Hyperfinite Equivalence Relations Research Articles

Hyperfiniteness of boundary actions of relatively hyperbolic groups

We show that if G is a finitely generated group hyperbolic relative to a finite collection of subgroups \mathcal{P} , then the natural action of G on the geodesic boundary of the associated relative Cayley graph induces a hyperfinite equivalence relation. As a corollary of this, we obtain that the natural action of G on its Bowditch boundary \partial (G,\mathcal{P}) also induces a hyperfinite equivalence relation. This strengthens a result of Ozawa obtained for \mathcal{P} consisting of amenable subgroups and uses a recent work of Marquis and Sabok.

Locally nilpotent groups and hyperfinite equivalence relations

Locally nilpotent groups and hyperfinite equivalence relations

Borel asymptotic dimension and hyperfinite equivalence relations

Borel asymptotic dimension and hyperfinite equivalence relations

The Shannon–McMillan–Breiman theorem beyond amenable groups

We introduce a new isomorphism-invariant notion of entropy for measure-preserving actions of arbitrary countable groups on probability spaces, which we call orbital Rokhlin entropy. It employs Danilenko’s orbital approach to entropy of a partition, and it is motivated by Seward’s recent generalization of Rokhlin’s characterization of entropy from amenable to general groups. A key ingredient in our approach is the use of an auxiliary probability-measure-preserving hyperfinite equivalence relation. Under the assumption of ergodicity of the auxiliary equivalence relation, our main result is a Shannon–McMillan–Breiman pointwise almost sure convergence theorem for the orbital entropy of partitions in measure-preserving group actions, the first such convergence result going beyond the realm of amenable groups. As a special case, we obtain a Shannon–McMillan–Breiman theorem for all strongly mixing actions of any countable group. Furthermore, we compare orbital Rokhlin entropy to Rokhlin entropy, and using an important recent result of Seward, we show that they coincide for free ergodic actions of any countable group. Finally, we consider actions of non-abelian free groups and demonstrate the geometric significance of the entropy equipartition property implied by the Shannon–McMillan–Breiman theorem. We show that the orbital entropy of a partition is the limit of the information functions of the sequence of partitions arising from refining any given finite partition along almost every horoball in the group.

Hyperfiniteness of boundary actions of hyperbolic groups

We prove that for every finitely generated hyperbolic group $G$, the action of $G$ on its Gromov boundary induces a hyperfinite equivalence relation.

Cardinal characteristics and countable Borel equivalence relations

AbstractBoykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we introduce a family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number . We analyze some of the basic behavior of these properties, showing, e.g., that the property corresponding to the splitting number coincides with smoothness. We then settle many of the implication relationships between the properties; these relationships turn out to be closely related to (but not the same as) the Borel Tukey ordering on cardinal characteristics.

Stable actions and central extensions

A probability-measure-preserving action of a countable group is called stable if its transformation-groupoid absorbs the ergodic hyperfinite equivalence relation of type II_1 under direct product. We show that for a countable group G and its central subgroup C, if G/C has a stable action, then so does G. Combining a previous result of the author, we obtain a characterization of a central extension having a stable action.

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.

Countable abelian group actions and hyperfinite equivalence relations

An equivalence relation E on a standard Borel space is hyperfinite if E is the increasing union of countably many Borel equivalence relations \(E_n\) where all \(E_n\)-equivalence classs are finite. In this article we establish the following theorem: if a countable abelian group acts on a standard Borel space in a Borel manner then the orbit equivalence relation is hyperfinite. The proof uses constructions and analysis of Borel marker sets and regions in the space \(2^{{\mathbb {Z}}^{<\omega }}.\) This technique is also applied to a problem of finding Borel chromatic numbers for invariant Borel subspaces of \(2^{{\mathbb {Z}}^n}\).

Stability in orbit equivalence for Baumslag–Solitar groups and Vaes groups

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 \mathrm {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.

On Constructing Ergodic Hyperfinite Equivalence Relations of Non-Product Type

AbstractProduct type equivalence relations are hyperfinitemeasured equivalence relations, which, up to orbit equivalence, are generated by product type odometer actions. We give a concrete example of a hyperfinite equivalence relation of non-product type, which is the tail equivalence on a Bratteli diagram. In order to show that the equivalence relation constructed is not of product type we will use a criterion called property A. This property, introduced by Krieger for non-singular transformations, is defined directly for hyperfinite equivalence relations in this paper.

Some remarks on topological full groups of Cantor minimal systems II

AbstractWe 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.

Characters on the full group of an ergodic hyperfinite equivalence relation

Let X be the space of all infinite 0 , 1 -sequences and E be the tail equivalence relation on X . In this paper we introduce a dense subgroup S ( 2 ∞ ) of the full group [ E ] and describe all indecomposable characters on S ( 2 ∞ ) . As result we obtain a description of indecomposable characters on [ E ] .

A coloring property for countable groups

AbstractMotivated by research on hyperfinite equivalence relations we define a coloring property for countable groups. We prove that every countable group has the coloring property. This implies a compactness theorem for closed complete sections of the free part of the shift action ofGon 2G. Our theorems generalize known results about.

On subrelations of ergodic measured type III equivalence relations

We discuss the classification up to orbit equivalence of inclusions S ⊂ R of measured ergodic discrete hyperfinite equivalence relations. In the case of type III relations, the orbit equivalence classes of such inclusions of finite index are completely classified in terms of triplets consisting of a transitive permutation group G on a finite set (whose cardinality is the index of S ⊂ R), an ergodic nonsingular R-flow V and a homomorphism of G to the centralizer of V . 0. Introduction. We consider nonsingular discrete ergodic hyperfinite equivalence relations on a standard measure space. Our concern is to classify pairs (R,S) where R is an ergodic equivalence relation and S ⊂ R is a subrelation of finite index (which means that the R-equivalence class of a.e. point consists of finitely many S-classes), up to orbit equivalence. This problem is closely related to the classification of subfactors in von Neumann algebras theory. For a single equivalence relation R the problem was solved by H. Dye [Dy] and W. Krieger [Kr] in terms of the associated flows. Then, in the case where R is of type II1, J. Feldman, C. Sutherland, and R. Zimmer [FSZ] provided a simple classification of ergodic R-subrelations of finite index and normal R-subrelations of arbitrary index. (We remark that in an earlier paper [Ge] M. Gerber classified R-subrelations of finite index in a different—but equivalent—context of finite extensions of ergodic probability preserving transformations.) These results were further extended in [Da1, §4] and [Da2], where quasinormal subrelations of type II1 were introduced and studied. Recently, T. Hamachi considered finite index subrelations of a type III0 equivalence relation R, introduced a system of invariants for orbit equiva2000 Mathematics Subject Classification: 28D99, 46L55. The research was supported in part by INTAS 97-1843.

Quasinormal subrelations of ergodic equivalence relations

We introduce a notion of quasinormality for a nested pair S ⊂ R \mathcal {S}\subset \mathcal {R} of ergodic discrete hyperfinite equivalence relations of type I I 1 II_{1} . (This is a natural extension of the normality concept due to Feldman-Sutherland-Zimmer.) Such pairs are characterized by an irreducible pair F ⊂ Q F\subset Q of countable amenable groups or rather (some special) their Polish closure F ¯ ⊂ Q ¯ \overline {F}\subset \overline {Q} . We show that “most” of the ergodic subrelations of R \mathcal {R} are quasinormal and classify them. An example of a nonquasinormal subrelation is given. We prove as an auxiliary statement that two cocycles of R \mathcal {R} with dense ranges in a Polish group are weakly equivalent.

The classification of hypersmooth Borel equivalence relations

This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which has received particular attention is the class of hyperfinite Borel equivalence relations. These can be defined as the increasing unions of sequences of Borel equivalence relations all of whose equivalence classes are finite or, as it turns out, equivalently those induced by the orbits of a single Borel auto-morphism. Hyperfinite equivalence relations have been classified in [DJK], under two notions of equivalence, Borel bi-reducibility, and Borel isomorphism.

A Borel Parametrlzatlon of Polish Groups

This paper constructs a standard space, PG, and a map p^PG—>G(p} such that each G(p) is a Polish group, and such that every Polish group is isomorphic to at least one of the groups G(p) ; PG thus serves as a parameter space for all Polish groups. We formulate the notion of a map from a standard B Space to Polish groups, and that of a from a standard groupoid to Polish groups; both are defined in terms of the existence of factorizations through PG. We apply these ideas to establish a general Cohomology Lemma, asserting that cocycles, with values in family of Polish groups, may be cobounded into a given family of dense, normal, subgroups, whenever the underlying groupoid is a hyperfinite equivalence relation. The purpose of this paper is to provide a parametrizatio n, by a standard space PG, for the space of Polish topological groups, i. e. those second countable topological groups whose underlying topology may be defined by a complete metric, and to present applications of this to the notion of Borel functor from a standard groupoid to Polish groups. The need for such concepts became apparent during the course of joint work with M. Takesaki on the classification of the possible actions (up to cocycle conjugacy) of a discrete amenable group on a hyperfinite, semifinite injective von Neumann algebra [12], and the paper can be viewed as preparatory to this work. However, the point of view adopted also reveals a definition of A. Connes, [4], of the notion of from a standard groupoid to standard measure spaces, as being very natural. A common situation in which the problems considered here arise is the following: if G is a Polish group and X a Polish G-space under

Foliations of polynomial growth are hyperfinite

We define a class of equivalence relations with polynomial growth and show that such relations always support finite invariant measures and are hyperfinite. In particular, foliations of polynomial growth define hyperfinite equivalence relations with respect to any family of finite invariant measures on transversals. We also extend a result of Dye for countable groups to show that if a locally compact second countable groupG acts freely on a Lebesgue spaceX with finite invariant measure, so that the orbit relation onX is hyperfinite, thenG is amenable.

The Rohlin tower theorem and hyper-finiteness for actions of continuous groups

We prove the Rohlin tower theorem for free measure preserving actions of locally compact second countable solvable groups and almost connected amenable groups. This theorem was known for l.c.s.c. abelian groups and was recently extended by Ornstein and Weiss to discrete solvable groups. We extend their methods to the continuous case, using the structure theory of the class of groups under consideration. As a corollary we obtain that free actions of such groups generate hyperfinite equivalence relations.