Non-orbit equivalent actions of 𝔽 n
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 4, pp. 675-696.

For any 2n, we construct a concrete 1-parameter family of non-orbit equivalent actions of the free group 𝔽 n . These actions arise as diagonal products between a generalized Bernoulli action and the action 𝔽 n (𝕋 2 ,λ 2 ), where 𝔽 n is seen as a subgroup of SL 2 ().

Pour tout 2n, nous construisons une famille concrète à un paramètre, des actions non orbitalement équivalentes du groupe libre 𝔽 n . Ces actions apparaissent comme produits diagonaux entre une action généralisée de Bernoulli et l’action 𝔽 n (𝕋 2 ,λ 2 ), où 𝔽 n est vu comme un sous-groupe de SL 2 ().

DOI: 10.24033/asens.2106
Classification: 37A20, 46L10
Keywords: free groups, orbit equivalence
Mot clés : groupes libres, équivalence orbitale
@article{ASENS_2009_4_42_4_675_0,
     author = {Ioana, Adrian},
     title = {Non-orbit equivalent actions of $\mathbb {F}_n$},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {675--696},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {4},
     year = {2009},
     doi = {10.24033/asens.2106},
     mrnumber = {2568879},
     zbl = {1185.37009},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2106/}
}
TY  - JOUR
AU  - Ioana, Adrian
TI  - Non-orbit equivalent actions of $\mathbb {F}_n$
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2009
SP  - 675
EP  - 696
VL  - 42
IS  - 4
PB  - Société mathématique de France
UR  - http://www.numdam.org/articles/10.24033/asens.2106/
DO  - 10.24033/asens.2106
LA  - en
ID  - ASENS_2009_4_42_4_675_0
ER  - 
%0 Journal Article
%A Ioana, Adrian
%T Non-orbit equivalent actions of $\mathbb {F}_n$
%J Annales scientifiques de l'École Normale Supérieure
%D 2009
%P 675-696
%V 42
%N 4
%I Société mathématique de France
%U http://www.numdam.org/articles/10.24033/asens.2106/
%R 10.24033/asens.2106
%G en
%F ASENS_2009_4_42_4_675_0
Ioana, Adrian. Non-orbit equivalent actions of $\mathbb {F}_n$. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 4, pp. 675-696. doi : 10.24033/asens.2106. http://www.numdam.org/articles/10.24033/asens.2106/

[1] S. I. Bezuglyĭ & V. Y. Golodets, Hyperfinite and II 1 actions for nonamenable groups, J. Funct. Anal. 40 (1981), 30-44. | MR | Zbl

[2] M. Burger, Kazhdan constants for SL (3,𝐙), J. reine angew. Math. 413 (1991), 36-67. | EuDML | MR | Zbl

[3] A. Connes, J. Feldman & B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems 1 (1981), 431-450. | MR | Zbl

[4] A. Connes & B. Weiss, Property T and asymptotically invariant sequences, Israel J. Math. 37 (1980), 209-210. | MR | Zbl

[5] H. A. Dye, On groups of measure preserving transformation. I, Amer. J. Math. 81 (1959), 119-159. | MR | Zbl

[6] I. Epstein, Orbit inequivalent actions of non-amenable groups, preprint arXiv:0707.4215, 2007.

[7] J. Feldman & C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. II, Trans. Amer. Math. Soc. 234 (1977), 325-359. | MR | Zbl

[8] D. Gaboriau, On orbit equivalence of measure preserving actions, in Rigidity in dynamics and geometry (Cambridge, 2000), Springer, 2002, 167-186. | MR | Zbl

[9] D. Gaboriau, Relative property (T) actions and trivial outer automorphism groups, preprint arXiv:0804.0358, 2008. | Zbl

[10] D. Gaboriau & R. Lyons, A measurable-group-theoretic solution to von Neumann's problem, Invent. Math. 177 (2009), 533-540. | MR | Zbl

[11] D. Gaboriau & S. Popa, An uncountable family of nonorbit equivalent actions of 𝔽 n , J. Amer. Math. Soc. 18 (2005), 547-559. | MR | Zbl

[12] S. L. Gefter & V. Y. Golodets, Fundamental groups for ergodic actions and actions with unit fundamental groups, Publ. Res. Inst. Math. Sci. 24 (1988), 821-847. | MR | Zbl

[13] G. Hjorth, A converse to Dye's theorem, Trans. Amer. Math. Soc. 357 (2005), 3083-3103. | MR | Zbl

[14] A. Ioana, Orbit inequivalent actions for groups containing a copy of 𝔽 2 , preprint arXiv:math/0701027, 2007. | MR | Zbl

[15] A. Ioana, A relative version of Connes’ χ(M) invariant and existence of orbit inequivalent actions, Ergodic Theory Dynam. Systems 27 (2007), 1199-1213. | MR | Zbl

[16] A. Ioana, Rigidity results for wreath product II 1 factors, J. Funct. Anal. 252 (2007), 763-791. | MR | Zbl

[17] A. Ioana, J. Peterson & S. Popa, Amalgamated free products of weakly rigid factors and calculation of their symmetry groups, Acta Math. 200 (2008), 85-153. | MR | Zbl

[18] V. F. R. Jones & K. Schmidt, Asymptotically invariant sequences and approximate finiteness, Amer. J. Math. 109 (1987), 91-114. | MR | Zbl

[19] Y. Kida, Classification of certain generalized Bernoulli actions of mapping class groups, preprint http://www.math.kyoto-u.ac.jp/~kida/papers/ber.pdf, 2008.

[20] N. Monod & Y. Shalom, Orbit equivalence rigidity and bounded cohomology, Ann. of Math. 164 (2006), 825-878. | MR | Zbl

[21] F. J. Murray & J. Von Neumann, On rings of operators, Ann. of Math. 37 (1936), 116-229. | JFM | MR | Zbl

[22] D. S. Ornstein & B. Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980), 161-164. | MR | Zbl

[23] D. S. Ornstein & B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math. 48 (1987), 1-141. | MR | Zbl

[24] K. Petersen, Ergodic theory, Cambridge Studies in Advanced Math. 2, Cambridge Univ. Press, 1983. | MR | Zbl

[25] S. Popa, On a class of type II 1 factors with Betti numbers invariants, Ann. of Math. 163 (2006), 809-899. | MR | Zbl

[26] S. Popa, Some computations of 1-cohomology groups and construction of non-orbit-equivalent actions, J. Inst. Math. Jussieu 5 (2006), 309-332. | MR | Zbl

[27] S. Popa, Strong rigidity of II 1 factors arising from malleable actions of w-rigid groups. I, Invent. Math. 165 (2006), 369-408. | MR | Zbl

[28] S. Popa, Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups, Invent. Math. 170 (2007), 243-295. | MR | Zbl

[29] S. Popa, Deformation and rigidity for group actions and von Neumann algebras, in International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, 2007, 445-477. | MR | Zbl

[30] S. Popa, On the superrigidity of malleable actions with spectral gap, J. Amer. Math. Soc. 21 (2008), 981-1000. | MR | Zbl

[31] S. Popa & S. Vaes, Actions of 𝔽 whose II 1 factors and orbit equivalence relations have prescribed fundamental group, preprint arXiv:0803.3351, 2008, to appear in J. Amer. Math. Soc. | MR | Zbl

[32] K. Schmidt, Amenability 1 (1981), 223-236. | MR | Zbl

[33] Y. Shalom, Measurable group theory, in European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, 391-423. | MR | Zbl

[34] R. J. Zimmer, Ergodic theory and semisimple groups, Monographs in Math. 81, Birkhäuser, 1984. | MR | Zbl

Cited by Sources: