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Fixed points and amenability in non-positive curvature

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Abstract

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.

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Notes

  1. Added in proof: Such amenable simple groups have recently been constructed by Juschenko–Monod, see [27].

  2. It is quite possible that a softer argument could be provided in the present setting by invoking Corollary 2 page 141 of [46].

References

  1. Adams, S., Ballmann, W.: Amenable isometry groups of Hadamard spaces. Math. Ann. 312(1), 183–195 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Albuquerque, P.: Patterson-Sullivan theory in higher rank symmetric spaces. Geom. Funct. Anal. 9(1), 1–28 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Auslander, L.: Discrete solvable matrix groups. Proc. Am. Math. Soc. 11, 687–688 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften 319. Springer, Berlin (1999)

  5. Burger, M., Mozes, S.: CAT(-1)-spaces, divergence groups and their commensurators. J. Am. Math. Soc. 9, 57–93 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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. Berberian

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caprace, P.-E.: Amenable groups and Hadamard spaces with a totally disconnected isometry group. Comment. Math. Helv. 84, 437–455 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Caprace, P.-E., Cornulier, Y., Monod, N., Tessera, R.: Amenable hyperbolic groups. Preprint (2011)

  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. Chabauty, C.: Limite d’ensembles et géométrie des nombres. Bull. Soc. Math. France 78, 143–151 (1950)

    MathSciNet  MATH  Google Scholar 

  13. Caprace, P.-E., Monod, N.: Isometry groups of non-positively curved spaces: structure theory. J. Topol. 2(4), 661–700 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Caprace, P.-E., Monod, N.: Isometry groups of non-positively curved spaces: discrete subgroups. J. Topol. 2(4), 701–746 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Caprace, P.-E., Sageev, M.: Rank rigidity for CAT(0) cube complexes. Geom. Funct. Anal. 21(4), 851–891 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cornulier, Y., Tessera, R.: Contracting automorphisms and \(L^p\)-cohomology in degree one. Arkiv för Matematik 49(2), 295–324 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. de Cornulier, Y.: On lengths on semisimple groups. J. Topol. Anal. 1(2), 113–121 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  19. Di Scala, A.J.: Minimal homogeneous submanifolds in Euclidean spaces. Ann. Global Anal. Geom. 21(1), 15–18 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Farley, D.S.: Actions of picture groups on CAT(0) cubical complexes. Geom. Dedicata 110, 221–242 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Furstenberg, H.: A Poisson formula for semi-simple Lie groups. Ann. Math. (2) 77, 335–386 (1963)

    Google Scholar 

  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, VIII

    Google Scholar 

  23. Hattori, T.: Geometric limit sets of higher rank lattices. Proc. Lond. Math. Soc. (3) 90(3), 689–710 (2005)

    Google Scholar 

  24. Heintze: E.: On homogeneous manifolds of negative curvature. Math. Ann. 211, 23–34 (1974)

    Google Scholar 

  25. Hochschild, G.: The Structure of Lie Groups. Holden-Day Inc., San Francisco (1965)

    MATH  Google Scholar 

  26. Iwasawa, K.: On some types of topological groups. Ann. Math. (2) 50, 507–558 (1949)

    Google Scholar 

  27. Juschenko, K., Monod, N.: Cantor systems, piecewise translations and simple amenable groups, preprint (2012)

  28. Kleiner, B.: The local structure of length spaces with curvature bounded above. Math. Z. 231(3), 409–456 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  31. Meier, D.: Non-Hopfian groups. J. Lond. Math. Soc. (2) 26(2), 265–270 (1982)

    Google Scholar 

  32. Meier, D.: Embeddings into simple free products. Proc. Am. Math. Soc. 93(3), 387–392 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  33. Milnor, J.: Growth of finitely generated solvable groups. J. Differ. Geometry 2, 447–449 (1968)

    MathSciNet  MATH  Google Scholar 

  34. Monod, N.: Superrigidity for irreducible lattices and geometric splitting. J. Am. Math. Soc. 19(4), 781–814 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Monod, N., Py, P.: An equivariant deformation of the hyperbolic space. Preprint (2012)

  36. Müller-Römer, P.R.: Kontrahierende Erweiterungen und kontrahierbare Gruppen. J. Reine Angew. Math. 283(284), 238–264 (1976)

    MathSciNet  Google Scholar 

  37. Montgomery, D., Zippin, L.: Topological Transformation Groups. Interscience Publishers, New York (1955)

    MATH  Google Scholar 

  38. Neumann, B.H.: An essay on free products of groups with amalgamations. Philos. Trans. Roy. Soc. Lond. Ser. A. 246, 503–554 (1954)

    Article  MATH  Google Scholar 

  39. Oliver, R.K.: On Bieberbach’s analysis of discrete Euclidean groups. Proc. Am. Math. Soc. 80(1), 15–21 (1980)

    MathSciNet  MATH  Google Scholar 

  40. Platonov, V.P.: Lokal projective nilpotent radicals in topological groups. Dokl. Akad. Nauk BSSR 9, 573–577 (1965)

    MathSciNet  MATH  Google Scholar 

  41. Raghunathan, M.S.: Discrete Subgroups of Lie Groups. Springer, New York (1972). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68

  42. Reiter, H.: \(L^{1}\)-Algebras and Segal Algebras. Springer, Berlin (1971). Lecture Notes in Mathematics, vol. 231

  43. Rosset, S.: A property of groups of non-exponential growth. Proc. Am. Math. Soc. 54, 24–26 (1976)

    MathSciNet  MATH  Google Scholar 

  44. Sălăjan, D.T.: CAT(0) geometry for the Thompson group. Ph.D. thesis, EPFL (2012)

  45. Schupp, P.E.: Embeddings into simple groups. J. Lond. Math. Soc. (2) 13(1), 90–94 (1976)

    Google Scholar 

  46. Šunkov, V.P.: On locally finite groups of finite rank. Algebra Logic 10, 127–142 (1971)

    Article  Google Scholar 

  47. Tits, J.: Free subgroups in linear groups. J. Algebra 20, 250–270 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  48. Ušakov, V.I.: Topological \(\overline{FC}\)-groups. Sibirsk. Mat. Ž. 4, 1162–1174 (1963)

    MathSciNet  Google Scholar 

  49. van Dantzig, D.: Studien over topologische algebra (proefschrift). Ph.D. thesis, Groningen (1931)

  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 76

  51. Yamabe, H.: A generalization of a theorem of Gleason. Ann. Math. (2) 58, 351–365 (1953)

    Google Scholar 

  52. Zassenhaus, H.: Beweis eines Satzes über diskrete Gruppen. Abh. Math. Semin. Hansische Univ. 12, 289–312 (1938)

    Article  MathSciNet  Google Scholar 

  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)

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Acknowledgments

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.

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Authors and Affiliations

  1. UCL-Math, Chemin du Cyclotron 2, 1348 , Louvain-la-Neuve, Belgium

    Pierre-Emmanuel Caprace

  2. EPFL, 1015 , Lausanne, Switzerland

    Nicolas Monod

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  1. Pierre-Emmanuel Caprace
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  2. Nicolas Monod
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Correspondence to Pierre-Emmanuel Caprace.

Additional information

P.-E. Caprace: F.R.S.-FNRS research associate. Supported in part by FNRS grant F.4520.11 and by the ERC grant #278469. N. Monod supported in part by the Swiss National Science Foundation and the ERC.

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Caprace, PE., Monod, N. Fixed points and amenability in non-positive curvature. Math. Ann. 356, 1303–1337 (2013). https://doi.org/10.1007/s00208-012-0879-9

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