We consider group measure space II$_1$ factors $M=L^{\infty}(X)\rtimesΓ$ arising from Bernoulli actions of ICC property (T) groups $Γ$ (more generally, of groups $Γ$ containing an infinite normal subgroup with relative property (T)) and prove a rigidity result for *--homomorphisms $θ:M\to M\bar{\otimes}M$. We deduce that the action $Γ\curvearrowright X$ is W$^*$--superrigid. This means that if $Λ\curvearrowright Y$ is {\bf any other} free, ergodic, measure preserving action such that the factors $M=L^{\infty}(X)\rtimesΓ$ and $L^{\infty}(Y)\rtimesΛ$ are isomorphic, then the actions $Γ\curvearrowright X$ and $Λ\curvearrowright Y$ must be conjugate. Moreover, we show that if $p\in M\setminus\{1\}$ is a projection, then $pMp$ does not admit a group measure space decomposition nor a group von Neumann algebra decomposition (the latter under the additional assumption that $Γ$ is torsion free).
We also prove a rigidity result for *--homomorphisms $θ:M\to M$, this time for $Γ$ in a larger class of groups than above, now including products of non--amenable groups. For certain groups $Γ$, e.g. $Γ=\Bbb F_2\times\Bbb F_2$, we deduce that $M$ does not embed in $pMp$, for any projection $p\in M\setminus\{1\}$, and obtain a description of the endomorphism semigroup of $M$.
We prove that if a countable group $Γ$ contains infinite commuting subgroups $H, H'\subset Γ$ with $H$ non-amenable and $H'$ ``weakly normal'' in $Γ$, then any measure preserving $Γ$-action on a probability space which satisfies certain malleability, spectral gap and weak mixing conditions (e.g. a Bernoulli $Γ$-action) is cocycle superrigid. If in addition $H'$ can be taken non-virtually abelian and $Γ\curvearrowright X$ is an arbitrary free ergodic action while $Λ\curvearrowright Y=\Bbb T^Λ$ is a Bernoulli action of an arbitrary infinite conjugacy class group, then any isomorphism of the asso…
We prove W$^*$-superrigidity for a large class of coinduced actions. We prove that if $Σ$ is an amenable almost-malnormal subgroup of an infinite conjugagy class (icc) property (T) countable group $Γ$, the coinduced action $Γ\curvearrowright X$ from an arbitrary probability measure preserving action $Σ\curvearrowright X_0$ is W$^*$-superrigid. We also prove a similar statement if $Γ$ is an icc non-amenable group which is measure equivalent to a product of two infinite groups. In particular, we obtain that any Bernoulli action of such a group $Γ$ is W$^*$-superrigid.
We generalize W*-superrigidity results about Bernoulli actions of rigid groups to general mixing Gaussian actions. We thus obtain the following: If Γ is any ICC group which is w-rigid (i.e. it contains an infinite normal subgroup with the relative property (T)) then any mixing Gaussian action σ of Γ is W*-superrigid. More precisely, if ρ is another free ergodic action of a group Λ such that the crossed-product von Neumann algebras associated with ρ and σ are isomorphic, then Λ and Γ are isomorphic, and the actions ρ and σ are conjugate. We prove a similar statement whenever Γ is a non-amenable…
We show that the class $\mathscr{B}$, of discrete groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli actions, is invariant under measure equivalence. We generalize this to the setting of discrete p.m.p. groupoids, and as a consequence we deduce that any nonamenable lattice in a product of two noncompact, locally compact second countable groups, must belong to $\mathscr{B}$. We also introduce a measure-conjugacy invariant called Weak Pinsker entropy and show that, if G is a group in the class $\mathscr{B}$, then Weak Pinsker entropy is an orbit-equivalence…
Consider a free ergodic measure preserving profinite action $Γ\curvearrowright X$ (i.e. an inverse limit of actions $Γ\curvearrowright X_n$, with $X_n$ finite) of a countable property (T) group $Γ$ (more generally of a group $Γ$ which admits an infinite normal subgroup $Γ_0$ such that the inclusion $Γ_0\subsetΓ$ has relative property (T) and $Γ/Γ_0$ is finitely generated) on a standard probability space $X$. We prove that if $w:Γ\times X\to Λ$ is a measurable cocycle with values in a countable group $Λ$, then $w$ is cohomologous to a cocycle $w'$ which factors through the map $Γ\times X\to Γ\t…
We prove cocycle and orbit equivalence superrigidity for lattices in SL(n,R) acting linearly on R^n, as well as acting projectively on certain flag manifolds, including the real projective space. The proof combines operator algebraic techniques with the property (T) in the sense of Zimmer for the action of SL(n,Z) on R^n, n \geq 4. We also show that the restriction of the associated orbit equivalence relation to a subset of finite Lebesgue measure, provides a II_1 equivalence relation with property (T) and yet fundamental group equal to R_+.
We prove a "unique crossed product decomposition" result for group measure space II_1 factors arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γin a fairly large family G, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T_n denotes the group of upper triangular matrices in PSL(n,Z), then any free, mixing p.m.p. action of the amalgamated free product of PSL(n,Z) with itself over T_n, is W*-superrigid, i.e. any isomorphism between L^\…
We prove that if a countable discrete group $Γ$ is {\it w-rigid}, i.e. it contains an infinite normal subgroup $H$ with the relative property (T) (e.g. $Γ= SL(2,\Bbb Z) \ltimes \Bbb Z^2$, or $Γ= H \times H'$ with $H$ an infinite Kazhdan group and $H'$ arbitrary), and $\Cal V$ is a closed subgroup of the group of unitaries of a finite von Neumann algebra (e.g. $\Cal V$ countable discrete, or separable compact), then any $\Cal V$-valued measurable cocycle for a measure preserving action $Γ\curvearrowright X$ of $Γ$ on a probability space $(X,μ)$ which is weak mixing on $H$ and {\it s-malleable} …
We prove a cocycle superrigidity theorem for a large class of coinduced actions. In particular, if $Λ$ is a subgroup of a countable group $Γ$, we consider a probability measure preserving action $Λ\curvearrowright X_0$ and let $Γ\curvearrowright X$ be the coinduced action. Assume either that $Γ$ has property (T) or that $Λ$ is amenable and $Γ$ is a product of non-amenable groups. Using Popa's deformation/rigidity theory we prove $Γ\curvearrowright X$ is $\mathcal U_{fin}$-cocycle superrigid, that is any cocycle for this action to a $\mathcal U_{fin}$ (e.g. countable) group $\mathcal V$ is coho…
We show that several important normal subgroups $Γ$ of the mapping class group of a surface satisfy the following property: any free, ergodic, probability measure preserving action $Γ\curvearrowright X$ is stably OE-superrigid. These include the central quotients of most surface braid groups and most Torelli groups and Johnson kernels. In addition, we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice-embeddings. Using these results in combination with previous results from [CIK13] we deduce that any free, ergodic, probability measure preservi…
Let $G_Γ\curvearrowright X$ and $G_Λ\curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_Γ$ and $G_Λ$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_Γ\curvearrowright X$ is stably orbit equ…
We prove that any isomorphism $θ:M_0\simeq M$ of group measure space II$_1$ factors, $M_0=L^\infty(X_0, μ_0) \rtimes_{σ_0} G_0$, $M=L^\infty(X, μ) \rtimes_σ G$, with $G_0$ containing infinite normal subgroups with the relative property (T) of Kazhdan-Margulis (i.e. $G_0$ {\it w-rigid}) and $G$ an ICC group acting by Bernoulli shifts $σ$, essentially comes from an isomorphism of probability spaces which conjugates the actions. Moreover, any isomorphism $θ$ of $M_0$ onto a ``corner'' $pMp$ of $M$, for $p\in M$ an idempotent, forces $p=1$. In particular, all group measure space factors associated…
We prove that if $G$ is a countable, discrete group having infinite, normal subgroups with the relative property (T), then the Bernoulli shift action of $G$ on ${\underset g \in G \to Π} (X_0, μ_0)_g$ for $(X_{0},μ_{0})$ an arbitrary probability space, has first cohomology group isomorphic to the character group of $G$.
Using an approach emerging from the theory of closable derivations on von Neumann algebras, we exhibit a class of groups CR satisfying the following property: given any groups G_1, G_2 in CR, then any free, ergodic, measure preserving action on a probability space G_1 x G_2 on X gives rise to a von Neumann algebra with unique group measure space Cartan subalgebra. Pairing this result with Popa's Orbit Equivalence Superrigidity Theorem we obtain new examples of W*-superrigid actions.
We survey Sorin Popa's recent work on Bernoulli actions. The paper was written on the occasion of the Bourbaki seminar. Using very original methods from operator algebras, Sorin Popa has shown that the orbit structure of the Bernoulli action of a property (T) group, completely remembers the group and the action. This information is even essentially contained in the crossed product von Neumann algebra, yielding the first von Neumann strong rigidity theorem in the literature. The same methods allow Popa to obtain II_1 factors with prescribed countable fundamental group.
We prove that for a weakly mixing algebraic action $σ: G\curvearrowright(X,ν)$, the $n$-cohomology group $H^n(G\curvearrowright X; \mathbb{T})$, after quotienting out the natural subgroup $H^n(G,\mathbb{T})$, contains $H^n(G,\widehat{X})$ as a natural subgroup for $n=1$. If we further assume the diagonal actions $σ^2, σ^4$ are $\mathbb{T}$-cocycle superrigid and $H^2(G, \widehat{X})$ is torsion free as an abelian group, then the above also holds true for $n=2$. Applying it for principal algebraic actions when $n=1$, we show that $H^2(G,\mathbb{Z}G)$ is torsion free as an abelian group when $G$…
We show that if G is a discrete group which does not have the Haagerup property but does have an unbounded cocycle into a C_0 representation and if G acts on a finite von Neumann algebra B such that the inclusion B \subset (B \rtimes G) has the Haagerup property from below then any group-measure space Cartan subalgebra must have a corner which embeds into B inside B \rtimes G. Taking the action to be trivial we produce examples of II_1 factors N such that N \otimes M is not a group-measure space construction whenever M is a finite factor with the Haagerup property. Taking the action on a proba…
We study actions of countable discrete groups which are amenable in the sense that there exists a mean on X which is invariant under the action of G. Assuming that G is nonamenable, we obtain structural results for the stabilizer subgroups of amenable actions which allow us to relate the first l2-Betti number of G with that of the stabilizer subgroups. An analogous relationship is also shown to hold for cost. This relationship becomes even more pronounced for transitive amenable actions, leading to a simple criterion for vanishing of the first l2-Betti number and triviality of cost. Moreover, …
A countable group Γis called shift-minimal if every non-trivial measure preserving action of Γweakly contained in the Bernoulli shift of Γon ([0,1]^Γ,λ^Γ) is free. We show that any group Γwhose reduced C^*-algebra admits a unique tracial state is shift-minimal, and that any group Γadmitting a free measure preserving action of cost>1 contains a finite normal subgroup N such that Γ/N is shift-minimal. Any shift-minimal group in turn is shown to have trivial amenable radical. Recurrence arguments are used in studying invariant random subgroups of a wide variety of shift-minimal groups. We also ex…
