Fixed points for bounded orbits  in Hilbert spaces

Gheysens, Maxime; Monod, Nicolas

doi:10.24033/asens.2317

Fixed points for bounded orbits in Hilbert spaces

Gheysens, Maxime; Monod, Nicolas

Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 50 (2017) no. 1, pp. 131-156.

Abstract
Résumé

Consider the following property of a topological group $G$ : every continuous affine $G$ -action on a Hilbert space with a bounded orbit has a fixed point. We prove that this property characterizes amenability for locally compact $σ$ -compact groups (e.g., countable groups).

Along the way, we introduce a “moderate” variant of the classical induction of representations and we generalize the Gaboriau-Lyons theorem to prove that any non-amenable locally compact group admits a probabilistic variant of discrete free subgroups. This leads to the “measure-theoretic solution” to the von Neumann problem for locally compact groups.

We illustrate the latter result by giving a partial answer to the Dixmier problem for locally compact groups.

Nous considérons la propriété suivante pour un groupe topologique $G$ : toute action affine continue de $G$ sur un espace hilbertien ayant une orbite bornée a un point fixe. Nous montrons qu'elle caractérise la moyennabilité des groupes localement compacts dénombrables à l'infini (en particulier des groupes discrets dénombrables).

Pour ce faire, nous introduisons une variante « modérée » de l'induction des représentations et nous généralisons le théorème de Gaboriau-Lyons pour montrer que tout groupe localement compact non moyennable admet, dans un sens probabiliste, des sous-groupes libres discrets. Ceci fournit une « solution au sens de la mesure » au problème de von Neumann pour les groupes localement compacts.

Nous illustrons ce dernier résultat en fournissant une réponse partielle au problème de Dixmier pour les groupes localement compacts.

Published online: 2017-11-12

MR Zbl

DOI: 10.24033/asens.2317

Classification: 47H10, 22D12, 22A05, 43A07, 20E05, 37A20
Keywords: Amenable group, fixed point theorem, von Neumann problem, Dixmier problem
Mot clés : Groupe moyennable, théorème du point fixe, problème de von Neumann, problème de Dixmier

@article{ASENS_2017__50_1_131_0,
     author = {Gheysens, Maxime and Monod, Nicolas},
     title = {Fixed points for bounded orbits  in {Hilbert} spaces},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {131--156},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 50},
     number = {1},
     year = {2017},
     doi = {10.24033/asens.2317},
     mrnumber = {3621428},
     zbl = {1373.43001},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2317/}
}

TY  - JOUR
AU  - Gheysens, Maxime
AU  - Monod, Nicolas
TI  - Fixed points for bounded orbits  in Hilbert spaces
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2017
SP  - 131
EP  - 156
VL  - 50
IS  - 1
PB  - Société Mathématique de France. Tous droits réservés
UR  - https://www.numdam.org/articles/10.24033/asens.2317/
DO  - 10.24033/asens.2317
LA  - en
ID  - ASENS_2017__50_1_131_0
ER  -

%0 Journal Article
%A Gheysens, Maxime
%A Monod, Nicolas
%T Fixed points for bounded orbits  in Hilbert spaces
%J Annales scientifiques de l'École Normale Supérieure
%D 2017
%P 131-156
%V 50
%N 1
%I Société Mathématique de France. Tous droits réservés
%U https://www.numdam.org/articles/10.24033/asens.2317/
%R 10.24033/asens.2317
%G en
%F ASENS_2017__50_1_131_0

Gheysens, Maxime; Monod, Nicolas. Fixed points for bounded orbits  in Hilbert spaces. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 50 (2017) no. 1, pp. 131-156. doi : 10.24033/asens.2317. https://www.numdam.org/articles/10.24033/asens.2317/

References
Cited by

Abels, H. Specker-Kompaktifizierungen von lokal kompakten topologischen Gruppen, Math. Z., Volume 135 (1973/74), pp. 325-361 (ISSN: 0025-5874) | DOI | MR | Zbl

Bergman, G. M. Generating infinite symmetric groups, Bull. London Math. Soc., Volume 38 (2006), pp. 429-440 (ISSN: 0024-6093) | DOI | MR | Zbl

Bader, U.; Furman, A.; Gelander, T.; Monod, N. Property (T) and rigidity for actions on Banach spaces, Acta Math., Volume 198 (2007), pp. 57-105 (ISSN: 0001-5962) | DOI | MR | Zbl

Bader, U.; Gelander, T.; Monod, N. A fixed point theorem for $L^{1}$ spaces, Invent. math., Volume 189 (2012), pp. 143-148 (ISSN: 0020-9910) | DOI | MR | Zbl

Burger, M.; Monod, N. Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal., Volume 12 (2002), pp. 219-280 (ISSN: 1016-443X) | DOI | MR | Zbl

Bourbaki, N., Actualités Sci. Ind., 1229, Hermann & Cie, Paris, 1955, 191 pages (; réimpression Springer, 2007) | MR | Zbl

Bourbaki, N., Actualités Sci. Ind., 1306, Hermann, Paris, 1963 (; réimpression Springer, 2007) | MR

Bühler, T. On the algebraic foundations of bounded cohomology, Mem. Amer. Math. Soc., Volume 214 (2011) (ISBN: 978-0-8218-5311-5, ISSN: 0065-9266) | DOI | MR | Zbl

Caprace, P.-E.; de Cornulier, Y. On embeddings into compactly generated groups, Pacific J. Math., Volume 269 (2014), pp. 305-321 (ISSN: 0030-8730) | DOI | MR | Zbl

Chifan, I.; Ioana, A. Ergodic subequivalence relations induced by a Bernoulli action, Geom. Funct. Anal., Volume 20 (2010), pp. 53-67 (ISSN: 1016-443X) | DOI | MR | Zbl

Day, M. M. Means for the bounded functions and ergodicity of the bounded representations of semi-groups, Trans. Amer. Math. Soc., Volume 69 (1950), pp. 276-291 (ISSN: 0002-9947) | DOI | MR | Zbl

Dixmier, J. Les moyennes invariantes dans les semi-groupes et leurs applications, Acta Sci. Math. Szeged, Volume 12 (1950), pp. 213-227 (ISSN: 0001-6969) | MR | Zbl

Dixmier, J., Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, 1996 reprint of the second (1969) edition | MR

Dunford, N.; Schwartz, J. T., With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, 7, Interscience Publishers, Inc., New York, 1958, 858 pages | MR | Zbl

de Sz. Nagy, B. On uniformly bounded linear transformations in Hilbert space, Acta Univ. Szeged. Sect. Sci. Math., Volume 11 (1947), pp. 152-157 | MR | Zbl

Epstein, I.; Monod, N. Nonunitarizable representations and random forests, Int. Math. Res. Not., Volume 2009 (2009), pp. 4336-4353 (ISSN: 1073-7928) | DOI | MR | Zbl

Ehrenpreis, L.; Mautner, F. I. Uniformly bounded representations of groups, Proc. Nat. Acad. Sci. U. S. A., Volume 41 (1955), pp. 231-233 (ISSN: 0027-8424) | DOI | MR | Zbl

Epstein, I. Some results on orbit inequivalent actions of non-amenable groups, ISBN: 978-0549-72398-1, ProQuest LLC, Ann Arbor, MI (2008) http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3322046 | MR

Feldman, J.; Moore, C. C. Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc., Volume 234 (1977), pp. 289-324 (ISSN: 0002-9947) | DOI | MR | Zbl

Furman, A., Geometry, rigidity, and group actions (Chicago Lectures in Math.), Univ. Chicago Press, Chicago, IL, 2011, pp. 296-374 | MR | Zbl

Gaboriau, D. Coût des relations d'équivalence et des groupes, Invent. math., Volume 139 (2000), pp. 41-98 (ISSN: 0020-9910) | DOI | MR | Zbl

Gaboriau, D. Invariant percolation and harmonic Dirichlet functions, Geom. Funct. Anal., Volume 15 (2005), pp. 1004-1051 (ISSN: 1016-443X) | DOI | MR | Zbl

Gaboriau, D.; Lyons, R. A measurable-group-theoretic solution to von Neumann's problem, Invent. math., Volume 177 (2009), pp. 533-540 (ISSN: 0020-9910) | DOI | MR | Zbl

Guivarc'h, Y. Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France, Volume 101 (1973), pp. 333-379 (ISSN: 0037-9484) | DOI | Numdam | MR | Zbl

Hjorth, G. A lemma for cost attained, Ann. Pure Appl. Logic, Volume 143 (2006), pp. 87-102 (ISSN: 0168-0072) | DOI | MR | Zbl

Hutchcroft, T.; Nachmias, A. Indistinguishability of trees in uniform spanning forests, Probab. Theory Relat. Fields (2016) ( doi:10.1007/s00440-016-0707-3 ) | MR | Zbl

Higman, G.; Neumann, B. H.; Neumann, H. Embedding theorems for groups, J. London Math. Soc., Volume 24 (1949), pp. 247-254 (ISSN: 0024-6107) | DOI | MR | Zbl

Houdayer, C. Invariant percolation and measured theory of nonamenable groups [after Gaboriau-Lyons, Ioana, Epstein], Séminaire Bourbaki, vol. 2010/2011, exposé n^o 1039, Astérisque, Volume 348 (2012), pp. 339-374 (ISBN: 978-2-85629-351-5, ISSN: 0303-1179) | Numdam | MR | Zbl

Haagerup, U. V.; Przybyszewska, A. Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces (2006) (preprint arXiv:math/0606794 )

Häggström, O.; Peres, Y. Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously, Probab. Theory Related Fields, Volume 113 (1999), pp. 273-285 (ISSN: 0178-8051) | DOI | MR | Zbl

Kechris, A. S., Graduate Texts in Math., 156, Springer, New York, 1995, 402 pages (ISBN: 0-387-94374-9) | DOI | MR | Zbl

Kakutani, S.; Kodaira, K. Über das Haarsche Mass in der lokal bikompakten Gruppe, Proc. Imp. Acad. Tokyo, Volume 20 (1944), pp. 444-450 | MR | Zbl

Klee, V. L. J. Some topological properties of convex sets, Trans. Amer. Math. Soc., Volume 78 (1955), pp. 30-45 (ISSN: 0002-9947) | DOI | MR | Zbl

Krickeberg, K. Convergence of martingales with a directed index set, Trans. Amer. Math. Soc., Volume 83 (1956), pp. 313-337 (ISSN: 0002-9947) | DOI | MR | Zbl

Kunze, R. A.; Stein, E. M. Uniformly bounded representations and harmonic analysis of the $2 \times 2$ real unimodular group, Amer. J. Math., Volume 82 (1960), pp. 1-62 (ISSN: 0002-9327) | DOI | MR | Zbl

Kechris, A. S.; Tsankov, T. Amenable actions and almost invariant sets, Proc. Amer. Math. Soc., Volume 136 (2008), pp. 687-697 (ISSN: 0002-9939) | DOI | MR | Zbl

Levitt, G. On the cost of generating an equivalence relation, Ergodic Theory Dynam. Systems, Volume 15 (1995), pp. 1173-1181 (ISSN: 0143-3857) | DOI | MR | Zbl

Lyons, R.; Peres, Y.; Schramm, O. Minimal spanning forests, Ann. Probab., Volume 34 (2006), pp. 1665-1692 (ISSN: 0091-1798) | DOI | MR | Zbl

Lyons, R.; Schramm, O. Indistinguishability of percolation clusters, Ann. Probab., Volume 27 (1999), pp. 1809-1836 (ISSN: 0091-1798) | DOI | MR | Zbl

Mazur, S. Über die kleinste konvexe Menge, die eine gegebene kompakte Menge enthält, Studia Math., Volume 2 (1930), pp. 7-9 | DOI | JFM

Monod, N.; Ozawa, N. The Dixmier problem, lamplighters and Burnside groups, J. Funct. Anal., Volume 258 (2010), pp. 255-259 (ISSN: 0022-1236) | DOI | MR | Zbl

Monod, N., Lecture Notes in Math., 1758, Springer, Berlin, 2001, 214 pages (ISBN: 3-540-42054-1) | DOI | MR | Zbl

Monod, N., International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1183-1211 | MR | Zbl

Monod, N.; Shalom, Y. Orbit equivalence rigidity and bounded cohomology, Ann. of Math., Volume 164 (2006), pp. 825-878 (ISSN: 0003-486X) | DOI | MR | Zbl

Nakamura, M.; Takeda, Z. Group representation and Banach limit, Tôhoku Math. J., Volume 3 (1951), pp. 132-135 (ISSN: 0040-8735) | DOI | MR | Zbl

Pisier, G., Lecture Notes in Math., 1618, Springer, Berlin, 2001, 198 pages (ISBN: 3-540-41524-6) | DOI | MR | Zbl

Pisier, G., Infinite groups: geometric, combinatorial and dynamical aspects (Progr. Math.), Volume 248, Birkhäuser, 2005, pp. 323-362 | DOI | MR | Zbl

Przybyszewska, A. Proper metrics, affine isometric actions and a new definition of group exactness (2005)

Pak, I.; Smirnova-Nagnibeda, T. On non-uniqueness of percolation on nonamenable Cayley graphs, C. R. Acad. Sci. Paris Sér. I Math., Volume 330 (2000), pp. 495-500 (ISSN: 0764-4442) | DOI | MR | Zbl

Rickert, N. W. Amenable groups and groups with the fixed point property, Trans. Amer. Math. Soc., Volume 127 (1967), pp. 221-232 (ISSN: 0002-9947) | DOI | MR | Zbl

Ripley, B. D. The disintegration of invariant measures, Math. Proc. Cambridge Philos. Soc., Volume 79 (1976), pp. 337-341 (ISSN: 0305-0041) | DOI | MR | Zbl

Rosendal, C. Global and local boundedness of Polish groups, Indiana Univ. Math. J., Volume 62 (2013), pp. 1621-1678 (ISSN: 0022-2518) | DOI | MR | Zbl

Srivastava, S. M., Graduate Texts in Math., 180, Springer, New York, 1998, 261 pages (ISBN: 0-387-98412-7) | DOI | MR | Zbl

Soardi, P. M.; Woess, W. Amenability, unimodularity, and the spectral radius of random walks on infinite graphs, Math. Z., Volume 205 (1990), pp. 471-486 (ISSN: 0025-5874) | DOI | MR | Zbl

Takesaki, M., Encyclopaedia of Math. Sciences, 124, Springer, Berlin, 2002, 415 pages Reprint of the first (1979) edition (ISBN: 3-540-42248-X) | MR | Zbl

Thom, A. The expected degree of minimal spanning forests (preprint arXiv:1306.0303, to appear in Combinatorica ) | MR

Thurston, W. Geometry and topology of 3-manifolds (1978) (Princeton notes)

Timár, Á. Indistinguishability of components of random spanning forests (preprint arXiv:1506.01370 ) | MR

Varadarajan, V. S. Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc., Volume 109 (1963), pp. 191-220 (ISSN: 0002-9947) | DOI | MR | Zbl

Cited by Sources: