arXiv subjects
Publications and source records attributed to Adrian Ioana.
We construct a sofic approximation of $\mathbb F_2\times\mathbb F_2$ that is not essentially a "branched cover" of a sofic approximation by homomorphisms. This answers a question of L. Bowen.
We develop a new method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C$^*$-algebras of countable groups with (relative) property (T). We derive that the full C$^*$-algebras of the groups $\mathbb Z^2\rtimes\text{SL}_2(\mathbb Z)$ and $\text{SL}_n(\mathbb Z)$, for $n\geq 3$, do not have the local lifting property (LLP). We also prove that the full C$^*$-algebras of a large class of groups $Γ$ with property (T), including those such that $\text{H}^2(Γ,\mathbb R)\not=0$ or $\text{H}^2(Γ,\mathbb ZΓ)\not=0$, do not have the lifting property (LP). More generally, we show that the same holds if $Γ$ admits a probability measure preserving action with non-vanishing second $\mathbb R$-valued cohomology. Finally, we prove that the full C$^*$-algebra of any non-finitely presented property (T) group fails the LP.
We study the notion of permutation stability (or P-stability) for countable groups. Our main result provides a wide class of non-amenable product groups which are not P-stable. This class includes the product group $Σ\timesΛ$, whenever $Σ$ admits a non-abelian free quotient and $Λ$ admits an infinite cyclic quotient. In particular, we obtain that the groups $\mathbb F_m\times\mathbb Z^d$ and $\mathbb F_m\times\mathbb F_n$ are not P-stable, for any integers $m,n\geq 2$ and $d\geq 1$. This implies that P-stability is not closed under the direct product construction, which answers a question of Becker, Lubotzky and Thom. The proof of our main result relies on a construction of asymptotic homomorphisms from $Σ\timesΛ$ to finite symmetric groups starting from sequences of finite index subgroups in $Σ$ and $Λ$ with and without property $(τ)$. Our method is sufficiently robust to show that the groups covered are not even flexibly P-stable, thus giving the first such non-amenable residually finite examples.
We provide a class of separable II$_1$ factors $M$ whose central sequence algebra is not the "tail" algebra associated to any decreasing sequence of von Neumann subalgebras of $M$. This settles a question of McDuff \cite{Mc69d}.
We introduce a wide class of countable groups, called properly proximal, which contains all non-amenable bi-exact groups, all non-elementary convergence groups, and all lattices in non-compact semi-simple Lie groups, but excludes all inner amenable groups. We show that crossed product II$_1$ factors arising from free ergodic probability measure preserving actions of groups in this class have at most one weakly compact Cartan subalgebra, up to unitary conjugacy. As an application, we obtain the first $W^*$-strong rigidity results for compact actions of $SL_d(\mathbb Z)$ for $d \geq 3$.
We provide a fairly large class of II$_1$ factors $N$ such that $M=N\bar{\otimes}R$ has a unique McDuff decomposition, up to isomorphism, where $R$ denotes the hyperfinite II$_1$ factor. This class includes all II$_1$ factors $N=L^{\infty}(X)\rtimesΓ$ associated to free ergodic probability measure preserving (p.m.p.) actions $Γ\curvearrowright (X,μ)$ such that either (a) $Γ$ is a free group, $\mathbb F_n$, for some $n\geq 2$, or (b) $Γ$ is a non-inner amenable group and the orbit equivalence relation of the action $Γ\curvearrowright (X,μ)$ satisfies a property introduced in \cite{JS85}. On the other hand, settling a problem posed by Jones and Schmidt in 1985, we give the first examples of countable ergodic p.m.p. equivalence relations which do not satisfy the property of \cite{JS85}. We also prove that if $\mathcal R$ is a countable strongly ergodic p.m.p. equivalence relation and $\mathcal T$ is a hyperfinite ergodic p.m.p. equivalence relation, then $\mathcal R\times\mathcal T$ has a unique stable decomposition, up to isomorphism. Finally, we provide new characterisations of property Gamma for II$_1$ factors and of strong ergodicity for countable p.m.p. equivalence relations.
Let $Γ\curvearrowright (X,μ)$ be a measure preserving action of a countable group $Γ$ on a standard probability space $(X,μ)$. We prove that if the action $Γ\curvearrowright X$ is not profinite and satisfies a certain spectral gap condition, then there does not exist a countable-to-one Borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets). As a consequence, we show that if $Γ$ is a countable dense subgroup of a compact non-profinite group $G$ such that the left translation action $Γ\curvearrowright G$ has spectral gap, then $Γ\curvearrowright G$ is antimodular and not orbit equivalent to any, {\it not necessarily free}, profinite action. This provides the first such examples of compact actions, partially answering a question of Kechris and answering a question of Tsankov.
We provide new examples of translation actions on locally compact groups with the "local spectral gap property" introduced in \cite{BISG15}. This property has applications to strong ergodicity, the Banach-Ruziewicz problem, orbit equivalence rigidity, and equidecomposable sets. The main group of study here is the group $\text{Isom}(\mathbb{R}^d)$ of orientation-preserving isometries of the euclidean space $\mathbb{R}^d$, for $d \geq 3$. We prove that the translation action of a countable dense subgroup $Γ$ on Isom$(\mathbb R^d)$ has local spectral gap, whenever the translation action of the rotation projection of $Γ$ on $\text{SO}(d)$ has spectral gap. Our proof relies on the amenability of $\text{Isom}(\mathbb{R}^d)$ and on work of Lindenstrauss and Varjú, \cite{LV14}.
We survey some of the progress made recently in the classification of von Neumann algebras arising from countable groups and their measure preserving actions on probability spaces. We emphasize results which provide classes of (W$^*$-superrigid) actions that can be completely recovered from their von Neumann algebras and II$_1$ factors that have a unique Cartan subalgebra. We also present cocycle superrigidity theorems and some of their applications to orbit equivalence. Finally, we discuss several recent rigidity results for von Neumann algebras associated to groups.
The goals of this paper are twofold. First, we generalize the result of Gaboriau and Lyons [GL07] to the setting of von Neumann's problem for equivalence relations, proving that for any non-amenable ergodic probability measure preserving (pmp) equivalence relation $\mathcal{R}$, the Bernoulli extension over a non-atomic base space $(K, κ)$ contains the orbit equivalence relation of a free ergodic pmp action of $\mathbb{F}_2$. Moreover, we provide conditions which imply that this holds for any non-trivial probability space $K$. Second, we use this result to prove that any non-amenable unimodular locally compact second countable group admits uncountably many free ergodic pmp actions which are pairwise not von Neumann equivalent (hence, pairwise not orbit equivalent).
We prove that if $Γ$ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the II$_1$ factor $L(Γ)$ is prime. In particular, we deduce that the II$_1$ factors associated to the arithmetic groups $\text{PSL}_2(\mathbb Z[\sqrt{d}])$ and $\text{PSL}_2(\mathbb Z[S^{-1}])$ are prime, for any square-free integer $d\geq 2$ with $d\not\equiv 1 mod{4}$ and any finite non-empty set of primes $S$. This provides the first examples of prime II$_1$ factors arising from lattices in higher rank semisimple Lie groups. More generally, we describe all tensor product decompositions of $L(Γ)$ for icc countable groups $Γ$ that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show that $L(Γ)$ is prime, unless $Γ$ is a product of infinite groups, in which case we prove a unique prime factorization result for $L(Γ)$.
We provide a fairly large family of amalgamated free product groups $Γ=Γ_1\ast_ΣΓ_2$ whose amalgam structure can be completely recognized from their von Neumann algebras. Specifically, assume that $Γ_i$ is a product of two icc non-amenable bi-exact (e.g., hyperbolic) groups, and $Σ$ is icc amenable and has trivial one-sided commensurator in $Γ_i$, for every $i\in\{1,2\}$. Then $Γ$ satisfies the following rigidity property: any group $\La$ such that $L(\La)$ is isomorphic to $L(\G)$ admits an amalgamated free product decomposition $\La=\La_1\ast_Δ\La_2$ such that the inclusions $L(Δ)\subseteq L(\La_i)$ and $L(Σ)\subseteq L(\G_i)$ are isomorphic, for every $i\in\{1,2\}$. This result significantly strengthens some of the previous Bass-Serre rigidity results for von Neumann algebras. As a corollary, we obtain the first examples of amalgamated free product groups which are W$^*$-superrigid.
We introduce a novel notion of {\it local spectral gap} for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action $Γ\curvearrowright G$, whenever $Γ$ is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group $G$. This extends to the non-compact setting recent works of Bourgain and Gamburd \cite{BG06,BG10}, and Benoist and de Saxcé \cite{BdS14}. We present several applications to the Banach-Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on $G$. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique $Γ$-invariant finitely additive measure defined on all bounded measurable subsets of $G$.
We prove that if a measure distal action $α$ of a countable group $Γ$ is weakly contained in a strongly ergodic probability measure preserving action $β$ of $Γ$, then $α$ is a factor of $β$. In particular, this applies when $α$ is a compact action. As a consequence, we show that the weak equivalence class of any strongly ergodic action completely remembers the weak isomorphism class of the maximal distal factor arising in the Furstenberg-Zimmer Structure Theorem.
We prove that there exist uncountably many separable II$_1$ factors whose ultrapowers (with respect to arbitrary ultrafilters) are non-isomorphic. In fact, we prove that the families of non-isomorphic II$_1$ factors originally introduced by McDuff \cite{MD69a,MD69b} are such examples. This entails the existence of a continuum of non-elementarily equivalent II$_1$ factors, thus settling a well-known open problem in the continuous model theory of operator algebras.
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. In particular, we obtain the first examples of compact actions of $\mathbb F_2\times\mathbb F_2$ which are W$^*$-superrigid.
We study equivalence relations $\mathcal R(Γ\curvearrowright G)$ that arise from left translation actions of countable groups on their profinite completions. Under the assumption that the action $Γ\curvearrowright G$ is free and has spectral gap, we describe precisely when $\mathcal R(Γ\curvearrowright G)$ is orbit equivalent or Borel reducible to another such equivalence relation $\mathcal R(Λ\curvearrowright H)$. As a consequence, we provide explicit uncountable families of free ergodic probability measure preserving (p.m.p.) profinite actions of $SL_2(\mathbb Z)$ and its non-amenable subgroups (e.g. $\mathbb F_n$, with $2\leqslant n\leqslant\infty$) whose orbit equivalence relations are mutually not orbit equivalent and not Borel reducible. In particular, we show that if $S$ and $T$ are distinct sets of primes, then the orbit equivalence relations associated to the actions $SL_2(\mathbb Z)\curvearrowright\prod_{p\in S}SL_2(\mathbb Z_p)$ and $SL_2(\mathbb Z)\curvearrowright\prod_{p\in T}SL_2(\mathbb Z_p)$ are neither orbit equivalent nor Borel reducible. This settles a conjecture of S. Thomas \cite{Th06}. Other applications include the first calculations of outer automorphism groups for concrete treeable p.m.p. equivalence relations, and the first concrete examples of free ergodic p.m.p. actions of $\mathbb F_{\infty}$ whose orbit equivalence relations have trivial fundamental group.
We study Cartan subalgebras in the context of amalgamated free product II$_1$ factors and obtain several uniqueness and non-existence results. We prove that if $Γ$ belongs to a large class of amalgamated free product groups (which contains the free product of any two infinite groups) then any II$_1$ factor $L^{\infty}(X)\rtimesΓ$ arising from a free ergodic probability measure preserving action of $Γ$ has a unique Cartan subalgebra, up to unitary conjugacy. We also prove that if $\mathcal R=\mathcal R_1*\mathcal R_2$ is the free product of any two non-hyperfinite countable ergodic probability measure preserving equivalence relations, then the II$_1$ factor $L(\mathcal R)$ has a unique Cartan subalgebra, up to unitary conjugacy. Finally, we show that the free product $M=M_1*M_2$ of any two II$_1$ factors does not have a Cartan subalgebra. More generally, we prove that if $A\subset M$ is a diffuse amenable von Neumann subalgebra and $P\subset M$ denotes the algebra generated by its normalizer, then either $P$ is amenable, or a corner of $P$ embeds into $M_1$ or $M_2$.