Abstract
We show that if \({\Gamma = \Gamma_1\times\dotsb\times \Gamma_n}\) is a product of \({{\rm n} \geq 2}\) non-elementary ICC hyperbolic groups then any discrete group \({\Lambda}\) which is \({W^*}\)-equivalent to \({\Gamma}\) decomposes as a direct product of n ICC groups and does not decompose as a direct product of k ICC groups when \({{\rm n} \not= {\rm k}}\). This gives a group-level strengthening of Ozawa and Popa’s unique prime decomposition theorem by removing all assumptions on the group \({\Lambda}\). This result in combination with Margulis’ normal subgroup theorem allows us to give examples of lattices in the same Lie group which do not generate stably equivalent II1 factors.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berbec M., Vaes S.: W*-superrigidity for group von Neumann algebras of left-right wreath products. Proc. Lond. Math. Soc. 108, 1116–1152 (2014)
R. Boutonnet and A. Carderi. Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups, preprint 2014
N.P. Brown and N. Ozawa. \({{\rm C}^\ast}\)-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, AMS, Providence.
Chifan I., Ioana A.: On a question of D Shylakhtenko. Proc. Am. Math. Soc. 139, 1091–1093 (2011)
I. Chifan, Y. Kida, and S. Pant, Structural results for the von Neumann algebras associated with surface braid groups, I, preprint. arXiv:1412.8025.
Chifan I., Sinclair T.: On the structural theory of \({{\rm II}_1}\) factors of negatively curved groups. Ann. Sci. Éc. Norm. Sup. 46, 1–33 (2013)
Chifan I., Sinclair T., Udrea B.: On the structural theory of \({{\rm II}_1}\) factors of negatively curved groups, II. Actions by product groups. Adv. Math. 245, 208–236 (2013)
Connes A.: Classification of injective factors. Ann. Math. 104, 73–115 (1976)
Cowling M., Haagerup U.: Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one. Invent. Math. 96, 507–549 (1989)
Dykema K.: Free products of hyperfinite von Neumann algebras and free dimension. Duke Math. J. 69, 97–119 (1993)
Furman A.: Gromov’s measure equivalence and rigidity of higher rank lattices. Ann. Math. 150(2), 1059–1081 (1999)
L. Ge. On maximal injective subalgebras of factors. Adv. Math. 118 no. 1 (1996), 34–70.
Ioana A.: Uniqueness of the group measure space decomposition for Popa’s \({\mathcal {H}\mathcal {T}}\) factors. Geom. Funct. Anal. 22, 699–732 (2012)
A. Ioana, S. Popa, and S. Vaes. A Class of superrigid group von Neumann algebras. Ann. Math. (2) 178 (2013), 231–286.
V.F.R. Jones. Index for subfactors. Invent. Math. 72 no. 1 (1983), 1–25.
Margulis G.A.: Finiteness of quotients of discrete groups. Func. Anal. Appl. 13, 178–187 (1979)
N. Monod and Y. Shalom. Orbit equivalence rigidity and bounded cohomology. Ann. Math. (2) 164 (2006), 825–878.
Ozawa N.: Solid von Neumann algebras. Acta Math. 192, 111–117 (2004)
N. Ozawa. A Kurosh type theorem for type II1 factors. Int. Math. Res. Not., Vol. 2006, Article ID97560, pp. 21.
Ozawa N.: Boundary amenability of relatively hyperbolic groups. Topol. Appl. 53, 2624–2630 (2006)
N. Ozawa and S. Popa. Some prime factorization results for type II1 factors. Invent. Math. 156 (2004), 223–234.
Ozawa N., Popa S.: On a class of II1 factors with at most one Cartan subalgebra. Ann. Math. 172, 713–749 (2010)
M. Pimsner and S. Popa. Entropy and index for subfactors. Ann. Sci. École Norm. Sup. 19 (1986), 57–106.
S. Popa. Strong rigidity of \({{\rm II}_1}\) factors arising from malleable actions of w-rigid groups I. Invent. Math. 165 (2006), 369–408.
S. Popa. Deformation and rigidity for group actions and von Neumann algebras, International Congress of Mathematicians. Vol. I, 445–477, Eur. Math. Soc., Zürich (2007).
Popa S., Vaes S.: Unique Cartan decomposition for \({{\rm II}_1}\) factors arising from arbitrary actions of hyperbolic groups. J. Reine Angew. Math. 694, 215–239 (2014)
S. Vaes. Explicit computations of all finite index bimodules for a family of II1 factors. Ann. Sci. Éc. Norm. Sup. 41 (2008), 743–788.
Vaes S.: One-cohomology and the uniqueness of the group measure space decomposition of a II1 factor. Math. Ann. 355, 661–696 (2013)
D.-V. Voiculescu. Circular and semicircular systems and free product factors, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), 45-60, Progr. Math. 92, Birkhäuser Boston, Boston, MA (1990).
R. J. Zimmer. Strong rigidity for ergodic actions of semisimple Lie groups. Ann. of Math. (2) 112 (1980), 511–529.
R.J. Zimmer. Ergodic Theory and Semisimple Groups, Monographs in Mathematics 81, Birkhäuser, Basel (1984).
Author information
Authors and Affiliations
Corresponding author
Additional information
I.C. was supported by NSF Grant DMS #1301370. R.dS. was supported in part by GAANN fellowship grants #P200A100028 and #P200A120058 and by a Sloan Center minigrant.
Rights and permissions
About this article
Cite this article
Chifan, I., de Santiago, R. & Sinclair, T. W*-rigidity for the von Neumann algebras of products of hyperbolic groups. Geom. Funct. Anal. 26, 136–159 (2016). https://doi.org/10.1007/s00039-016-0361-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-016-0361-z