Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-24T23:16:33.316Z Has data issue: false hasContentIssue false
  • English
  • Français

Actions of amenable equivalence relations on CAT(0) fields

Published online by Cambridge University Press:  30 October 2012

MARTIN ANDEREGG  and
PHILIPPE HENRY
MARTIN ANDEREGG
Affiliation:
University of Geneva, 2-4 rue du Lièvre, Case postale 64, CH-1211 Genève 4, Switzerland (email: martin.anderegg@unige.ch)
PHILIPPE HENRY
Affiliation:
EPFL, SB MATHGEOM EGG, Station 8, CH-1015 Lausanne, Switzerland (email: philippe.henry@epfl.ch)
Get access Rights & Permissions [Opens in a new window]

Abstract

We present the general notion of Borel fields of metric spaces and show some properties of such fields. Then we make the study specific to the Borel fields of proper CAT(0) spaces and we show that the standard tools we need behave in a Borel way. We also introduce the notion of the action of an equivalence relation on Borel fields of metric spaces and we obtain a rigidity result for the action of an amenable equivalence relation on a Borel field of proper finite dimensional CAT(0) spaces. This main theorem is inspired by the result obtained by Adams and Ballmann regarding the action of an amenable group on a proper CAT(0) space.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 1 , February 2014 , pp. 21 - 54
Copyright
Copyright © 2012 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adams, S. and Ballmann, W.. Amenable isometry groups of Hadamard spaces. Math. Ann. 312 (1998), 183–195.Google Scholar
[2]Adams, S. and Lyons, R.. Amenability, Kazhdan’s property and percolation for trees, groups and equivalence relations. Israel J. Math. 75 (1991), 341–370.Google Scholar
[3]Alvarez, A.. Une théorie de Bass–Serre pour les relations d’équivalence et les groupoïdes boréliens. Thèse, Université de Lyon–École Normale Supérieure de Lyon, 2008.Google Scholar
[4]Anantharaman-Delaroche, C. and Renault, J.. Amenable groupoids. Monographie de l’enseignement mathématique. Genève, 2000.Google Scholar
[5]Anderegg, M.. Moyennabilité et courbure: $G$-espaces boréliens et espaces CAT(0). Thèse no 4790, École polytechnique fédérale de Lausanne, 2010.Google Scholar
[6]Avez, A.. Variétés riemanniennes sans points focaux. C. R. Acad. Sci. Paris Sér. A–B 270 (1970), 188–191.Google Scholar
[7]Ballmann, W.. Lectures on Spaces of Nonpositive Curvature. Birkhäuser, Basel, 1995.Google Scholar
[8]Bridson, M. R. and Heafliger, A.. Metrics Spaces of Non-Positive Curvature. Springer, Berlin, 1999.CrossRefGoogle Scholar
[9]Burger, M. and Schroeder, V.. Amenable groups and stabilizers of measures on the boundary of a Hadamard manifold. Math. Ann. 276(3) (1987), 505–514.Google Scholar
[10]Castaing, C.. Sur les multi-applications mesurables. Rev. Française Informat. Recherche Opérationnelle 1 (1967), 91–126.Google Scholar
[11]Connes, A., Feldman, J. and Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam. Sys. 1 (1981), 431–450.Google Scholar
[12]Caprace, P.-E. and Lytchak, A.. At infinity of finite-dimensional CAT(0) spaces. Math. Ann. 346(1) (2010), 1–21.Google Scholar
[13]Castaing, C. and Valadier, M.. Convex Analysis and Measurable Multifunctions (Lecture Notes in Mathematics, 580). Springer, Berlin, 1977.CrossRefGoogle Scholar
[14]Delode, C., Arino, O. and Penot, J.-P.. Champs mesurables et multisections. Ann. Inst. H. Poincaré Sect. B 12(1) (1976), 11–42.Google Scholar
[15]Dixmier, J.. Les algèbres d’opérateurs dans l’espace hilbertien (Algèbres de Von Neumann). Gauthier-Villars, Paris, 1969.Google Scholar
[16]Dougherty, R., Jackson, S. and Kechris, A. S.. The structure of hyperfinite borel equivalence relations. Trans. Amer. Math. Soc. 341(1) (1994), 193–225.Google Scholar
[17]Doob, J. L.. Measure Theory. Springer, New York, 1994.Google Scholar
[18]Duchesne, B.. Infinite dimensional non-positively curved symmetric spaces of finite rank. Preprint, 2011, arXiv:1109.0441. Int. Math. Res. Not. http://imrn.oxfordjournals.org/content/early/2012/03/06/imrn.rns093.full.pdf?keytype=ref&ijkey=2f9cf5LGkTS8w9R.Google Scholar
[19]Engelking, R.. General Topology. Heldermann, Berlin, 1989.Google Scholar
[20]Fell, J. M. G. and Doran, R. S.. Representations of \ast-Algebras, Locally Compact Groups and Banach \ast-Algebraic Bundles (two volumes). Academic Press, London, 1988.Google Scholar
[21]Feldman, J. and Moore, C.. Ergodic equivalence relations, cohomology and Von Neumann algebras, part I. Trans. Amer. Math. Soc. 234(2) (1977), 289–324.Google Scholar
[22]Feldman, J. and Moore, C.. Ergodic equivalence relations, cohomology and Von Neumann algebras, part II. Trans. Amer. Math. Soc. 234(2) (1977), 325–359.Google Scholar
[23]Fujiwara, K., Nagano, K. and Shioya, T.. Fixed point sets of parabolic isometries of CAT(0)-spaces. Comment. Math. Helv. 81(2) (2006), 305–335.Google Scholar
[24]Gaboriau, D.. Coût des relations d’équivalence et des groupes. Invent. Math. 139(1) (2000), 41–98.Google Scholar
[25]Giles, R. J.. Introduction to the Analysis of Metric Spaces. Cambridge University Press, Cambridge, 1987.Google Scholar
[26]Gromov, M.. Structures métriques pour les variétés riemanniennes (Textes Mathématiques, 1). Eds. Lafontaine, J. and Pansu, P.. CEDIC, Paris, 1981, pp. 1–120.Google Scholar
[27]Henry, Ph.. Actions de relations d’équivalence sur les champs d’espaces métriques CAT(0). Thèse no. 4825, École polytechnique fédérale de Lausanne, 2010.Google Scholar
[28]Himmelberg, C. J.. Measurable relations. Fund. Math. 87 (1975), 53–72.Google Scholar
[29]Jackson, S., Kechris, A. S. and Louveau, A.. Countable Borel equivalence relations. J. Math. Log. 2(1) (2002), 1–80.CrossRefGoogle Scholar
[30]Kanovei, V. G.. Borel Equivalence Relations: Structure and Classification. American Mathematical Society, Providence, 2008.Google Scholar
[31]Kechris, A. S.. Classical Descriptive Set Theory. Springer, New York, 1995.Google Scholar
[32]Kechris, A. S. and Miller, B. D.. Topics in Orbit Equivalence (Lecture Notes in Mathematics, 1852). Springer, Berlin, 2004.Google Scholar
[33]Lang, U. and Schroeder, V.. Kirszbraun’s theorem and metric spaces of bounded curvature. Geom. Funct. Anal. 7(3) (1997), 535–560.Google Scholar
[34]Monod, N.. Superrigidity for irreductible lattices and geometric splitting. J. Amer. Math. Soc. 19(4) (2006), 781–814.Google Scholar
[35]Urysohn, P.. Sur un espace métrique universel. Bull. Sci. Math. 51 (1927), 43–64 & 74–90.Google Scholar
[36]Valadier, M.. Sur le plongement d’un champ mesurable d’espaces métriques dans un champ trivial. Ann. Inst. H. Poincaré Sect. B 14(2) (1978), 165–168.Google Scholar
[37]Zimmer, R. J.. Hyperfinite factors and amenable ergodic actions. Invent. Math. 41 (1977), 23–31.Google Scholar
[38]Zimmer, R. J.. Amenable ergodic group actions and an application to Poisson boundaries of random walks. J. Funct. Anal. 27 (1978), 350–372.Google Scholar
[39]Zimmer, R. J.. Curvature of leaves in amenable foliations. Amer. J. Math. 105(4) (1983), 1011–1022.CrossRefGoogle Scholar