A Characterization of Amenable Groups by Besicovitch Pseudodistances

. The Besicovitch pseudodistance deﬁned in [BFK99] for one-dimensional conﬁgurations is invariant by translations. We generalize the deﬁnition to arbitrary countable groups and study how properties of the pseudodistance, including invariance by translations, are determined by those of the sequence of ﬁnite sets used to deﬁne it. In particular, we recover that if the Besicovitch pseudodistance comes from a nonde-creasing exhaustive Følner sequence, then every shift is an isometry. For non-Følner sequences we prove that some shifts are not isometries, and the Besicovitch pseudodistance with respect to some subsequence even makes them non-continuous.


Introduction
The Besicovitch pseudodistance was proposed by Blanchard, Formenti and Kůrka in [BFK99] as an "antidote" to sensitivity of the shift map in the prodiscrete (Cantor) topology of the space of 1D configurations over a finite alphabet.The idea is to take a window on the integer line, which gets larger and larger, and compute the probability that in a point under the window, chosen uniformly at random, two configurations will take different values.The upper limit of this sequence of probabilities behaves like a distance, except for taking value zero only on pairs of equal configurations: this defines an equivalence relation, and the resulting quotient space is a metric space on which the shift is an isometry, or equivalently, the distance is shift-invariant.
The original choice of windows is X n = [−n : n], the set of integers from −n to n included.This notion can be easily extended to arbitrary dimension d ≥ 1, taking a sequence of hypercubic windows.If we allow arbitrary shapes, the notion of Besicovitch space can be extended to configurations over arbitrary groups; in this case, however, the properties of the group and the choice of the windows can affect the the distance being or not being shift-invariant.An example of a Besicovitch pseudodistance which is not shift-invariant is given in [Cap09], where it is also proved that, if a countable group is amenable (cf.[CGK13] and [CSC10, Chapter 4]), then the Besicovitch distance with respect to any nondecreasing exhaustive Følner sequence is shift-invariant.The class of amenable groups is of great interest and importance in group theory, symbolic dynamics, and cellular automata theory.
In this paper, we explore the relation between the properties of Besicovitch pseudodistances over configuration spaces with countable base group and those of the sequence of finite sets used to define it.We introduce a notion of synchronous Følner equivalence between sequences, and a related order relation where one sequence comes before another sequence if it is synchronously Følner-equivalent to a subsequence of the latter.This notion, on the one hand, generalizes that of Følner sequences, and on the other hand, allows us to compare the Besicovitch distances and submeasures associated to different sequences.In particular, we prove that an increasing sequence of finite sets is Følner if and only if every shift is an isometry for the corresponding Besicovitch distance: this provides the converse of [Cap09, Theorem 3.5].Finally, we give conditions for absolute continuity and Lipschitz continuity of Besicovitch submeasures with respect to each other.

Background
We use the notation X Y to mean that X is a finite subset of Y .We denote the symmetric difference of two sets X and Y as X∆Y .We write

Submeasures
The following definition is classical (see for instance [Sab06]).
If G and A are two sets, the difference set of two functions x, y : Any submeasure over G gives rise to an associated pseudodistance over A G : Remark 1.The topological space corresponding to such a pseudodistance is homogeneous in the following sense: the balls around every two points y and z are isometric.Indeed, identify A with the additive group Z/ |A| Z. Then for every y, z ∈ A G the map ψ y,z : A G → A G defined by ψ y,z (x)(i) = x(i) − y(i) + z(i) for every x ∈ A G and i ∈ G is an isometry between any ball around y and the corresponding one around z.
Consequently, the identity map, from space A G endowed with d ν onto space A G endowed with d µ , is continuous (resp.α-Lipschitz) if and only if µ is absolutely continuous (resp.α-Lipschitz) with respect to ν.In that case the identity is even absolutely continuous.

Shifts and translations
If A is an alphabet, G is a group, and g ∈ G, the shift by g is the function σ g : A G → A G defined by σ g (x)(i) = x(g −1 i), for every x ∈ A G and i ∈ G.A map ψ from A G to itself is shift-invariant if ψσ g = σ g ψ for every g ∈ G.Note that ∆(σ g (x), σ g (y)) = g∆(x, y) for every x, y ∈ A G and g ∈ G.
Since the maps ψ y,z from Remark 1 are shift-invariant, one can see that the shift is continuous, Lipschitz, etc in every x if and only if it is in one x.
Given g ∈ G, let gµ(X) = µ(g −1 X) for every X ⊂ G. Then d µ (σ g (x), σ g (y)) = d g −1 µ (x, y), that is, the shift by g, within space A G endowed with d µ , is topologically the same as the identity map, from A G endowed with d µ onto space A G endowed with d g −1 µ .Remark 1 can then be rephrased into the following.
Remark 3. If G is a group, g ∈ G, and A G is endowed with d µ , then σ g is continuous (resp.α-Lipschitz) if and only if g −1 µ is absolutely continuous (resp.α-Lipschitz) with respect to µ.In that case, the shift by g is even absolutely continuous.

Besicovitch submeasure and pseudodistance
Among classical examples of submeasures are the ones that induce the Cantor topology, the shift-invariant Besicovitch pseudodistance, the Weyl pseudodistance (see [HM17, Def 4.1.1]. . .We will focus on the Besicovitch topology.Let X and Y be nonempty sets and let (X n ) be a nondecreasing sequence of finite subsets of X.We may or may not require that (X n ) be exhaustive, that is, For example, if X = N, Y = {0, 1}, and x(i) = 0 for every i ∈ N and y ∈ {0, 1} N is the characteristic function of the prime numbers, then d (Xn) (x, y) = 0.The topology of the quotient space is very different from the prodiscrete topology.
We will now concentrate on the case of nondecreasing sequences (X n ).

Følner equivalence
Let (X n ) and (Y n ) be nondecreasing sequences of finite subsets of G.We say that they are synchronously Følner-equivalent if Corollary 1.The synchronous Følner equivalence is an equivalence relation.

The proofs are left to the reader (see [CGN] for details).
We also denote ( To be convinced of the equivalence, note that the minimum is reached by some m n for each n ∈ N, because (Y m ) is nondecreasing and X n is finite.Thanks to symmetry of synchronous equivalence, we also have that ( = 0. We say that they are Følner-equivalent, and write ( . This is the case if they are synchronously Følner equivalent, but the converse is false.As counterexamples, one can consider twice the same sequence, but with repetitions on both sides that are longer and longer, and not synchronized.If one wants to obtain strictly increasing sequences, repetitions can be replaced by very slowly increasing sequences (point by point).
Remark 4. It is easy to see that is a preorder relation.In turn, Følnerequivalence, being defined as the equivalence corresponding to the preorder , is an equivalence relation.

Comparing Besicovitch submeasures
A basic tool in our set constructions will be the following elementary remark.
Remark 5.If (X n ) is nondecreasing and exhaustive, then for every finite set W and every ε > 0, there exists We deduce the following, which will be useful in our constructions.
Lemma 1.Let (X n ) be a nondecreasing exhaustive sequence of an infinite group G. Let W = i∈N W i where ∅ = W i G for each i ∈ N, such that, for every n ∈ N, there are at most finitely many i's such that W i ∩X n = ∅ (this is the case, for example, if the W i 's are pairwise disjoint); in that case j n = max Wj ∩Xn =∅ j is well-defined for every n.Then: 3. In general, there exists a nondecreasing integer sequence l such that, denoting Proof.
We know that this sequence goes to infinity (even though it may not be nondecreasing), because only finitely many W i 's intersect each X m , but they all intersect at least one.Hence, µ (Xn) (W ) ≥ lim sup i→∞ P ( W | X mi ).We get the desired inequality by noting that W i ⊂ W . 2. Point 1 already gives one inequality.For the converse: The last inequality comes from the fact that the sequence (j n ) is nondecreasing (because (X n ) is nondecreasing), and not upper-bounded (because the W i 's are nonempty), so it goes to infinity.3. Let us define some sequence l by recurrence, from any seed l 0 ∈ N. Assume that l n is defined, and write k n = n (Xn) ( j≤n W lj ).Choose any l n+1 such that for every m ≥ l n+1 , W m does not intersect X kn−1 (this is possible by assumption).If j n = max W l j ∩Xn =∅ j, then n (Xn) ( j<jn W lj ) = k jn−1 .By definition, W lj n does not intersect X kj n −1−1 .Since W lj n intersects X n , we can deduce that n > k jn−1 −1.This means that (W li ) satisfies the hypothesis of Point 2. Replacing the lim sup by a lim can be achieved by taking a subsequence.

For every
If m n realizes the maximum for each n ∈ N, and if ε < 1, then these properties imply that In particular, the properties imply that δ ≤ ε.

Proof.
Let 2⇒1 If property 2 is satisfied and µ (Yn) (W ) ≥ ε, then: The W i satisfy the hypotheses of Lemma 1, so that Point 3 gives l ∈ N N , with µ (Xn) (W l ) = lim i→∞ max m∈N P ( W li | X m ).By construction, we have: , so that: Taking the limit, we get that µ (Xn) (W l ) < δ.On the other hand, applying now Point 1 of Lemma 1 to sequence (Y n ): The previous lemma now allows to characterize the main properties of interest for comparing two Besicovitch submeasures.
Theorem 1.Let (X n ) and (Y n ) be nondecreasing and exhaustive.

µ
One can even see from the proof that (Y n ) (X n ) if and only if there exists

Proof.
1. Just note that the λ-Lipschitz property of µ (Yn) is equivalent to the properties in Lemma 2, for every δ and ε = λδ, and hence to: 2. From Lemma 2, µ (Yn) is absolutely continuous with respect to µ (Xn) if and only if From Point 1, this is equivalent to the existence of some λ such that µ (Yn) is λ-Lipschitz with respect to µ (Xn) .

Let (m
We can conclude by Point 1. Conversely, suppose that By the last inequalities in Lemma 2, we know that lim n∈N By Point 3 of Proposition 1, we obtain that (Y n ) (X n ). 4. This is direct from the definitions and the Point 3.
The following is direct from Theorem 1 and Remark 2.
if and only if the identity map from A G endowed with d (Xn) onto A G endowed with d (Yn) is 1-Lipschitz (resp.an isometry).
Here are particular classes of sequences, where the proposition can be applied.
Corollary 3. Let (X n ) and (Y n ) be nondecreasing and exhaustive.
1 =⇒ 2 If (X n ) and (Y n ) are synchronously Følner equivalent, then so are (X ln ) and (Y ln ) for every increasing (l n ) ∈ N N .We conclude thanks to Theorem 1. 2 =⇒ 3 This is obvious. 1 =⇒ 3 If (X n ) and (Y n ) are not synchronously Følner-equivalent, then there exists an infinite set I ⊂ N and a real number α > 0 such that ∀n ∈ I, |Xn∆Yn| |Xn| ≥ α.This implies that for every increasing sequence (l n ) ∈ I N , (X ln ) and (Y ln ) are not synchronously Følner-equivalent.We can take an increasing sequence (l n ) ∈ I N such that n (Ym) (X ln , ε ln ) = l n+1 , for some real sequence (ε n ) converging to 0. Then (X ln ) and (Y ln ) satisfy the assumptions for Point 3 of Corollary 3.There are nondecreasing non-Følner sequences for which the shift is Lipschitz (but not an isometry) in Z d .Here's an example: X n = ( −n, n ∪ 2 −n, n ) d .Indeed, for every n, 1 + X n ⊂ X 2n and |X2n| |Xn| = (8n−1) d (4n−1) d , which converges to 2 d when n goes to infinity.We conclude by Point 1 of Corollary 3, with m n = 2n and α = 2 d .But the shift is not an isometry because the sequence is not Følner: µ((2Z) d ) = 2 d /3 d > µ((2Z + 1) d ) = 1/3 d .

Shift
"Dually" to shifts, we can define the propagation π g : A G → A G by π g (x)(i) = x(ig).A block map (see [LM95] for G = Z) is, in essence, a composition of a radius-0 function with a product of propagations.The same characterizations are true for propagations as for shift maps, to which we can derive the following: Corollary 7. A nondecreasing exhaustive sequence (X n ) of finite subsets of a f.g. group G is right Følner if and only if for every increasing sequence (l n ) ∈ N N , every block map with neighborhood size k is k-Lipschitz for d (X ln ) .

Conclusions
We have presented a way to compare Besicovitch submeasures (in terms of absolute continuity, Lipschitz continuity, equality) thanks to the sequences of finite sets which describe them.In a shift space (with respect to a finitely generated group) endowed with the Besicovitch topology, we have derived conditions on the defining sequence for the shift maps to be continuous, Lipschitz or isometries.As part of this, we gave another characterization of f.g.amenable groups.
Future work will involve the study of other topological and dynamical properties (cf.[CGN]) or extension to configuration spaces on possibly uncountable groups.The latter would require the use of the more general notions of directed set and of net, and although the definition of Besicovitch pseudodistance and submeasure would be immediate to extend, the techniques used to prove the main lemmas could need a major revision.

Corollary 4 .
Let (X n ) and (Y n ) be nondecreasing and exhaustive.Assume that |X n | ∼ n→∞ |Y n |.Then the following are equivalent.1. (X n ) and (Y n ) are synchronously Følner-equivalent.
us start by proving the final inequalities.Suppose lim inf n∈N Then on the one hand, it is clear that lim inf n∈N ε|Yn| |Xm n | is even bigger, which gives the first inequality.On the other hand, since |Y n \ X mn | ≥ |Y n | − |X mn |, we can see that lim inf n∈N (ε − 1) Yn and in this case, (X n ) and (gX n ) are even synchronously Følner-equivalent.A (left) Følner sequence for a countable group G is a g-Følner sequence for every g ∈ G.A countable group is amenable if and only if it admits a Følner sequence: see [CSC10, Chapter 4] also for many equivalent definitions.The following is a rephrasing of Corollary 2. (X n ) is g-Følner if and only if µ (Xn) = µ (g −1 Xn) if and only if the shift by g is an isometry.2. (X n ) is Følner if and only if every shift is an isometry.3.If G is finitely generated (see below) then it is amenable if and only if there exists a nondecreasing exhaustive sequence (X n ) of finite subsets of G such that every shift is an isometry.Note that one implication of Point 3 was already stated in [Cap09, Theorem 3.5], but the proof contains a confusion between left and right Følner.A group G is finitely generated (briefly, f.g.) if E G exists such that for every g ∈ G there exists e 1 , . . ., e n ∈ E ∪ E −1 such that e 1 • • • e n = g.Remarkably (cf.[Pet, Lemma 5.3]) if a f.g. group is amenable, then it has a nondecreasing exhaustive Følner sequence.In addition, if the size of the balls grows polynomially with the radius, then they form a Følner sequence, so Point 3 of Corollary 5 generalizes [HM17, Cor 4.1.4].Corollary 6.Let G be a finitely generated group.1.If (X n ) is the sequence of balls with respect to some generating set of cardinality α, then every shift is α-Lipschitz.2. If g ∈ G, a nondecreasing exhaustive sequence is g-Følner if and only if all of its subsequences yield a Besicovitch pseudodistance for which the shift by g is continuous.3. G is amenable if and only if it admits a nondecreasing exhaustive sequence of finite subsets of which all subsequences yield a Besicovitch distance for which every shift is continuous.|E| + 1) n .We can apply Point 2 of Corollary 3. 2. This comes from Corollary 4. 3.This comes from Point 2.
1.If E is the generating set and E n the corresponding radius-n ball, then E 0 = {e} where e is the identity of G andE n+1 = (E ∪ E −1 ) • E n , so |E n | ≤ (2