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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References
Adams, S., Ballmann, W.: Amenable isometry groups of Hadamard spaces. Math. Ann. 312(1), 183–195 (1998)
Albuquerque, P.: Patterson-Sullivan theory in higher rank symmetric spaces. Geom. Funct. Anal. 9(1), 1–28 (1999)
Auslander, L.: Discrete solvable matrix groups. Proc. Am. Math. Soc. 11, 687–688 (1960)
Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften 319. Springer, Berlin (1999)
Burger, M., Mozes, S.: CAT(-1)-spaces, divergence groups and their commensurators. J. Am. Math. Soc. 9, 57–93 (1996)
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)
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
Burger, M., Schroeder, V.: Amenable groups and stabilizers of measures on the boundary of a Hadamard manifold. Math. Ann. 276(3), 505–514 (1987)
Caprace, P.-E.: Amenable groups and Hadamard spaces with a totally disconnected isometry group. Comment. Math. Helv. 84, 437–455 (2009)
Caprace, P.-E., Cornulier, Y., Monod, N., Tessera, R.: Amenable hyperbolic groups. Preprint (2011)
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)
Chabauty, C.: Limite d’ensembles et géométrie des nombres. Bull. Soc. Math. France 78, 143–151 (1950)
Caprace, P.-E., Monod, N.: Isometry groups of non-positively curved spaces: structure theory. J. Topol. 2(4), 661–700 (2009)
Caprace, P.-E., Monod, N.: Isometry groups of non-positively curved spaces: discrete subgroups. J. Topol. 2(4), 701–746 (2009)
Caprace, P.-E., Sageev, M.: Rank rigidity for CAT(0) cube complexes. Geom. Funct. Anal. 21(4), 851–891 (2011)
Cornulier, Y., Tessera, R.: Contracting automorphisms and \(L^p\)-cohomology in degree one. Arkiv för Matematik 49(2), 295–324 (2011)
de Cornulier, Y.: On lengths on semisimple groups. J. Topol. Anal. 1(2), 113–121 (2009)
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)
Di Scala, A.J.: Minimal homogeneous submanifolds in Euclidean spaces. Ann. Global Anal. Geom. 21(1), 15–18 (2002)
Farley, D.S.: Actions of picture groups on CAT(0) cubical complexes. Geom. Dedicata 110, 221–242 (2005)
Furstenberg, H.: A Poisson formula for semi-simple Lie groups. Ann. Math. (2) 77, 335–386 (1963)
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
Hattori, T.: Geometric limit sets of higher rank lattices. Proc. Lond. Math. Soc. (3) 90(3), 689–710 (2005)
Heintze: E.: On homogeneous manifolds of negative curvature. Math. Ann. 211, 23–34 (1974)
Hochschild, G.: The Structure of Lie Groups. Holden-Day Inc., San Francisco (1965)
Iwasawa, K.: On some types of topological groups. Ann. Math. (2) 50, 507–558 (1949)
Juschenko, K., Monod, N.: Cantor systems, piecewise translations and simple amenable groups, preprint (2012)
Kleiner, B.: The local structure of length spaces with curvature bounded above. Math. Z. 231(3), 409–456 (1999)
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)
Lennox, J.C., Robinson, D.J.S.: The theory of infinite soluble groups. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2004)
Meier, D.: Non-Hopfian groups. J. Lond. Math. Soc. (2) 26(2), 265–270 (1982)
Meier, D.: Embeddings into simple free products. Proc. Am. Math. Soc. 93(3), 387–392 (1985)
Milnor, J.: Growth of finitely generated solvable groups. J. Differ. Geometry 2, 447–449 (1968)
Monod, N.: Superrigidity for irreducible lattices and geometric splitting. J. Am. Math. Soc. 19(4), 781–814 (2006)
Monod, N., Py, P.: An equivariant deformation of the hyperbolic space. Preprint (2012)
Müller-Römer, P.R.: Kontrahierende Erweiterungen und kontrahierbare Gruppen. J. Reine Angew. Math. 283(284), 238–264 (1976)
Montgomery, D., Zippin, L.: Topological Transformation Groups. Interscience Publishers, New York (1955)
Neumann, B.H.: An essay on free products of groups with amalgamations. Philos. Trans. Roy. Soc. Lond. Ser. A. 246, 503–554 (1954)
Oliver, R.K.: On Bieberbach’s analysis of discrete Euclidean groups. Proc. Am. Math. Soc. 80(1), 15–21 (1980)
Platonov, V.P.: Lokal projective nilpotent radicals in topological groups. Dokl. Akad. Nauk BSSR 9, 573–577 (1965)
Raghunathan, M.S.: Discrete Subgroups of Lie Groups. Springer, New York (1972). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68
Reiter, H.: \(L^{1}\)-Algebras and Segal Algebras. Springer, Berlin (1971). Lecture Notes in Mathematics, vol. 231
Rosset, S.: A property of groups of non-exponential growth. Proc. Am. Math. Soc. 54, 24–26 (1976)
Sălăjan, D.T.: CAT(0) geometry for the Thompson group. Ph.D. thesis, EPFL (2012)
Schupp, P.E.: Embeddings into simple groups. J. Lond. Math. Soc. (2) 13(1), 90–94 (1976)
Šunkov, V.P.: On locally finite groups of finite rank. Algebra Logic 10, 127–142 (1971)
Tits, J.: Free subgroups in linear groups. J. Algebra 20, 250–270 (1972)
Ušakov, V.I.: Topological \(\overline{FC}\)-groups. Sibirsk. Mat. Ž. 4, 1162–1174 (1963)
van Dantzig, D.: Studien over topologische algebra (proefschrift). Ph.D. thesis, Groningen (1931)
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
Yamabe, H.: A generalization of a theorem of Gleason. Ann. Math. (2) 58, 351–365 (1953)
Zassenhaus, H.: Beweis eines Satzes über diskrete Gruppen. Abh. Math. Semin. Hansische Univ. 12, 289–312 (1938)
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)
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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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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DOI: https://doi.org/10.1007/s00208-012-0879-9