We give an elementary proof that there are two topological generators for the full group of every aperiodic hyperfinite probability measure preserving Borel equivalence relation. Our proof explicitly constructs topological generators for the orbit equivalence relation of the irrational rotation of the circle, and then appeals to Dye's theorem and a Baire category argument to conclude the general case.
A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we show that this question has a positive answer when the acting group is locally nilpotent. This extends previous results obtained by Gao-Jackson for abelian groups and by Jackson-Kechris-Louveau for finitely generated nilpotent-by-finite groups. Our proof is based on a mixture of coarse geometric properties of locally nilpotent groups together with an adaptation of the Gao-Jackson machine…
We show that for any Polish group $G$ and any countable normal subgroup $Γ\triangleleft G$, the coset equivalence relation $G/Γ$ is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite.
A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we prove that this question always has a positive answer when the acting group is polycyclic, and we obtain a positive answer for all free actions of a large class of groups including the lamplighter group and all virtually solvable groups having finite Prüfer rank. This marks the first time that a group of exponential volume-growth has been verified to have this property. In obtaining this…
Almost forty years ago, Connes, Feldman and Weiss proved that for measurable equivalence relations the notions of amenability and hyperfiniteness coincide. In this paper we define the uniform version of amenability and hyperfiniteness for measurable graphed equivalence relations of bounded vertex degrees and prove that these two notions coincide as well. Roughly speaking, a measured graph $\cG$ is uniformly hyperfinite if for any $\eps>0$ there exists $K\geq 1$ such that not only $\cG$, but all of its subgraphs of positive measure are $(\eps,K)$-hyperfinite. We also show that this condition is…
We give a formula relating the topological rank of the full group of an aperiodic pmp equivalence relation to the cost of its ergodic components. Furthermore, we obtain examples of full groups having 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 give an algebraic characterization of aperiodicity for pmp equivalence relations,…
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 unimodula…
Consider a measured equivalence relation acting on a bundle of hyperbolic metric spaces by isometries. We prove that every aperiodic hyperfinite subequivalence relation is contained in a {\em unique} maximal hyperfinite subequivalence relation. We classify elements of the full group according to their action on fields on boundary measures (extending earlier results of Kaimanovich), study the existence and residuality of different types of elements and obtain an analogue of Tits' alternative.
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.
We use edge slidings and saturated disjoint Borel families to give a conceptually simple proof of Hjorth's theorem on cost attained: if a countable p.m.p. ergodic equivalence relation $E$ is treeable and has cost $n \in \mathbb{N} \cup \{\infty\}$ then it is induced by an a.e. free p.m.p. action of the free group $\mathbb{F}_n$ on $n$ generators. More importantly, our techniques give a significant strengthening of this theorem: the action of $\mathbb{F}_n$ can be arranged so that each of the $n$ generators alone acts ergodically. The existence of an ergodic action for the first generator immed…
We associate to every action of a Polish group on a standard probability space a Polish group that we call the orbit full group. For discrete groups, we recover the well-known full groups of pmp equivalence relations equipped with the uniform topology. However, there are many new examples, such as orbit full groups associated to measure preserving actions of locally compact groups. In fact, we show that such full groups are complete invariants of orbit equivalence. We give various characterizations of the existence of a dense conjugacy class for orbit full groups, and we show that the ergodic …
We show that a Borel action of a standard Borel group which is isomorphic to a sum of a countable abelian group with a countable sum of real lines and circles induces an orbit equivalence relation which is hypersmooth, i.e., Borel reducible to eventual agreement on sequences of reals, and it follows from this result along with the structure theory for locally compact abelian groups that Borel actions of Polish LCA groups induce orbit equivalence relations which are essentially hyperfinite, extending a result of Gao and Jackson and answering a question of Ding and Gao.
After a review of some of the main results about hyperfinite equivalence relations and their cocycles in the measured setting, we give a definition of a topological AF-equivalence relation. We show that every cocycle is cohomologous to a quasi-product cocycle. We then study the problem of determining the quasi-invariant probability measures admitting a given cocycle as their Radon-Nikodym derivative.
An ergodic p.m.p. equivalence relation $ \mathcal{R}$ is said to be stable if $\mathcal{R} \cong \mathcal{R} \times \mathcal{R}_0$ where $\mathcal{R}_0$ is the unique hyperfinite ergodic type $\mathrm{II}_1$ equivalence relation. We prove that a direct product $\mathcal{R} \times \mathcal{S}$ of two ergodic p.m.p. equivalence relations is stable if and only if one of the two components $\mathcal{R}$ or $\mathcal{S}$ is stable. This result is deduced from a new local characterization of stable equivalence relations. The similar question on McDuff $\mathrm{II}_1$ factors is also discussed and so…
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…
A measure-preserving action of a discrete countable group on a standard probability space is called stable if the associated equivalence relation is isomorphic to its direct product with the ergodic hyperfinite equivalence relation of type II_1. We show that any Baumslag-Solitar group has such an ergodic, free and stable action. It follows that any Baumslag-Solitar group is measure equivalent to its direct product with any amenable group. The same property is obtained for the inner amenable groups of Vaes.
We prove that commutator subgroups of topological full groups arising from minimal subshifts have exponential growth. We also prove that the measurable full group associated to the countable, measure-preserving, ergodic and hyperfinite equivalence relation is topologically generated by two elements.
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.
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\curv…
In this work, we study pmp actions of countable groups on arbitrary diffuse probability spaces under the point of view of weak equivalence. We will show that any such an action is weakly equivalent to an action on a standard probability space. We also propose a metric on the space of actions modulo weak equivalence which is equivalent to the topology of Abért and Elek. We will give a simpler proof of the compactness of the space, showing that convergence is characterized by ultraproducts. Using this topology, we will show that a profinite action is weakly equivalent to an ultraproduct of finit…
Let $E$ be a measure preserving equivalence relation, with countable equivalence classes, on a standard Borel probability space $(X,B,μ)$. Let $([E],d_{u})$ be the the (Polish) full group endowed with the uniform metric. If $F_r = \langle s_1, \ldots, s_r \rangle$ is a free group on $r$-generators and $α\in \operatorname{Hom}(F_r,[E])$ then the stabilizer of a $μ$-random point $α(F_r)_x$ is a random subgroup of $F_r$ 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 $F_r$ is obtained by t…
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.
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 $i\geq 1$.
In this paper we study the dynamics of Bernoulli flows and their subflows over general countable groups from the symbolic and topological perspectives. We study free subflows (subflows in which every point has trivial stabilizer), minimal subflows, disjointness of subflows, the problem of classifying subflows up to topological conjugacy, and the differences in dynamical behavior between pairs of points which disagree on finitely many coordinates. We call a point hyper aperiodic if the closure of its orbit is a free subflow and we call it minimal if the closure of its orbit is a minimal subflow…
