Ergodic Equivalence Relations Research Articles

Solid Ergodicity and Orbit Equivalence Rigidity for Coinduced Actions

Abstract We prove that the solid ergodicity property is stable with respect to taking coinduction for a fairly large class of coinduced action. More precisely, assume that $\Sigma <\Gamma $ are countable groups such that $g\Sigma g^{-1}\cap \Sigma $ is finite for any $g\in \Gamma \setminus \Sigma $. Then any measure preserving action $\Sigma \curvearrowright X_0$ gives rise to a solidly ergodic equivalence relation if and only if the equivalence relation of the associated coinduced action $\Gamma \curvearrowright X$ is solidly ergodic. We also obtain orbit equivalence rigidity for such actions by showing that the orbit equivalence relation of a rigid or compact measure preserving action $\Sigma \curvearrowright X_0$ of a property (T) group is “remembered” by the orbit equivalence relation of $\Gamma \curvearrowright X$.

Representing interpolated free group factors as group factors

We construct a one parameter family of ICC groups \{G_t\}_{t > 1} , with the property that the group factor L(G_t) is isomorphic to the interpolated free group factor L(\mathbb F_t):=L(\mathbb{F}_2)^{1/\sqrt{t-1}} , for all t . Moreover, the groups G_t have fixed cost t , are strongly treeable and freely generate any treeable ergodic equivalence relation of same cost.

Highly faithful actions and dense free subgroups in full groups

In this paper, we show that every measure-preserving ergodic equivalence relation of cost less than m comes from a “rich” faithful invariant random subgroup of the free group on m generators, strengthening a result of Bowen which had been obtained by a Baire category argument. Our proof is completely explicit: we use our previous construction of topological generators for full groups and observe that these generators induce a totally non free action. We then twist this construction so that the action is moreover amenable onto almost every orbit and highly faithful. In particular, we obtain that the full group of a measure-preserving ergodic equivalence of cost less than m contains a dense free subgroup on m generators.

Strongly ergodic equivalence relations: spectral gap and type III invariants

We obtain a spectral gap characterization of strongly ergodic equivalence relations on standard measure spaces. We use our spectral gap criterion to prove that a large class of skew-product equivalence relations arising from measurable $1$-cocycles with values in locally compact abelian groups are strongly ergodic. By analogy with the work of Connes on full factors, we introduce the Sd and $\unicode[STIX]{x1D70F}$ invariants for type $\text{III}$ strongly ergodic equivalence relations. As a corollary to our main results, we show that for any type $\text{III}_{1}$ ergodic equivalence relation ${\mathcal{R}}$, the Maharam extension $\text{c}({\mathcal{R}})$ is strongly ergodic if and only if ${\mathcal{R}}$ is strongly ergodic and the invariant $\unicode[STIX]{x1D70F}({\mathcal{R}})$ is the usual topology on $\mathbb{R}$. We also obtain a structure theorem for almost periodic strongly ergodic equivalence relations analogous to Connes’ structure theorem for almost periodic full factors. Finally, we prove that for arbitrary strongly ergodic free actions of bi-exact groups (e.g. hyperbolic groups), the Sd and $\unicode[STIX]{x1D70F}$ invariants of the orbit equivalence relation and of the associated group measure space von Neumann factor coincide.

Generic IRS in free groups, after Bowen

Let E E be a measure preserving equivalence relation, with countable equivalence classes, on a standard Borel probability space ( X , B , μ ) (X,\mathcal {B},\mu ) . Let ( [ E ] , d u ) ([E],d_{u}) be the (Polish) full group endowed with the uniform metric. If F r = ⟨ s 1 , … , s r ⟩ \mathbb {F}_r = \langle s_1, \ldots , s_r \rangle is a free group on r r -generators and α ∈ H o m ( F r , [ E ] ) \alpha \in \mathrm {Hom}(\mathbb {F}_r,[E]) , then the stabilizer of a μ \mu -random point \alpha (\mathbb {F}_r)_x \ltcc \mathbb {F}_r is a random subgroup of F r \mathbb {F}_r whose distribution is conjugation invariant. Such an object is known as an invariant random subgroup or an IRS for short. Bowen’s generic model for IRS in F r \mathbb {F}_r is obtained by taking α \alpha to be a Baire generic element in the Polish space H o m ( F r , [ E ] ) {\mathrm {Hom}}(\mathbb {F}_r, [E]) . The lean aperiodic model is a similar model where one forces α ( F r ) \alpha (\mathbb {F}_r) to have infinite orbits by imposing that α ( s 1 ) \alpha (s_1) be aperiodic. In Bowen’s setting we show that for r > ∞ r > \infty the generic IRS \alpha (\mathbb {F}_r)_x \ltcc \mathbb {F}_r is of finite index almost surely if and only if E = E 0 E = E_0 is the hyperfinite equivalence relation. For any ergodic equivalence relation we show that a generic IRS coming from the lean aperiodic model is co-amenable and core free. Finally, we consider the situation where α ( F r ) \alpha (\mathbb {F}_r) is highly transitive on almost every orbit and in particular the corresponding IRS is supported on maximal subgroups. Using a result of Le Maître we show that such examples exist for any aperiodic ergodic E E of finite cost. For the hyperfinite equivalence relation E 0 E_0 we show that high transitivity is generic in the lean aperiodic model.

Connected Polish groups with ample generics

In this paper, we give the first examples of connected Polish groups that have ample generics, answering a question of Kechris and Rosendal. We show that any Polish group with ample generics embeds into a connected Polish group with ample generics and that full groups of type III hyperfinite ergodic equivalence relations have ample generics. We also sketch a proof of the following result: the full group of any type III ergodic equivalence relation has topological rank 2.

On full groups of non-ergodic probability-measure-preserving equivalence relations

This article generalizes our previous results [Le Maître. The number of topological generators for full groups of ergodic equivalence relations. Invent. Math. 198 (2014), 261–268] to the non-ergodic case by giving a formula relating the topological rank of the full group of an aperiodic probability-measure-preserving (pmp) equivalence relation to the cost of its ergodic components. Furthermore, we obtain examples of full groups that have a dense free subgroup whose rank is equal to the topological rank of the full group, using a Baire category argument. We then study the automatic continuity property for full groups of aperiodic equivalence relations, and find a connected metric for which they have the automatic continuity property. This allows us to provide an algebraic characterization of aperiodicity for pmp equivalence relations, namely the non-existence of homomorphisms from their full groups into totally disconnected separable groups. A simple proof of the extreme amenability of full groups of hyperfinite pmp equivalence relations is also given, generalizing a result of Giordano and Pestov to the non-ergodic case [Giordano and Pestov. Some extremely amenable groups related to operator algebras and ergodic theory. J. Inst. Math. Jussieu6(2) (2007), 279–315, Theorem 5.7].

The number of topological generators for full groups of ergodic equivalence relations

We completely elucidate the relationship between two invariants associated with an ergodic probability measure-preserving (pmp) equivalence relation, namely its cost and the minimal number of topological generators of its full group. It follows that for any free pmp ergodic action of the free group on $n$ generators, the minimal number of topological generators for the full group of the action is $n+1$, answering a question of Kechris.

Geometric covering arguments and ergodic theorems for free groups

We present a new approach to the proof of ergodic theorems for actions of non-amenable groups, and give here a complete self-contained account of it in the case of free groups. Our approach is based on direct geometric covering arguments and asymptotic invariance arguments generalizing those developed in the ergodic theory of amenable groups. The results we describe go beyond those previously established for measure-preserving actions of free groups, and demonstrate the significant role the boundary action of the free group plays in the ergodic theory of its measure-preserving actions. Furthermore, our approach suggests the possibility of putting the ergodic theory of amenable groups and nonamenable groups on an equal footing : both can be viewed as special cases in the general ergodic theory of amenable ergodic equivalence relations with finite invariant measure.

Amalgamated free product type III factors with at most one Cartan subalgebra

AbstractWe investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras ${\mathop{M{}_{1} \ast }\nolimits}_{B} {M}_{2} $ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $({M}_{1} , {\varphi }_{1} )\ast ({M}_{2} , {\varphi }_{2} )$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established by A. Ioana [Cartan subalgebras of amalgamated free product ${\mathrm{II} }_{1} $factors, arXiv:1207.0054]. Next, we prove that any countable nonsingular ergodic equivalence relation $ \mathcal{R} $ defined on a standard measure space and which splits as the free product $ \mathcal{R} = { \mathcal{R} }_{1} \ast { \mathcal{R} }_{2} $ of recurrent subequivalence relations gives rise to a nonamenable factor $\mathrm{L} ( \mathcal{R} )$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors ${\mathrm{L} }^{\infty } (X)\rtimes \Gamma $ arising from nonsingular free ergodic actions $\Gamma \curvearrowright (X, \mu )$ on standard measure spaces of amalgamated groups $\Gamma = {\mathop{\Gamma {}_{1} \ast }\nolimits}_{\Sigma } {\Gamma }_{2} $ over a finite subgroup $\Sigma $.

Fuglede-Kadison determinants for operators in the von Neumann algebra of an equivalence relation

We calculate the Fuglede-Kadison determinant for operators of the form ∑n i=1 MfiLgi where Lgi are unitaries or partial isometries coming from Borel (partial) isomorphisms gi on a probability space which generate an ergodic equivalence relation, and Mfi are multiplication operators. We obtain formulas for the cases when the relation is treeable or the fi’s and gi’s satisfy some restrictions.

Topological properties of full groups

AbstractWe study full groups of countable, measure-preserving equivalence relations. Our main results include that they are all homeomorphic to the separable Hilbert space and that every homomorphism from an ergodic full group to a separable group is continuous. We also find bounds for the minimal number of topological generators (elements generating a dense subgroup) of full groups allowing us to distinguish between full groups of equivalence relations generated by free, ergodic actions of the free groups Fm and Fn if m and n are sufficiently far apart. We also show that an ergodic equivalence relation is generated by an action of a finitely generated group if an only if its full group is topologically finitely generated.

Borel complexity of isomorphism between quotient Boolean algebras

In response to a question of Farah, “How many Boolean algebras are there?” [Far04], one of us (Oliver) proved that there are continuum-many nonisomorphic Boolean algebras of the form with I a Borel ideal on the natural numbers, and in fact that this result could be improved simultaneously in two directions:(i) “Borel ideal” may be improved to “analytic P-ideal”(ii) “continuum-many” may be improved to “E0-many”; that is, E0 is Borel reducible to the isomorphism relation on quotients by analytic P-ideals.See [Oli04].In [AdKechOO], Adams and Kechris showed that the relation of equality on Borel sets (and therefore, any Borel equivalence relation whatsoever) is Borel reducible to the equivalence relation of Borel bireducibility. (In somewhat finer terms, they showed that the partial order of inclusion on Borel sets is Borel reducible to the quasi-order of Borel reducibility.) Their technique was to find a collection of, in some sense, strongly mutually ergodic equivalence relations, indexed by reals, and then assign to each Borel set B a sort of “direct sum” of the equivalence relations corresponding to the reals in B. Then if B1, ⊆ B2 it was easy to see that the equivalence relation thus induced by B1 was Borel reducible to the one induced by B2, whereas in the opposite case, taking x to be some element of B / B2, it was possible to show that the equivalence relation corresponding to x, which was part of the equivalence relation induced by B1, was not Borel reducible to the equivalence relation corresponding to B2.

A characterization of coactions whose fixed-point algebras contain special maximal abelian $\ast$-subalgebras

It is shown that, for the von Neumann algebra $A$ obtained from a principal measured groupoid $\mathcal{R}$ with the diagonal subalgebra $D$ of $A$ , there exists a natural ‘bijective’ correspondence between coactions on $A$ that fix $D$ pointwise and Borel 1-cocycles on $\mathcal{R}$ . As an application of this result, we classify a certain type of coactions on approximately finite-dimensional type II factors up to cocycle conjugacy. By using our characterization of coactions mentioned above, we are also able to generalize to some extent those results of Zimmer concerning 1-cocycles on ergodic equivalence relations into compact groups.

On the normalizing groupoids and the commensurability groupoids for inclusions of factors associated to ergodic equivalence relations–subrelations

It is shown that for the inclusion of factors ( B ⊆ A ) : = ( W ∗ ( S , ω ) ⊆ W ∗ ( R , ω ) ) corresponding to an inclusion of ergodic discrete measured equivalence relations S ⊆ R , S is normal in R in the sense of Feldman–Sutherland–Zimmer [J. Feldman, C.E. Sutherland, R.J. Zimmer, Subrelations of ergodic equivalence relations, Ergodic Theory Dynam. Systems 9 (1989) 239–269] if and only if A is generated by the normalizing groupoid of B. Moreover, we show that there exists the largest intermediate equivalence subrelation N R ( S ) which contains S as a normal subrelation. We further give a definition of “commensurability groupoid” as a generalization of normality. We show that the commensurability groupoid of B in A generates A if and only if the inclusion B ⊆ A is discrete in the sense of Izumi–Longo–Popa [M. Izumi, R. Longo, S. Popa, A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras, J. Funct. Anal. 155 (1998) 25–63]. We also show that there exists the largest equivalence subrelation Comm R ( S ) such that the inclusion B ⊆ W ∗ ( Comm R ( S ) , ω ) is discrete. It turns out that the intermediate equivalence subrelations N R ( S ) and Comm R ( S ) ⊆ R thus defined can be viewed as groupoid-theoretic counterparts of a normalizer subgroup and a commensurability subgroup in group theory.

A lemma for cost attained

A treeable ergodic equivalence relation of integer cost is generated by a free action of the free group on the corresponding number of generators. Every countable treeable ergodic equivalence relation is induced by the free action of some countable group.

L2-homology for von Neumann algebras

The aim of this paper is to introduce a notion of L-homology in the context of von Neumann algebras. Finding a suitable (co)homology theory for von Neumann algebras has been a dream for several generations (see [KR71a, KR71b, JKR72, SS95] and references therein). One’s hope is to have a powerful invariant to distinguish von Neumann algebras. Unfortunately, little positive is known about the Kadison-Ringrose cohomology H∗ b (M,M), except that it vanishes in many cases. Furthermore, there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von Neumann algebra. Our interest in developing an L-cohomology theory was revived by the introduction of Lcohomology invariants in the field of ergodic equivalence relations in the paper of Gaboriau [Gab02]. His results in particular imply that L-Betti numbers β (2) i (Γ) of a discrete group are the same for measure-equivalent groups (i.e., for groups that can generate isomorphic ergodic measure-preserving equivalence relations). Parallels between the “worlds” of von Neumann algebras and measurable equivalence relations have been noted for a long time (starting with the parallel between the work of Murray and von Neumann [MvN] and that of H. Dye [Dy]). Thus there is hope that an invariant of a group that “survives” measure equivalence will survive also “von Neumann algebra equivalence”, i.e., will be an invariant of the von Neumann algebra of the group. The original motivation for our construction comes from the well understood analogy between the theory of II1-factors and that of discrete groups, based on the theory of correspondences [Con, Con94]. This analogy has been remarkably efficient to transpose analytic properties such as “amenability” or “property T” from the group context to the factor context [Con80] [CJ] and more recently in the breakthrough work of Popa [Popa] [Con03]. We use the theory of correspondences together with the algebraic description of L-Betti numbers given by Luck. His definition involves the computation of the algebraic group homology with coefficients in the group von Neumann algebra. Following the guiding idea that the category of bimodules over a von Neumann algebra is the analogue of the category of modules over a group, we are led to the following algebraic definition of L-homology of a von Neumann algebra M :

Outer automorphism groups of some equivalence relations

Let \mathsf R a be countable ergodic equivalence relation of type {\rm II}_1 on a standard probability space (X,\mu) . The group \mathsf{Out}\ \mathsf R of outer automorphisms of \mathsf R consists of all invertible Borel measure preserving maps of the space which map \mathsf R -classes to \mathsf R -classes modulo those which preserve almost every \mathsf R -class. We analyze the group \mathsf{Out}\ \mathsf R for relations \mathsf R generated by actions of higher rank lattices, providing general conditions on finiteness and triviality of \mathsf{Out}\ \mathsf R and explicitly computing \mathsf{Out}\ \mathsf R for the standard actions. The method is based on Zimmer's superrigidity for measurable cocycles, Ratner's theorem and Gromov's Measure Equivalence construction.

A converse to Dye’s theorem

Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of F 2 \mathbb {F}_2 on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel equivalence relation which is not universal for treeable in the ≤ B \leq _B ordering.

Fundamental groups of some ergodic equivalence relations of type II∞

Automorphisms of equivalence relations generated by translations of dense arithmetic subgroups and groups of Q-rational points on semisimple noncompact Lie groups (real andp-adic) are studied. These automorphisms are proved to preserve a Haar measure.