On the superrigidity of malleable actions with spectral gap

By Sorin Popa. 2006

223 citations recorded by the source corpus.

Some of the most interesting aspects of the dynamics of measure preserving actions of countable groups on probability spaces, V rx (X, /?), are revealed by the study of group measure space von Neumann algebras L??(X) x T ([MvNl]) and the classification of actions up to orbit equivalence {OE), i.e. up to isomorphism of probability spaces carrying the orbits of actions onto each other. Although one is in von Neumann algebras and the other in ergodic theory, the two problems are closely related, as an OE of actions V rx X, A rxY has been shown to implement an algebra isomorphism L??(X) x T ~ L?? (y ) x A taking L??(X) onto L??(Y), and vice-versa ([Si], [Dy], [FM]). In particular, the isomorphism class of L??(X) xi T only depends on the equivalence relation 7?r = {{t,gt) 1t X,g ? T}. Thus, von Neumann equivalence (vNE) of group actions, requiring isomorphism of their group measure space algebras, is weaker than OE. Since there are exam ples of non-OE actions whose associated von Neumann algebras are all isomorphic ([CJ1]), it is in general strictly weaker. On the other hand, OE is manifestly weaker than classical conjugacy, which for free actions r rx X, A rx Y requires isomor phism of probability spaces A : (X,p) ~ (Y,v) satisfying Ar A-1 = A (so in particular r ~ A). How much weaker vNE and OE can be with respect to conju gacy is best seen in the amenable case, where by a celebrated theorem of Connes all free ergodic actions of all (infinite) amenable groups give rise to the same Hi factor ([Cl]) and by ([Dy], [OW], [CFW]) they are undistinguishable under OE as well. Also, any embedding of algebras L??(X) xi T C L??(Y) x A with A amenable forces T to be amenable. But the non-amenable case is extremely complex, and for many years progress has been slow ([MvN2], [Dy], [Me], [C2], [CW], [Sc]), even after the discovery of the first rigidity phenomena by Connes in von Neumann algebras ([C3, C4]) and by Zimmer in OE ergodic theory ([ZI, Z2]). This changed dramatically over the last 7-8 years, with the advent of a variety of striking rigidity results ([Ful], [Gl], [G2], [MoSh], [H], [HK], [P1]-[P8]; see [P9] for a survey; also [Sh] for a survey on OE rigidity). Our aim in this paper is to investigate the most "extreme" such phenomena, called strong rigidity, which show that for certain classes of source group actions

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e2fee7eaf5ac76608c0b16b9af43625cde00514f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Classification and rigidity for von Neumann algebras (2012, background): “cluding: (1) the standard action SLn(Z) y(Tn,λn), for n >3 [Fu98] (2) Bernoulli actions Γ y(X,µ) = (X0,µ0)Γ of many groups, including property (T) groups [Po05] and products of non-amenable groups [Po06a] (3) arbitrary free ergodic pmp actions of certain mapping class groups [Ki06] (4) free ergod”
Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors (2010, unknown): “Proposition 4.2 (Popa, [31]) .”
Bass-Serre rigidity results in von Neumann algebras (2008, methodology): “In [30], Popa showed that in many previous “deformation/rigidity” arguments (e.”
W*-superrigidity for Bernoulli actions of property (T) groups (2010, methodology): “Since [u, x] = 0, by using an idea from [39], proof of Theorem 4.”
Continuous orbit equivalence rigidity (2015, background): “Indeed, impressive orbit equivalence rigidity results have been obtained in [28, 7, 15, 16, 17, 14, 13].”
Cartan subalgebras of amalgamated free product II$_1$ factors (2012, methodology): “amated free product groups have this property. In [Io10] we proved that Bernoulli actions of icc property (T) groups are W∗-superrigid. By combining Theorem 1.1 with the cocycle superrigidity theorem [Po06a] we derive the following. Corollary 1.2. Let Γ = Γ1 ∗Λ Γ2 and Γ′ = Γ′ 1 ∗Λ′ Γ ′ 2 be two amal”
Group measure space decomposition of II1 factors and W*-superrigidity (2009, methodology): “ugh these techniques, one needs some amount of rigidity for one of the group actions and a deformation property for the other (like for example in 6.2 of [Po01], 7.1 of [Po04], 7.7 of [IPP05], 1.5 of [Po06a], etc). Since in W∗-superrigidity statements all assumptions must be on the side of the sourc”
Orbit inequivalent actions for groups containing a copy of $\mathbb{F}_{2}$ (2006, background): “Moreover, recently, the following classes of non-amenable groups have been shown to admit uncountably many non-OE actions: property (T) groups ([H]), free groups ([GP], see also [T],[I2]), weakly rigid groups ([P2]), non-amenable products of infinite groups ([ P4 ], see also [MSh],[I1]) and mapping ”
RIGIDITY FOR VON NEUMANN ALGEBRAS (2017, result): “Generalizing this result, Chifan and Sinclair showed that in fact the von Neumann algebra of any icc hyperbolic group is strongly solid Chifan and Sinclair [2013]. For further examples of strongly solid factors, see Ozawa and Popa [2010b], Houdayer [2010], Houdayer and Shlyakhtenko [2011], and Sincl”
Ergodic Subequivalence Relations Induced by a Bernoulli Action (2008, methodology): “free ultrafilter on N. Proof. By the proofs of [ Po4 , Lem.”
W*‐superrigidity for group von Neumann algebras of left–right wreath products (2012, methodology): “In [25], Popa introduced his spectral gap methods to prove W∗-rigidity theorems for A0 G in the case where G is a direct product of nonamenable groups.”
On cocycle superrigidity for Gaussian actions (2009, methodology): “([38, 40] (for Bernoulli shift actions)) Let 0 be a group which contains an infinite wq-normal subgroup 00 such that the pair (0, 00) has relative property (T), or such that 00 is the direct product of an infinite group and a non-amenable group, and let (X0, μ0) be a standard probability space.”