Coût des relations d’équivalence et des groupes

We study a new dynamical invariant for dicrete groups: the cost. It is a real number in {1−1/n}∪[1,∞], bounded by the number of generators of the group, and it is well behaved with respect to finite index subgroups. Namely, the quantities 1 minus the cost are related by multiplying by the index. The cost of every infinite amenable group equals 1. We compute it in some other situations, including free products, free products with amalgamation and HNN-extensions over amenable groups and for direct product situations. For instance, the cost of the free group on n generators equals n. We prove that each possible finite value of the cost is achieved by a finitely generated group. It is dynamical because it relies on measure preserving free actions on probability Borel spaces. In most cases, groups have fixed price, which implies that two freely acting groups which define the same orbit partition must have the same cost. It enables us to distinguish the orbit partitions of probability-preserving free actions of free groups of different ranks. At the end of the paper, we give a mercuriale, i.e. a list of costs of different groups. The cost is in fact an invariant of ergodic measure-preserving equivalence relations and is defined using graphings. A treeing is a measurable way to provide every equivalence class (=orbit) with the structure of a simplicial tree, this an example of graphing. Not every relation admits a treeing: we prove that every free action of a cost 1 non-amenable group is not treeable, but we prove that subrelations of treeable relations are treeable. We give examples of relations which cannot be produced by an action of any finitely generated group. The cost of a relation which can be decomposed as a direct product is shown to be 1. We define the notion for a relation to be a free product or an HNN-extension and compute the cost for the resulting relation from the costs of the building blocks. The cost is also an invariant of the pairs von Neumann algebra/Cartan subalgebra. Oblatum 27-I-1999 & 4-IV-1999 / Published online: 22 September 1999     

Localized cohomology and some applications of Popa’s cocycle superrigidity theorem

We prove that orbit equivalence of measure preserving ergodic a.e. free actions of a countable group with the relative property (T) is a complete analytic equivalence relation.     

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 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 ordering.

     

Groups generating transversals to semisimple lie group actions

We describe those discrete groups with finite measure preserving actions that are stably orbit equivalent to such an action of a higher rank simple Lie group. This is applied to obtain information on the question of when ergodic equivalence relations are generated by a free action of a group. Research partially supported by the National Science Foundation and the Israel-U.S. Binational Science Foundation.     

Measure equivalence rigidity and bi-exactness of groups

We get three types of results on measurable group theory; direct product groups of Ozawa's class S groups, wreath product groups and amalgamated free products. We prove measure equivalence factorization results on direct product groups of Ozawa's class S groups. As consequences, Monod-Shalom type orbit equivalence rigidity theorems follow. We prove that if two wreath product groups A?G, B?Γ of non-amenable exact direct product groups G, Γ with amenable bases A, B are measure equivalent, then G and Γ are measure equivalent. We get Bass-Serre rigidity results on amalgamated free products of non-amenable exact direct product groups.     

Finite entropy actions of free groups,rigidity of stabilizers,and a Howe-Moore type phenomenon

We study a notion of entropy, called f-invariant entropy, introduced by Lewis Bowen for probability measure preserving actions of finitely generated free groups. In the degenerate case, the f-invariant entropy is -∞. In this paper, we investigate the qualitative consequences of an action having finite f-invariant entropy. We find three main properties of such actions. First, the stabilizers occurring in factors of such actions are highly restricted. Specifically, the stabilizer of almost every point must be either trivial or of finite index. Second, such actions are very chaotic in the sense that when the space is not essentially countable, every non-identity group element acts with infinite Kolmogorov-Sinai entropy. Finally, we show that such actions display behavior reminiscent of the Howe-Moore property. Specifically, if the action is ergodic, there exists an integer n such that for every non-trivial normal subgroup K, the number of K-ergodic components is at most n. Our results are based on a new formula for f-invariant entropy.     

The space of stable weak equivalence classes of measure-preserving actions

The concept of (stable) weak containment for measure-preserving actions of a countable group Γ is analogous to the classical notion of (stable) weak containment of unitary representations. If Γ is amenable then the Rokhlin lemma shows that all essentially free actions are weakly equivalent. However if Γ is non-amenable then there can be many different weak and stable weak equivalence classes. Our main result is that the set of stable weak equivalence classes naturally admits the structure of a Choquet simplex. For example, when $\mathrm{\Gamma }=\mathbb{Z}$ this simplex has only a countable set of extreme points but when Γ is a nonamenable free group, this simplex is the Poulsen simplex. We also show that when Γ contains a nonabelian free group, this simplex has uncountably many strongly ergodic essentially free extreme points.     

On the classification problem of free ergodic actions of nonamenable groups

We show that, for any countable discrete nonamenable group Γ, the relations of conjugacy, orbit equivalence, stable orbit equivalence, von Neumann equivalence, and stable von Neumann equivalence of free ergodic pmp actions of Γ on the standard atomless probability space are not Borel. This answers a question of Kechris.     

The automorphism group of the Gaussian measure cannot act pointwise

Classical ergodic theory deals with measure (or measure class) preserving actions of locally compact groups on Lebesgue spaces. An important tool in this setting is a theorem of Mackey which provides spatial models for BooleanG-actions. We show that in full generality this theorem does not hold for actions of Polish groups. In particular there is no Borel model for the Polish automorphism group of a Gaussian measure. In fact, we show that this group as well as many other Polish groups do not admit any nontrivial Borel measure preserving actions.     

One-cohomology and the uniqueness of the group measure space decomposition of a II1 factor

We provide a unified and self-contained treatment of several of the recent uniqueness theorems for the group measure space decomposition of a II₁ factor. We single out a large class of groups Γ, characterized by a one-cohomology property, and prove that for every free ergodic probability measure preserving action of Γ the associated II₁ factor has a unique group measure space Cartan subalgebra up to unitary conjugacy. Our methods follow closely a recent article of Chifan–Peterson, but we replace the usage of Peterson’s unbounded derivations by Thomas Sinclair’s dilation into a malleable deformation by a one-parameter group of automorphisms.     

Normal ergodic actions

The von Neumann-Halmos theory of ergodic transformations with discrete spectrum makes use of the duality theory of locally compact abelian groups to characterize those transformations preserving a probability measure, which are defined by a rotation on a compact abelian group. We use the recently developed duality between general locally compact groups and Hopf-von Neumann algebras to characterize those actions of a locally compact group, preserving a σ-finite measure, which are defined by a dense embedding in another group. They are characterized by the property of normality, previously introduced by the author, and motivated by Mackey's theory of virtual groups. The discrete spectrum theory is readily seen to come out as the special case in which the invariant measure is finite.     

Orbit equivalence rigidity for ergodic actions of the mapping class group

We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping class groups as well.      

Ergodic flows of positive entropy can be time changed to becomeK-flows

It is shown that every Kakutani equivalence class of ergodic measure preserving transformations of positive entropy containsK-automorphisms. Also, every ergodic flow of positive entropy can be time changed to become aK-flow and every ergodic automorphism of positive entropy is a cross-section of someK-flow.     

Recurrence of matrix cocycles

This paper considers measurable cocycles with values in the subgroup of SL(2, ℂ) of diagonal and skew-diagonal matrices over an ergodic transformation preserving the probability measure. We prove the recurrence of such cocycles under certain conditions as well as the equivalence of two definitions of the recurrence.     

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. Work supported in part by NSF grant MCS 74-19876. A02.     

On simplicity concepts for ergodic actions

A weakly mixing measure preserving action of a locally compact second countable group on a standard probability space is called 2-fold near simple if every ergodic joining of it with itself is either product measure or is supported on a ‘convex combination’ of graphs. A similar definition can be given for near simplicity of higher order. This generalizes the Veech-del Junco-Rudolph notion of simplicity. Our main results include the following. An analogue of Veech theorem on factors holds for the 2-fold near simple actions. A weakly mixing group extension of an action with near MSJ is near simple. The action of a normal co-compact subgroup is near simple if and only if the whole action is near simple. The subset of all 2-fold near simple transformations (i.e., ℤ-actions) is meager in the group of measure preserving transformations endowed with the weak topology. Via the (C, F)-construction, we produce a near simple quasi-simple transformation which is disjoint from any simple map. The work was supported in part by CRDF, grant UM1-2546-KH-03. Dedicated to V. Ya. Golodets on the occasion of his 70-th birthday.     

Realizations of rotations on an indecomposable compact monothetic group

By the classical Halmos‐von Neumann theorem, each compact monothetic group corresponds to an ergodic dynamical system with discrete spectrum. For such groups we prove two results. We first construct a compact monothetic group which does not split into a direct product of a connected and a totally disconnected compact monothetic group. Then we present a measure preserving dynamical system on the unit square being isomorphic to a rotation on this indecomposable group.     

Ergodicity of Rokhlin cocycles

We develop a general study of ergodic properties of extensions of measure preserving dynamical systems. These extensions are given by cocycles (called here Rokhlin cocycles) taking values in the group of automorphisms of a measure space which represents the fibers. We use two different approaches in order to study ergodic properties of such extensions. The first approach is based on properties of mildly mixing group actions and the notion of complementary algebra. The second approach is based on spectral theory of unitary representations of locally compact Abelian groups and the theory of cocycles taking values in such groups. Finally, we examine the structure of self-joinings of extensions. We partially answer a question of Rudolph on lifting mixing (and multiple mixing) property to extensions and answer negatively a question of Robinson on lifting Bernoulli property. We also shed new light on some earlier results of Glasner and Weiss on the class of automorphisms disjoint from all weakly mixing transformations. Answering a question asked by Thouvenot we establish a relative version of the Foiaş—Stratila theorem on Gaussian—Kronecker dynamical systems. Research partially supported by KBN grant 2 P03A 002 14 (1998).     

Compact quantum metric spaces and ergodic actions of compact quantum groups

We show that for any co-amenable compact quantum group A=C(G) there exists a unique compact Hausdorff topology on the set EA(G) of isomorphism classes of ergodic actions of G such that the following holds: for any continuous field of ergodic actions of G over a locally compact Hausdorff space T the map T→EA(G) sending each t in T to the isomorphism class of the fibre at t is continuous if and only if the function counting the multiplicity of γ in each fibre is continuous over T for every equivalence class γ of irreducible unitary representations of G. Generalizations for arbitrary compact quantum groups are also obtained. In the case G is a compact group, the restriction of this topology on the subset of isomorphism classes of ergodic actions of full multiplicity coincides with the topology coming from the work of Landstad and Wassermann. Podle? spheres are shown to be continuous in the natural parameter as ergodic actions of the quantum SU(2) group. We also introduce a notion of regularity for quantum metrics on G, and show how to construct a quantum metric from any ergodic action of G, starting from a regular quantum metric on G. Furthermore, we introduce a quantum Gromov-Hausdorff distance between ergodic actions of G when G is separable and show that it induces the above topology.     

Free continuous actions on zero-dimensional spaces

We show that every countably infinite group admits a free, continuous action on the Cantor set having an invariant probability measure. We also show that every countably infinite group admits a free, continuous action on a non-homogeneous compact metric space and the action is minimal (that is to say, every orbit is dense). In answer to a question posed by Giordano, Putnam and Skau, we establish that there is a continuous, minimal action of a countably infinite group on the Cantor set such that no free continuous action of any group gives rise to the same equivalence relation.