Mathematische Annalen

, Volume 356, Issue 4, pp 1303–1337 | Cite as

Fixed points and amenability in non-positive curvature

  • Pierre-Emmanuel CapraceEmail author
  • Nicolas Monod


Consider a proper cocompact CAT(0) space \(X\). We give a complete algebraic characterisation of amenable groups of isometries of \(X\). For amenable discrete subgroups, an even narrower description is derived, implying \(\mathbf{Q}\)-linearity in the torsion-free case. We establish Levi decompositions for stabilisers of points at infinity of \(X\), generalising the case of linear algebraic groups to \(\text{ Is}(X)\). A geometric counterpart of this sheds light on the refined bordification of \(X\) (à la Karpelevich) and leads to a converse to the Adams–Ballmann theorem. It is further deduced that unimodular cocompact groups cannot fix any point at infinity except in the Euclidean factor; this fact is needed for the study of CAT(0) lattices. Various fixed point results are derived as illustrations.


Compact Group Compactible Subgroup Closed Subgroup Soluble Group Geodesic Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The final writing of this paper was partly accomplished when both authors were visiting the Mittag-Leffler Institute, whose hospitality was greatly appreciated. Thanks are also due to Ami Eisenmann for pointing out Corollary E.


  1. 1.
    Adams, S., Ballmann, W.: Amenable isometry groups of Hadamard spaces. Math. Ann. 312(1), 183–195 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Albuquerque, P.: Patterson-Sullivan theory in higher rank symmetric spaces. Geom. Funct. Anal. 9(1), 1–28 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Auslander, L.: Discrete solvable matrix groups. Proc. Am. Math. Soc. 11, 687–688 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften 319. Springer, Berlin (1999)Google Scholar
  5. 5.
    Burger, M., Mozes, S.: CAT(-1)-spaces, divergence groups and their commensurators. J. Am. Math. Soc. 9, 57–93 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bourbaki, N.: Éléments de mathématique. Première partie. (Fascicule III.) Livre III; Topologie générale. Chap. 3: Groupes topologiques. Chap. 4: Nombres réels, Troisième édition revue et augmentée, Actualités Sci. Indust., No. 1143. Hermann, Paris (1960)Google Scholar
  7. 7.
    Bourbaki, N.: Integration. II. Chapters 7–9, Elements of Mathematics (Berlin). Springer-Verlag, Berlin (2004). Translated from the 1963 and 1969 French originals by Sterling K. BerberianGoogle Scholar
  8. 8.
    Burger, M., Schroeder, V.: Amenable groups and stabilizers of measures on the boundary of a Hadamard manifold. Math. Ann. 276(3), 505–514 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Caprace, P.-E.: Amenable groups and Hadamard spaces with a totally disconnected isometry group. Comment. Math. Helv. 84, 437–455 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Caprace, P.-E., Cornulier, Y., Monod, N., Tessera, R.: Amenable hyperbolic groups. Preprint (2011)Google Scholar
  11. 11.
    Cannon, J.W., Floyd, W.J., Parry, W.R.: Introductory notes on Richard Thompson’s groups. Enseign. Math. (2) 42(3–4), 215–256 (1996)Google Scholar
  12. 12.
    Chabauty, C.: Limite d’ensembles et géométrie des nombres. Bull. Soc. Math. France 78, 143–151 (1950)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Caprace, P.-E., Monod, N.: Isometry groups of non-positively curved spaces: structure theory. J. Topol. 2(4), 661–700 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Caprace, P.-E., Monod, N.: Isometry groups of non-positively curved spaces: discrete subgroups. J. Topol. 2(4), 701–746 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Caprace, P.-E., Sageev, M.: Rank rigidity for CAT(0) cube complexes. Geom. Funct. Anal. 21(4), 851–891 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Cornulier, Y., Tessera, R.: Contracting automorphisms and \(L^p\)-cohomology in degree one. Arkiv för Matematik 49(2), 295–324 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    de Cornulier, Y.: On lengths on semisimple groups. J. Topol. Anal. 1(2), 113–121 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    de la Harpe, P., Guyan Robertson, A., Valette, A.: On exactness of group \(C^*\)-algebras. Q. J. Math. Oxford Ser. (2) 45(180), 499–513 (1994)Google Scholar
  19. 19.
    Di Scala, A.J.: Minimal homogeneous submanifolds in Euclidean spaces. Ann. Global Anal. Geom. 21(1), 15–18 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Farley, D.S.: Actions of picture groups on CAT(0) cubical complexes. Geom. Dedicata 110, 221–242 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Furstenberg, H.: A Poisson formula for semi-simple Lie groups. Ann. Math. (2) 77, 335–386 (1963)Google Scholar
  22. 22.
    Hall, P.: On the embedding of a group in a join of given groups. J. Austral. Math. Soc. 17, 434–495 (1974). Collection of articles dedicated to the memory of Hanna Neumann, VIIIGoogle Scholar
  23. 23.
    Hattori, T.: Geometric limit sets of higher rank lattices. Proc. Lond. Math. Soc. (3) 90(3), 689–710 (2005)Google Scholar
  24. 24.
    Heintze: E.: On homogeneous manifolds of negative curvature. Math. Ann. 211, 23–34 (1974)Google Scholar
  25. 25.
    Hochschild, G.: The Structure of Lie Groups. Holden-Day Inc., San Francisco (1965)zbMATHGoogle Scholar
  26. 26.
    Iwasawa, K.: On some types of topological groups. Ann. Math. (2) 50, 507–558 (1949)Google Scholar
  27. 27.
    Juschenko, K., Monod, N.: Cantor systems, piecewise translations and simple amenable groups, preprint (2012)Google Scholar
  28. 28.
    Kleiner, B.: The local structure of length spaces with curvature bounded above. Math. Z. 231(3), 409–456 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Leeb, B.: A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry. Bonner Mathematische Schriften, 326. Universität Bonn Mathematisches Institut, Bonn (2000)Google Scholar
  30. 30.
    Lennox, J.C., Robinson, D.J.S.: The theory of infinite soluble groups. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2004)Google Scholar
  31. 31.
    Meier, D.: Non-Hopfian groups. J. Lond. Math. Soc. (2) 26(2), 265–270 (1982)Google Scholar
  32. 32.
    Meier, D.: Embeddings into simple free products. Proc. Am. Math. Soc. 93(3), 387–392 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Milnor, J.: Growth of finitely generated solvable groups. J. Differ. Geometry 2, 447–449 (1968)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Monod, N.: Superrigidity for irreducible lattices and geometric splitting. J. Am. Math. Soc. 19(4), 781–814 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Monod, N., Py, P.: An equivariant deformation of the hyperbolic space. Preprint (2012)Google Scholar
  36. 36.
    Müller-Römer, P.R.: Kontrahierende Erweiterungen und kontrahierbare Gruppen. J. Reine Angew. Math. 283(284), 238–264 (1976)MathSciNetGoogle Scholar
  37. 37.
    Montgomery, D., Zippin, L.: Topological Transformation Groups. Interscience Publishers, New York (1955)zbMATHGoogle Scholar
  38. 38.
    Neumann, B.H.: An essay on free products of groups with amalgamations. Philos. Trans. Roy. Soc. Lond. Ser. A. 246, 503–554 (1954)zbMATHCrossRefGoogle Scholar
  39. 39.
    Oliver, R.K.: On Bieberbach’s analysis of discrete Euclidean groups. Proc. Am. Math. Soc. 80(1), 15–21 (1980)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Platonov, V.P.: Lokal projective nilpotent radicals in topological groups. Dokl. Akad. Nauk BSSR 9, 573–577 (1965)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Raghunathan, M.S.: Discrete Subgroups of Lie Groups. Springer, New York (1972). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68Google Scholar
  42. 42.
    Reiter, H.: \(L^{1}\)-Algebras and Segal Algebras. Springer, Berlin (1971). Lecture Notes in Mathematics, vol. 231Google Scholar
  43. 43.
    Rosset, S.: A property of groups of non-exponential growth. Proc. Am. Math. Soc. 54, 24–26 (1976)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Sălăjan, D.T.: CAT(0) geometry for the Thompson group. Ph.D. thesis, EPFL (2012)Google Scholar
  45. 45.
    Schupp, P.E.: Embeddings into simple groups. J. Lond. Math. Soc. (2) 13(1), 90–94 (1976)Google Scholar
  46. 46.
    Šunkov, V.P.: On locally finite groups of finite rank. Algebra Logic 10, 127–142 (1971)CrossRefGoogle Scholar
  47. 47.
    Tits, J.: Free subgroups in linear groups. J. Algebra 20, 250–270 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Ušakov, V.I.: Topological \(\overline{FC}\)-groups. Sibirsk. Mat. Ž. 4, 1162–1174 (1963)MathSciNetGoogle Scholar
  49. 49.
    van Dantzig, D.: Studien over topologische algebra (proefschrift). Ph.D. thesis, Groningen (1931)Google Scholar
  50. 50.
    Wehrfritz, B.A.F: Infinite linear groups. An account of the group-theoretic properties of infinite groups of matrices. Springer, New York (1973). Ergebnisse der Matematik und ihrer Grenzgebiete, Band 76Google Scholar
  51. 51.
    Yamabe, H.: A generalization of a theorem of Gleason. Ann. Math. (2) 58, 351–365 (1953)Google Scholar
  52. 52.
    Zassenhaus, H.: Beweis eines Satzes über diskrete Gruppen. Abh. Math. Semin. Hansische Univ. 12, 289–312 (1938)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Zelmanov, E.I.: On the restricted Burnside problem. In: Proceedings of the International Congress of Mathematicians, vols. I, II (Kyoto, 1990), pp. 395–402. Math. Soc. Japan (1991)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.UCL-MathLouvain-la-NeuveBelgium
  2. 2.EPFLLausanneSwitzerland

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