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Adrian Ioana

Publications and source records attributed to Adrian Ioana.

At least 55 records · Page 3Linked to original sources
arXiv2014

Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions

We study equivalence relations that arise from translation actions $Γ\curvearrowright G$ which are associated to dense embeddings $Γ<G$ of countable groups into second countable locally compact groups. Assuming that $G$ is simply connected and the action $Γ\curvearrowright G$ is strongly ergodic, we prove that $Γ\curvearrowright G$ is orbit equivalent to another such translation action $Λ\curvearrowright H$ if and only if there exists an isomorphism $δ:G\rightarrow H$ such that $δ(Γ)=Λ$. If $G$ is moreover a real algebraic group, then we establish analogous rigidity results for the translation actions of $Γ$ on homogeneous spaces of the form $G/Σ$, where $Σ<G$ is either a discrete or an algebraic subgroup. We also prove that if $G$ is simply connected and the action $Γ\curvearrowright G$ has property (T), then any cocycle $w:Γ\times G\rightarrowΛ$ with values into a countable group $Λ$ is cohomologous to a homomorphism $δ:Γ\rightarrowΛ$. As a consequence, we deduce that the action $Γ\curvearrowright G$ is orbit equivalent superrigid: any free nonsingular action $Λ\curvearrowright Y$ which is orbit equivalent to $Γ\curvearrowright G$, is necessarily conjugate to an induction of $Γ\curvearrowright G$.

math.DS↗
arXiv2013

W*-superrigidity for arbitrary actions of central quotients of braid groups

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 of the direct product $\tilde B_{n_1}\times\cdots\times\tilde B_{n_k}$, for some $n_1,\ldots,n_k\geqslant 4$. The proof uses the dichotomy theorem for normalizers inside crossed products by free groups from \cite{PV11} in combination with the OE superrigidity theorem for actions of mapping class groups from \cite{Ki06}.

math.OA↗
arXiv2012

Unbounded derivations, free dilations and indecomposability results for II$_1$ factors

We give sufficient conditions, in terms of the existence of unbounded derivations satisfying certain properties, which ensure that a II$_1$ factor $M$ is prime or has at most one Cartan subalgebra. For instance, we prove that if there exists a real closable unbounded densely defined derivation $δ:M\rightarrow L^2(M)\bar{\otimes}L^2(M)$ whose domain contains a non-amenability set, then $M$ is prime. If $δ$ is moreover "algebraic" (i.e. its domain $M_0$ is finitely generated, $δ(M_0)\subset M_0\otimes M_0$ and $δ^*(1\otimes 1)\in M_0$), then we show that $M$ has no Cartan subalgebra. We also give several applications to examples from free probability. Finally, we provide a class of countable groups $Γ$, defined through the existence of an unbounded cocycle $b:Γ\rightarrow \mathbb C(Γ/Λ)$, for some subgroup $Λ<Γ$, such that the II$_1$ factor $L^{\infty}(X)\rtimesΓ$ has a unique Cartan subalgebra, up to unitary conjugacy, for any free ergodic probability measure preserving (pmp) action $Γ\curvearrowright (X,μ)$.

math.OA↗
arXiv2012

Classification and rigidity for von Neumann algebras

In this talk, I will survey recent progress made on the classification of von Neumann algebras arising from countable groups and their actions on probability spaces. In particular, I will present the first results which provide classes of (superrigid) groups and actions that can be entirely reconstructed from their von Neumann algebras.

math.OA↗
arXiv2012

A class of superrigid group von Neumann algebras

We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra L(G) completely "remembers" the group G. More precisely, if L(G) is isomorphic to the von Neumann algebra L(Λ) of an arbitrary countable group Λ, then Λ must be isomorphic to G. This represents the first superrigidity result pertaining to group von Neumann algebras.

math.OA↗
arXiv2012

Uniqueness of the group measure space decomposition for Popa's $\Cal H\Cal T$ factors

We prove that every group measure space II$_1$ factor $L^{\infty}(X)\rtimesΓ$ coming from a free ergodic rigid (in the sense of [Po01]) probability measure preserving action of a group $Γ$ with positive first $\ell^2$--Betti number, has a unique group measure space Cartan subalgebra, up to unitary conjugacy. We deduce that many $\Cal H\Cal T$ factors, including the II$_1$ factors associated with the actions $Γ\curvearrowright \Bbb T^2$ and $Γ\curvearrowright$ SL$_2(\Bbb R)$/SL$_2(\Bbb Z)$, where $Γ$ is a non--amenable subgroup of SL$_2(\Bbb Z)$, have a unique group measure space Cartan subalgebra.

math.OA↗
arXiv2011

Rigid actions need not be strongly ergodic

A probability measure preserving action of Γon (X,μ) is called rigid if the inclusion of L^\infty(X) into the crossed product L^\infty(X) \rtimes Γhas the relative property (T) in the sense of Popa. We give examples of rigid, free, probability measure preserving actions that are ergodic but not strongly ergodic. The same examples show that rigid actions may admit non-rigid quotients.

math.OA↗
arXiv2011

W*-superrigidity for Bernoulli actions of property (T) groups

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 $M$.

math.OA↗
arXiv2010

Rigidity for equivalence relations on homogeneous spaces

We study Popa's notion of rigidity for equivalence relations induced by actions on homogeneous spaces. For any lattices $Γ,Λ$ in a semisimple Lie group $G$ with finite center and no compact factors we prove that the action $Γ\curvearrowright G/Λ$ is rigid. If in addition $G$ has property (T) then we derive that the von Neumann algebra $L^{\infty}(G/Λ)\rtimesΓ$ has property (T). We also show that if the adjoint action of $G$ on the Lie algebra of $G$ - $\{0\}$ is amenable (e.g. if $G=SL_2(\Bbb R)$), then any ergodic subequivalence relation of the orbit equivalence relation of the action $Γ\curvearrowright G/Λ$ is either hyperfinite or rigid.

math.DS↗
arXiv2009

On a Question of D. Shlyakhtenko

In this short note we construct two countable, infinite conjugacy class groups which admit free, ergodic, probability measure preserving orbit equivalent actions, but whose group von Neumann algebras are not (stably) isomorphic.

math.OA↗
arXiv2009

On Relative Property (T) and Haagerup's Property

We consider the following three properties for countable discrete groups $Γ$: (1) $Γ$ has an infinite subgroup with relative property (T), (2) the group von Neumann algebra $LΓ$ has a diffuse von Neumann subalgebra with relative property (T) and (3) $Γ$ does not have Haagerup's property. It is clear that (1) $\Longrightarrow$ (2) $\Longrightarrow$ (3). We prove that both of the converses are false.

math.GR↗
arXiv2009

Non-Orbit Equivalent Actions of $\Bbb F_n$

For any $2\leq n\leq \infty$, we construct a concrete 1-parameter family of non-orbit equivalent actions of the free group $\Bbb F_n$. These actions arise as diagonal products between a generalized Bernoulli action and the action $\Bbb F_n\curvearrowright (\Bbb T^2,λ^2)$, where $\Bbb F_n$ is seen as a subgroup of SL$_2(\Bbb Z)$.

math.GR↗
arXiv2009

Relative Property (T) for the Subequivalence Relations Induced by the Action of SL$_2(\Bbb Z)$ on $\Bbb T^2$

Let $\Cal S$ be the equivalence relation induced by the action SL$_2(\Bbb Z)\curvearrowright (\Bbb T^2,λ^2)$, where $λ^2$ denotes the Haar measure on the 2-torus, $\Bbb T^2$. We prove that any ergodic subequivalence relation $\Cal R$ of $\Cal S$ is either hyperfinite or rigid in the sense of S. Popa ([Po06]). The proof uses an ergodic-theoretic criterion for rigidity of countable, ergodic, probability measure preserving equivalence relations. Moreover, we give a purely ergodic-theoretic formulation of rigidity for free, ergodic, probability measure preserving actions of countable groups.

math.OA↗
arXiv2008

Cocycle Superrigidity for Profinite Actions of Property (T) Groups

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 $Γ\times X\to Γ\times X_n$, for some $n$. As a corollary, we show that any orbit equivalence of $Γ\curvearrowright X$ with any free ergodic measure preserving action $Λ\curvearrowright Y$ comes from a (virtual) conjugacy of actions.

math.GR↗
arXiv2008

Ergodic Subequivalence Relations Induced by a Bernoulli Action

Let $Γ$ be a countable group and denote by $\Cal S$ the equivalence relation induced by the Bernoulli action $Γ\curvearrowright [0,1]^Γ$, where $[0,1]^Γ$ is endowed with the product Lebesgue measure. We prove that for any subequivalence relation $\Cal R$ of $\Cal S$, there exists a partition $\{X_i\}_{i\geq 0}$ of $[0,1]^Γ$ with $\Cal R$-invariant measurable sets such that $\Cal R_{|X_0}$ is hyperfinite and $\Cal R_{|X_i}$ is strongly ergodic (hence ergodic), for every $i\geq 1$.

math.DS↗
arXiv2006

Rigidity results for wreath product II$_1$ factors

We consider II$_1$ factors of the form $M=\bar{\bigotimes}_{G}N\rtimes G$, where either i) $N$ is a non-hyperfinite II$_1$ factor and $G$ is an ICC amenable group or ii) $N$ is a weakly rigid II$_1$ factor and $G$ is ICC group and where $G$ acts on $\bar{\bigotimes}_{G}N$ by Bernoulli shifts. We prove that isomorphism of two such factors implies cocycle conjugacy of the corresponding Bernoulli shift actions. In particular, the groups acting are isomorphic. As a consequence, we can distinguish between certain classes of group von Neumann algebras associated to wreath product groups.

math.OA↗