Non-orbit equivalent actions of ${𝔽}_n$

Ioana, Adrian

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 4, pp. 675-696

For any $2\le n\le \infty$, 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 ${𝔽}_n\curvearrowright ({𝕋}^2,{\lambda }^2)$, where ${𝔽}_n$ is seen as a subgroup of ${\mathrm{SL}}_2\left(\mathbb{Z}\right)$.

Pour tout $2\le n\le \infty$, 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 ${𝔽}_n\curvearrowright ({𝕋}^2,{\lambda }^2)$, où ${𝔽}_n$ est vu comme un sous-groupe de ${\mathrm{SL}}_2\left(\mathbb{Z}\right)$.

Classification : 37A20, 46L10
Keywords: free groups, orbit equivalence
Mots-clés : groupes libres, équivalence orbitale

[1] S. I. Bezuglyĭ & V. Y. Golodets, Hyperfinite and ${\mathrm{II}}_1$ actions for nonamenable groups, J. Funct. Anal. 40 (1981), 30-44. | Zbl | MR

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

[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. | Zbl | MR

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

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

[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. | Zbl | MR

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

[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. | Zbl | MR

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

[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. | Zbl | MR

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

[14] A. Ioana, Orbit inequivalent actions for groups containing a copy of ${𝔽}_2$, preprint arXiv:math/0701027, 2007. | Zbl | MR

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

[16] A. Ioana, Rigidity results for wreath product ${\mathrm{II}}_1$ factors, J. Funct. Anal. 252 (2007), 763-791. | Zbl | MR

[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. | Zbl | MR

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

[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. | Zbl | MR

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

[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. | Zbl | MR

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

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

[25] S. Popa, On a class of type ${\mathrm{II}}_1$ factors with Betti numbers invariants, Ann. of Math. 163 (2006), 809-899. | Zbl | MR

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

[27] S. Popa, Strong rigidity of ${\mathrm{II}}_1$ factors arising from malleable actions of $w$-rigid groups. I, Invent. Math. 165 (2006), 369-408. | Zbl | MR

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

[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. | Zbl | MR

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

[31] S. Popa & S. Vaes, Actions of ${𝔽}_{\infty }$ whose II${}_1$ factors and orbit equivalence relations have prescribed fundamental group, preprint arXiv:0803.3351, 2008, to appear in J. Amer. Math. Soc. | Zbl | MR

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

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

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