1996)Google Scholar
  18. 18.
    Fathi, A., Laudenbach, F., Poénaru V. (eds.): Travaux de Thurston sur les surfaces. Séminaire Orsay, Astérisque, vol. 66–67. Société Mathématique de France (1979)Google Scholar
  19. 19.
    Goldman O., Iwahori N.: The space of p-adic norms. Acta Math. 109, 137–177 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Goldman W.M., Millson J.J.: Local rigidity of discrete groups acting on complex hyperbolic space. Invent. Math. 88, 495–520 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Gromov M.: Asymptotic Invariants of Infinite Groups. Cambridge University Press, Cambridge (1991)Google Scholar
  22. 22.
    Gromov, M.: Structures métriques pour les variétés riemanniennes, édité par J. Lafontaine et P. Pansu, Cedic, Fernand Nathan (1981)Google Scholar
  23. 23.
    Helgason S.: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York (1978)zbMATHGoogle Scholar
  24. 24.
    Hitchin N.J.: Lie groups and Teichmüller space. Topology 31, 449–473 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Johnson, D., Millson, J.J.: Deformation spaces associated to compact hyperbolic manifolds. Discrete Groups in Geometry and Analysis (New Haven, CT, 1984). Progress in Mathematics, vol. 67, pp. 48–106. Birkhäuser, Basel (1987)Google Scholar
  26. 26.
    Kapovich M., Leeb B.: On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds. Geom. Funct. Anal. 5, 582–603 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Kleiner B., Leeb B.: Rigidity of quasi-isometries for symmetric spaces of higher rank. Publ. Math. IHES 86, 115–197 (1997)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Labourie F.: Anosov flows, surface groups and curves in projective space. Invent. Math. 165, 51–114 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Landvogt E.: Some functorial properties of the Bruhat–Tits building. J. Reine Angew. Math. 518, 213–241 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Lubotzky, A., Mozes, S., Raghunathan, M.S.: The word and Riemannian metrics on lattices of semisimple Lie groups. Inst. Hautes Études Sci. Publ. Math. No. 91, pp. 5–53 (2000)Google Scholar
  31. 31.
    Luna D.: Sur certaines opérations différentiables des groupes de Lie. Am. J. Math. 97, 172–181 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Margulis G.: Discrete Subgroups of Semi-Simple Groups. Ergebnisse der Mathematik und ihrer Grenz, vol. 17. Springer, Berlin (1991)Google Scholar
  33. 33.
    Morgan J.: Group actions on trees and the compactification of the space of classes of SO(n, 1)-representations. Topology 25, 1–33 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Morgan J., Morgan J.: Valuations, trees and degeneration of hyperbolic structures I. Ann. Math. 122, 398–476 (1985)Google Scholar
  35. 35.
    Mostow G.D.: Strong Rigidity Of Locally Symetric Spaces. Annals of Mathematical Studies, vol. 78. Princeton University Press, Princeton (1973)Google Scholar
  36. 36.
    Parreau, A.: Dégénérescences de sous-groupes discrets de groupes de Lie semisimples et actions de groupes sur des immeubles affines. thèse de doctorat, University of Orsay (2000)Google Scholar
  37. 37.
    Parreau, A.: Immeubles affines: construction par les normes et étude des isométries. Crystallographic Groups and Their Generalizations (Kortrijk, 1999). Contemporary Mathematics, vol. 262, pp. 263–302. American Mathematical Society, Providence (2000)Google Scholar
  38. 38.
    Parreau A.: Sous-groupes elliptiques de groupes linéaires sur un corps valué. J. Lie Theory 13, 271–278 (2003)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Parreau, A.: La distance vectorielle dans les espaces symétriques et les immeubles affines. en préparationGoogle Scholar
  40. 40.
    Espaces de représentations complètement réductibles. J. Lond. Math. Soc. (2011). doi: 10.1112/jlms/jdq076
  41. 41.
    Paulin F.: Topologie de Gromov équivariante, structures hyperboliques et arbres réels. Invent. Math. 94, 53–80 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Paulin, F.: De la géométrie et la dynamique des groupes discrets. Mémoire d’habilitation, ENS Lyon (Juin 1995)Google Scholar
  43. 43.
    Paulin F.: Dégénérescence de sous-groupes discrets des groupes de Lie semi-simples. C. R. Acad. Sci. Paris Sér. I Math. 324(11), 1217–1220 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Richardson R.W.: Conjugacy classes of n-tuples in Lie algebras and algebraic groups. Duke Math. J. 57, 1–35 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Richardson R.W., Slodowy P.J.: Minimum vectors for real reductive algebraic groups. J. Lond. Math. Soc. 42, 409–429 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Rousseau, G.: Euclidean buildings. In: Bessières, A.P.L., Rémy, B. (eds.) Géométries à courbure négative ou nulle, groupes discrets et rigidités. Séminaires et Congrès, vol. 18. Société mathématique de France (2008)Google Scholar
  47. 47.
    Serre, J.P.: Complète Reductibilite. Séminaire Bourbaki 2003–2004, Astérisque, vol. 299, pp. 195–217, Exp. No. 932Google Scholar
  48. 48.
    Tits, J.: Immeubles de type affine, dans “Buildings and the Geometry of Diagrams”, In: Rosati, L. (ed.) Proc. CIME Como 1984, Lecture Notes, vol. 1181, pp. 159–190. Springer, Berlin (1986)Google Scholar
  49. 49.
    Tits J.: Reductive groups over local fields. Proc. Symp. Pure Math. Am. Math. Soc. 33, 29–69 (1977)Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institut Fourier, Université Grenoble I et CNRSSaint-Martin-d’Hères CedexFrance

Personalised recommendations