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 associated II$_1$ factors $L^\infty X \rtimes Γ\simeq L^\infty Y \rtimes Λ$ comes from a conjugacy of the actions.
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 present a general setting to investigate U_fin-cocycle superrigidity for Gaussian actions in terms of closable derivations on von Neumann algebras. In this setting we give new proofs to some U_fin-cocycle superrigidity results of S. Popa and we produce new examples of this phenomenon. We also use a result of K. Schmidt to give a necessary cohomological condition on a group representation in order for the resulting Gaussian action to be U_fin-cocycle superrigid.
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…
In this article, we prove that if a finitely generated group $G$ is not torsion then a necessary and sufficient condition for every full shift over $G$ has (continuous) cocycle superrigidity is that $G$ has one end. It is a topological version of the well known Popa's measurable cocycle superrigidity theorem. For the proof of sufficient condition, we introduce a new specification property for shifts over general groups which play a similar role as malleable property in the measurable setting. This new specification property is good enough for us to extend the method using homoclinic equivalenc…
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 $Γ\times X\to Γ\t…
We consider cross-product II$_1$ factors $M = N\rtimes_σ G$, with $G$ discrete ICC groups that contain infinite normal subgroups with the relative property (T) and $σ: G \to {\text{\rm Aut}}N$ trace preserving actions of $G$ on finite von Neumann algebras $N$ that are ``malleable'' and mixing. Examples are the weighted Bernoulli and Bogoliubov shifts. We prove a rigidity result for such factors, showing the uniqueness of the position of $L(G)$ inside $M$. We use this to calculate the fundamental group $\mycal F(M)$ in terms of the weights of the shift, for certain arithmetic groups $G$ such as…
Let $G$ be either a profinite or a connected compact group, and $Γ, Λ$ be finitely generated dense subgroups. Assuming that the left translation action of $Γ$ on $G$ is strongly ergodic, we prove that any cocycle for the left-right translation action of $Γ\timesΛ$ on $G$ with values in a countable group is virtually cohomologous to a group homomorphism. Moreover, we prove that the same holds if $G$ is a (not necessarily compact) connected simple Lie group provided that $Λ$ contains an infinite cyclic subgroup with compact closure. We derive several applications to OE - and W$^*$- superrigidity…
We prove that certain coinduced actions for an inclusion of finitely generated commensurated subgroups with relative one end are continuous cocycle superrigid actions. We also show the necessity for the relative end assumption.
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 $Γ$ be an irreducible lattice in a product of two locally compact groups and assume that $Γ$ is densely embedded in a profinite group $K$. We give necessary conditions which imply that the left translation action $Γ\curvearrowright K$ is "virtually" cocycle superrigid: any cocycle $w:Γ\times K\rightarrowΔ$ with values in a countable group $Δ$ is cohomologous to a cocycle which factors through the map $Γ\times K\rightarrowΓ\times K_0$, for some finite quotient group $K_0$ of $K$. As a corollary, we deduce that any ergodic profinite action of $Γ=\text{SL}_2(\mathbb Z[S^{-1}])$ is virtually c…
For any $n\geqslant 4$ let $\tilde B_n=B_n/Z(B_n)$ be the quotient of the braid group $B_n$ through its center. We prove that any free ergodic probability measure preserving (pmp) action $\tilde B_n\curvearrowright (X,μ)$ is W$^*$-superrigid in the following sense: if $L^{\infty}(X)\rtimes\tilde B_n\cong L^{\infty}(Y)\rtimesΛ$, for an arbitrary free ergodic pmp action $Λ\curvearrowright (Y,ν)$, then the actions $\tilde B_n\curvearrowright X,Λ\curvearrowright Y$ are stably (or, virtually) conjugate. Moreover, we prove that the same holds if $\tilde B_n$ is replaced with a finite index subgroup …
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…
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 geometric superrigidity for actions of cocompact lattices in semisimple Lie groups of higher rank on infinite dimensional Riemannian manifolds of nonpositive curvature and finite telescopic dimension.
We define a bounded cohomology class, called the {\em median class}, in the second bounded cohomology -- with appropriate coefficients --of the automorphism group of a finite dimensional CAT(0) cube complex X. The median class of X behaves naturally with respect to taking products and appropriate subcomplexes and defines in turn the {\em median class of an action} by automorphisms of X. We show that the median class of a non-elementary action by automorphisms does not vanish and we show to which extent it does vanish if the action is elementary. We obtain as a corollary a superrigidity result …
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\curvearro…
In this expository article, we give a survey of Adrian Ioana's cocycle superrigidity theorem for profinite actions of Property (T) groups, and its applications to ergodic theory and set theory. In addition to a statement and proof of Ioana's theorem, this article features: * An introduction to rigidity, including a crash course in Borel cocycles and a summary of some of the best-known superrigidity theorems; * Some easy applications of superrigidity, both to ergodic theory (orbit equivalence) and set theory (Borel reducibility); and * A streamlined proof of Simon Thomas's theorem that the clas…
Finite rank median spaces are a simultaneous generalisation of finite dimensional ${\rm CAT}(0)$ cube complexes and real trees. If $Γ$ is an irreducible lattice in a product of rank one simple Lie groups, we show that every action of $Γ$ on a complete, finite rank median space has a global fixed point. This is in sharp contrast with the behaviour of actions on infinite rank median spaces. The fixed point property is obtained as corollary to a superrigidity result; the latter holds for irreducible lattices in arbitrary products of compactly generated groups. In previous work, we introduced "Rol…
We prove general superrigidity results for actions of irreducible lattices on CAT(0) spaces; first, in terms of the ideal boundary, and then for the intrinsic geometry (including for infinite-dimensional spaces). In particular, one obtains a new and self-contained proof of Margulis' superrigidity theorem for uniform irreducible lattices in non-simple groups. The proofs rely on simple geometric arguments, including a splitting theorem which can be viewed as an infinite-dimensional (and singular) generalization of the Lawson-Yau/Gromoll-Wolf theorem.
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…
Kac-Moody groups over finite fields are finitely generated groups. Most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of non-positively curved twinned buildings: those are the most important (but not the only) examples of twin building lattices. We prove that these lattices are simple if and only if the corresponding buildings are (irreducible and) not of affine type (i.e. they are not Bruhat-Tits buildings). In fact, many of them are finitely presented and enjoy property (T). Our arguments explain geometrically why simplicity fails to ho…
We single out a large class of groups ${\mathscr{M}}$ for which the following unique prime factorization result holds: if $Γ_1,\dots,Γ_n\in {\mathscr{M}}$ and $Γ_1\times\dots\timesΓ_n$ is measure equivalent to a product $Λ_1\times\dots\timesΛ_m$ of infinite icc groups, then $n \ge m$, and if $n = m$ then, after permutation of the indices, $Γ_i$ is measure equivalent to $Λ_i$, for all $1\leq i\leq n$. This provides an analogue of Monod and Shalom's theorem \cite{MS02} for groups that belong to ${\mathscr{M}}$. Class ${\mathscr{M}}$ is constructed using groups whose von Neumann algebras admit an…
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 prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, in certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure-preserving actions, and also answers a question of Connes for such groups.
This paper includes a series of structural results for von Neumann algebras arising from measure preserving actions by product groups on probability spaces. Expanding upon the methods used earlier by the first two authors \cite{CS}, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. For instance we show that every II$_1$ factor associated with a weakly amenable group in the class $\mathcal S$ of Ozawa is strongly solid, \cite{OzSolid}. There is also the following product version of this result: any maximal abelian $\star$-subal…
Ozawa showed that for any i.c.c., hyperbolic group, the associated group factor is solid. Developing a new approach that combines some methods of Peterson, Ozawa and Popa, and Ozawa, we strengthen this result by showing that these factors are strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana, we show that profinite actions of lattices in Sp(n,1), n>1, are virtually W*-superrigid.
We prove a rigidity result for cocycles from higher rank lattices to $\mathrm{Out}(F_N)$ and more generally to the outer automorphism group of a torsion-free hyperbolic group. More precisely, let $G$ be either a product of connected higher rank simple algebraic groups over local fields, or a lattice in such a product. Let $G\curvearrowright X$ be an ergodic measure-preserving action on a standard probability space, and let $H$ be a torsion-free hyperbolic group. We prove that every Borel cocycle $G\times X\to\mathrm{Out}(H)$ is cohomologous to a cocycle with values in a finite subgroup of $\ma…
In this paper we study characters on special linear groups SL_n(R), where R is either an infinite field or the localization of an order in a number field. We give several applications to the theory of measure preserving actions, operator-algebraic superrigidity, and almost homomorphisms.
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 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 $G$…
This paper concerns a study of three families of non-compact type symmetric spaces of infinite dimension. Although they have infinite dimension they have finite rank. More precisely, we show they have finite telescopic dimension. We also show the existence of Furstenberg maps for some group actions on these spaces. Such maps appear as a first step toward superrigidity results.
We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space $X$ has a non-empty intersection in the visual bordification $ \bar{X} = X \cup \partial X$. Using this fact, several results known for proper CAT(0) spaces may be extended to finite-dimensional spaces, including the existence of canonical fixed points at infinity for parabolic isometries, algebraic and geometric restrictions on amenable group actions, and geometric superrigidity for non-elementary actions of irreducible uniform lattices in products of locally compact groups.
We analyse volume-preserving actions of product groups on Riemannian manifolds. To this end, we establish a new superrigidity theorem for ergodic cocycles of product groups ranging in linear groups. There are no a priori assumptions on the acting groups, except a spectral gap assumption on their action. Our main application to manifolds concerns irreducible actions of Kazhdan product groups. We prove the following dichotomy: Either the action is infinitesimally linear, which means that the derivative cocycle arises from unbounded linear representations of all factors. Otherwise, the action is …
For every hyperbolic group and more general hyperbolic graphs, we construct an equivariant ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant which implies that several Measure Equivalence and Orbit Equivalence rigidity results established by Monod-Shalom hold for all non-elementary hyperbolic groups and their non-elementary subgroups. We also derive superrigidity results for actions of general irreducible lattices on a large class of hyperbolic metric spaces.
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, …
