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Comptes Rendus Mathématique
Volume 354, n° 11
pages 1114-1118 (novembre 2016)
Doi : 10.1016/j.crma.2016.09.011
Received : 6 July 2015 ;  accepted : 27 September 2016
Approximations of standard equivalence relations and Bernoulli percolation at p u
Approximations de relations d'équivalence standard et percolation de Bernoulli à p u
 

Damien Gaboriau a , Robin Tucker-Drob b
a CNRS, Unité de mathématiques pures et appliquées, ENS-Lyon, Université de Lyon, France 
b Department of Mathematics, Texas A&M University, College Station, TX, USA 

Abstract

The goal of this note is to announce certain results in orbit equivalence theory, especially concerning the approximation of p.m.p. standard equivalence relations by increasing sequences of sub-relations, with application to the behavior of the Bernoulli percolation on Cayley graphs at the threshold  .

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Résumé

Le but de cette note est d'annoncer certains résultats d'équivalence orbitale, concernant notamment la notion d'approximation de relations d'équivalence standard préservant la mesure de probabilité par suites croissantes de sous-relations, avec application au comportement en   de la percolation de Bernoulli sur les graphes de Cayley.

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Version française abrégée

La notion de relation d'équivalence standard hyperfinie (i.e. réunion croissante de sous-relations standard finies) joue un rôle fondamental en théorie de l'équivalence orbitale. Plus généralement, on considère la notion d'approximation d'une relation d'équivalence mesurée standard  , i.e. la possibilité d'écrire   comme une réunion croissante d'une suite de sous-relations d'équivalence standard  . Une telle approximation est dite triviale s'il existe une partie borélienne A de mesure non nulle sur laquelle les restrictions coïncident à partir d'un certain rang :  . Nous établissons des conditions sous lesquelles les approximations de certaines relations d'équivalence préservant la mesure de probabilité (p.m.p. ) sont nécessairement triviales.

Théorème 0.1

Soit G un groupe engendré par deux sous-groupes infinis de type fini H et K qui commutent. Considérons une action p.m.p.   sur l'espace borélien standard telle que H agit de manière fortement ergodique et K de manière ergodique. Alors, toute approximation de la relation d'équivalence engendrée   est nécessairement triviale.

Puisque les actions par décalage de Bernoulli des groupes non moyennables sont automatiquement fortement ergodiques, ce résultat a des conséquences en théorie de la percolation de Bernoulli sur les graphes de Cayley. Pour des informations concernant les liens entre équivalence orbitale et percolation, on peut consulter [[3]]. En fait, le couplage standard permet de traduire l'étude relative aux variations du paramètre de rétention   de la percolation en l'étude d'une famille croissante de relations d'équivalence standard p.m.p.  , telle que pour tout  , on a  . Le paramètre critique   (cf. [[5]]) est l'infimum des p pour lesquels on peut trouver une partie borélienne non négligeable A sur laquelle les restrictions   et   coïncident (de tels p sont dits appartenir à la phase d'unicité ). Pour les groupes dont les actions Bernoulli n'admettent pas d'approximation non triviale, le paramètre   lui-même n'appartient pas à la phase d'unicité. C'est le cas des groupes qui apparaissent dans le Théorème 0.1. Des conditions d'exhaustion par des sous-groupes distingués (en un sens faible) nous permettent d'élargir encore la famille de nouveaux exemples.

Les notions de dimension géométrique et de dimension approximative d'une relation d'équivalence mesurée ont été introduites dans [[2]], où il est démontré qu'une non-annulation du d -ième nombre de Betti   fournit une minoration par d de ces deux notions de dimension. La première est analogue à la notion de dimension géométrique pour un groupe et la deuxième est le minimum des   des dimensions géométriques le long des suites approximantes. Bien entendu, pour les relations non approximables, les deux notions de dimension coïncident. On peut alors exhiber des familles de groupes possédant des actions de dimensions approximatives variables.

English version

Bernoulli bond percolation

Let   be a Cayley graph for a finitely generated group G . The Bernoulli bond percolation on  , with retention parameter  , considers the i.i.d. assignment to each edge in E of the value 1 (open) with probability p and of the value 0 (closed) with probability  . The number of infinite clusters (connected components of open edges), for the resulting probability measure   on  , is  -a.s. either 0, 1 or ∞. There are two critical values,  , depending on the graph, which govern three regimes, as summarized in the following picture (see [[5]]):



While it is far from being entirely understood, there are some partial results concerning the situation at the threshold  , and our Theorem 1.1 contributes to this study.

For groups with infinitely many ends,   [[8]]; thus the percolation at   belongs to the uniqueness phase. On the other hand, the percolation at the threshold   does not belong to the uniqueness phase (and thus  ) for all Cayley graphs of infinite groups with Kazhdan's property (T) [[9]]. Y. Peres [[11]] proved that for a non-amenable direct product of infinite groups  , and for any Cayley graph associated with a generating system   with   and  , the percolation at   does not belong to the uniqueness phase. We extend this result to a larger family of groups than direct products, and to any of their Cayley graphs.

Theorem 1.1

Let G be a non-amenable group generated by two commuting infinite and finitely generated subgroups H and K. Then for every Cayley graph   of G, the percolation at   does not belong to the uniqueness phase.

The same result holds when G admits an infinite normal subgroup H such that the pair   has the relative property (T). This has also been observed by C. Houdayer (personal communication). Using some weak forms of normality, we can extend the scope of our theorem, for instance when G is a nonamenable (generalized) Baumslag–Solitar group (see Theorem 2.6), or a nonamenable HNN-extension of   relative to an isomorphism between two finite index subgroups.

Theorem 1.1 follows from a general result on approximations of standard probability measure preserving equivalence relations (Théorème 0.1). We refer to [[3]] and references therein for general information concerning connections between equivalence relations and percolation on graphs.

Approximations of standard equivalence relations

Let   be a standard probability measure preserving (p.m.p. ) equivalence relation on the atomless probability standard Borel space  . See [[1]] for a general axiomatization of this notion.

Definition 2.1

An approximation   to   is an exhausting increasing sequence of standard sub-equivalence relations:  . An approximation is trivial if there is some n and a non-negligible Borel subset   on which the restrictions coincide:  . We say that   is non-approximable if every approximation is trivial. An action   is approximable if its orbit equivalence relation   is approximable.

For instance, all free p.m.p. actions of a non-finitely generated group are approximable. Finite standard equivalence relations are non-approximable.

Proposition 2.2

The following are examples of approximable equivalence relations.

(1)
Every aperiodic p.m.p. action of an (infinite) amenable group is approximable by a sequence of sub-equivalence relations with finite classes.
(2)
Every ergodic non-strongly ergodic p.m.p. equivalence relation admits an approximation by   with diffuse ergodic decompositions.
(3)
Any free product   of aperiodic p.m.p. equivalence relations is approximable.

Item (1) follows from the Ornstein–Weiss theorem [[10]]. Item (2) relies heavily on results of Jones–Schmidt [[7]]. Recall that strong ergodicity , a reinforcement of ergodicity introduced by K. Schmidt, requires that: for every sequence   of Borel subsets of X such that   for each  , we must have  . Item (3) will be developed in [[4]].

Proposition 2.3

The following are examples of non-approximable equivalence relations.

(1)
Every p.m.p. action of a Kazdhan property (T) group is non-approximable.
(2)
Every p.m.p. action of  , where   acts ergodically, is non-approximable. More generally, this is the case for free actions of finitely generated relative property (T) pairs  , where H is normal, infinite, and acts ergodically.

We prove the following effective version of Théorème 0.1.

Theorem 2.4

Let G be a countable group generated by two commuting subgroups H and K. Consider a p.m.p. action   of G in which H acts strongly ergodically and K acts ergodically. Let   be any Borel sub-equivalence relation of  . For each  , set  . Let S and T be generating sets for H and K respectively. Then, for every  , there exists   such that if   satisfies:

(i)
  for all  , and
(ii)
  for all  ,
then there exists a Borel set  , with  , where the restrictions coincide:  .

Sketch of proof. Since the action of H is strongly ergodic, for every  , we may find   such that if   is any Borel set satisfying  , then either   or  .

Given  , we choose   such that  . Strong ergodicity for H delivers  . We then choose δ satisfying the condition  .

By the commuting assumption, for every k in the group K , for every s in the generating set  , we have that  . Hence, by property (i),  , so that for each  
(1)either μ(Ak)<ϵ0 or μ(Ak)>1−ϵ0. Consider now the subset   of K .

Property (ii) along with ((1)) and  , imply  .
Since  , then   is a subgroup of K . Indeed, clearly  , and if   then   hence   by ((1)), and thus  .
It follows that  . We have shown that   for all  .

Theorem 2.7 of [[6]] then implies that  , for every element   of the full group of the orbit equivalence relation   of K . Thus, by Lemma 2.14 of [[6]], there exists an  -invariant Borel set   with   such that  . Indeed,   is relatively non-approximable in   (see below). We now claim that
(2)for each g∈G, either μ(Ag)<8ϵ0, or g−1B∩B⊆Ag (thus in this case μ(Ag)>1−8ϵ0). If   for some  . Then the set   is a non-null subset of B , so it meets almost every   equivalence class since   is ergodic. For each   we can find some   such that  . Then   and  , so  , whence  .

Let  .

Since   and  , then properties (i) and (ii) and Claim ((2)) imply that  .
Since   then   is a subgroup of G : It is clear that   (since  ). If   then   and likewise  , so that   and hence   by Claim ((2)).
Therefore,  . This shows that  .  □

Consider a pair   of p.m.p. standard equivalence relations. A standard sub-relation   of p.m.p. standard equivalence relations is relatively non-approximable if for every approximation   of  , there is some n and a non-negligible A with  . This notion is useful through several variants of the following proposition.

Proposition 2.5

If   contains a sub-equivalence relation   and if   is generated by a family   of isomorphisms of the space such that   is ergodic for each i, then every approximation   for which there is a non-negligible A with   has to be trivial.

Consider such an approximation. We introduce the Window Trick :

Let   be the sub-relation of   generated by   and  . We claim that:

(a)
 , and
(b)
  is an approximation of  .

Now, the set   is  -invariant: if   and  , then  . So  . Thus   has full measure as soon as it is non-negligible, and this happens for large enough n since   is an approximation. Taking an n that is suitable for all i , we obtain  . So that  .  □

Let   be a Baumslag–Solitar group . The kernel N of the modular map  ,  ,   consists of the elements w of G that commute with a certain power   of a .

Theorem 2.6

If the kernel N of the modular map acts strongly ergodically and all the (non-trivial) powers of a act ergodically, then the action of   is non-approximable.

Indeed, one can find a finitely generated subgroup   of N that already acts strongly ergodically. There is a common power   that commutes with  . Applying Théorème 0.1, we obtain that   is non-approximable. Thus the sub-relation generated by   is relatively non-approximable. Proposition 2.5 applied to the pair of relations generated by   and   with  , first; and then, the same proposition applied to the pair generated by   with  , proves the result. □

We also obtain similar results for (most) inner amenable groups and various related families of groups.

Approximate and geometric dimensions

Besides consequences in Bernoulli bond percolation, Theorem 2.4 allows us to obtain some information about the approximate dimension .

A standard p.m.p. equivalence relation  , when considered as a measured groupoid, may act on bundles (fields) of simplicial complexes   over X . The action is proper if its restriction to the 0-skeleton   of the sub-bundle is smooth. The dimension of such a bundle is the maximum dimension of a fiber  , and the bundle is said to be contractible if (almost) every fiber is contractible. The geometric dimension of   is the minimum of the dimensions of the  -bundles which are proper and contractible. The approximate dimension of   is the minimum of the dimensions d such that   admits an approximation   by sub-relations of dimension d . These notions were introduced in [[2]].

For instance, smooth equivalence relations have geometric dimension = 0. Aperiodic treeable equivalence relations are exactly those with geometric dimension =1. Their approximate dimension is = 0 if and only if they are hyperfinite and is = 1 otherwise. One can show the general inequalities: approx-dim ≤ geom-dim ≤ approx-dim + 1. It is unknown whether there are groups admitting free p.m.p. actions with different geometric dimensions. As for approximate dimension, various situations may occur. For instance, we obtain:

Proposition 3.1

Let   be the direct product of d copies of the free group   and one copy of  . All its free p.m.p. actions have geometric dimension =  . It admits both free p.m.p. actions with approximate dimension = d and =  .

As already mentioned, free products of equivalence relations are always approximable. This is no longer the case for free actions of amalgamated free products over an infinite central subgroup   when the common subgroup has indices greater than 3 in the factors (apply Théorème 0.1 to, say, the Bernoulli shift action with   and  ). This allows us to produce examples of group actions that are amalgamated free products of treeable over amenable, but which are not approxi-treeable (approximable by treeable): take for instance   and   abelian.


Acknowledgements

The first author was supported by the ANR project GAMME (ANR-14-CE25-0004) and by the CNRS. The second author was supported by NSF grant DMS 1600904.

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