On a measurable analogue of small topological full groups II
François Le Maître

 














































 
  
  
  
  
  
  
  
  
  

  
  
  
    
 
 
  
  
  
  
  
  
    
  
  
    

  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  



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A measurable analogue of topological full groups II]On a measurable 
analogue of small topological full groups II



Université de Paris, Sorbonne Université, CNRS,
 
	Institut de Mathématiques de Jussieu-Paris Rive Gauche,

	F-75013 Paris, France



[
    François Le Maître
    Received: 28 September 2018 / Accepted: 5 February 2019
===========================================================







We pursue the study of ^1 full groups of graphings and of the closures of their derived groups, which we call derived ^1 full groups. Our main result shows that aperiodic probability measure-preserving actions of finitely generated groups have finite Rokhlin entropy if and only if their derived ^1 full group has finite topological rank.  We further show that a graphing is amenable if and only if its ^1 full group is, and explain why various examples of (derived) ^1 full groups fit very well into Rosendal's geometric framework for Polish groups. As an application, we obtain that every abstract group isomorphism between ^1 full groups of amenable ergodic graphings must be a quasi-isometry for their respective ^1 metrics. We finally show that ^1 full groups of rank one transformations have topological rank 2. 







§ INTRODUCTION


In his pioneering work <cit.>, Dye defined full subgroups of the group (X,μ) of all measure-preserving transformations of a standard probability space (X,μ) as those subgroups 𝔾 which are stable under cutting and pasting their elements along a countable measurable partition: for every measurable partition (A_n)_n∈  of X and every sequence (T_n)_∈ of elements of 𝔾 such that (T_n(A_n))_n∈ is also a partition of X, the measure preserving transformation T defined by T(x)=T_n(x) for all x∈ A_n also belongs to 𝔾. 
Then, given a measure-preserving action of a countable group Γ on a standard probability space (X,μ) (i.e. a homomorphism α:Γ→(X,μ)), the smallest full subgroup of (X,μ) containing α(Γ) is called the full group [α(Γ)] of the action α. 

 The full group of the action α can be concretely understood as the group of all measure-preserving transformations T such that T(x) belongs to the Γ-orbit of x for all x∈ X. Moreover, if two actions are conjugate, then their full groups are isomorphic. It was then realized that the full group was exactly capturing the partition of the space into orbits induced by the action, leading to the notion of orbit equivalence which is by now a fairly large subject on its own (see for instance the survey <cit.>).

The present paper is about a family of smaller subgroups which one can associated to any measure-preserving action of any finitely generated group. These were defined in <cit.> and are called ^p full groups, although they are not full as subgroups of (X,μ), but only finitely full, which means that they are stable under cutting and pasting their elements along a finite measurable partition.


 Their definition uses the extra geometric structure provided by the Schreier graph of the action. To be more precise, given a measure preserving action α:Γ→(X,μ) of a finitely generated group Γ and a finite symmetric generating set S for Γ, let us first denote by d_S the path metric associated to the Schreier graph of the action with respect to the set S. For p∈[1,+∞[, we then define the ^p full group of (α,S) be the group of all T∈ [α(Γ)] such that the map 

    x↦ d_S(x,T(x))

is in ^p(X,μ). Changing the generating set S does not change the ^p full group, so the ^p full group of the action α itself is well-defined, and we denote it by [α(Γ)]_p. If two actions are conjugate, then their ^p full groups are isomorphic. Moreover, the natural ^p metric on ^p full groups is complete, separable, right-invariant and endows the ^p full groups with a group topology which does not depend on the choice of the finite generating set. It is thus natural to ask the following.

	 How are properties of the action reflected in the ^p full group as a 
	 topological group ?	 
  
One of the natural invariants of topological groups is the topological rank, namely the smallest number of elements needed to generated a dense subgroup, so it is natural to ask how the action influences the topological rank of the associated ^p full group. 




 In <cit.> we showed that the topological rank of the ^1 full group of an ergodic -action was finite if and only if the action had finite entropy.
The main goal of the present paper is to extend this result to general 
measure-preserving actions of finitely generated groups. In order to do so, the 
right notion of entropy is Rokhlin entropy, which was recently 
introduced by Seward <cit.>. It is the infimum 
of the entropies of generating partitions and, for -actions, it coincides 
with the usual definition of entropy of a measure-preserving transformation by 
a result of Rokhlin <cit.>. A crucial result 
of Seward is that the finiteness of the Rokhlin entropy characterizes 
measure-preserving subshifts, generalizing Krieger's theorem to the widest 
imaginable setup (see Section <ref> for more on this).


Our proof that ergodic -actions of finite entropy have a topologically finitely generated ^1 full group used crucially the derived ^1 full group , defined as the closure of the  subgroup generated by commutators in the ^1 full group of the action[The quotient of the ^1 full group by the derived ^1 full group is the largest abelian continuous quotient of the ^1 full group, and is thus called the topological abelianization of the ^1 full group.]. Indeed, the latter is far easier to understand, in particular it is topologically generated by naturally defined involutions arising from the generating set of Γ (see Theorem <ref>  below). The main result from <cit.> is then a direct consequence of the following two statements:


	
  * The derived ^1 full group is topologically finitely generated as soon as the action has finite entropy.
	
  * The topological abelianization of the ^1 full group of an ergodic -action is isomorphic to , in particular it always has finite topological rank. 


 In the present paper, we generalize the first item from the above list as follows (see Section <ref> for the proof). 


	Let Γ be a finitely generated group, let α:Γ→(X,μ) be a measure-preserving aperiodic Γ-action on (X,μ). The following are equivalent.
	
		
  * The action has finite Rokhlin entropy.
		
  * The derived ^1 full group of the action has finite topological rank. 
	


Apart from Seward's characterization of subshifts, another key ingredient in the proof is Nekrashevych's work on alternating topological full groups <cit.>, which provides dense finitely generated subgroups in the derived ^1 full groups. The reader is referred to Section <ref> for the definition of alternating topological full group and Section <ref> for the proof of their density in ^1 full groups. 


In the above result, it is natural to ask whether one can remove the “derived” adjective in item (<ref>), as in the -case. However, we have no idea what the topological abelianization of the ^1 full group of a general measure-preserving ergodic action of a finitely generated group looks like, and in particular we do not know when it happens to be topologically finitely generated. We don't have any example or non example, even for ^2-actions. 


Let us now consider another invariant of topological groups: amenability. In <cit.> we showed that if the ^1 full group of a free action of a finitely generated group Γ was amenable, then Γ itself was amenable, and left the question of the converse open. Here we show that the converse holds so as to obtain the following result (see section <ref> for definitions and a more general statement in terms of graphings).


Let Γ be a finitely generated group acting freely on a standard probability space (X,μ). Then the following are equivalent:

	
  * Γ is amenable;
	
  * The ^1 full group of the action is amenable;
	
  * The derived ^1 full group of the action is amenable;
	
  * The derived ^1 full group of the action is extremely amenable;
	
  * The derived ^1 full group of the action is whirly amenable.



Note that the whole ^1 full group of any free ergodic -action factors onto  <cit.> and hence cannot be extremely amenable. So we cannot have a stronger notion of amenability for the whole ^1 full group in the above equivalences.


We also note that ^1 full groups provide a rich playground for Rosendal's extension of geometric group theory to Polish groups: the ^1 metric can often canonically be obtained (up to quasi-isometry) purely from the topological group structure. We refer the reader to Section <ref> for precise statements, and single out the following application, proved in Section <ref>.


	Let Γ and Λ be two finitely generated amenable groups acting in a measure-preserving and ergodic manner on (X,μ). Then every abstract group isomorphism between their ^1 full groups must be a quasi-isometry for their respective ^1 metrics. 


The conclusion of the theorem is somehow optimal: if we replace quasi-isometry by bilipschitz equivalence, the theorem fails (see Example <ref>). We don't know how to generalize Theorem <ref> to the nonamenable setup, but we conjecture that the following stronger statement holds.  


	Let Φ be a graphing. Then the ^1 metric on the ^1 full group of Φ is maximal. 


Moreover, the following natural problem arises.


	Classify (derived) ^1 full groups up to quasi-isometry.


A simpler instance of the above problem is to decide whether the derived ^1 full groups of the 2-odometer and of the 3-odometer are quasi-isometric, which has a natural reformulation in terms of quasi-isometry of the dyadic and triadic symmetric groups for an ^1-like metric (see the paragraph following Question <ref> for details).


Finally, we make progress on the computation of the exact topological rank of ^1 full groups of measure-preserving ergodic transformations. As reminded before, we had shown in <cit.> that the topological rank of ^1 full groups of a measure-preserving ergodic transformations is finite if and only if the transformation has finite entropy. Using Marks' work <cit.>, we moreover had obtained that the topological rank of the ^1 full group of irrational rotations is equal to two. 

Here, we extend this result to the class of rank one transformations, which we will properly define in section <ref>. Let us for now mention that every compact transformation is rank one (in particular every irrational rotation is rank one), and that every rank one transformation has entropy zero.


	Let T be a rank-one measure-preserving transformation. Then the topological rank of its ^1 full group is equal to 2.


It would be very interesting to have an example of a measure-preserving ergodic transformation such that the topological rank of its ^1 full group is finite, but greater than 2. 


The paper ends with further remarks and questions related to these results. We introduce the notion of ^p-full orbit equivalence and ask whether it is equivalent to ^p orbit equivalence, we mention a natural extension of Theorem <ref> to symmetric ^p full groups. We finally make a few more remarks on the topological abelianization of ^1 full groups. 




	I am very grateful to Alessandro Carderi, Nicolás Matte Bon and Todor 
	Tsankov for many enriching conversations around the topics of the present 
	paper. I would also like to thank Sébastien Gouëzel for allowing me to 
	include Theorem <ref> in this paper, and the 
	anonymous 
	referee for their helpful remarks, especially for suggesting that the 
	answer to Question 5.5 from the first version of the paper should be 
	negative.

	Research supported by Projet ANR-14-CE25-0004 GAMME and Projet 
	ANR-17-CE40-0026 AGRUME.


	






§ PRELIMINARIES


In what follows Γ and Λ will always denote countable infinite groups. 



 §.§ Standard probability spaces


Recall that a measurable space is standard if its σ-algebra comes from the Borel σ-algebra induced by the topology of some Polish space.  
Measurable maps between standard Borel spaces are simply called Borel maps. One of the highlights of standard Borel spaces is the fact that the range of any injective Borel map  is Borel so that Borel bijections automatically have a Borel inverse <cit.>. Another important fact is that in any standard Borel space we have a countable family of Borel subsets (C_n) which separates points: for every distinct x,y∈ X there is n∈ such that x∈ C_n ⇎ y∈ C_n.

Given a countable group Γ, a Γ-action on X is called Borel if the bijections induced by the elements of Γ are Borel maps. 
Two Borel actions Γ X and Γ Y are conjugate if there is a Borel Γ-equivariant bijection between X and Y.

The support of a Borel bijection T:X→ X is the Borel set T:={x∈ X: T(x)≠ x}.
Here are two useful lemmas on Borel bijections of standard Borel spaces which only rely on the existence of a countable family of Borel subsets which separates points.

[see e.g. <cit.>]
Let T be a Borel bijection of a standard Borel space X. Then there is a partition of T in three Borel subsets  A_1, A_2, A_3 such that for all i∈{1,2,3} the set T(A_i) disjoint from A_i. 













By applying the above lemma n times we have.

 
Let X be a standard Borel space, let (T_i)_i=1^n be a finite family of Borel bijections of X. 
Then there is a finite Borel partition (A_j)_j=1^m of ⋂_i=1^n T_i such that for each i∈{1,...,n} and each j∈{1,...,m} we have that T_i(A_j) is disjoint from A_j. 
 Recall that given a measurable map f:X→ Y between measurable spaces, if μ is a probability measure on X then one can define the pushforward measure f_*μ on Y by f_*μ(A)=μ(f(A)) for all every A⊆ Y.

We will  work most of the time in  a standard probability space (X,μ), which means that X is still a standard Borel space but now equipped with a nonatomic Borel probability measure μ. Given two standard probability spaces (X, μ) and (Y,ν), there is a Borel bijection S: X→ Y such that S_*μ=ν <cit.>, so we may as well work in a fixed standard probability space (X,μ). 





 §.§ The measure algebra and its automorphisms


Let (X,μ) be a standard probability space, we denote by (X,μ) the 
quotient of the algebra ℬ(X) of Borel subsets of X by the ideal of 
measure zero sets; in other words in the measure algebra we identify two Borel 
sets A and B as soon as μ(A△ B)=0. Unless specified 
otherwise, we will always be working in the measure algebra when considering 
Borel sets, so for instance when we write A⊆ B we mean 
μ(A∖ B)=0. 

A measure-preserving transformation of (X,μ) is a Borel bijection T:X→ X such that T_*μ=μ. 
We denote by (X,μ) be the group of all measure-preserving transformations, two such transformations being identified if they coincide on a full measure subset of X. We say that a measure-preserving transformation is periodic if all its orbits are finite.

 Every element of (X,μ) defines a measure-preserving automorphism of the measure algebra, and conversely every automorphism of the measure algebra comes from a measure-preserving transformation which is unique up to measure zero <cit.>. In particular, since every measure-preserving Borel bijection between full measure Borel subsets of (X,μ) defines an automorphism of its measure algebra, it also defines a measure-preserving transformation. This will be used implicitly in many proofs. 
 
 Given T∈(X,μ) and A⊆ X of positive measure, we define the T-return time to A as the function n_T,A:X→∖{0} given by: for all x∈ A, n_A,T(x) is the smallest n∈ such that T^n(x)∈ A. The Poincaré recurrence theorem ensures us that the return time is well defined up to measure zero, also called the return time to A. We can then define the measure-preserving transformation T_A induced by T on A by: 
 
    T_A(x)={[ T^n_A,T(x)    if x∈ A;          x      else. ].


The  measure algebra (X,μ) can be equipped with a complete metric d_μ defined by d_μ(A,B)=μ(A△ B).
The following lemma which was used implicitly in <cit.>.


	Let (X,μ) be a standard probability space, let 𝒟 be a set of Borel subsets of X such that the image of 𝒟 in (X,μ) is dense. Then there is a full measure Borel subset X'⊆ X such that 𝒟 separates the points of X'.
 

	Let (C_n)_n∈ separate the points of X, by changing our enumeration we may as well assume each C_n appears infinitely often. Now for each n∈ we pick D_n∈𝒟 such that μ(C_n△ D_n)<2^-n. By the Borel-Cantelli lemma, there is a full measure Borel subset X'⊆ X such that for all x∈ X', there is N∈ such that for all n≥ N, x∉C_n△ D_n. By construction the set X' is as wanted. 





 §.§ Ergodic theory


An action of a countable group Γ on a standard probability space (X,μ) is measure-preserving if it is a Borel action and for every γ∈Γ we have γ_*μ=μ. Every such action yields a homomorphism Γ→(X,μ) which only depends on the action up to measure zero and conversely, every such homomorphism comes from a measure-preserving action which is unique up to measure zero (see <cit.>). 

Two measure-preserving actions of Γ are conjugate if they are conjugate as Borel actions up to restricting to full measure Γ-invariant Borel subsets. Viewing actions as homomorphisms α,β: Γ→(X,μ), this is equivalent to the existence of T∈(X,μ) such that for all γ∈Γ we have β(γ)=Tα(γ)T.
The ^1 full groups that we are interested in only depend on the action up to measure zero, so we can and will forget about what happens on measure zero sets most of the time. By doing so we will implicitly use the following : if we have a Borel set A of full measure, then it contains the full measure Γ-invariant set ⋂_γ∈Γγ A.

Two basic invariants of conjugacy of measure preserving actions are aperiodicity (the Γ-orbit of almost every x∈ X is infinite) and the stronger notion of ergodicity (every Γ-invariant Borel subset has measure 0 or 1). We will only be interested in aperiodic actions, and sometimes we will restrict to ergodic ones. Another important invariant is freeness, which means that for almost all x∈ X and all γ∈Γ we have γ x≠ x. Note that aperiodicity follows from freeness since all the groups we consider are infinite.

Let I be a countable set and let ν be a probability measure 
on I which is not a Dirac measure, then the measure preserving action 
Γ (I^Γ,ν^⊗Γ) given by (γ· 
x)_g=x_γ g is free and ergodic. Such an action is called a 
Bernoulli shift.







 §.§ Subshifts and Rokhlin entropy
 

Let X be a standard Borel space, let Γ X be a Borel action. An observable on X is a Borel map φ: X→ I where I is a countable set. To every such observable we can associate a Borel partition of X indexed by I defined as (φ{i})_i∈ I, and conversely if 𝒫 is a countable partition of X then the map φ: X→𝒫 which sends every x∈ X to the unique P∈𝒫 such that x∈ P is an observable. 
An observable φ  is called  (dynamically) generating if for all x,y∈ X with x≠ y there is γ∈Γ such that φ(γ x)≠φ(γ y). 

A Borel action Γ X on a standard Borel space is called a subshift if it admits a generating observable with finite range. 
A measure-preserving Γ-action on a standard probability space is called a measure-preserving subshift if there is a full measure Borel Γ-invariant set Y such that the restriction of the action to Y is a subshift. Note that Bernoulli shifts over finite spaces are examples of measure-preserving subshifts.

Finally, an action by homeomorphisms on a compact Hausdorff space is called a topological subshift if it has a continuous generating observable (such an observable must then take values in a finite set by compactness and continuity).

We have the following well-known characterization of subshifts up to conjugacy (in the respective Borel  and topological categories). 


The following hold. 


  * A Borel action Γ X is a subshift if and only if it is conjugate to a restriction of Γ I^Γ to an invariant Borel subset where I is a finite set.

  * A Γ-action by homeomorphisms on a compact zero-dimensional topological space K is a topological subshift if and only if it is conjugate (via a homeomorphism) to the restriction of  Γ I^Γ  to an invariant closed subset, where I is a finite set. 



For (1), let φ: X→ I be a generating observable where I is finite, and consider the Γ-equivariant map Φ_I:X→ I^Γ given by x↦(φ_I(γ x))_γ∈Γ. Since φ is generating and Borel, Φ_I is an injective Borel map. Its range is thus a Borel Γ-invariant subset of I^Γ. 

For (2), consider the same map and observe that it now has to be a homeomorphism onto its image.


If we are now given a measure-preserving Γ-action on (X,μ), we say that an observable φ is  generating if we can find a Borel Γ-invariant subset X'⊆ X of full measure such that the restriction of φ to X' is generating for Γ X'. The following analogue of Proposition <ref> is immediate.


	A measure-preserving Γ-action on (X,μ) is a measure-preserving subshift if and only if it admits a generating observable whose range is finite. 


Let (X,μ) be standard probability space. 
The entropy of an observable φ: X→ I is the non-negative real number H(φ) defined by the equation 

    H(φ)   =-∑_i∈ Iμ(φ ({i})) lnμ(φ({i})),

where we make the convention that 0ln(0)=0.
Here are two fundamental properties of entropy:


  * Subadditivity: for any two observables φ and ψ we have 

    H(φ×ψ)≤(̋φ)+H(ψ)

 where (φ,ψ) is the observable x↦ (φ(x),ψ(x)). 
 
  * Concavity: if we fix k∈, the observables taking value in {1,...,k} which maximize entropy are exactly those which give the same probability to every i∈{1,...,k} (and their entropy is thus ln k)


[Seward]
	The Rokhlin entropy of a measure-preserving aperiodic Γ-action is the infimum of the entropies of its generating observables. 


Every aperiodic Borel action admits a generating observable by <cit.>, so the above quantity is an element of [0,+∞]. The following remarkable result of Seward is a far-reaching generalization of Krieger's generator theorem which will allow us to work with arbitrary countable groups rather than . 

[Seward <cit.>]
Let Γ be a countable group, let Γ(X,μ) be a measure-preserving action. The action is a measure-preserving subshift if and only if it has finite Rokhlin entropy.


Seward also has a quantitative version of the above result, yielding a 
generating observable whose range is as small as possible (see 
<cit.>, where he also defines Rokhlin entropy).



 §.§ Full groups




If (X,μ) is a standard probability space, and A,B are Borel subsets of X, a partial isomorphism of (X,μ) of domain A and range B is a Borel bijection f: A→ B which is measure-preserving for the measures induced by μ on A and B respectively. We denote by f=A its domain, and by f=B its range. Given two partial isomorphisms φ_1: A→ B and φ_2: C→ D, we define their composition φ_2∘φ_1 as the map φ_1(B∩ C)→φ_2(B∩ C) given by φ_2∘φ_1(x)=φ_1φ_2(x). We also define the inverse φ of a partial isomorphism φ: A→ B as the unique map φ: B→ A such that φ∘φ=id_A.

By definition a graphing is a countable set of partial isomorphisms. Every graphing Φ generates a measure-preserving equivalence relation ℛ_Φ, defined to be the smallest equivalence relation containing (x,φ(x)) for every φ∈Φ and x∈φ. Given a graphing Φ, a Φ-word is a composition of finitely many elements of Φ or their inverses. Observe that (x,y)∈ℛ_Φ iff and only if there exists a Φ-word w such that y=w(x). Moreover we have a natural path metric d_Φ on the ℛ_Φ-classes: for (x,y)∈ℛ_Φ we let d_Φ(x,y) be the minimum length of a Φ-word w such that y=w(x).

A graphing is called aperiodic when the ℛ_Φ-class of μ-almost every x∈ X is  infinite.
Given a Borel set A, we say that A is ℛ_Φ-invariant if whenever x∈ A and (x,y)∈ℛ_Φ, we have y∈ A. Equivalently, this means that for all x∈ A and all φ∈Φ, we have φ(x)∈ A and φ(x)∈ A. A graphing Φ is ergodic if every ℛ_Φ-invariant Borel set has measure 0 or 1. Every ergodic graphing is automatically aperiodic.

The full group of a measure-preserving equivalence relation ℛ is the group [ℛ] of automorphisms of (X,μ) which induce a 
permutation on every ℛ-class, that is

    [ℛ]={φ∈(X,μ): ∀ x∈ X, φ(x) ℛ  x}.

It is a separable group when equipped with the complete metric d_u defined by 
    d_u(T,U)=μ({x∈ X: T(x)≠ U(x)}.
 The metric d_u is called the uniform metric and the topology it induces is called the uniform topology. One also defines the pseudo full group of ℛ, denoted by [[ℛ]], which consists of all partial isomorphisms φ such that φ(x) ℛ  x for all x∈φ. 



 §.§ L1 full groups


[<cit.>]
Let Φ be a graphing on a standard probability space (X,μ). Its ^1 full group [Φ]_1 is the group of all T∈[ℛ_Φ] such that 

    ∫_Xd_Φ(x,T(x))dμ(x)<+∞.



The ^1 full group of Φ is a Polish group for the topology induced by the right-invariant complete metric d^1_Φ defined by 

    d^1_Φ(T,U)=∫_X d_Φ(T(x),U(x))dμ(x).

When Γ is a finitely generated group acting on a standard probability space (X,μ) by measure-preserving transformations, we can also define the associated ^1 full group as follows: let S be a finite generating set of the group Γ, then S is a graphing whose associated equivalence relation is the equivalence relation ℛ_Γ whose equivalence classes are the Γ-orbits. 
Observe that up to bilipschitz equivalence, the associated metric d_S on the Γ-orbits does not depend on S, so that two finite generating sets of Γ yield the same ¹ full group with the same topology. This is by definition the ^1 full group of the Γ-action, which we  will often simply denote by [Γ]_1 without referring explicitly to the action. 

Here is another way to think of the ^1 full group of a Γ-action, similar to what was done in <cit.>. Consider the Polish space ^1(X,μ,Γ) of integrable maps X→Γ, i.e. maps f:X→Γ such that ∫_X d_S(1,f(x))dμ(x)<+∞. To each element f of ^1(X,μ,Γ) we associate a map T_f: X→ X defined by T_f(x)=f(x)· x. The subspace

    [Γ]_1={f∈^1(X,μ,Γ): T_f∈ [ℛ_Γ]}

is closed, and we define on it a law * which is compatible with the group law on [ℛ_Γ] by letting
f*g(x)=f(T_g(x))g(x). Then ([Γ]_1,*) is a Polish group, and [Γ]_1 is topologically isomorphic to its quotient by the closed normal subgroup {f∈[Γ]_1: T_f=id_X} via the map f↦ T_f. 

If we are given T∈ [Γ]_1, a lift of T to [Γ]_1 is called a cocycle associated to T. When the Γ action is free, f↦ T_f is injective and so each T∈[Γ]_1 has a unique associated cocycle c_T: X→Γ given by the equation 

    T(x)=c_T(x)· x.


An important feature of cocycles associated to elements of ^1 full groups is that they have finite entropy. 


Let Γ be a finitely generated group. Then every element of ^1(X,μ,Γ) has finite entropy. 


We finally note the following lemma about cocycles which are measurable with 
respect to an invariant sub-σ-algebra.


	Let Γ(X,μ) be a measure-preserving action of a finitely 
	generated 
	group Γ, suppose 𝒜 is a Γ-invariant sub-
	σ-algebra of the Borel subsets of X. Let [Γ]_1^𝒜 
	denote the set consisting of all elements of 
	[Γ]_1 which admit an 𝒜-measurable cocycle. Then 
	[Γ]_1^𝒜 is a group.


	Let T,U∈ [Γ]_1^𝒜, and
	let c_T and c_U be cocycles for T and U which are 𝒜-measurable. Then the map c:X→Γ defined by 
	
    c(x)=c_T(U(x))c_U(x)

	is a cocycle for TU, and c(x)=γ if and only if there are 
	γ_1,γ_2∈Γ such that γ_1γ_2=γ, 
	c_U(x)=γ_2, c_T(γ_2 x)=γ_1. So c is clearly 𝒜-measurable as wanted.



	The same proof yields a similar statement for the whole full group of 
	the action.






 §.§ Derived L1 full groups


Contrarily to many Polish groups that have been studied, ^1 full groups are not topologically simple in general, and can actually happen to have a non trivial topological abelianization. The closure of their derived group (i.e. the group generated by commutators) is thus distinct from them a priori. 
Let us give it a shorter name.

The derived ^1 full group of a graphing 
Φ is the closure of the derived group of the ^1 full group of Φ. 
It is denoted by [Φ]_1'. 


Given φ∈Φ and a Borel set A⊆ X, let us define an 
involution 
I_φ,A by: 

    I_φ,A(x)={[          φ(x) if x∈ 
    A∖φ(A);          φ(x)  if x∈φ(A)∖ A;             x         else. ].

We can now give two fundamental properties of the derived ^1 full group. 
The second one suggests to rather call it the symmetric ^1 full 
group (see Section <ref>).

[<cit.>]
	Let Φ be an aperiodic graphing. Then the following hold:
		
  * every periodic element of the ^1 full group of Φ actually belongs to the derived ^1 full group of Φ;
		
  * the derived ^1 full group of Φ is topologically 
		generated by the set of involutions {I_φ,A:φ∈Φ, 
		A⊆ X}.
	

Observe that since I_φ,A depends continuously on A∈(X,μ) 
and (X,μ) is connected, we can deduce that [Φ]_1' is connected 
(cf. Corollary 3.14 in <cit.>). We still 
don't know whether it is simply connected. Another important question is to 
understand what is the topological abelianization of ^1 full groups, i.e. 
what is [Φ]_1/[Φ]_1'. The only known case is that of ergodic 
-actions, for which we then always have []_1/[]_1'= 
<cit.>. See the last section of the paper 
for more on this. 

Using Item (b) from the above statement along with its full group version 
<cit.>, we also 
have the following.


	Let Φ be an aperiodic graphing. Then the derived ^1 full group of 
	Φ is dense in the full group [ℛ_Φ] for the uniform 
	topology.


Given an aperiodic graphing Φ and an ℛ_Φ-invariant Borel set A, consider the subgroup [Φ_A]'_1 consisting of all the elements of [Φ]'_1 supported in A. It is a closed normal subgroup of [Φ]'_1, and the following result says that these are the only ones.

[<cit.>]
Let Φ be an aperiodic graphing. Then every closed normal subgroup of [Φ]_1' is of the form [Φ_A]'_1, where A is ℛ_Φ-invariant. 


As a corollary, one can show that a graphing is ergodic if and only if its 
derived ^1 full group is topologically simple <cit.>. 






 §.§ Conditional measures


Given a measure preserving action Γ(X,μ) of a finitely generated group Γ, it will be important for us to know for which Borel sets A and B we can find an element of T of the ^1 full group of the action such that T(A) is close to B. Because the ^1 full group is dense in the full group [ℛ_Γ], it suffices to understand the [ℛ_Γ]-orbits in the measure algebra. It is a well-known fact that when the action is ergodic, there is T∈[ℛ_Γ] such that T(A)=B if and only if μ(A)=μ(B). In the non ergodic case, one needs to refine the measure μ to a conditional measure with respect to the σ-algebra of Γ-invariant sets. 


	Let ℛ be a measure-preserving equivalence relation. Denote by M_ℛ the σ-algebra of ℛ-invariant Borel subsets. The conditional measure of A∈(X,μ) is the M_ℛ-measurable function μ_ℛ defined by 
	
    μ_ℛ(A)=π_ℛ(χ_A),

	where π_ℛ is orthogonal projection onto the closed subspace of ^2(X,μ) consisting of the M_ℛ-measurable functions. 


Note that μ_ℛ is well-defined up to measure zero. We now have the following. 

Let ℛ be a measure-preserving equivalence relation and A,B∈(X,μ). Then μ_ℛ(A)=μ_ℛ(B) if and only if there is an involution T∈[ℛ] such that T(A)=B.


Aperiodicity has an important reformulation in terms of conditional measures, which is also due to Dye who calls it Maharam's lemma. For a proof in our setup, see <cit.>.


	A measure-preserving equivalence relation ℛ is aperiodic if and only if for any A∈(X,μ), and for any M_ℛ-measurable function f such that 0≤ f≤μ_ℛ(A), there exists B⊆ A such that the ℛ-conditional measure of B equals f.





 §.§ Alternating topological full groups


Alternating topological full groups were defined by Nekrashevych and have been the source of outstanding new examples of finitely generated groups <cit.>. They provide the right setup for generalizing Matui's results on finite generatedness of derived groups of topological full groups of homeomorphisms of the Cantor space <cit.>. Topological full groups were first studied and defined by Giordano, Putnam and Skau <cit.>. 
Here we give a restricted definition of (alternating) topological full groups which will be sufficient for our purposes.  

Let 2^ be the Cantor space and let ν be a Borel probability measure on it. Recall from <cit.> that a continuous graphing on the measured Cantor space (2^,ν) is a graphing Φ such that for all φ∈Φ, the sets φ, φ are clopen and φ: φ→φ is a homeomorphism. 


Let Φ be a continuous graphing on (2^,ν). The 
topological full group of Φ, denoted by [Φ]_c, is the group 
of all homeomorphisms T of 2^ such that for all x_0∈ 2^ there is a 
neighborhood U of x_0 and a Φ-word w such that for all x∈ U, 
T(x)=w(x). 



Say that an element T of the topological full group of Φ is a basic 3-cycle if there are a three clopen subsets U_1, U_2, U_3 which partition the support of T and two Φ-words w_1,w_2 such that w_1(U_1)=U_2 and w_2(U_2)=U_3 and for all x∈ X we have 

    T(x)={[     w_1(x)  if x∈ U_1;     w_2(x)  if x∈ U_2; w_1 w_2(x)  if x∈ U_3;          x otherwise. ].

The alternating topological full group of a continuous graphing Φ is the group 𝔄[Φ] generated by basic 3-cycles. One can check that the alternating topological full group is a normal subgroup contained in the derived group of the topological full group. 

Let us observe that when Γ is a countable group acting by homeomorphisms on the Cantor space, then if S is any generating set for Γ, the elements of S define a continuous graphing whose topological full group is the topological full group of the groupoid of germs of the Γ-action in the sense of <cit.> (see also <cit.> and the two paragraphs which follow it). Moreover, the alternating topological full group of the continuous graphing S is equal to the alternating topological full group of the groupoid of germs of the Γ-action. 



	Since we do not require the action to be free or essentially free in the topological sense, the etale groupoid of germs of the action is not Hausdorff in general, in particular its unit space may not be closed, although it is compact. As a consequence, Lemma 2.2 from <cit.> does not apply and elements of the topological full group may very well have a non clopen support[In the topological setup, the support of a homeomorphism T:X→ X is rather defined as the closure as the set of x∈ X such that T(x)≠ x.], as opposed to elements of the alternating topological full group.



We have the following straightforward consequence of <cit.> and <cit.> which will be crucial in our proof of Theorem <ref>.

[Nekrashevych]
	Let Γ be a finitely generated group, suppose Γ acts on the Cantor space by homeomorphism and that every Γ-orbit has cardinality at least 5. Then the action is a topological subshift if and only if its alternating topological full group is finitely generated.






§ DENSITY OF THE ALTERNATING TOPOLOGICAL FULL GROUP


In this section, we show that the alternating topological full group of a 
continuous graphing is always dense in its derived ^1 full group. We first 
need to work a bit on the general setup, so (X,μ) is still a standard 
probability space.

A measure-preserving transformation is called a 3-cycle when it has order 3, or equivalently when all its nontrivial orbits have cardinality 3. Given a 3-cycle T, there is A⊆ X such that T=A⊔ T(A)⊔ T^2(A), and the identity (1   2   3)=(1   2)(1   3)(1   2)(1   3) yields that T is a commutator, so the 3-cycles of the ^1 full group actually belong to the derived ^1 full group.  

We need to ensure that there are many 3-cycles. This is a consequence of the 
following two lemmas, which are straightforward generalizations of <cit.> and <cit.> respectively. 

Recall the following notation: given a measure-preserving transformation T 
and A⊆ X, we denote by T_A the transformation induced by T on A 
(see Section <ref>). In what follows, we 
only use this construction when A is T-invariant, and then T_A is the map 
which acts as T on A and trivially outside of A.


Let Φ be a graphing. Then for every periodic U∈[ℛ_Φ], there exists an increasing sequence of U-invariant sets A_n⊆ U such that U=⋃_n∈ A_n and for all n∈, U_A_n∈[Φ]_1.


Let U∈[ℛ_Φ] be periodic and for all n∈, let A_n={x∈ U: d_Φ(x,U^k(x))<n for all k>0}. By definition each A_n is U-invariant, and since U is periodic, ⋃_n∈ A_n= U, so μ( U∖ A_n)→ 0. By the definition of A_n, each U_A_n belongs to [Φ]_1.



Let Φ be an aperiodic graphing. Then for every Borel A⊆ X, there 
is a 3-cycle U∈ [ℛ_Φ] whose support is equal to A. 


By Maharam's Lemma (Proposition <ref>), one can write A=A_1⊔ A_2⊔ A_3 where μ_ℛ_Φ(A_1)=μ_ℛ_Φ(A_2)=μ_ℛ_Φ(A_3)=μ_ℛ_Φ(A)/3. By a result of Dye (see Proposition <ref>), there are T_1,T_2∈ [ℛ_Φ] such that T_1(A_1)=A_2 and T_2(A_2)=A_3). We then define T∈[ℛ_Φ] by

    T(x)=
    {[     T_1(x)  if x∈ A_1;     T_2(x)  if x∈ A_2; T_1 T_2(x)  if x∈ A_3;          x otherwise. ].

The transformation T is the desired 3-cycle. 

As an immediate consequence of the two previous lemmas, we have:


Let Φ be an aperiodic graphing. Then for every Borel A⊆ X, there 
is a sequence of 3-cycles U_n∈ [Φ]_1 such that ( U_n) is an 
increasing sequence of subsets of A and μ(A∖ U_n)→ 0. 



	
Let Φ be a graphing. Then [Φ]_1' is topologically generated by 3-cycles. 

	
Let X_p be the periodic part of the graphing, i.e. the set of x whose ℛ_Φ-class is finite, and let X_a=X∖ X_p. Then if one lets Φ_p
 (resp. Φ_a) be the restriction of Φ to X_p (resp. X_a), the derived ^1 full group of Φ splits naturally as a direct product [Φ_p]_1'× [Φ_a]_1'. So it suffices to establish the theorem in the aperiodic and in the periodic case. 
 
Let us start by supposing Φ is aperiodic. The group generated by 3-cycles is normal in [Φ]_1' so its closure N is a closed normal subgroup of [Φ]_1'. By <cit.> we conclude that N is equal to the group of T∈[Φ]'_1 whose support is contained in the reunion of the supports of the 3-cycles. Using the previous proposition, we see that the reunion of such supports is equal to X, so N=[Φ]'_1, which end the proof of the theorem in the aperiodic case.

Now suppose Φ is periodic. For every n∈ let X_n be the set of x∈ X whose ℛ_Φ-class has d_Φ-diameter less than n. Then X=⋃_n∈ X_n, the X_n's are Φ-invariant and non-decreasing so if we let G_n={T∈ [Φ]_1:  T⊆ X_n} then the reunion of the G_n's is dense in [Φ]_1. It thus suffices to show that commutator of elements of G_n is a product of 3-cycles belonging to G_n, which comes from the fact that by boundedness of the orbits, each G_n is a full group and every periodic element of a full group which is a commutator is a product of 3-cycles. 



Let Φ be a continuous graphing on (2^,ν). Then its alternating topological full group is dense in its derived ^1 full group. 


By the previous proposition, it suffices to show that every 3-cycle in [Φ]_1 can be approximated by an element of the alternating topological full group. So let T be a 3-cycle in the ^1 full group of Φ. Let A⊆ X such that T=A⊔ T(A)⊔ T^2(A). 

Since the set of Φ-words is countable, we have a partition (A_n) of A and Φ-words w_1,n, w_2,n such that for all x∈ X we have

    T(x)=
    {[                    w_1,n(x)      if x∈ A_n  for some n∈;                    w_2,n(x)   if x∈ T(A_n)  for some n∈;              w_1,n w_2,n(x) if x∈ T^2(A_n)  for some n∈;                           x                  otherwise. ].

In particular, for every n∈ the set A_n⊔ T(A_n)⊔ T^2(A_n) is T-invariant, and if we let T_n:=T_A_n⊔ T(A_n)⊔ T^2(A_n), then by the dominated convergence theorem we conclude T=lim_m→+∞∏_i=0^m T_m. So it suffices to prove that each T_n is a limit of elements of the alternating topological full group. 

Let n∈, then there is a sequence of clopen sets (U_i)_i∈ such that μ(A_n△ U_i)→ 0 [i→+∞]. If we let V_i=U_i∖ (T(U_i)∪ T^2(U_i)), then by continuity we also have μ(A_n△ V_i)→ 0. We now define T_n,i by 

    T_n,i(x)=
    {[       w_1,n(x)      if x∈ V_i;       w_2,n(x)   if x∈ T(V_i); w_1,n w_2,n(x) if x∈ T^2(V_i);              x     otherwise. ].

Then T_n,i is clearly a basic 3-cycle, and hence belongs to the alternating topological full group of Φ. Moreover by the dominated convergence theorem we have T_n,i→ T_n [n→+∞], which finishes the proof. 










§ TOPOLOGICAL RANK OF DERIVED L1 FULL GROUPS


In this section, we prove that the derived ^1 full group of a measure-preserving Γ-action has finite topological rank if and only if it is a subshift (Theorem <ref>). We want to use Nekrashevych's result on finite generatedness of alternating topological full groups, and the next section prepares the ground for this. 








 §.§ From measure-preserving subshifts to topological subshifts


In this preparatory section we will show that every aperiodic measure-preserving subshift is conjugate to a topological subshift on the Cantor space where each orbit has cardinality at least 5. For this, we will freely use the characterization of subshifts provided by Proposition <ref>.

Given a group Γ and a generating set S for Γ, we equip Γ with the graph metric induced by its Cayley graph with respect to S. The closed ball of radius n around the identity for this metric is denoted by B_S(e,n). 


Let Γ be a finitely generated group, suppose S⊆Γ is a finite generating set and let Γ X be a subshift with every orbit of cardinality at least 5. Then we can find a finite generating Borel partition 𝒫 such that for every P∈𝒫 there are γ_1,...,γ_4∈ B_S(e,4) such that P, γ_1 P,...,γ_4 P are disjoint. 


Let us first show that for every x∈ X there are γ_1,...,γ_4∈ B_S(e,4) such that x,γ_1 x,...,γ_4x are all distinct. Observe that if for some n∈ we have B_S(e,n) x=B_S(e,n+1)x then B_S(e,n)x is a finite set which is invariant under each generator of Γ, hence Γ-invariant. In particular we must have  B_S(e,4) x≥ 5 so for every x∈ X there are γ_1,...,γ_4∈ B_S(e,4) such that x,γ_1 x,...,γ_4x are all distinct.

Now apply lemma <ref> to the family (γλ)_γ,λ∈ B_S(e,4) to get a partition 𝒫 such that for every P∈𝒫 there are γ_1,...,γ_4∈ B_S(e,4) such that P, γ_1 P,...,γ_4 P are disjoint. The desired partition is obtained by taking the join of 𝒫 with a finite generating partition. 



Let Γ be a finitely generated group acting on (X,μ) by an aperiodic measure-preserving subshift. Then the action is conjugate to a topological subshift on the Cantor space where every orbit has cardinality at least 5.


Let 𝒫 be as in the previous lemma and consider Ω=⋁_γ∈ B_S(e,4)γ𝒫. Observe that for μ-almost every x∈ X, there are γ_1,...,γ_4∈ B_S(e,4) such that x, γ_1 x,...,γ_4 x all belong to distinct elements of Ω. Let φ_Ω:X→Ω be the associated observable. 

Now consider the Γ-equivariant map Φ_Ω:X→Ω^Γ given by x↦(φ_Ω(γ x))_γ∈Γ. Observe that the set of sequences (ω_γ)_γ∈Γ such that for all γ∈Γ there are γ_1,...,γ_4∈ B_S(e,4) such that ω_γ,ω_γγ_1,...,ω_γγ_4 are all distinct is a closed Γ-invariant set containing the essential range of Φ_Ω, so the Γ-action on the essential range of Φ_Ω is a topological subshift all whose orbits have cardinality at least 5. 

Moreover the measure μ has no atoms because the Γ-orbits are almost all infinite. So the essential range of Φ_Ω contains no isolated points and thus must be homeomorphic to the Cantor space by a theorem of Brouwer (see <cit.>).
We conclude that the Γ-action on the essential range of Φ_Ω is a topological subshift on the Cantor space which has the desired properties. 



	I do not know whether one can improve the above result so as to get an 
	aperiodic topological subshift.



 §.§ Proof of Theorem <ref>


Let us recall the statement that we are aiming at.

Let Γ be a finitely generated group and consider a 
measure-preserving aperiodic Γ-action on a standard probability space 
(X,μ). Then the following are equivalent:


  * the Γ-action has finite Rokhlin entropy and

  * the derived ^1 full group of the Γ-action has finite topological rank.



Let us start by the direct implication (<ref>)(<ref>). Suppose the measure-preserving Γ-action has finite Rokhlin entropy. By Seward's theorem (Theorem <ref>), we may assume that the action is a measurable subshift, and by Proposition <ref> we can further assume that it is a topological subshift all whose orbits have cardinality at least 5. Nekrashevych's result (Theorem <ref>) yields that the associated alternating topological full group is finitely generated. So the alternating topological full group is the desired dense finitely generated subgroup of the derived ^1 full group of the action by Proposition <ref>.


We now prove the reverse implication (<ref>)(<ref>). Suppose that for some n∈ we have T_1,..., T_n ∈ [Γ]'_1 which generate a dense subgroup of [Γ]'_1. By definition for each i∈{1,...,n} the transformation T_i admits an associated cocycle c_i∈^1(X,μ,Γ). By Lemma <ref> each cocycle c_i has finite entropy. Finally, let c be the observable associated to a partition (A_1,A_2) of X into two disjoint sets such that for all x∈ X, μ_ℛ_Γ(A_1)(x)=μ_ℛ_Γ(A_2)(x)=1/2, as provided by Maharam's lemma (cf. Proposition <ref>).

The observable c_1×⋯× c_n× c has finite entropy by 
subadditivity, let us show that it is generating so that (<ref>) holds. Let 𝒜̃ be the σ-algebra generated by 
the Γ-translates of the partition associated to the observable 
c_1×⋯× c_n× c, we will show that it separates the points 
of a full measure subset of X, thus finishing the proof that the action has 
finite Rokhlin entropy.

First, by Lemma <ref>, every
element of the group generated by T_1,...,T_n admits a cocycle which is 
𝒜̃-measurable. Furthermore, the following holds.
 

The σ-algebra 𝒜̃ is invariant under every element of the group generated by T_1,...,T_n.

 
We just saw that every element of the group generated by T_1,...,T_n admits a 
cocycle which is 𝒜̃-measurable. 
But such elements must preserve 𝒜̃ because if A∈𝒜̃ and U is an element of the full group of ℛ_Γ admitting a cocycle which is 𝒜̃-measurable, we find a partition (A_γ)_γ∈Γ of A such that each A_γ belongs to 𝒜̃ and for each x∈ A_γ, U(x)=γ x.
 So U(A)=_γ∈Γγ(A_γ) and since 𝒜̃ is a Γ-invariant σ-algebra which contains each A_γ we conclude that U(A)∈𝒜̃. 

Let 𝒜 be the closure of the image of 𝒜̃ in 
(X,μ)[Actually 𝒜̃ is automatically closed 
because 𝒜̃ is a σ-algebra, but we won't need that.], 
which is then also invariant under every element of the group generated by 
T_1,...,T_n. By Lemma <ref>, in order to show 
that 𝒜̃ separates the points of a full measure subset of 
X,  we only need to show that 𝒜=(X,μ).
We first prove that 𝒜 satisfies the following property with respect to the ℛ_Γ-conditional measure μ_ℛ_Γ: 

    ∀ A∈𝒜 ∀ B∈(X,μ),  μ_ℛ_Γ(A)=μ_ℛ_Γ(B)  B∈𝒜.

Indeed let A∈𝒜, and suppose B∈(X,μ) satisfies μ_ℛ_Γ(A)=μ_ℛ_Γ(B). By Proposition <ref> we may find an involution U∈[ℛ_Γ] such that U(A)=B. Using Lemma <ref>, we then find a sequence (U_k)_k∈ of elements of the group generated by T_1,...,T_n such that μ(U_k(A)△ B)→ 0. Since 𝒜 is closed and invariant under elements of the group generated by T_1,...,T_n, we conclude that B∈𝒜. 


Now observe recall that by the definition of our observable c, the algebra 𝒜 contains a set B whose ℛ_Γ-conditional measure is constant equal to 1/2. 
By Maharam's lemma, whenever A∈(X,μ) satisfies μ_ℛ_Γ(A)(x)≤ 1/2 for all x∈ X, we can write A as the intersection 
of two elements of (X,μ) whose ℛ_Γ-conditional measure 
is constant equal to 1/2. So by (<ref>), 
𝒜 contains all sets whose ℛ_Γ-conditional measure is 
everywhere at most 1/2. Using Maharam's lemma once more, we see that every 
element of (X,μ) can be written as the union of two elements of 
(X,μ) whose ℛ_Γ-conditional measure is everywhere at 
most 1/2, so we conclude 𝒜=(X,μ) as wanted.







§ AMENABILITY AND L1 FULL GROUPS


Recall that a Polish group G is extremely amenable when every continuous G-action on a compact Polish space has a fixed point, while G is amenable if every continuous G-action on a compact Polish space admits an invariant Borel probability measure. Following <cit.> we say that a Polish group G is whirly amenable if it is amenable and, given any continuous G-action on a compact Polish space K, the support of every invariant Borel probability measure is contained in the set of fixed points. 

One has the following implications: whirly amenability implies extreme 
amenability which implies amenability. The first examples of whirly amenable 
groups were found by Y. Glasner, B. Tsirelson and B. Weiss, who showed that 
Lévy groups are  whirly amenable <cit.>. Note that if G is a Polish group 
which contains a dense increasing union of whirly amenable groups, then G is 
whirly amenable. 


Let Φ be an amenable graphing, our aim is to show that its derived ^1 
full group is whirly amenable. Recall that a measure-preserving equivalence 
relation is called  finite if all its classes are finite. The two 
following lemmas are essentially contained in 
<cit.>, but we include a full proof for the 
reader's convenience.

Let Φ be an aperiodic 
graphing, suppose ℛ_Φ is written as an increasing union of 
equivalence relations ℛ_n. Then
⋃_n∈[ℛ_n]∩ [Φ]_1'
is dense in [Φ]_1'. 


By Theorem <ref> we only need to be able to approximate involutions from  [Φ]_1' by elements from ⋃_n∈[ℛ_n]∩ [Φ]_1'. Let U∈ [Φ]_1' be an involution, then the U-invariant sets A_n={x∈ X: xℛ_n U(x)} satisfy ⋃_n∈ A_n= X since ⋃_n ℛ_n=ℛ_Φ. By the dominated convergence theorem, we then have U_A_n→ U. Moreover U_A_n is an involution so U_A_n∈ [Φ]_1' and by construction U_A_n∈[ℛ_n] so U_A_n∈ [ℛ_n]∩ [Φ]_1' and the lemma is proved.


Let Φ be an aperiodic graphing, 
let ℛ be a finite subequivalence relation of ℛ_Φ whose 
equivalence classes have a uniformly bounded Φ-diameter. Then [ℛ] is a closed subgroup of [Φ]_1'. 


The fact that there is a uniform bound M on the diameter of the ℛ-classes ensures us that [ℛ] is a subgroup of [Φ]_1. Moreover, the inclusion map is M-lipschitz (for the uniform distance on [ℛ] and d^1_Φ on [Φ]_1) and its inverse is clearly 1-lipschitz, so [ℛ] is closed in [Φ]_1. Finally, [ℛ] only consists of periodic elements, and since these belong to [Φ]_1' (see <cit.>), we conclude that [ℛ] is a closed subgroup of [Φ]_1'.


Let Φ be an amenable aperiodic graphing. Then [Φ]_1' is 
whirly amenable and [Φ]_1 is amenable. 


Since [Φ]_1 is an extension of [Φ]_1' by an abelian (hence amenable) group, the second statement follows from the first, so we only need to prove that [Φ]_1' is whirly amenable. 

By the Connes-Feldmann-Weiss theorem <cit.>, we may write ℛ_Φ as an increasing union of finite equivalence relations ℛ_n. Let us now change these so that their classes have uniformly bounded diameter. 

We first find an increasing sequence of integers (φ(n))_n∈ such that for all n∈, 
    μ({x∈ X: _Φ([x]_ℛ_n)>φ(n)})<1/2^n.

Then by the Borel-Cantelli lemma, for almost all x∈ X there are only finitely many n∈ such that _Φ([x]_ℛ_n)>φ(n). So if we define new equivalence relations 𝒮_n by (x,y)∈𝒮_n if (x,y)∈ℛ_n and ∀ m≥ n, _Φ([x]_ℛ_m)≤φ(n), we still have ⋃_n∈𝒮_n=ℛ and the 𝒮_n-classes have a uniformly bounded diameter as wanted.

By Lemma <ref>, ⋃_n∈[𝒮_n]∩ [Φ]_1' is dense in [Φ]_1'. Then Lemma <ref> yields that [𝒮_n]∩ [Φ]_1'=[𝒮_n] and that the induced topology on [𝒮_n] is the uniform topology. Now recall that full groups of finite equivalence relations are whirly amenable for the uniform topology (they are products of groups of the form ^0(Y,ν,𝔖_n) where (Y,ν) is non-atomic, and those are Levy groups by work of Y. Glasner, generalised by T. Giordano and V. Pestov <cit.>). We conclude that each [𝒮_n] is whirly amenable so that [Φ]_1' also is.



In <cit.> it was shown that 
non amenable graphings have non amenable ^1 full groups and non amenable 
derived ^1 full groups. So Theorem <ref> 
is established, and more generally the following holds.


Let Φ be an aperiodic graphing on a standard probability space (X,μ). 
The following are equivalent:


  * the graphing Φ is amenable;

  * the ^1 full group [Φ]_1 is amenable;

  * the derived ^1 full group [Φ]_1' is amenable.

  * the derived ^1 full group [Φ]_1' is extremely amenable.

  * the derived ^1 full group [Φ]_1' is whirly amenable. 



The above argument can be refined so as to show that derived ^1 full 
groups of amenable graphings are actually Lévy groups, and that one can find 
Levy families made of finite subgroups. The details are however tedious and the 
increasing chain of  finite subgroups is not explicit so we won't provide a 
proof here. It was asked in a first version of this paper whether the following 
metrics (which are induced by the ^1 metric on the ^1 full 
group of a 2-odometer on the subgroups 𝔖_2^n) could provide a 
concrete Lévy 
family for the derived 
^1 full group of the odometer.


Let 𝔖_n be the symmetric group over {1,...,n}. Define 
Spearman's footrule metric on it by 

    d^n_^1(σ,τ)=1/n∑_i=1^n σ(i)-τ(i).



Let us however point out that, as suggested by the referee, the family 
(𝔖_n,d^n_^1) is not a 
Lévy family. Indeed, by a result of Diaconis and Graham, the random variable 
d^n_^1(𝕀,σ) where σ is a uniform element of 𝔖_n has asymptotically mean n/3 and variance  2n/45 
<cit.> (moreover, suitably 
normalized, its 
law converges to the normal distribution). In particular, for n large enough, 
the 
ϵ-neighborhood of the ball of radius n/3 will be very close in 
measure to that same ball, thus violating concentration of measure.











§ LARGE SCALE GEOMETRY AND L1 FULL GROUPS




 §.§ Large scale geometry of Polish groups


In this preliminary section, we recall how Rosendal's framework allow one to study Polish groups as geometric objects. First, a map f:X→ Y between two metric spaces (X,d_X) and (Y,d_Y) is called a quasi-isometric embedding if there is a constant C>0 such that for all x_1,x_2∈ X, 

    -C+1/C d_X(x_1,x_2)≤ d_Y(f(x_1),f(x_2))≤ C+ Cd_Y(f(x_1),f(x_2)).

It is coarsely surjective if there is C>0 such that every element of Y is at distance at most C from f(X). Finally f is a quasi-isometry if it is a coarsely surjective quasi-isometric embedding. We say that two metric spaces (X,d_X) and (Y,d_Y) are quasi-isometric if there is a quasi-isometry between them. It is not hard to check that quasi-isometry an equivalence relation between metric spaces. The equivalence class of a metric space is called its quasi-isometry type.

Two metrics d and d' on the same set X are called quasi-isometric if the identity map is a quasi isometry between (X,d) and (X,d'). Note that the coarse surjectivity condition is then automatic so this just means that there is C>0 such that for all x_1, x_2∈ X, 

    -C+1/C d(x_1,x_2)≤ d'(x_1,x_2)≤ C+ Cd(x_1,x_2).



A continuous right-invariant metric d on a Polish group G is called maximal if for every continuous right-invariant metric d' on G, there is C>0 such that for all g_1,g_2∈ G, d'(g_1,g_2)≤ C+Cd(g_1,g_2). It follows from the definition that any two maximal right-invariant metrics on a Polish group are quasi-isometric, so if a Polish group admits such a metric, then it has a well-defined quasi-isometry type.

We finally give two conditions on a right-invariant metric which together are equivalent to the maximality of the  metric by <cit.>.


A metric d on a set X is called large-scale geodesic if there is K>0 such that for any x,y∈ X, there are x_0,...,x_n∈ X with x_0=x and x_n=y such that d(x_i,x_i+1)≤ K and

    ∑_i=0^n-1d(x_i,x_i+1)≤ K d(x,y).




A right-invariant metric d on a topological group G is called coarsely proper if for every neighborhood of the identity V and every N∈, there are a finite subset F⊆ G and n∈ such that 

    B_d(e,N)⊆(FV)^n.






 §.§ Derived L1 full groups  of amenable graphings


In this section, we show that derived ^1 full groups of amenable graphings fit rather well into C. Rosendal's framework,  enabling their study as geometric objects. 

To be more precise, we show that the ^1 metric on derived ^1 full groups of amenable graphings is maximal among left-invariant continuous metrics up to quasi-isometry, which implies that derived ^1 full groups of amenable graphings have a well defined quasi-isometry type which is moreover induced by the ^1 metric. 


Let Φ be an amenable graphing. Then every element of [Φ]_1' is a limit of periodic elements. 


Using the Connes-Feldmann-Weiss theorem <cit.>, we write ℛ_Φ as an increasing union of finite equivalence relations ℛ_n. Then every element of [ℛ_n] is periodic and by Lemma <ref> the union ⋃_n∈[ℛ_n]∩ [Φ]_1'
is dense in [Φ]_1' so we are done.





The next lemma is key to our results on the geometry of ^1 full groups. 

	Let Φ be a graphing. Let T∈[Φ]_1 be periodic. Then for every N∈, there are periodic elements V_1,...,V_N∈ [Φ]_1 such that T=V_1⋯ V_N and for every i∈{1,...,N}, 
	
    d^1_Φ(V_i,𝕀_X)=1/N d^1_Φ(T,𝕀_X).



Let M=d^1_Φ(T,𝕀_X).
Let (A_t)_t∈[0,M] be a continuous increasing path from ∅ to T, and consider B_t=⋃_n∈(TU)^n(A_t). Since TU is periodic, the map t↦ B_t is still a continuous increasing path from ∅ to T now consisting of T-invariant sets.
 
Define a continuous increasing map ψ:[0,M]→[0,M] by  
    ψ(t)=∫_B_td_Φ(x,T(x)).
 We have ψ(0)=0 and ψ(M)=M, so  by the intermediate value theorem we find t_0=0<t_1<...<t_N-1<t_N=M such that for all i∈{0,...,N}, ψ(t_i)=iM/N. We then let A_i=B_t_i∖ B_t_i-1 for each i∈{1,...,N}. By construction each A_i is T-invariant and T=_i=1^n A_i. So if we define V_i=T_A_i, we get T=∏_i=1^n V_i. Finally for each i∈{1,...,n} the equality A_i=B_t_i∖ B_t_i-1 yields 
    d^1_Φ(𝕀_X,V_i)=∫_B_t_id_Φ(x,T(x))-∫_B_t_i-1d_Φ(x,T(x))=ψ(t_i)-ψ(t_i-1)=M/N

as wanted.



	Observe that the above proof also yields the existence of a geodesic 
	segment between any periodic element and the identity map. Indeed, if 
	ψ is a right inverse of ψ defined above, we let 
	W_t=T_B_ψ(t), and then (W_t)_t∈[0,M] is a geodesic 
	segment from 𝕀_X to T.



	Let Φ be an amenable graphing, let K>0 and let N∈. Then we have the following inclusions in the derived ^1 full group of Φ:
	
    B_d^1_Φ(1,K/N)^N⊆ B_d^1_Φ(1,K)⊆ B_d^1_Φ(1,K/N)^N+1.



	The first inclusion is a straightforward consequence of the triangle inequality and right invariance. For the second one, let T∈ B_d^1_Φ(1,K), then since periodic elements are dense we can find U∈ B_d^1_Φ(1,K/N) such that TU is periodic and TU∈ B_d^1_Φ(𝕀_X,K). 
	
	By the previous proposition, the are V_1,...,V_N∈ B_d^1_Φ(𝕀_X,K/N) such that TU =V_1⋯ V_n. We conclude that T=TU U=(∏_k=1^NV_k) U so T∈ B_d^1_Φ(1,K/N)^N+1 as wanted. 



Let Φ be an amenable graphing, then the ^1 metric on the derived ^1 full group of Φ is maximal.


By the previous proposition, the ^1 metric is both large-scale geodesic and coarsely proper. So <cit.> yields that the ^1 metric is maximal. 



We finally show that although derived ^1 full groups of amenable graphings do have a geometry, the latter is fundamentally infinite-dimensional.


Suppose Γ is an amenable group containing an element of infinite order, and let Γ(X,μ) be a free measure-preserving action. Then for every n∈, the derived ^1 full group of the Γ action contains a quasi-isometric copy of ^n. 


It suffices to produce a quasi-isometric copy of ^n since the latter is quasi-isometric to ^n. 
Let T be the aperiodic measure-preserving transformation which corresponds to the element of Γ of infinite order. 
Using Rokhlin's lemma for T, we find A⊆ X  which intersect almost every T-orbit and such that all sets A,T(A),...,T^2n-1(A) are disjoint. 
For each i∈{0,...,n-1} let T_i=T_T^2i(A)T_T^2i+1(A).  
Observe that since TT_T^2i(A)T=T_T^2i+1(A), each T_i is a commutator and thus belongs to the derived ^1 full group. 
It is now straightforward to check that the marked group generated by T_1,...,T_n is quasi-isometric to ^n. 


As pointed out by the referee, it would be interesting to know whether some 
infinite dimensional metric spaces such as ℓ^1 can be coarsely embedded 
into derived ^1 full groups.
We end this section with a basic open question on the geometry of derived 
^1 full groups.


Are the derived ^1 full groups of the 2-odometer and of the 3-odometer quasi-isometric ?


Note that by <cit.>, the derived ^1 full 
groups of the 2-odometer and of the 3-odometer are not isomorphic as 
abstract groups. Also, the countable group of dyadic  permutations 𝔖_2^∞ (resp. 𝔖_3^∞) is dense in the derived ^1 
full group of the 2-odometer (resp. 3-odometer), and the induced metric is 
the inductive limit of the metrics from Definition <ref>, so this is really a question about quasi-isometry for the non finitely 
generated countable groups 𝔖_2^∞ and 𝔖_3^∞ equipped with these natural metrics. 



 §.§ L1 full groups of ergodic Z-actions
 

For measure-preserving -actions, we have a much better understanding of elements of the ^1 full group. This will allow us to show that the ^1 metric on the whole ^1 full group is maximal, and hence defines its quasi isometry type. Since a measure-preserving -action is nothing but a single measure-preserving transformation, we rather speak of ^1 full groups of measure-preserving transformations. Let us start by reminding Belinskaya's decomposition results for elements of the ^1 full group <cit.>. 

Given a measure-preserving transformation T, we denote by [T]_1 its ^1 full group. If T is aperiodic then every T-orbit is endowed with a linear ordering ≤_T defined by x≤_Ty if and only if there is n≥ 0 such that y=T^n(x). An element U of the full group of T is almost positive if for all x∈ X, there is N∈ such that for all n≥ N we have U^n(x)≥_Tx. It is almost negative if for all x∈ X, there is N∈ such that for all n≥ N we have U^n(x)≤_T x. Every element of the ^1 full group has a natural decomposition with respect to such elements. 


Let T be a measure-preserving aperiodic transformation, let U∈ [T]_1. Then there is a partition X=X_-⊔ X_per⊔ X_+ of X into U-invariant sets such that U_X_- is almost negative, U_X_+ is almost positive and U_X_per is periodic. 


	This is a direct consequence of <cit.>.


Observe that since the sets from the above decomposition are U-invariant and partition X, we then have U=U_X_-U_X_perU_X_+ and

    d^1_T(𝕀_X,U)=d^1_T(𝕀_X,U_X_-)+
    d^1_T(𝕀_X,U_X_per)+d^1_T(𝕀_X,U_X_+).

Recall that if T is an aperiodic measure-preserving transformation, every U∈[T]_1 has an associated cocycle c_U:X→ uniquely defined by the equation U(x)=T^c_U(x)(x). We can then defined the index map I:[T]_1→ by I(U)=∫_X c_U(x)dμ(x). In <cit.>, we prove that if T is ergodic, the kernel of the index map is the derived ^1 full group of T and the index map takes values into . The conjunction of Proposition 4.15 and Proposition 4.17 from <cit.> implies the following result.




Let U∈ [T]_1 be almost positive, let n=I(U). Then there is a family (A_i)_i=1^n of subsets of X  such that U=T_A_1⋯ T_A_n V, where V is periodic. 




Let T be an ergodic measure-preserving transformation.
Then the ^1 metric on the ^1 full group of T is maximal. 


We first show that for every N∈, every element of [T]_1 of ^1 norm[By ^1 norm, we mean the ^1 distance to 𝕀_X.] less than N can be written as a product of at most 11N elements, all of which are either T, T, or periodic of ^1 norm less than 1. 

Let X=X_per⊔ X_+⊔ X_- be the partition of X into U-invariant sets provided by Lemma <ref>. We start by noting that d^1_T(U_X_+,𝕀_x)≤ d^1_T(U,𝕀_x) and thus I(U_X_+)<N. We may thus write

    U_X_+=T_A_1⋯ T_A_n V,

where n=I(U_X_+)<N and V is periodic. By Kac's return time theorem <cit.>, each T_A_i is at distance 1 from 𝕀_X, so by the triangle inequality, we have d^1_T(V,𝕀_X)<2N. 

Now for each i∈{1,...,n} we have T_A_i=(T_A_iT)T. Moreover 
T_A_iT is periodic and d^1_T(T_A_iT,𝕀_X)<2. Since U_X_+ 
is equal to the product (T_A_1T)T⋯ (T_A_nT)T V and 
d^1_T(V,𝕀_X)<2N, Lemma <ref> applied to both V and 
the T_A_iT yields that U_X_+ is a product of at most 5N elements, 
all of which are either periodic and in the open unit ball or equal to T. 

The same reasoning for U_X_- yields that U_X_- can be written as a product of at most 5N elements, all of which are either periodic of norm <1 or equal to T.

Finally, Lemma <ref> yields that U_X_per can be written as the product of at most N periodic elements of norm <1. 
We can thus conclude that U=U_X_+U_X_perU_X_- can be written as the product of at most 11N elements, all of which are either periodic of norm <1 or belong to {T,T}.

This fact already yields that the ^1 metric is large-scale geodesic, and we now need to show that d^1_T is coarsely proper. 
So let N∈, let V be a neighborhood of the identity.
We find k∈ such that B_d^1_T(𝕀_X,1/k)⊆ V and we deduce from the above fact along with  Lemma <ref> that

    B_d^1_T(𝕀_X,N)⊆({T,T} V)^11kN.

The metric d^1_T is thus both large-scale geodesic and coarsely proper, so using <cit.> we conclude that d^1_T is maximal as wanted. 




 §.§ Automatic quasi-isometry: proof of Theorem <ref>


In this section we prove the following result, which directly yields Theorem <ref>.

	Let Φ and Ψ be two amenable ergodic graphings on a standard probability space  (X,μ). Then every abstract group isomorphism between their ^1 full groups must be a quasi-isometry for their respective ^1 metrics. 


	Let ρ: [Φ]_1→[Ψ]_1 be an abstract group isomorphism. 
	First observe that by <cit.>, there is a measure-preserving transformation S such that for all ∈[Φ]_1, ρ(U)=SUS. In other words, we can assume that ρ is the conjugacy by some S∈(X,μ), and it follows that ρ is d_u-isometric, in particular it is continuous for the uniform topology. 
	On [Φ]_1 the uniform topology is refined by the ^1 topology, and since it separates points we get that it generates the Borel σ-algebra of [Φ]_1 induced by the ^1 topology. We conclude that ρ is a Borel group isomorphism, and it follows from Pettis' lemma (see e.g. <cit.>) that ρ is a topological group isomorphism. 
	
	In particular, ρ induces a topological group isomorphism between [Φ]_1' and [Ψ]_1', and since their respective ^1 metrics are maximal (Corollary <ref>), we conclude that ρ induces a quasi-isometry for their respective ^1 metrics: there is C>0 such that for all T_1,T_2∈[Φ]'_1, 
	
    -C+1/Cd^1_Φ(T_1,T_2)≤ d^1_Ψ(ρ(T_1),ρ(T_2))≤ Cd^1_Φ(T_1,T_2)+C

	
	In order to establish the same inequality for elements of the whose ^1 full group, first note that by right-invariance, one may as well assume T_1=𝕀_X, and then note that by symmetry, it suffices to find C'>0 such that for all T∈[Φ]_1, 
	
    d^1_Φ(𝕀_X,ρ(T)≤ C'd^1_Ψ(𝕀_X,T)+C'.

	We will show that C'=6C works. 
	
	Let T∈[Φ]_1, then by a well-known lemma (see e.g. <cit.>), there is a partition (A_1,A_2,B_1,B_2,B_3) of T such that 
	
    T(A_1)=A_2, T(B_1)=B_2 and  T(B_2)=B_3.

	We use this decomposition to define three involutions U_1,U_2,U_3 as follows:
	
    U_1(x)   ={[           T(x) if x∈ A_1⊔ B_1;           T(x) if x∈ A_2⊔ B_2;              x     otherwise, ].
    
    	U_2(x)   ={[          T(x)     if x∈ B_2;          T(x)     if x∈ B_3;             x otherwise and ].
    
    
    	U_3(x)   ={[                 T(x)       if x∈ A_2⊔ B_3;                 T(x) if x∈ T(A_2)⊔ T(B_3);                    x           otherwise. ].

	By construction for each i=1,2,3 we have d^1_Φ(𝕀_X,U_i)≤ 2d^1_Φ(𝕀_X,T), moreover since U_1,U_2 and U_3 are involutions they actually belong to the derived ^1 full group of Φ. We can then decompose T as 
	
    T=U_1 A_1⊔ B_1⊔ U_2 B_2⊔ U_3 A_2⊔ B_3⊔𝕀_X∖ T.

	Because ρ is the conjugacy by the transformation S, we then have 
	
    ρ(T)=ρ(U_1)_ S(A_1⊔ B_1)⊔ρ(U_2)_ S(B_2)⊔ρ(U_3)_ S(A_2⊔ B_3)⊔𝕀_X∖ S( T).

	We thus have d^1_Ψ(𝕀_X,ρ(T))≤∑_i=1^3d^1_Ψ(𝕀_X,U_i).
	Since for i=1,2,3, 
    d^1_Ψ(𝕀_X,ρ(U_i))≤ C d^1_Φ(𝕀_X,U_i)+C≤ 2Cd^1_Φ(𝕀_X,T)+C,

	we conclude that 
	
    d^1_Ψ(𝕀_X,ρ(T))≤∑_i=1^3d^1_Ψ(𝕀_X,U_i)≤ 6C d^1(Φ(𝕀_X,T))+3C

	as wanted.


Note that if we knew that the ^1 metric on ^1 full groups of amenable graphings is maximal, then the first two paragraphs of the above proof would suffice to get the conclusion of Theorem <ref>. As explained in the introduction, we actually conjecture that the following is true, which would then imply that one can remove both the amenability and the ergodicity hypothesis in Theorem <ref> . 


	Let Φ be a graphing. Then the ^1 metric on the ^1 full group of Φ is maximal. 


We end this section by explaining why in Theorem <ref> the conclusion cannot be strengthened to bilipschitzness. Recall that in a topological space, a subset is G_δ if it can be written as a countable intersection of open sets, and that by the Baire category theorem, in a complete metric space every countable intersection of dense open sets has to be dense.


	Let T be an aperiodic transformation. Then the set of all A∈(X,μ) such that the return time to A is unbounded is dense G_δ in (X,μ).


	The set ℬ of all A∈(X,μ) such that the return time to A is unbounded is the intersection over n∈ of the sets 
	
    ℬ_n:={A∈(X,μ): μ(T^-n(A)∖⋃_i=0^n-1T^-i(A))>0},

	so it is G_δ. Let us now fix n∈ and show that ℬ_n is dense so as to finish the proof. Let ϵ>0, let N∈ such that n/N<ϵ. By aperiodicity we find B⊆ X with positive measure such that B,T(B),...,T^Nn(B) are disjoint, and so in particular μ(B)<1/Nn. Now observe that for any A∈(X,μ), the set 
	
    (A∪ B∪ T^n(B))∖_i=1^n-1T^i(B)

	is in ℬ_n and is nμ(B)<ϵ-close to A, which shows that ℬ_n is dense.



 
	Let T be an aperiodic measure-preserving transformation, and consider A⊆ X with unbounded return time, as provided by the previous lemma. Then the conjugacy by the induced transformation T_A is not bilipschitz: for each n we may find B⊆ A such that  T(B), ..., T^n(B) are all disjoint from A, and then if U is the involution defined by U=T_ B⊔ T_ T(B)⊔𝕀_X∖ (B⊔ T(B), we have d^1_T(𝕀_X,T_AUT_A)≥ n d^1_T(𝕀_X,U).	







§ TOPOLOGICAL RANK TWO FOR RANK ONE TRANSFORMATIONS


In this section we prove that every rank one measure-preserving transformation has an ^1 full group whose topological rank is equal to two. Let us recall the definition of such transformations and first introduce some useful terminology.


Let T be a measure-preserving transformation and A⊆ X Borel. Let N_T(A)∈ be the greatest integer such that the sets A, T(A),..., T^N_T(A)-1(A) are all disjoint. We then have a Rokhlin partition 𝒫_A,T=(A,T(A),...,T^N_T(A)-1(A)) of Y_A,T:=⋃_i=0^N_T(A)-1T^i(A) and we let M_A,T be the sets of finite unions of elements of 𝒫_A,T. 


Note that Rokhlin's lemma guarantees the existence of Rokhlin partitions 𝒫_A,T with arbitrarily small atoms and μ(Y_A,T) arbitrarily close to 1. 


A measure-preserving transformation T is rank one if for every ϵ>0 and every finite family (B_i)_i=1^k of Borel sets there is a Borel set A  and Borel sets A_1,...,A_k∈ M_A,T such that for all i∈{1,...,k}, 

    μ(A_i△ B_i)<ϵ.



It is not hard to check from the definition that rank one transformations are ergodic and form a G_δ set in (X,μ) for the weak topology (see <cit.> for a detailed proof). An easy example of a rank one transformation is provided by the odometer, and so by Halmos' conjugacy lemma (see <cit.>), rank one transformations form a dense G_δ set in (X,μ).

The following theorem of Baxter says that rank one transformations are actually quite close to odometers. 

[<cit.>]Let T be a rank one 
transformation. Then there is a decreasing sequence (A_n) of Borel subsets of 
X such that M_A_n,T is  increasing and its union is μ-dense in 
the Borel σ-algebra. 

A sequence (A_n) as in the previous theorem will be called a rank one basis of T, so in our terminology a measure-preserving is rank one if and only if it admits a rank one basis. Observe that if (A_n) is a rank one basis for T then the measure of A_n then tends to zero and N_T(A_n)→+∞.

	
Let T_0 be the dyadic odometer on X={0,1}^, and for every n∈ we let A_n denote the set of sequences (x_i)∈{0,1}^ such that x_0=⋯=x_n=0. Then (A_n) is a rank one basis for T_0 with N_T_0(A_n)=2^n and the additional property that Y_A_n,T=X for every n∈.



We will now generalize to rank one transformations the fact that dyadic transformations are dense in the derived ^1 full group of the odometer. Let us first see how Definition <ref> provides us natural embeddings of the symmetric group in the derived ^1 full group. 


Let T be a measure-preserving transformation and A⊆ X be Borel. We then define the embedding 

    ρ_A: 𝔖({0,...,N_T(A)-1})→ [T]_1'

by: for all σ∈𝔖({0,...,N(A)-1}) and all x∈ Y_A,T, if i∈{0,...,N(A)-1} is such that x∈ T^i(A), then 

    ρ_A(σ)(x)=T^σ(i)-i(x),

and ρ_A(σ)(x) for x∉Y_A,T. Observe that by construction for all i∈{0,...,N(A)-1} we have ρ_A(σ)(T^i(A))=T^σ(i)(A).
We then  let 
    𝔖_A,T:=ρ_A(𝔖({0,...,N(A)-1})) and 𝔄_A,T:=ρ_A(𝔄({0,...,N(A)-1})).

Observe that if A⊆ B and M_A,T⊆ M_B,T then 𝔖_A,T≤𝔖_B,T and 𝔄_A,T≤𝔄_B,T. 
The following generalizes to ^1 full groups a result from the thesis of the author <cit.>.


Let (A_n) be a rank one basis of a measure-preserving transformation T. Then 

    ⋃_n∈𝔄_A_n,T

is dense in [T]_1'. 


Let U_n be the involution defined by U_n=φ_A_n((0  1)), then d^1_T(U_n,1)→ 0. Since  𝔖_A_n,T=𝔄_A_n,T∪ U_n𝔄_A_n,T, we conclude that  the density of ⋃_n∈𝔄_A_n,T will follow from that of ⋃_n∈𝔖_A_n,T which will thus prove instead. 

Since [T]_1' is topologically generated by involutions of the form I_T,A (cf. Theorem <ref>), we only need to approximate such involutions by elements of ⋃_n∈𝔖_A_n,T. So let A⊆ X, we then have a sequence (B_n) of elements of M_A_n,T with B_n→ A. Furthermore we may assume that B_n is disjoint from A_n⊔ T^N(A_n)-1 since the measure of A_n⊔ T^N(A_n)-1 tends to zero as n tends to infinity. This implies I_B_n,T∈𝔖_A_n,T and since  I_B_n,T→ I_A,T we get the desired conclusion. 













Let T be a measure preserving transformation, let A⊆ X and let n∈ such that 3≤ n ≤ N_T(A)-2. Consider the element U=ρ_A((0  1  ⋯   n-1)). Then the group generated by T and U contains 𝔄_A,T. 


By conjugating U by powers of T, we see that the group generated by T and U contains all the 
ρ_A((i   i+1  ⋯   i+n-1)) for i+n-1≤ N(A). The lemma then follows from the fact that 𝔄({0,...,N(A)-1}) is contained in the group generated by the (i   i+1  ⋯   i+n-1) for 0≤ i≤ N(A)-n. 
We briefly recall the proof of the latter fact for the reader's convenience. 

Denote by G the group generated by the (i   i+1  ⋯   i+n-1) for 0≤ i≤ N(A)-n. First note that G contains the permutation 
    (n   n-1  ⋯   1)(0   1⋯    n-1)=(0   n    n-1)
 so it also contains (0   n    n-1)^2=(0   n-1   n). Conjugating by a power of (1   2  ⋯   n), we obtain that G contains the element (0   1   2). Conjugating again by appropriate cycles we get that G contains all the (i   i+1   i+2) for 0≤ i≤ n-3, and since the latter generate the group 𝔄({0,...,N(A)-1}) the result follows.  


The usual way of disjointifying a countable family of sets (B_n) is to let B'_n:=B_n∖⋃_m<nB_m, but note that letting B'_n=B_n∖⋃_m>nB_m also works. We will need to apply this idea to measure-preserving transformations so as to make their support disjoint. 

For p∈, let us call a p-cycle a transformation all whose orbits have cardinality either 1 or p. Given a countable family (U_n)_n∈ where each U_n is a p_n-cycle, we explain how to change them so that they have disjoint support. For each n, we  let V_n be the transformation induced by U_n on the biggest U_n-invariant set contained in
U_n∖⋃_m>n U_m.
Note that since U_n is a p_n-cycle, we can easily compute the support of V_n which is

    V_n= U_n∖(⋃_i=0^p_n-1U_n^i(⋃_m>n U_m)).

Of course this support might be empty, but observe that if the supports of the remaining U_m's are very small then V_n will actually be very close to U_n. The sequence (V_n) obtained this way is called the disjointification of the sequence (U_n).

We are now ready to prove the main result of this section.


Let T be a rank one transformation. Then the topological rank of its ^1 full group [T]_1 is equal to 2. 


Let (p_n)_n∈ be the increasing enumeration of prime numbers. Let ϵ_n be a sequence of positive real numbers decreasing to zero, e.g. ϵ_n=2^-n. Let (A_n) be a one-dimensional ranking of T. 


Using lemma <ref>, one can now build by induction an increasing sequence  of integers (k_n) and positive real numbers (δ_n) such that for all n, if we let 
    U_n:=φ_A_k_n((0   1⋯ p_n-1))
 then the following conditions are satisfied.


  *  Any element V_n which is δ_n-close to U_n satisfies that the group generated by T and V_n contains every element of 𝔄_A_k_n,T up to an ϵ_n error.

  *  For  every m<n, we have p_m∑_k=m+1^nμ( U_k)< δ_m.

  *  We have k_n≥ 4^n2^n+2^n.


Consider the sequence (V_n) defined as the disjointification of the U_n's. Thanks to condition (<ref>) we have that V_n is δ_n-close to U_n. By condition (<ref>) the group T,V_n contains every element of 𝔄_A_k_n,T up to an ϵ_n error.

Now let V:=∏_n∈ V_n. We first establish a general inequality which will yield that V∈[T]_1 and moreover the closed group generated by V contains every V_n. 


For all n∈ we have 
    ( ∏_i<np_i )(∑_i≥ np_i/k_i)≤2^-n2^n.



We first need an explicit bound on p_n. A well-known way to get such a bound is as follows: Euclid's argument that there are infinitely many prime numbers actually shows that  p_n+1≤ p_0⋯ p_n+1. From this, a straightforward induction yields the bound 
    p_n≤ 2^2^n.

Now by condition (<ref>) we have k_n≥ 4^n2^n+2^n, so for every n≥ 1 we have 

    ∑_i≥ np_i/k_i   ≤ 4^-n2^n∑_i≥ n2^2^i/4^2^i
       ≤ 4^-n2^n

Now ∏_i<np_i≤ (p_n)^n≤ 2^n2^n so combining this with the previous inequality we see that (<ref>) holds.

Now observe that for every n∈ we have d^1_T(V_n,𝕀_X)_1≤ d^1_T(U_n,id_X)≤2p_n/k_n. The inequality (<ref>) for n=0 yields ∑_i≥ 02p_i/k_i<+∞ so that V∈[T]_1 as announced.

Fix n∈. For every m≥ n+1 let P_m=∏_i< m, i≠ n p_i and let t_m∈{1,...,p_n-1} such that P_mt_m=1 p_n. Now observe that

    V^P_mt_m=V_n∏_i≥ mV_i^P_mt_m.


Since t_m≤ p_n, we have P_mt_m≤∏_i< m p_i so that  

    d^1_T(∏_i> mV_i^P_mt_m,𝕀_X)≤( ∏_i< mp_i )∑_i≥ kd^1_T(V_i,𝕀_X).

But for every i we have V_i_1≤2p_i/k_i so we conclude by inequality (<ref>) that we have d^1_T(∏_i> mV_i^P_mt_m,𝕀_X)→ 0 and hence 
    V^P_mt_m→ V_n  [m→+∞].

So for every n∈ the closed subgroup generated by V contains V_n. 

Since the group T,V_n contains every element of 𝔄_A_k_n,T up to an ϵ_n error and ϵ_n→ 0, we conclude by Proposition <ref> that the group generated by T and V is dense in [T]_1 which has thus topological rank at most 2. Since the ^1 full group [T]_1 is not abelian, we conclude that [T]_1 has topological rank 2 as wanted. 













§ FURTHER REMARKS AND QUESTIONS





 §.§ Lp orbit equivalence and Lp full orbit equivalence
 


It is  an instructive exercise to show that given two measure preserving actions α:Γ→(X,μ), β:Λ→(X,μ) and a measure-preserving transformation T, the following are equivalent:

	
  * T sends α(Γ)-orbits to β(Λ)-orbits ;
	
  * Tα(Γ)T is a subgroup of  [β(Λ)] and 
	Tβ(Λ)T is a subgroup of [α(Γ)];
	
  * T[α(Γ)]T=[β(Γ)].

If T satisfies any of the above three conditions, one says that T is an orbit equivalence between the actions α and β, and that the latter are orbit equivalent.


We then have two natural ^p versions of orbit equivalence, which are 
direct translations of item (ii) and (iii) from the above equivalence. The 
first one was defined by Austin in <cit.> and 
the second appears to be new. 




	Let p∈[1,+∞]. Two measure-preserving actions of finitely generated groups  α:Γ→(X,μ) and β:Λ→(X,μ) are called  ^p-orbit equivalent if there is T∈(X,μ) such that 
	
    Tα(Γ)T≤ [β(Λ)]_p and Tβ(Λ)T≤[α(Γ)]_p.


 For p=+∞, the group 
[α(Γ)]_p is  
the group of all T∈[α(Γ)] such that the distance from T(x) to 
x is essentially bounded. The ^∞ metric on this group is the 
discrete metric, so it is not a Polish group.



	Let p∈[1,+∞]. Two measure-preserving actions of finitely generated groups  α:Γ→(X,μ) and β:Λ→(X,μ) are called  ^p-fully orbit equivalent if there is T∈(X,μ) such that 
	
    T[α(Γ)]_pT=[β(Λ)]_p.





It is clear that ^p full orbit equivalence implies ^p orbit equivalence, and that ^∞ orbit equivalence is equivalent to ^∞ full orbit equivalence. Moreover, for all p<q we have that ^q orbit equivalence implies ^p orbit equivalence, but is is unclear that ^q full orbit equivalence implies ^p full orbit equivalence. Also note that all these notions collapse to flip-conjugacy for ergodic -actions by Belinskaya's theorem. 

A natural intermediate question is to ask what it means for two graphings to share the same ^1 full group, hoping that the identity map has to be induce an ^∞ orbit equivalence between them. However, this is not so, as the following example of Sébastien Gouëzel shows. 


[Gouëzel] Given an aperiodic 
T∈(X,μ), there is T'∈(X,μ) such that [T]_1=[T']_1 but 
T' does not belong to the ^∞ full group of T.


First note that for every S∈(X,μ) we have [STS]_1=S[T]_1S, so for every S∈ [T]_1 we have [STS]_1=[T]_1. 
So it suffices to find S∈ [T]_1 such that  T'=STS does not belong to the ^∞ full group of T. 

Now observe that given any S∈ [T]_1, the function x↦ d_T(x,STS(x)) is bounded if and only if the function x↦ d_T(S(x),ST(x)) is bounded. 
It thus suffices to find S∈[T]_1 such that the function x↦ d_T(S(x),ST(x)) is unbounded. 
This can be achieved by taking A⊆ X with unbounded return time via Lemma <ref>, and considering the induced transformation S=T_A. Indeed if x∈ A has return time n_T,A(x)≥ 2 then ST(x)=T(x) and so d_T(S(x),ST(x))=n_T,A(x)-1. 




Gouëzel's example arises as a conjugate of T by some S∈[T]_1, but it is natural to ask more generally which S∈(X,μ) satisfy [STS]_1=[T]_1. Note that such an S has to be an automorphism of the equivalence relation ℛ_T. Two sources of such S are provided by the group of measure-preserving transformations which conjugate T to T^±1, and elements of the ^1 full group of T. Still, we do not have a good understanding of the group of S∈(X,μ) satisfying [STS]_1=[T^±1]_1, and the following question is open.

Let T be an ergodic measure-preserving transformation. Is the 
group of S∈(X,μ) satisfying [T]_1=[STS]_1 Borel ? If so, is it 
Polishable ?
 

Note that by Fremlin's reconstruction theorem, every abstract automorphism of [T]_1 has to be the conjugacy by some S∈(X,μ), so the above question is actually about the automorphism group of [T]_1. Also note that the automorphism group of the full group of any measure preserving equivalence relation is Polishable, see <cit.>.











 §.§ Symmetric Lp full groups



The notions of symmetric (and alternating) topological full groups have natural analogues in our setup. Recall from <cit.> that for any p∈[1,+∞[ and any graphing Φ, the ^p full group of Φ is defined as the group of all elements T of the full group of ℛ_Φ such that 

    ∫_Xd_Φ(x,T(x))^pdμ(x)<+∞.

It is a Polish group for the metric d^p_Φ defined by 

    d^p_Φ(T_1,T_2)=[∫_X d_Φ(T_1(x),T_2(x))^pdμ(x)]^1/p,
 and we  define the following closed normal subgroups.


	The symmetric ^p full group of a graphing Φ is the closed subgroup of [Φ]_p generated by all n-cycles for n≥ 2. The alternating ^p full group is defined as the closed subgroup generated by all 3-cycles. 


Note that if p=1, both subgroups coincide with the derived ^1 full group of Φ, and I don't know whether the same holds for p>1. Nevertheless,
if Φ is aperiodic, the same proof as Proposition <ref>	shows that the alternating ^p full group of Φ coincides with its symmetric ^p full group. One can also adapt the arguments from <cit.> to show that in the aperiodic case, closed normal subgroups of the symmetric ^p full group correspond to invariant sets; in particular the symmetric ^p full group of Φ is topologically simple iff Φ is ergodic. 
Moreover, the proof of Theorem <ref> can easily be adapted to show the following.

 The topological rank of the symmetric ^p full group of a 
measure-preserving aperiodic action of a finitely generated group is finite if 
and only if the action has finite Rokhlin entropy.


In a work in progress with C. Conley and R. Tucker-Drob, we show that for ergodic -actions, the symmetric ^p full group coincides with the derived ^p full group, and that the topological abelianization of the whole ^p full group is , just as in the ^1 case. In particular, we can deduce that the whole ^p full group of an ergodic measure-preserving transformation has finite topological rank if and only if the transformation has finite entropy, and answer positively question 5.5 from <cit.> in the case the graphing comes from a -action. 






 §.§ Topological rank of (derived) L1 full groups


Theorem <ref> is not satisfactory in two ways: first, as in the  case, a more quantitative statement would be desirable. In particular, establishing a lower bound on the topological rank in terms of the Rokhlin entropy would be very nice. Second, it would be more natural to have a similar statement for the whole ^1 full group, but in order to do so one has to understand the topological abelianization of ^1 full groups. We would like to ask the following question, noting that a positive answer would imply that Theorem <ref> also holds for the whole ^1 full group.


	Let Γ be a finitely generated group. Is the topological abelianization of the ^1 full group of any ergodic measure-preserving Γ-action always topologically finitely generated ? 


At the moment, the question is open even for Γ=^2. It was answered in 
the affirmative for Γ= in <cit.>, 
and with Conley and Tucker-Drob we have a proof that the answer is also yes 
when Γ is the infinite dihedral group (we actually show that in this 
case the topological abelianization is always trivial, thus obtaining examples 
of topologically simple ^1 full groups). Also note that the topological 
rank of the topological abelianization can be greater than the rank of the acting 
group: for a free ergodic ^2 action, if the two canonical generators have 
diffuse ergodic decomposition then ^2 can be obtained as a quotient of the 
topological abelianization by integrating the cocycles, but it has topological 
rank 3>2. 





cdraifplain



 