J Ér

Émie Brieussel

AMENABILITY AND NON UNIFORM GROWTH OF SOME DIRECTED AUTOMORPHISM GROUPS OF A ROOTED TREE

A result of amenability of some automorphism groups of a spherically homogeneous rooted tree of bounded valency is given. It is used to construct uncountably many amenable groups of non uniform exponential growth. Their Cayley graphs can be made arbitrary close to that of some groups of intermediate growth. Yet those groups are not in the class SG of subexponentially amenable groups.

Introduction

Given a finitely generated group Γ endowed with a generating set S the growth function, b Γ,S (r) is defined as the number of group elements which are products of less than a given number r of generators and their inverses. The growth of Γ is qualified exponential when the exponential growth rate h S (Γ) = lim r b Γ,S (r) strictly exceeds 1 for some, hence for all, generating set S. The growth is said intermediate if h S (Γ) = 1 and the growth function is not polynomial, that is when the group is not virtually nilpotent [START_REF] Gromov | Groups of polynomial growth and expanding maps[END_REF]). The growth is qualified uniform when the infimum of the exponential growth rates over all generating sets strictly exceeds 1, non uniform when exponential but: inf S h S (Γ) = 1.

The question of existence of groups of non uniform exponential growth was asked by Gromov in 1981 in the little green book [START_REF] Gromov | Structures métriques pour les variétés riemanniennes[END_REF]. It has been shown that such groups do not occur in several classes such as hyperbolic groups (see [Kou]), linear groups (see [EMO]), elementary amenable groups (see [START_REF] Osin | Algebraic entropy of elementary amenable groups[END_REF]). A pleasant exposition is given in [dlH]. The first examples of such groups have been provided by Wilson in [START_REF] Wilson | On exponential growth and uniformly exponential growth for groups[END_REF] and [START_REF] Wilson | Further groups that do not have uniformly exponential growth[END_REF]. They contain free subgroups. Another example is due to Bartholdi in [Bar]. The main object of this paper is the following: Theorem 1.1. There exist uncountably many pairwise non isomorphic amenable groups of non uniform exponential growth. These groups will appear as subgroups of the group Aut(T d) of automorphisms of a spherically homogeneous rooted tree, which is described. In Section 3 a subgroup of Aut(T d) is proved to be amenable in case of bounded valency of the tree. This Main Theorem 3.1 implies in particular that the group considered in [Bar] is amenable. Sections 4 and 5 are devoted to the proof of this Main Theorem. In Section 6, using specific generating sets of the alternate group of permutation, some groups of intermediate growth are introduced. These groups are proved to be dense in Date: 12 February, 2008. the profinite group of alternate automorphism of the rooted tree. The groups of Theorem 1.1 are constructed in Section 7, using results of Wilson ([Wil2]). Some part of Wilson Theorem 7.1, namely the convergence to 1 of the exponential growth rates associated to different generating sets, is reinterpreted as a convergence of the Cayley graphs to Cayley graphs of the groups of intermediate growth introduced in the previous section. The last Section 8 deals with the question of subexponential amenability. The groups of non uniform exponential growth constructed are proved not to be in the class SG.

Automorphisms of spherically homogeneous rooted trees

2.1. Spherically homogeneous rooted tree. Given a sequence d = {d j } j≥0 of integers d j ≥ 2, the associated spherically homogeneous rooted tree denoted T d is defined as follows: the vertices are indexed by all finite sequences v = (i 1 i 2 . . . i k ) with i j in {1, 2, . . . , d j-1 }, including the empty sequence ∅ called the root, and the edges link the pairs {(i 1 i 2 . . . i k ), (i 1 i 2 . . . i k i k+1 )}. Note that the sequence d need not be infinite in which case the tree is finite.

The distance (each edge has length 1) from a vertex to the root is called the level of the vertex. The vertices of level l(v) = n form the nth layer (or level) of cardinality d 0 d 1 . . . d n-1 .

Each vertex v of level n gives rise to a spherically homogeneous rooted subtree T v when restricting to vertices of the form (vi n i n+1 . . . i n+k ). The tree T v is isomorphic to the tree T σ n d associated to the sequence σ n d = {d j } j≥n (with σ denoting the usual shift σ : (d 0 d 1 d 2 . . . ) → (d 1 d 2 d 3 . . . )).

2.2. Automorphism group. An automorphism of T d is a graph automorphism, that is a bijection of the set of vertices mapping edges to edges, which fixes the root. These properties imply that the layers are preserved, and an automorphism acts on a layer by permutation. The group of all such automorphisms will be denoted Aut(T d). Spherical homogeneity ensures that Aut (T d) and Aut(T σ d) are related by an isomorphism:

Aut(T d) ≃ Aut(T σ d) ≀ S d 0 . (2.1) Recall that G ≀ S d ≃ (G × • • • × G) ⋊ S d
where S d (the group of permutation of the set {1, 2, . . . , d}) acts on the d copies of G by permutation. This identification will allow to write extensively f = (f 1 , f 2 , . . . , f d 0 )σ with f in Aut(T d), the f i in Aut(T σ d) and σ in S d 0 . The product rule is f g = (f 1 , f 2 , . . . , f d 0 )σ(g 1 , g 2 , . . . , g d 0 )τ = (f 1 g σ(1) , . . . , f d 0 g σ(d 0 ) )στ . In particular, there is a projection p : Aut(T d) → S d 0 called restriction to the first level. The kernel of this projection is called the stabilizer of the first level, denoted St 1 (Aut(T d)), easily checked to be isomorphic to the direct product Aut(T σ d) × • • • × Aut(T σ d) with d 0 factors.

More generally for each integer n, there is an isomorphism:

Aut(T d) ≃ Aut(T σ n d) ≀ Aut(T d 0 ...d n-
(Aut(T d)) ≃ Aut(T σ n d) × • • • × Aut(T σ n d),
the elements of which will occasionally be written g = (g 1...1 , . . . , g d 0 ...d n-1 ) n .

The full group of automorphism can be viewed as a profinite group via:

Aut(T d) = lim ← - n→∞ Aut(T d 0 ...d n-1 ) = lim ← - n→∞ (S d n-1 ≀ S d n-2 ≀ • • • ≀ S d 0 ). (2.3)
A basis of open sets for the profinite topology associated is {St n (Aut(T d))} n≥0 . This topology can also be defined as associated to any of the following metrics δλ on Aut(T d). Given a decreasing sequence λ = {λ n } n≥0 of positive numbers tending to zero, set:

δλ(g, h) = inf{λ n |g(v) = h(v) for all vertices v of level ≤ n}.
A nice description of automorphisms of a rooted tree is to draw portraits. A portrait is a function g from the set of all vertices v of the tree T d taking permutation values g(v) ∈ S d l(v) . A portrait gives rise to a unique automorphism via the formula:

g(i 1 i 2 i 3 . . . i k ) = (g(∅)i 1 )(g(i 1 )i 2 )(g(i 1 i 2 )i 3 ) . . . (g(i 1 . . . i k-1 )i k ).
Conversely, every automorphism has a unique portrait. The metrics δλ are such that two automorphisms are n-close if their portraits coincide on the n first layers.

An automorphism is said to be even (or alternate) if all the permutations g(v) ∈ S d l(v) involved in the portrait are alternate permutations g(v) ∈ A d l(v) . The group of alternate automorphisms will be denoted Aut e (T d). It satisfies:

Aut e (T d) = lim ← - n→∞ Aut e (T d 0 ...d n-1 ) = lim ← - n→∞ A d n-1 ≀ A d n-2 ≀ • • • ≀ A d 0 , (2.4)
the profinite topology, the distances associated and the stabilizers of levels are defined in the same way as for the full automorphism group. Note that if T 2 is a 2-regular rooted tree, then Aut e (T 2 ) is the trivial group. d), those directed by a given infinite geodesic of the tree T d starting from the root. Such a geodesic can always be chosen to be that passing at all vertices indexed by 11 . . . 1 (the leftmost geodesic in the illustrations). First introduce actions of some permutation groups on T d. The group S d 0 acts on the rooted tree by permuting the subtrees of the first layer:

ι 0 : S d 0 ֒→ Aut(T d).
More precisely, ι 0 is defined by ι 0 (σ)(i 1 i 2 . . . i k ) = σ(i 1 )i 2 . . . i k . For simplicity of notation, we will identify σ = ι 0 (σ) = (id T σ d , . . . , id T σ d )σ and call those rooted automorphisms (their portrait is trivial outside of the root).

The infinite direct product

H = S d 1 ו • •×S d 1 ×S d 2 ו • •×S d 2 ×.
. . of permutation groups where S d k appears d k-1 -1 times also acts in a canonical way (once a geodesic is chosen) on the rooted tree T d:

ι : H ֒→ Aut(T d).
Indeed, consider the vertices 1 k i = 1 . . . 1i with k ones and i in {2, . . . , d k }. They form the set P of vertices at distance exactly 1 of the leftmost geodesic 111 . . . . Each permutation group S d k acts on a subtree T 1 k i via the above homomorphism ι 0 (corresponding to the rooted tree T σ k d). More precisely, the action is recursively defined through the wreath product by: ι(σ 2 , . . . , σ d 0 , σ 12 , . . . , σ 1d 1 , . . . ) = (ι ′ (σ 12 , . . . , σ 1d 1 , . . . ), σ 2 , . . . , σ d 0 ), where ι ′ (σ 12 , . . . , σ 1d 1 , . . . ) represents the action of the restriction H ։ H1 via:

ι ′ : H1 = S d 1 -1 d 2 × S d 2 -1 d 3 × . . . ֒→ Aut(T 1 ) ≃ Aut(T σ d)
The geometry of the set P ensures that the action of different factors commute, thus ι is a well defined injection. This is best understood by Figure 2, showing the portrait is non trivial only on P . The automorphisms obtained in ι( H) are said to be directed by the geodesic 111 . . . . Given a subgroup A of S d 0 and a subgroup H of H, denote by G(A, H) the subgroup of Aut(T d) generated by ι 0 (A) and ι(H). Such a group will be called a directed group of automorphisms. Note that the group H might not be countable as H is not. The group G(S d 0 , H) will be called full group of directed automorphisms. Note that the isomorphism (2.1) induces an isomorphism:

G(S d 0 , H) ≃ G(S d 1 , H1 ) ≀ S d 0 .
(2.5)

The class of groups of the form G(A, H) has been considered in [START_REF] Grigorchuk | Just Infinite Branch Groups[END_REF]. It gathers many famous examples such as the family of Aleshin-Grigorchuk groups known to be torsion (see [START_REF] Aleshin | Finite automata and Burnside problem on periodic groups[END_REF]) and of intermediate growth (see [START_REF] Grigorchuk | Degrees of growth of finitely generated groups, and the theory of invariant means[END_REF]). Other interesting examples are some groups of non uniform growth constructed by Wilson ([Wil1], [START_REF] Wilson | Further groups that do not have uniformly exponential growth[END_REF]) and Bartholdi ([Bar]), to which Section 7 is devoted.

The main theorem

In this section the Main theorem on full directed automorphism groups is stated and its proof is reduced to the proof of the a priori weaker Theorem 3.2.

Theorem 3.1 (Main Theorem). Let d = (d i ) i≥0 be a sequence of integers d i ≥ 2, let S d 0 , H and G(S d 0 , H) be the full directed subgroup of Aut(T d), then: 1) if the sequence d is bounded, the group G(S d 0 , H) is amenable.
2) if the sequence d is unbounded, the group G(S d 0 , H) contains a free group F 2 on two generators.

The proof of part 1) of the Main Theorem 3.1 reduces to proving the following, which will be the object of Sections 4 and 5.

Theorem 3.2. Let d = (d i ) i≥0 be a bounded sequence of integers 2 ≤ d i ≤ D, let H < H be a finite saturated subgroup, then the directed subgroup G(S d 0 , H) of Aut(T d) is amenable.
Proof of part 2). The second part of the Main Theorem is an immediate consequence of the following lemma stated in [START_REF] Wilson | Further groups that do not have uniformly exponential growth[END_REF] (see also [TW]).

Lemma 3.3 ([Wil2]

). Let F be the free product of two non-trivial finite groups which are not both of order 2, and S be any infinite subset of N. Then the alternate permutation group A d is a homomorphic image of F for all sufficiently large d and the intersection of the kernels of all epimorphisms from F to groups A d with d ∈ S is the trivial subgroup.

This implies that if d is unbounded then the group H already contains a free group F 2 on two generators. Indeed, let F = Z/2Z ⋆ Z/3Z the free group generated by elements x of order 2 and y of order 3. Let D be such that there is an onto homomorphism ϕ d : F → A d when d ≥ D. Define:

h 1 = (ϕ d 1 (x), . . . , ϕ d 1 (x), ϕ d 2 (x), . . . , ϕ d 2 (x), . . . ) ∈ H h 2 = (ϕ d 1 (y), . . . , ϕ d 1 (y), ϕ d 2 (y), . . . , ϕ d 2 (y), . . . ) ∈ H where ϕ d (x) = ϕ d (y) = 1 ∈ A d ⊂ S d if d < D. Then Lemma 3.3 ensures that the subgroup h 1 , h 2 < H is isomorphic to F which contains F 2 as a subgroup of finite index.
If the sequence d is bounded then the properties of the group H are much different.

Fact 3.4. Let H = T 1 × T 2 × . . . where the groups T i belong to a finite family F = {F 1 , . . . F D } of finite groups, then every finitely generated subgroup H ′ of H is finite.

Proof. Let h 1 , . . . , h k be generators of H ′ , they are of the form h j = (h 1 j , h 2 j , . . . ) with h i j ∈ T i . There are at most

M = D.(max{#F i }) k different k + 1-tuples (h i 1 , h i 2 , . . . , h i k , F i ).
Let I be a subset of N of size less than M such that all different (k + 1)-tuples appear when i describes I. Then the projection π I : H → × i∈I T i is injective, so that H ′ is finite. Definition 3.5. A finite subgroup H of the group H = T 1 × T 2 × . . . where the T i belong to a finite family F of finite group is said to be saturated if the equidistributed probability measure q H on H projects on each coordinate i to the equidistributed probability measure

q T i on T i , that is if h = (h 1 , h 2 , . . . ) ∈ H then q H (h i = t) = q T i (t) = 1 #T i . Fact 3.6. Every finite subgroup H ′ of H is included in a finite saturated group H.
Proof. With the above notations set for each i in I:

J i = {j ∈ N|(h j 1 , h j 2 , . . . , h j k , F j ) = (h i 1 , h i 2 , . . . , h i k , F i )}.
There is a diagonal embedding T i → × j∈J i T j and as ∪ i∈I J i = N we get a diagonal injection:

× i∈I T i ֒→ H
the image H of which contains H ′ and is saturated by construction, knowing a finite direct product is always saturated.

Proof that Theorem 3.2 implies the Main Theorem. To prove the group G(S d 0 , H) is amenable, it is sufficient to prove amenability for every finitely generated subgroup G f (Theorem 1.2.7. in [Gre]), which reduces, assuming Theorem 3.2, to show that G f is included in some G(S d 0 , H) for H finite saturated. Indeed, let s 1 , . . . , s k be generators of G f , each s j is of the form s j = a 1 j h 2 j a 3 j . . . h n j j , with a i j ∈ S d 0 and (h i j ) i,j < H finitely generate a subgroup H ′ which is included in some finite saturated subgroup H by Facts 3.4 and 3.6.

Scheme of the proof of Theorem 3.2

This section is devoted to the scheme of the proof of Theorem 3.2 which implies the Main Theorem 3.1. The details are given in Section 5. Groups of the form G(S d 0 , H) share similarities with the Basilica group defined by a three state automaton introduced by Grigorchuk and Zuk in [GZ]. The Basilica group was shown to be amenable by Bartholdi and Virag (see [BV]) using selfsimilarity of some random walks. This method, called the "Münchhausen trick", has been used to show amenability of a few other groups (see [Kai] and [Muc]). We proceed with the same methods, using Kesten's criterion on symmetric random walks.

As H is a finite saturated subgroup of H = S d 0 -1

d 1 × S d 1 -1 d 2 × . . . , let us denote H k its restriction to Hk = S d k -1 d k+1 ×S d k+1 -1 d k+2 ×.
. . which is also a finite saturated subgroup and it follows from (2.5) that G(S d 0 , H) ֒→ G(S d 1 , H 1 ) ≀ S d 0 , and more generally the group G(S d k , H k ) is a directed subgroup of Aut(T σ k d) satisfying the crucial:

G(S d k , H k ) ֒→ G(S d k+1 , H k+1 ) ≀ S d k .
The word metric does not behave appropriately enough through this wreath product embedding, rather use: Proposition 4.1 (A fractal family of pseudo norms of exponential growth). There exists a family of pseudo norms ν k on G(S d k , H k ) (which means symmetric positive functions ν k : G(S d k , H k ) → R + satisfying the triangle inequality) such that:

a) if g belongs to G(S d k , H k ) and has image g = (g 1 , . . . , g d k )σ in G(S d k+1 , H k+1 ) ≀ S d k , then ν k (g) ≤ ν k+1 (g 1 ) + • • • + ν k+1 (g d k ), and b) if B ν k (r) = {g ∈ G(S d k , H k )|ν k (g) ≤ r}, then #B ν k (r) ≤ C r
where C is a constant depending only on the bound D on the valencies of the tree and the size of the finite group H (which contains H k for every k).

Let p denote the symmetric probability measure on the finite generating set S d 0 ∪ H of G(S d 0 , H) defined by p(a) = 1 2#S d 0 for a ∈ S d 0 and p(h) = 1 2#H for h ∈ H. The random walk associated is Z N = s 1 . . . s N where the s i are independent random variables identically p-distributed. The set of random sequences (Z N ) N ∈N is endowed with the product measure (defined on the sigma algebra generated by cylinders) P = p ⊗∞ . The drift of this random walk with respect to the pseudo norm ν = ν 0 vanishes: Proposition 4.2. The random walk (Z N ) satisfies:

ν(Z N ) N -→ N →+∞
0, P a.s..

To prove this proposition, another (non symmetric) random walk is usefull. Let us define Y n = t 0 t 1 t 2 . . . t n where t 2i are random variables equidistributed on S d 0 and t 2i+1 are equidistributed on H and all the t i are independent. Denote Q = (q S d 0 ⊗ q H ) ⊗∞ the associated measure on the set of sequences (Y n ) n∈N (with respect to the cylindrical sigma algebra), then:

Proposition 4.3. The random walk (Y n ) satisfies: ν(Y n ) n -→ n→+∞ 0, Q a.s..
The key argument to prove Proposition 4.3 is the next Lemma 4.6 together with Proposition 4.1 a).

Remark 4.4 (On the dependence on t 0 ). The pseudo norm ν = ν 0 satisfies ν(ah) = ν(h) for every a in S d 0 and h in H

(Proposition 5.2 (2)), which ensures ν(Y n ) = ν(t -1 0 Y n ) = ν(t 1 t 2 . . . t n ), showing that ν(Y n ) is independent of t 0 .
This will be of importance and justifies the: Definition 4.5. Two random variables U and V on G(S d 0 , H) are said ν-equivalent if ν(U) and ν(V ) have the same distribution law on N, which will be denoted:

U ∼ ν-law V.
Consider the random walk (Y n ) n∈N and its image in the wreath product of the form

Y n = (Y 1 n , . . . , Y d 0 n )σ
Y n = (Y 1 n , . . . , Y d 0 n )σ n its image in the wreath product. For each coordinate (Y t n ) n the sequence (Y n ) n defines a sequence of random integers (m t (n)) n and a random sequence (ε t (n)) n taking values in {0, 1} such that: (1) For every integer n the values of m t (n) and ε t (n) depend only on (Y n ′ ) n ′ ≤n . (2) For every integer n the coordinate Y t n belonging to G(S d 1 , H 1 ) has the same ν 1 -distribution law as the random variable Y ′ mt(n)+εt(n) .
More precisely the conditional law:

(Y t n |m t (n), ε t (n)) ∼ ν 1 -law Y ′ mt(n)+εt(n) . (3) The random sequence (m t (n)) n satisfies: m t (n) ∼ n→+∞ d 0 -1 d 0 n d 0 , Q a.s..
Propositions 4.1 and 4.2 are sufficient to apply the:

Theorem 4.7 (Kesten criterion of amenability [Kes]). Let Γ be a finitely generated group and (Z N ) a symmetric random walk on Γ. The group Γ is amenable if and only if the sequence (P (Z 2N = id Γ )) N does not decay exponentially fast with N.

The following fact is also usefull:

Fact 4.8. Let (Z N ) a symmetric random walk on a finitely generated group Γ, then for any fixed integer N the function Γ → [0, 1] :

g → P (Z 2N = g) is maximal for g = id Γ .
Proof of the Fact 4.8. Let p k (x, y) denote the probability to go from x to y in k steps, let δ x denote the function on Γ taking values 1 on x and 0 elsewhere and M the symmetric random walk operator on the space l 2 (Γ). Then Cauchy inequality implies:

p 2N (id, x) 2 = M 2N δ id , δ x 2 = M N δ id , M N δ x 2 ≤ ||M N δ id ||.||M N δ x || = p 2N (id, id).p 2N (x, x) = p 2N (id, id) 2 .
Note that Theorem 4.7 and Fact 4.8 only apply to symmetric random walks.

Proof of Theorem 3.2. Given an arbitrary positive ε the previous Fact 4.8 applied to the symmetric random walk (Z N ) constructed above raises:

P (ν(Z 2N ) ≤ ε2N) = ν(g)≤ε2N P (Z 2N = g) ≤ P (Z 2N = id G(S d 0 ,H) )#B ν (ε2N),
and the Propositions 4.1 b) and 4.2 ensure:

P (Z 2N = id) ≥ P ν(Z 2N ) 2N ≤ ε C -ε2N ∼ N →∞ C -ε2N .
Thus P (Z 2N = id) does not decrease exponentially fast and Kesten's criterion proves Theorem 3.2 and thus the Main Theorem.

5. Details of the proof of Theorem 3.2 5.1. Fractal pseudo norms of exponential growth (proof of Proposition 4.1). To the symmetric generating set S = (S

d 0 ∪ H)\{1} of G(S d 0 , H) is associated the word norm on G(S d 0 , H) by: |g| = min{r|g = z 1 . . . z r , z i ∈ S}.
Denote B S (r) the ball of radius r associated to this norm (that is the set of all g such that |g| ≤ r), then #B S (r) ≤ (#S) r .

Note that since G(S d 0 , H) is a quotient of the free product S d 0 * H a word z 1 . . . z r is a minimal representative of g (that is r = |g|) only in the following cases: either z 2j ∈ S d 0 \{1} and z 2j+1 ∈ H\{1}, or conversely. This brings another definition:

||g|| 0 = min{r|g = a 1 h 1 a 2 h 2 . . . h r a r+1 , a i ∈ S d 0 , h j ∈ H}.
(5.1)

The following is straightforward:

Properties 5.1. The function ||.|| 0 is a norm when restricted to the stabilizer of the first level St 1 (G(S d 0 , H)), namely it satisfies:

(1)

||gh|| 0 ≤ ||g|| 0 + ||h|| 0 for all g, h in G(S d 0 , H), (2) ||g -1 || 0 = ||g|| 0 for all g in G(S d 0 , H), ( 3
) ||g|| 0 = 0 if and only if g ∈ S d 0 ,
This function ||.|| 0 is related to the usual word norm since for g in G(S d 0 , H):

2||g|| 0 -1 ≤ |g| ≤ 2||g|| 0 + 1, which implies that if B ||.|| 0 (r) is the ball of radius r associated to ||.|| 0 in G(S d 0 , H), then: #B ||.|| 0 (r) ≤ (#S) 2r+1 .
Following [BV], let us introduce a new function ν on G(S d 0 , H) which is to be thought of as a fractal distance. For g ∈ G(S d 0 , H) and a vertex v on layer k = l(v) of T d, denote by g v the action of g on the descendant subtree A subtree T of T d is said to be rooted if it contains the root ∅ of T d. It is said regular if for every vertex v ∈ T , either T contains the d l(n) descendant of v, either it contains none of them.

T v ≃ T σ k d of T
Given a finite regular rooted subtree T of T d with set of leaves ∂T , define a function ν T on G(S d 0 , H) by:

ν T (g) = v∈∂T (1 + ||g v || l(v) ).
and a function ν : G(S d 0 , H) → N as:

ν(g) = min{ν T (g)|T is a finite regular rooted subtree of T d}.
(5.

2)

The construction (5.2) defines similarly a function

ν k : G(S d k , H k ) → N for the subgroup G(S d k , H k ) < Aut(T σ k d) ≃ Aut(T v )
for any vertex v on the kth layer. Note that ν = ν 0 and that the following proposition is still true replacing ν by ν k and ν 1 by ν k+1 .

Proposition 5.2. The function ν satisfies:

(1) Let g in G(S d 0 , H) and g = (g 1 , . . . , g d 0 )σ be its embedded image in the wreath product G(S d 0 , H) ֒→ G(S d 1 , H 1 ) ≀ S d 0 , then:

ν(g) = min{ν 1 (g 1 ) + • • • + ν 1 (g d 0 ), 1 + ||g|| 0 }.
(2) Let g in G(S d 0 , H), then ν(g) = ν(g -1 ).

(3) Let g, g ′ be in G(S d 0 , H), then ν(gg ′ ) ≤ ||g|| 0 + ν(g ′ ).

(4) Let g, g ′ be in G(S d 0 , H), then ν(gg ′ ) ≤ ν(g) + ν(g ′ ).

In particular, this function ν is a pseudo-norm on G(S d 0 , H).

The use of induction in the proof of Proposition 5.2 requires the:

Property 5.3. Let g in G(S d k , H k ) have image g = (g 1 , . . . , g d k )σ in the wreath product G(S d k+1 , H k+1 ) ≀ S d k and assume ||g|| k ≥ 2 then ||g t || k+1 < ||g|| k for any coordinate t.
Proof of Property 5.3. An element g admits a minimal representative of the form g = h σ 1 1 . . . h σr r σ r+1 with σ i in S d k and h i in H k (remind x y = yxy -1 ). Moreover by construction h = (h ′ , a 2 , . . . , a d k ) with h ′ in H k+1 and a i in S d k+1 and the conjugate h σ is the same d k -tuple where the coordinates are σ permuted. This ensures

||g 1 || k+1 + • • • + ||g d 0 || k+1 ≤ ||g|| k . It is sufficient to prove the property for ||g|| k = 2, that is g = h σ 1 1 h σ 2 2 . If σ 1 (1) = σ 2 (1) the property is obvious. If σ 1 (1) = σ 2 (1) then ||g i || k+1 = 0 if i = σ 1 (1) and ||g σ 1 (1) || k+1 = ||h ′ 1 h ′ 2 || k+1 = 1 because h ′ 1 h ′ 2 is an element of H k+1 .
Proof of Proposition 5.2. Note that 1 + ||g|| 0 = ν {∅} (g) and assume the minimum in definition (5.2) is obtained for a finite regular rooted tree T = {∅}. Clearly

∂T = ∂T (1) ∪ • • •∪ ∂T (d 0 )
where T (v) denotes the intersection of T with the subtree T v of T d hung on vertex v, thus:

ν T (g) = v∈∂T (1 + ||g v || l(v) ) = d 0 t=1 v∈∂T (t) (1 + ||g v || l(v) ) = d 0 t=1 ν T (t) (g t ),
which is minimal if and only if ν T (t) (g t ) = ν 1 (g t ) is minimal for all t. This implies part (1).

It follows that if ||g||

0 = 1 then ν(g) = 2 = ν(g -1 ). Similarly if ||g|| k = 1 for g ∈ G(S d k , H k ) then ν k (g) = 2 = ν k (g -1
). Assume by induction on r that ν k (g) = ν k (g -1 ) if ||g|| k ≤ r and this jointly for every level k, then the inverse formula g -1 = σ -1 (g -1 1 , . . . , g -1 d 0 ) and the induction hypothesis ensuring ν 1 ((g -1 ) 1 ) +

• • •+ν 1 ((g -1 ) d 0 ) = ν 1 (g 1 )+• • •+ν 1 (g d 0 ) (as ||g t || k+1 < ||g|| k by Property 5.3) together with part (1) show part (2).
To prove part (3), note first that ν(ag) = ν(g) for all a ∈ S d 0 . Indeed, a only permutes the subtrees of the first level and does not increase any of the ||g v || l(v) . To conclude, it is sufficient to show that when h is in H, we have ν T (hg) ≤ 1 + ν T (g) for any finite regular subtree T . Proceed by induction on the size of T . Indeed, this is true for T = {∅} by Property 5.1 (1). More generally, denoting g = (g 1 , . . . ,

g d 0 )σ 0 and h = (h 1 , a 2 , . . . , a d 0 ) with g t in G(S d 1 , H 1 ), h 1 in H 1 and a t in S d 1 , we get hg = (h 1 g 1 , a 2 g 2 , . . . , a d 0 g d 0 )σ 0 and: ν T (hg) = ν T (1) (h 1 g 1 ) + d 0 t=2 ν T (t) (a t g t ) ≤ 1 + ν T (1) (g 1 ) + d 0 t=2 ν T (t) (g t ) = 1 + ν T (g)
using the induction hypothesis on T (1).

Part (4) is implied by part (3) in case ν(g) = 1 + ||g|| 0 or ν(g ′ ) = 1 + ||g ′ || 0 . Otherwise: ν(gg ′ ) ≤ d 0 t=1 ν 1 ((gg ′ ) t ) = d 0 t=1 ν 1 (g t g ′ σ(t) ) ≤ d 0 t=1 ν 1 (g t ) + ν 1 (g σ(t) ) = ν(g) + ν(g ′ ),
where the second inequality comes by joint induction on ||g|| k using Property 5.3. In order to prove this proposition, recall classical estimates on the number of rooted subtrees of a rooted tree. The formula below can be found in [PR], the equivalent is derived from Stirling's formula.

Let B ν (r) = {g ∈ G(S d 0 , H)|ν(g) ≤ r}
Proposition 5.5. The number of (not necessarily regular) rooted subtrees of a Dregular tree T D containing r vertices is:

s (D) r = 1 r C r-1 Dr ∼ r→+∞ 1 D -1 D 2(D -1)π r -3 2 D D (D -1) (D-1) r .
More precisely the following is sufficient:

Corollary 5.6. The number t (D) r of regular rooted subtrees of T d (with d bounded by D) containing at most r leaves satisfies:

t (D) r ≤ (K D ) r , for K D = D 2D (D -1) 2(D-1) ,
provided r is sufficiently large.

Proof. It is well known that a subtree with at most r leaves contains at most 2r -1 vertices and the asymptotic equivalent of s (D) r gives the corollary.

Proof of Proposition 5.4. If ν(g) ≤ r then there exists a regular rooted subtree T such that ν T (g) ≤ r. In particular, such a subtree has less than r leaves so that there are at most (K D ) r choices for T (corollary 5.6). Given T , the element g is described by all g(v) ∈ S d l(v) where v ∈ T , which allow at most (D!) # T ≤ (D!) r choices, and all g v ∈ G(S d l(v) , H l(v) ) with v ∈ ∂T , which satisfy:

v∈∂T ||g v || l(v) ≤ r.
The number of possibilities for this last choice is less than (M + 1) 2r where M = max{#B ||.|| k (1)} (finite because the size of the generating set S d k ∪ H k on layer k depends only on d k ≤ D and #H k ≤ #H) bounds the number of symbols which represent an automorphism of norm 1 on a given leaf. An extra symbol (a coma) is added to denote passing to the next leaf. All in all, taking C = K D D!(M + 1) 2 gives the desired result. 5.2. Similarity of random walks (proof of Lemma 4.6). First recall elementary probabilistic facts which will be usefull.

Fact 5.7. Let (z i ) i≥1 be independent random variables equidistributed on a finite group F . Then the sequence (X k ) k≥1 of products X k = z 1 . . . z k is a family of independent random variables equidistributed on F .

Proof of Fact 5.7. Denote by q F the equidistribution measure on the finite group F . It is sufficient to prove by induction that:

q ⊗∞ F (X i = f i , i ≤ k) = k i=1 q ⊗∞ F (X i = f i ) = k i=1 q F (X i = f i ),
for arbitrary f 1 , . . . , f k in F , which comes from:

q ⊗∞ F (X i = f i , i ≤ k) = q ⊗∞ F (X k = f k |X i = f i , i ≤ k -1)q ⊗∞ F (X i = f i , i ≤ k -1) = q ⊗∞ F (z k = f -1 k-1 f k ) k-1 j=1 q ⊗∞ F (X j = f j ) = q F (z k = f -1 k-1 f k ) k-1 j=1 q F (X j = f j ).
Fact 5.8. Let z be a random variable equidistributed on a finite group F acting transitively on a finite set A, then q F (z(t) = t ′ ) = 1 #A for all t, t ′ in A.

Proof of Fact 5.8. The quotient

F/Stab F (t) is of size #A. If z 0 (t) = t ′ (transitivity) then z 0 Stab F (t) = {z|z(t) = t ′
} has the same size as Stab F (t) by injectivity of left translation in F .

Fact 5.9. Let (u i ) i∈N be independent random Bernoulli variables on {0, 1} (say p(u i = 0) = p and p(u i = 1) = 1p for some p in ]0, 1[). Let f (w N ) be the number of alternations in the subsequence w N = u 1 . . . u N , that is the number of indexes i such that u i = u i+1 . Equivalently, 1 + f (w N ) is the number of maximal packs of constant successive terms. Then:

f (w N ) ∼ N →+∞ 2p(1 -p)N, P = p ⊗∞ a.s..
Proof of Fact 5.9. Apply the law of large numbers to f (w

N ) = N -1 i=1 1 {u i =u i+1 } knowing that E(1 {u i =u i+1 } ) = 2p(1 -p)
and that the terms are independent.

Proof of Lemma 4.6. Consider the random walk Y n at step n as:

Y n = t 0 t 1 . . . t n = a 1 h 1 a 2 h 2 . . . a s h s a s+1 with s = [ n 2 ] (a s+1 empty if n even),
where the terms a i (resp. h i ) are random variables equidistributed in S d 0 (resp. in H), all being independent. This can be rewritten Y n = h σ 1 1 . . . h σs s σ s+1 (remind the conjugate notation h σ = σhσ -1 ) where the σ i = a 1 a 2 . . . a i are independent random variables equidistributed in S d 0 by Fact 5.7.

Using coordinates in the wreath product an element h of H has the form h = (h 1 , a 2 , . . . , a d 0 ) with h 1 in H 1 and a i in S d 1 and each of them is equidistributed for h equidistributed in H by saturation (note that the coordinates are not independent). Conjugating by a rooted automorphism σ raises h σ = (a σ(1) , . . . , a σ(d 0 ) ) with h 1 in position σ(1).

Consider now the random walk

Y n = (Y 1 n , . . . , Y d 0 n
)σ n at time n and focus on coordinate t, which is a product Y t n = u 1 . . . u s of s independent terms such that u i belongs to and is equidistributed in S d 1 (resp. H 1 ) if σ i (t) belongs to {2, . . . , d 0 } (resp. σ i (t) = 1). Since the σ i are equidistributed in S d 0 the probability that u i is in S d 1 (resp. H 1 ) for a given i is d 0 -1 d 0 (resp. 1 d 0 ) by Fact 5.8. This is summarized in:

Q(u i = g) = d 0 -1 d 0 1 #S d 1 if g ∈ S d 1 , 1 d 0 1 #H 1 if g ∈ H 1 ,
and the terms u i are independent. Define m t (n) to be the number of maximal packs of successive u i belonging either to S d 1 , or to H 1 in the sequence Y t n = u 1 . . . u s . Fact 5.9 ensures that:

m t (n) ∼ n→+∞ 2 1 d 0 1 - 1 d 0 s ∼ n→+∞ d 0 -1 d 0 n d 0 .
Given an integer n, assume we know the distribution D of which terms u i are in S d 1 and H 1 , then the kth pack of terms v k = u i k u i k +1 . . . u j k of constant belonging is a product of equidistributed independent elements in the finite group

S d 1 or H 1 hence is equidistributed. In this situation Y t n = v 0 v 1 . . . v mt(n)
where two cases are possible:

either u 1 belongs to S d 1 (set ε t (n) = 0), the terms v 2k+1 are equidistributed in H 1 and v 2k are equidistributed in S d 1 , which is of the form Y ′ mt(n) ; or u 1 belongs to H 1 (set ε t (n) = 1), then re index the v i as Y t n = id S d 1 v 1 . . . v mt(n)+1 which is of the form Y ′ mt(n)+1
except for v 0 which follows the Dirac law on id S d 1 ; this has no influence on the ν-distribution of the sequences (Remark 4.4). In both cases:

(Y t n |D) ∼ ν-law Y ′ mt(n)+εt(n)
, where the condition depends only on the number of alternations m t (n) and the starting condition ε t (n) of the distribution D.

5.3.

Zero drift of (Y n ) (proof of Proposition 4.3). First note that the Kolmogorov 01-law implies almost sure constance of lim sup ν(Yn) n .

Lemma 5.10. For every integer k denote (Y

(k) n ) n the random walk on G(S d k , H k ) which is taking independent equidistributed increments alternatively in S d k and H k , in particular (Y n ) = (Y (0) n ) and (Y ′ n ) = (Y (1)
n ). Then there exists l k in [0, 1 2 ] such that:

lim sup n→+∞ ν k (Y (k) n ) n = l k , Q k = (q S d k ⊗ q H k ) ⊗∞ a.s.. Proof. Proposition 5.2 (1) implies ν k (Y (k) n ) ≤ 1 + ||Y (k) n || k ≤ n+1 2 so that the lim sup is ≤ 1 2 . Given l in [0, 1 2 ] the event E l = {lim sup ν k (Y (k) n ) n
≤ l} is a tail event, that is an event which is independent of any finite subsequence (Y (k) n ) n≤N , hence has probability 0 or 1 by the 01-Kolmogorov law. The function l → Q k (E l ) is increasing, right continuous and takes values in {0, 1}, so that there exists l k such that Q k (E l ) = 0 for l < l k and Q k (E l ) = 1 for l ≥ l k . Then:

Q k {lim sup ν k (Y (k) n ) n = l k } = Q k (E l k \ ∪ n≥1 E l k -1 n ) = 1.
Proof of Proposition 4.3. To show l 0 = 0, prove l k ≤ (D-1) D l k+1 where D is the bound on the valencies of the spherically homogeneous rooted tree T d. This is sufficient as l k ≤ 1 2 for every k. To ease notations, compute for k = 0. Proposition 4.1 (a) ensures:

lim sup n→+∞ ν(Y n ) n ≤ lim sup n→+∞ d 0 t=1 ν 1 (Y t n ) n ≤ d 0 t=1 lim sup n→+∞ ν 1 (Y t n ) n .
(5.3)

To compute the right side introduce the condition (m t (n)):

lim sup n→+∞ ν 1 (Y t n ) n = lim sup n→+∞ ν 1 (Y t n ) m t (n) m t (n) n ≤ lim sup n→+∞ ν 1 (Y t n ) m t (n) lim sup n→+∞ m t (n) n ,
where Lemma 4.6 gives lim sup mt(n

) n = ( d 0 -1 d 0 ) 1 d 0 , Q a.s. and lim sup ν 1 (Y t n ) m t (n) = lim sup ν 1 (Y ′ mt(n)+ε t n ) m t (n) = l 1 , Q a.s. because m t (n) → +∞ Q a.s.
. The last estimates gathered together on a Q probability one event show that:

l 0 ≤ d 0 t=1 l 1 d 0 -1 d 0 1 d 0 = d 0 -1 d 0 l 1 ≤ D -1 D l 1 .
5.4. Zero drift of (Z N ) (proof of Proposition 4.2). Recall the:

Fact 5.11. Let (a i ) i∈N be a random sequence in {0, 1} N endowed with a probability measure µ. Assume that there exists an infinite subset I of N such that µ(a i = 1) ≥ δ > 0 for all i ∈ I, then µ(a i = 1 for infinitely many i) ≥ δ.

Proof of Fact 5.11. Let E = {(a i )|a i = 1 infinitely often} and assume by contradiction µ(E) = δ ′ < δ, this implies µ(E c ∩ {a i = 1}) ≥ δδ ′ for all i in I. However the complement of E is the infinite increasing union:

E c = ∪ n∈N {(a i )|a i = 0 for i ≥ n} = ∪ n∈N F n , so that µ(F N ) ≥ 1 -δ+δ ′
2 for some N. But the case i ≥ N raises the contradiction:

µ(E c ∩ {a i = 1}) = µ((E c \ F N ) ∩ {a i = 1}) ≤ µ(E c \ F N ) ≤ δ + δ ′ 2 -δ ′ = δ -δ ′ 2 .
Proposition 4.3 will be used in the (a priori) weaker form:

Corollary 5.12. For every positive ε and α, there exists N 0 such that for n ≥ N 0 :

Q ν(Y n ) n ≤ ε ≥ 1 -α.
Proof. Assume the statement does not hold, then there exists ε 0 , α 0 and infinitely many integers

n k with Q ν(Yn k ) n k ≥ ε 0 ≥ α 0 and then Q lim sup ν(Yn)
n ≥ ε 0 ≥ α 0 by Fact 5.11, contradicting Proposition 4.3.

The random walks (Z N ) and (Y n ) are closely related by: Fact 5.13. Let N be a fixed integer. To each walk Z N = s 1 . . . s N is associated the number of alternations a(N) from s i in S d 0 to s i+1 in H or vice versa. Then the conditional law of Z N satisfies:

(Z N |a(N)) ∼ ν-law Y a(N ) .
Proof. Conditioning by the distribution D of which terms s i are in S d 0 and in H, the walk is rewritten: ) where t 2j = s i 2j-1 . . . s i 2j are equidistributed in S d 0 (except maybe t 0 which could be empty) and t 2j+1 = s i 2j . . . s i 2j+1 are equidistributed in H, all factors being independent, which is the definition of the random walk Y a(N ) . The condition matters only on a(N) and not D. This Fact 5.13 allows us to show a weak form: Lemma 5.14. For every positive ε and α, there exists M such that for N ≥ M:

Z N = s 1 . . . s i 0 s i 0 +1 . . . s i 1 . . . s i a(N) = t 0 t 1 . . . t a(N
P ν(Z N ) N ≤ ε ≥ 1 -α.
Proof. Fact 5.9 ensures that the conditioning term a(N) satisfies lim N →∞ a(N ) N = 1 2 , P almost surely. In particular for every positive α there exists an integer N 1 such that P (a(N) ≥ N

3 ) ≥ 1α for all N ≥ N 1 . Now compute under the condition a(N):

P ν(Z N ) N ≤ ε = a(N ) P ν(Z N ) N ≤ ε|a(N) P (a(N)), but if N ≥ N 1 then P (a(N) ≤ N 3 ) ≤ α.
Moreover for N ≥ 3N 0 (defined by Corollary 5.12) the condition a(N) ≥ N 3 ≥ N 0 ensures via Fact 5.13:

P ν(Z N ) N ≤ ε|a(N) = Q ν(Y a(N ) ) a(N) a(N) N ≤ ε ≥ 1 -α, because a(N ) N ≤ 1. All in all, when N ≥ max{N 1 , 3N 0 }: P ν(Z N ) N ≤ ε ≥ a(N )≥ N 3 (1 -α)P (a(N)) ≥ (1 -α) 2 ,
which proves Lemma 5.14.

The previous Lemma ensures that P almost surely: lim inf ν(Z N ) N = 0 (Fact 5.11). To get Proposition 4.2 use: Theorem 5.15 (Kingman subadditive Theorem ( [Kal] 9.14)). Let (X m,n ) be random variables such that:

(1) X 0,n ≤ X 0,m + X m,n for all 0 < m < n, (2) (X m+1,n+1 ) has the same law as (X m,n ), (3) E(X + 0,1 ) < +∞, then the random sequence ( X 0,n n ) converges almost surely.

Applying this to X n,m = ν(Z -1 m Z n ) shows that the inferior limit is in fact a limit, proving Proposition 4.2. The interested reader will remark that Lemma 5.14 is sufficient for our purpose and thus the Main Theorem does not rely on Kingman's Theorem.

6. Groups of intermediate growth 6.1. Generating pairs for alternate groups. In his paper [START_REF] Wilson | Further groups that do not have uniformly exponential growth[END_REF] (Proposition 2.1), Wilson constructs interesting generating pairs of alternate groups A d : Proposition 6.1 (Wilson [Wil2]). Let d ≥ 29, then the alternate group of permutation A d of the finite set {1, . . . , d} contains an eligible (see [START_REF] Wilson | Further groups that do not have uniformly exponential growth[END_REF] for the full definition) pair of elements x d , y d . In particular:

1) the pair is generating: x d , y d = A d , the elements have order 2 and 3:

x 2 d = y 3 d = 1
, and a fixed point property that there exists α and β in {1, . . . , d} such that:

x d (α) = y d x d y -1 d (α) = α and y d (β) = β (up to re index we assume α = 1 and β = 2).
2) let x = (u, 1, . . . , 1)x d 0 and ŷ = (1, v, 1, . . . , 1)y d 0 belong to Aut(T d) with d 0 ≥ 29 and u, v in Aut(T σ d) with u 2 = v 3 = 1, then the group generated by x and ŷ contains the whole group of alternate rooted automorphisms A d 0 . More precisely:

x, ŷ ≃ u, v ≀ A d 0 .
Given a (not necessarily bounded) sequence d of integers ≥ 29, the above Proposition 6.1 allows to define recursively the following pair of automorphisms of the spherically homogeneous rooted tree T d (remind the assumption on fixed points α = 1 and β = 2):

x d = (x σ d, 1, . . . , 1)x d 0 , y d = (1, y σ d, 1, . . . , 1)y d 0 . (6.1)
This definition is best understood by looking at the portraits on Figure 3. The automorphism subgroup generated is denoted H d = x d, y d . Note that in the case d = σ d is a constant sequence the group H d is generated by a two (non trivial) state automaton.

Property 6.2. The alternate automorphism x d has order 2, and y d has order 3.

Proof. Show by joint (on x σ i d for i in N) induction on k that x 2 σ i d acts trivially on the k first levels of T σ i d. This implies it acts trivially on the whole tree hence is trivial automorphism. Proposition 6.1 1) ensures:

x 2 σ i d = (x 2 σ i d, 1, . . . , 1)x 2 d i = (x 2 σ i d, 1, . . . , 1
), which initiates the induction. Moreover x 2 σ i d acts trivially on the subtrees T 2 , . . . , T d i of T σ i d and as x 2 σ i+1 d on T 1 which acts trivially on the k first level of T 1 by induction hypothesis. This proves x 2 σ i d acts trivially on the k + 1 first levels of T σ i d.

x d 0

x d 1 x d 2 . . . 1 . . . 1 1 . . . 1 1 . . . 1 y d 0 1 y d 1 1 y d 2 1 . . . 1 . . . 1 1 . . . 1 1 . . . 1
Figure 3. Portraits of the elements x d and y d.

Density properties.

Proposition 6.3. The subgroup d) endowed with the profinite topology from (2.4).

H d = x d, y d < Aut e (T d) is dense in Aut e (T
Proof. It is sufficient to show that the subgroup d) of alternate automorphisms of portrait supported on the k firsts levels is included in H d for arbitrary k. Proceed by joint (on H σ i d for i ∈ N) induction on k to show:

A d k ≀• • •≀A d 0 < Aut e (T
H σ i d ≃ H σ i+k d ≀ A d i-1 ≀ • • • ≀ A d i , (6.2)
which will be sufficient taking i = 0 and the trivial subgroup of H σ k d. The case k = 0 follows from Proposition 6.1 2):

H σ i d = x σ i d, y σ i d ≃ x σ i+1 d, y σ i+1 d ≀ A d i = H σ i+1 d ≀ A d i . (6.3)
Assuming isomorphism (6.2) then isomorphism (6.3) for i + k proves step k + 1:

H σ i d ≃ H σ i+k d ≀ A d i+k-1 ≀ • • • ≀ A d i ≃ H σ i+k+1 d ≀ A d i+k ≀ A d i+k-1 ≀ • • • ≀ A d i .
This density property is in contrast with the case of the full (non alternate) automorphism group of a rooted tree: Proposition 6.4. The group Aut(T d) endowed with the profinite topology from (2.3) admits no finitely generated dense subgroup.

Proof. Denote sgn : S d → Z/2Z the signature morphism of permutations. Given an element g in Aut(T d), recall that g(v) is the permutation in S d l(v) associated to vertex v in the portrait of g.

(Recall g = (g 1...1 , . . . , g v , . . . , g d 0 ...d l(v)-1 )τ l(v)-1 with g v in Aut(T v ) ≃ Aut(T σ l(v) d) and τ l(v)-1 ∈ Aut(T d 0 ...d l(v)-1 ), then g v has image g v = (g v1 , . . . , g vd l(v) )g(v) via the isomorphism Aut(T σ l(v) d) ≃ Aut(T σ l(v)+1 d) ≀ S d l(v) .)
Similarly to Lemma 1. in [START_REF] Aleshin | A free group of finite automata[END_REF], define for each integer k the following morphism (of products of signatures of permutations on level k in the portraits):

R k : Aut(T d) → Z/2Z g → R k (g) = v∈Level(k) sgn(g(v)).
The computations via the isomorphism (2.2) show this is a group morphism. The product morphism ϕ : Aut(T d) → (Z/2Z) ∞ defined as ϕ(g) = (R 0 (g), R 1 (g), . . . ) is then a surjective group morphism continuous for the profinite topologies. Assume now there exists a finitely generated dense subgroup G of Aut(T d), then ϕ(G) is a finitely generated dense subgroup of (Z/2Z) ∞ . This is impossible since any finitely generated subgroup of (Z/2Z) ∞ is finite thanks to Fact 3.4.

Density in Aut e (T d) of a finitely generated subgroup implies superpolynomial growth: Proposition 6.5. Let d = (d i ) i∈N a sequence of integers d i ≥ 3, then any dense finitely generated subgroup of Aut e (T d) has superpolynomial growth.

Proof. Let G be such a group and k an arbitrary integer, then the level

k stabilizer St k (G) ≃ G 1...1 × • • • × G d 0 ...d k-1 is a direct product of d 0 . . . d k-1 subgroups of Aut e (T σ k d)
each of which inherits the property to be dense and finitely generated. In particular each of the groups G v is infinite (d i ≥ 3) and thus has at least linear growth, so that the subgroup St k (G) of finite index and thus G have growth function at least b(r) r d 0 ...d k-1 , hence superpolynomial.

Intermediate growth.

Proposition 6.6. The group H d < Aut(T d) has intermediate growth.

Proof of Proposition 6.6. Superpolynomial growth follows from Propositions 6.3 and 6.5, so there remains to prove subexponential growth. Proceed as in [START_REF] Grigorchuk | Degrees of growth of finitely generated groups, and the theory of invariant means[END_REF]. Denote B k (r) the ball of radius r in H σ k d for the word metric |.| k associated with the generating set

x σ k d, y σ k d , denote b k (r) its cardinal and c k = lim r b k (r) = h {x σ k d,y σ k d} (H σ k d)
its exponential growth rate. The fixed point condition on eligible pairs ensures:

x dy dx dy -1 d x d = (x σ d, 1, . . . , y σ d, y -1 σ d , . . . , 1)x d 0 y d 0 x d 0 y -1 d 0 x d 0 , (6.4)
with y σ d in positions x d 0 (2) and x d 0 y d 0 x d 0 (2) and the second and third x σ d cancel out. As the generators are of order 2 and 3 every element g = (g 1 , . . . , g d 0 )σ in B 0 (r) admits a minimal representative word of the form g = x dy ε 1 d x dy ε 2 d . . . x dy εn d x d, with ε i in {-1, 1}. Given g (more precisely given a fixed minimal representative word), denote a(g) the number of alternations in the sequence (ε i ), equality (6.4) implies:

|g 1 | 1 + • • • + |g d 0 | 1 ≤ |g| 0 -a(g).
(6.5)

Given any parameter t ≥ 2, split the ball B 0 (r) into:

B + 0 (r) = {g ∈ B 0 (r)|a(g) ≥ r t }, B - 0 (r) = {g ∈ B 0 (r)|a(g) ≤ r t }.
The size of the first part of the ball is bounded by:

b + 0 (r) ≤ #A d 0 r 1 +•••+r d 0 ≤(1-1 t )r b 1 (r 1 ) . . . b 1 (r d 0 ). (6.6)
Indeed, each element g = (g 1 , . . . , g d 0 )σ of B 0 (r) is injectively described by the permutation σ in A d 0 and the coordinates g 1 , . . . , g d 0 the sum of the |.| 1 length is bounded by ra(g) ≤ (1 -1 t )r thanks to computation (6.5). The size of the second part of the ball is bounded by (recall notation C k n for the number of subsets of size k in {1, . . . , n}):

b - 0 (r) ≤ 4 s≤ r t C s r ≤ 4 r t C r t
r .

(6.7)

Indeed the term 4 corresponds to choosing the start of the representative word (y, y -1 , xy or xy -1 ), s represents the number of alternation a(g) and C s r the number of choice for the positions of such alternations.

The size is estimated by b 0 (r) ≤ b + 0 (r) + b - 0 (r) ≤ max{2b + 0 (r), 2b - 0 (r)}, and taking limits of r-roots raises c 0 ≤ max{c

1-1 t 1 , t 1 t (1-1 t ) ( 1 t -1) }, since Stirling formula ensures: (4 r t C r t r ) 1 r ∼ r→∞   4 r t (2πr) 2π r t 2π(1 -1 t )r   1 r r e ( r et ) 1 t ((1 -1 t ) r e ) 1-1 t ∼ r→∞ t 1 t (1- 1 t ) 1 t -1 .
The estimate is valid for any level k so that for all parameter t ≥ 2:

c k ≤ max{c 1-1 t k+1 , t 1 t (1 - 1 t ) 1 t -1 }.
In particular, this shows the sequence (c k ) k increases (note

t 1 t (1 -1 t ) 1 t -1 → 1 for t → ∞).
Moreover the sequence is bounded by 2 (the groups are quotients of Z/2Z * Z/3Z), hence admits a limit c ∞ , which satisfies by continuity:

c ∞ ≤ max{c 1-1 t ∞ , t 1 t (1 - 1 t ) 1 t -1 }
for any parameter t ≥ 2, which is impossible unless c ∞ = 1 (otherwise take t large enough). This shows subexponential growth of the groups H σ k d.

Remark 6.7. When the tree has bounded valency, set f (r) = max{b k (r)|k ∈ N} the estimate (6.6) can be made homogeneous on d k ≤ D. This together with estimate (6.7) applied for a parameter t of the form t = K log(r) raises inequality:

f (r) ≤ K    r 1 +•••+r D ≤(1-K log(r) )r D i=1 f (r i )    + KC K log(r) r .
A computation due to Erschler (Lemma 6.4 in [Ers]) gives the explicit upper bound on the growth:

b 0 (r) ≤ f (r) ≤ exp K log(log(r))r log(r) .
7. Groups of non uniform growth

7.1. A Theorem of Wilson.
The first examples of groups of non uniform exponential growth have been constructed by Wilson in [Wil1]. The following Theorem from [START_REF] Wilson | Further groups that do not have uniformly exponential growth[END_REF] is a generalization.

Theorem 7.1 (Wilson [Wil2]). Let k be a positive integer and χ k a class of groups with the two properties:

(

1) each group G in χ k is perfect (that is G = [G, G]
) and can be generated by

k involutions; (2) each group G in χ k is isomorphic to a permutational wreath product G 1 ≀ A d with G 1 ∈ χ k and d ≥ 29.
Then each group G in χ k contains two sequences of elements (a (n) ), (b (n) ) such that:

(a) (a (n) ) 2 = (b (n) ) 3 = 1 and a (n) , b (n) = G for each n and, (b) h {a (n) ,b (n) } (G) → 1 as n → ∞.
In section 4. of [START_REF] Wilson | Further groups that do not have uniformly exponential growth[END_REF], Wilson constructs subgroups of Aut e (T d) for unbounded sequences d = (d i ) i in the classes χ k . Unboundedness of the sequence permits to construct such groups with a subgroup isomorphic to the free group F 2 on two generators. This ensures exponential growth, but prevents amenability.

In the next section groups in the class χ k are constructed similarly but acting on bounded valency rooted tree. The Main Theorem 3.1 will apply to show amenability. Exponential growth is due to the presence of free semigroups. Note however that in [START_REF] Wilson | On exponential growth and uniformly exponential growth for groups[END_REF] Wilson constructs groups of automorphism of a regular (in particular bounded valency) rooted tree which have non uniform growth and contain a free group. 7.2. Amenable groups of non uniform growth. Let d = (d i ) i be a bounded sequence of integers 5 ≤ d i ≤ D, define a subgroup of the group H (constructed in section 2.3) as Ā < H = S d 0 -1

d 1 × S d 1 -1 d 2 × . . . where Ā = A d 1 × A d 2 × .
. . as an abstract group and each group A d k is acting as a rooted automorphism on T 1 k-1 2 ; this is best understood by Figure 4. Now for each integer d in {5, . . . , D}, denote E d = {i ≥ 1|d i = d}. There is a diagonal injection:

j d : A d ֒→ i∈E d A d i < Ā,
and the diagonal product of those injections:

j : A d = D d=5 A d ֒→ Ā.
To ease notation the image subgroup of A d is still denoted A d. It is a finite saturated subgroup of Ā. The subgroup of Aut e (T d) generated by alternate rooted automorphisms Proof. Show this Proposition simultaneously for all groups

A d 0 and A d is denoted G 0 = G(A d 0 , A d) < Aut e (T d). Note that when d = σ d is a constant sequence, the group G(A d , A d) is generated by a finite automaton. . . . a 3 1 . . . 1 a 2 1 . . . 1 a 1 1 . . . 1
G i = G(A d i , A σ i d) < Aut e (T σ i d).
The group G i is perfect because generated by copies of the groups A d i , A d , d ∈ {5, . . . , D} which are perfect (even simple). Moreover, those groups (hence G i ) are generated by double transpositions, in particular by involutions the number of which depends only on D, so that the condition (1) of definition of groups in the class χ k is satisfied for some k depending only on D.

To check condition (2), note first that the injection in the wreath product (2.1) has image in: (v, v, v, . . . ). The following Lemma due to Bartholdi (Proposition 2.3 in [Bar]) ensures that ūu, vv ≃ S 2 is a free semigroup. More precisely: Lemma 7.4 (Bartholdi [Bar]). The quotient semigroup ūu, ūv, vu, vv /(ūu = ūv, vu = vv) ≃ S 2 is freely generated by {ūu, vv}.

G i = G(A d i , A σ i d) ֒→ G(A d i+1 , A σ i+1 d) ≀ A d i = G i+1 ≀ A d i . ( 7
A d i , so that G i contains b σ 2 = σb 2 σ -1 = (1, a 2 , b ′ 2 , 1, . . . , 1), hence [b 1 , b σ 2 ] = (1, [a 1 , a 2 ], 1, . . . , 1) and then 1 × A d i+1 × • • • × 1 by perfection. Similarly given any two b ′ 1 , b ′ 2 in A σ i+1 d, the group G i contains [b 1 , b τ 2 ] = ([b ′ 1 , b ′ 2 ], 1, . . . , 1) where τ = (23)(45), hence A σ i+1 d × 1 × • • • × 1. Since A
This ensures exponential growth of the group G 0 .

(0) i = G i such that (a (0) i ) 2 = (b (0) i ) 3 = 1.
Out of this first generating pair, Wilson constructs a sequence of generating pairs for G i , defined inductively as (also see Figure 5 and compare with Figure 3):

a (n+1) i = (a (n) i+1 , 1, . . . , 1)x d i , b (n+1) i = (1, b (n+1) i+1 , 1, . . . , 1)y d i . (7.2)
The fact that a (n) i and b

(n) i have order 2 and 3 and that they generate G i is a direct consequence of the properties of the generating pairs x d i , y d i of the alternate group A d i (see Proposition 6.1).

Definition 7.6 (Distance between Cayley graphs). Let (Γ, S) and (∆, T ) be two groups with generating sets, denote B Γ,S (R) the restriction of the Cayley graph of Γ relatively to the generating set S to vertices at distance less than R of the neutral element (for the word distance in S). The distance between (Γ, S) and (∆, T ) is defined as:

d((Γ, S), (∆, T )) = inf{ 1 R |B Γ,S (R) ∼ G B ∆,T (R)},
where Gr 1 ∼ G Gr 2 if Gr 1 and Gr 2 are isometric as colored graphs.

Non uniform growth of the group G 0 comes from the two next propositions, since intermediate growth of Note that this Proposition is true independently of the amenability or not of the group G 0 in a class χ k . In particular, such a convergence is also true for the non amenable groups constructed by Wilson in [Wil1], [START_REF] Wilson | Further groups that do not have uniformly exponential growth[END_REF].

H d implies h {x d ,y d} (H d) = 1. x d 0 x d 1 x d 2 x d n-1 a (0) n 1 . . . 1 1 . . . 1 1 . . . 1 y d 0 1 y d 1 1 y d 2 y d n-1 1 b (0) n 1 . . . 1 1 . . . 1
Proposition 7.8. If d((Γ, S n ), (∆, T )) → 0, then:

lim sup n→∞ h Sn (Γ) ≤ h T (∆).
Proof of Proposition 7.8. Given a positive ε the definition of h T (∆) ensures that for R ≥ R 0 large enough the ball B ∆,T (R) has size #B ∆,T (R) ≤ (h T (∆) + ε) R . Now the convergence of Cayley graphs shows that for n ≥ N large enough #B Γ,Sn (R) ≤ (h T (∆) + ε) R , and by subadditivity #B Γ,Sn (kR

) ≤ #B Γ,Sn (R) k ≤ (h T (∆) + ε) kR , so that: h Sn (Γ) = lim kR #B Γ,Sn (kR) ≤ h T (∆) + ε,
which was required.

The proof of Proposition 7.7 uses the following:

Lemma 7.9 (of contraction). If x = (u, 1, . . . , 1)x d and ŷ = (1, v, 1, . . . , 1)y d are as in Proposition 6.1, then for elements g = (g 1 , . . . , g d )σ in the wreath product isomorphism x, ŷ ≃ u, v ≀ A d , one has for each coordinate t:

|g t | {u,v} ≤ 1 2 (|g| {x,ŷ} + 1),
where |.| S denotes the word norm associated to the generating set S (inverses of elements of S have length 1).

Proof. It is sufficient to check that xŷ ε = (u, 1, . . . , v ε , 1, . . . , 1)x d y d with v ε on coordinate x d (2) = 1.

Proof of Proposition 7.7. Introduce other relations depending on integer l ≥ 1 on groups with generating sets: (Γ, S) ∼ l (∆, T ) if for every free word w of length less than l in S (elements and inverses) one has w(S) = id Γ if and only if w(T ) = id ∆ (for l = 1 the relation ∼ 1 just means S ∪ S -1 and T ∪ T -1 have the same size).

If the relation (Γ, S) ∼ 2l+1 (∆, T ) is satisfied then d((Γ, S), (∆, T )) ≤ 1 l because to describe B Γ,S (R) it is sufficient to know when g ′ g -1 = s for every g, g ′ in B Γ,S (R) and s in S ∪ S -1 .

To ease notations set S

(n) i = {a (n) i , b (n) i } and T i = {x σ i d, y σ i d}. It is sufficient to show for all integers i: (G i , S (n) i ) ∼ ln (H σ i d, T i ) with a sequence l n → ∞.
Proceed by induction on n, using:

w(S (n+1) i ) = (w 1 (S (n) i+1 ), . . . , w d i (S (n) i+1 ))w(x d i , y d i ), w(T i ) = (w 1 (T i+1 ), . . . , w d i (T i+1 ))w(x d i , y d i ),
where for each coordinate t the elements w t (S (n) i+1 ) and w t (T i+1 ) involve the same word w t because the permutations on the first level are the same for generators in S (n+1) i or in T i (namely x d i and y d i ). Lemma 7.9 ensures that |w

t (T i+1 )| ≤ 1 2 (|w(T i )| + 1) (and |w t (S (n) i+1 )| ≤ 1 2 (|w(S (n+1) i
)| + 1)) so that if w has length less than l n+1 = 2l n -1 one has w(S (n+1) i ) = id G i if and only if w(T i ) = id H σ i d . The result follows since the sequence (l n ) starts with l 0 = 1 and l 1 = 2.

Corollary 7.10 (of Proposition 7.7). The group H d of intermediate growth is not finitely presented.

Proof. Assume the contrary H d = x d, y d|r 1 , . . . , r k . Let R be bigger than the maximal length of the relations r 1 , . . . , r k , and n large enough so that:

d((G 0 , {a (n) 0 , b (n) 0 }), (H d, {x d, y d})) ≤ 1 R .
Then the automorphisms a [Neu]). Let (G i ) i∈N be a sequence of finitely generated perfect groups such that for each i there exists an integer

d i ≥ 5 such that G i ≃ G i+1 ≀ A d i . Consider the isomorphisms: G 0 ≃ G i ≀ A d i-1 ≀ • • • ≀ A d 0 ≃ (G i × • • • × G i ) d 0 ...d i-1 times ⋊Aut e (T d 0 ...d i-1 ), then the subgroups K i = (G i × • • • × G i ) for i ∈ N are
the only normal subgroups of G 0 of finite index. Moreover if one (hence all) of the groups G i is residually finite, then (K i ) i∈N are the only non trivial normal subgroups of G 0 ; in particular G 0 is just infinite.

Proposition 8.1 (as well as Lemma 8.3) is a slight generalization of Theorem 5.1. in [Neu]. The proof is given here for the sake of completeness and to avoid the reader multiple references and notations. The second part is also similar to Theorem 4. in [START_REF] Grigorchuk | Just Infinite Branch Groups[END_REF]. Note that all examples in this paper are groups of automorphism of a rooted tree. In particular they satisfy the assumption of residual finiteness.

Corollary 8.2. Two groups G 0 and H 0 satisfying the hypothesis of Proposition 8.1 (in particular groups in a class χ k ) for two different sequences of integers (d i ) i and (e i ) i are non isomorphic.

Proof. The index of K i in G 0 has value:

[G 0 : K i ] = #Aut e (T d 0 ...d i-1 ) = #(A d i-1 ≀ • • • ≀ A d 0 ) = a(d i-1 ) d i-2 ...d 0 . . . a(d 1 ) d 0 a(d 0 ), where a(d) = d! 2 = #A d .
In particular the sequence of index of subgroups ([G 0 : K i ]) i is an isomorphism invariant from which the sequence (d i ) i can be recovered.

Lemma 8.3. Under the hypothesis of Proposition 8.1, the only normal subgroups of G 0 containing K m are K 0 , K 1 , . . . , K m .

The proof of this lemma will use the: Fact 8.4. Given a finite group Γ, assume ∆ ⊳ Γ is a minimal normal subgroup (minimal means the only normal subgroup of Γ strictly contained in ∆ is trivial) and that the centralizer Cent Γ (∆) of ∆ is trivial. Then ∆ is the unique minimal normal subgroup of Γ.

Proof. Assume ∆ ′ is another such subgroup, then ∆ ∩ ∆ ′ is trivial by minimality. In particular, for every δ ∈ ∆ and δ

′ ∈ ∆ ′ the commutator ∆ ∩ ∆ ′ ∋ [δ, δ ′ ] = 1, which ensures ∆ ′ ⊂ Cent Γ (∆) = {1}, contradiction.
Proof of Lemma 8.3. By induction on m and using Fact 8.4, it is sufficient to prove that:

A (1...1) d m-1 × • • • × A (d 0 ...d m-2 ) d m-1 ≃ K m-1 /K m ⊳ G 0 /K m ≃ Aut e (T d 0 ...d m-1 )
is minimal and has trivial centralizer, which shows K m-1 is the only minimal subgroup of G 0 containing K m .

Let U a non trivial subgroup normal in G 0 /K m and included in K m-1 /K m . Then 1 = y ∈ U can be written y = (y 1...1 , . . . , y d 0 ...d m-1 ) in the wreath product

G 0 /K m ≃ A d m-1 ≀ Aut e (T d 0 ...d m-2 ), with some coordinate 1 = y v ∈ A (v)
d m-1 × • • • × A (d 0 ...d m-2 ) d m-1 = K m-1 /K m , proving minimality. Transitivity also shows that the centralizer Cent G 0 /Km (K m-1 /K m ) is included in St m-1 (G 0 /K m ) = K m-1 /K m ,
which has trivial center, hence the centralizer is trivial.

Proof of Proposition 8.1. Let X be a finite group and f : G 0 → X a homomorphism. Restricting to factors of the subgroups

K m = G (1...1) m × • • • × G (v) m × • • • × G (d 0 ...d m-1 ) m
, it appears that for m large enough there exists v = v ′ such that the associated factors have the same image f (G

(v) m ) = f (G (v ′ ) m ) = Y , which must be abelian because [G (v) m , G (v ′ ) m ] = 1, hence Y = {1} because G (v) m ≃ G m is perfect. This shows G (v) m ⊂ Ker(f ). Moreover for each coordinate v ′ there exists ϕ ∈ Aut e (T d 0 ...d m-1 ) such that ϕ(v) = v ′ , so that ϕG (v) m ϕ -1 = G (v ′ )
m ⊂ Ker(f ) and consequently K m lies in the kernel of f . Applying Lemma 8.3 shows Ker(f ) = K i for some i ≤ m, which proves the first part. Now assume G 0 is residually finite, and N ⊳ G 0 is an arbitrary normal subgroup. The description of the first part ensures that ∩ m≥0 K m = {1}, and as the sequence of subgroups (K m ) m is strictly decreasing there exists an integer n such that N ≤ K n and N K n+1 . To get the second part, it is sufficient to prove N ≥ K n+1 since the first part will force K n = N.

Consider x ∈ N \ K n+1 and its image x = (x 1...1 , . . . x d 0 ...d n-1 ) n in the factor decomposition of K n . There exists v such that:

x v = (x v1 , . . . , x vdn )σ v ∈ G (v) n ≃ G n+1 ≀ A dn , with a non trivial permutation σ v , and in particular there are s = t in {1, . . . d n } with σ v (s) = t. Now given any two elements ξ, η in G n+1 , define f, g in K n = (G n × • • • × G n ) as: f = (1, . . . , 1, f v , 1 . . . , 1) n , f v = (1, . . . , 1, ξ, 1, . . . , 1) ∈ G n+1 ≀ A dn , g = (1, . . . , 1, g v , 1 . . . , 1) n , g v = (1, . . . , 1, η, 1, . . . , 1) ∈ G n+1 ≀ A dn , with ξ, η on coordinate s. The normal subgroup N contains the commutator [f, x] = f xf -1 x -1 = (1, . . . , 1, [f v , x v ], 1, . . . , 1) n , where: [f v , x v ] = (1, . . . , ξ, . . . , 1)(x v1 , . . . , x vdn )σ v (1, . . . , ξ -1 , . . . , 1)σ -1 v (x -1 v1 , . . . , x -1 vdn ) = (1, . . . , 1, ξ, 1, . . . , 1, x vt ξ -1 x -1 vt , 1, . . . , 1), with ξ in coordinate s and x vt ξ -1 x -1 vt in coordinate t. Taking another commutator, the subgroup N contains [g, [f, x]] = (1, . . . , 1, [g v , [f v , x v ]], 1, . . . , 1) n with: 8.2. Non subexponential amenability. Denote SG 0 (respectively EG 0 ) the class of groups such that all finitely generated subgroups have subexponential growth (respectively are abelian). Assume that for an ordinal α > 0 the classes SG β and EG β are defined for every ordinal β < α. When α is a limit ordinal, set SG α = ∪ β<α SG β (respectively EG α = ∪ β<α EG β ). When α is a successor ordinal, define SG α (respectively EG α ) to be the class of groups that can be obtained from groups in the class SG α-1 (respectively EG α-1 ) either by taking direct limits, or by taking extension by a group from the class SG 0 (respectively EG 0 ).

Each class SG α (respectively EG α ) is closed under taking quotients and subgroups. Moreover, the class SG = ∪ α SG α (respectively EG = ∪ α EG α ) where the union runs over all ordinals α, is the smallest class of groups containing SG 0 (respectively EG 0 ) which is closed under the operations of taking subgroups, quotients, extensions and direct limits. As these operations preserve amenability, which is satisfied in SG 0 (respectively EG 0 ), the class SG (respectively EG) is called class of subexponentially (respectively elementary) amenable groups.

This construction of classes of groups is detailled in [START_REF] Osin | Elementary Classes of Groups[END_REF]. It is obvious that EG α is a subclass of SG α for each ordinal α and that the class SG contains the class EG. This inclusion is strict (see [START_REF] Grigorchuk | Degrees of growth of finitely generated groups, and the theory of invariant means[END_REF]) and the Basilica group introduced in [GZ] was the first example of an amenable group out of SG. Osin has shown in [START_REF] Osin | Algebraic entropy of elementary amenable groups[END_REF] that the class EG contains no group of non uniform growth. In particular, groups in the class χ such as the groups G(A d 0 , A d) of non uniform exponential growth introduced in section 7.2 are not in EG. The following Proposition shows these groups are not even in SG, providing uncountably many pairwise non isomorphic examples of amenable groups outside SG.

Proposition 8.5. Consider a residually finite group G belonging to a class χ k (see section 7), then one of the two following holds:

1) either G belongs to the class SG 0 of groups of subexponential growth, 2) or G does not belong to the class SG of subexponentially amenable groups.

In particular, residually finite groups of exponential growth in a class χ k are not in SG.

Recall an elementary property of ordinals:

Fact 8.6 (Theorem 7.3 (5) in [Kun]). Let C be a non empty set of ordinals, then there exists x ∈ C such that for every y ∈ C, one has x ≤ y. In other words, C has a minimum.

Proof of Proposition 8.5. The proof is similar to that in [GZ]. Let G a group in a class χ k having exponential growth, in particular not in the class SG 0 . Denote G i the group in the class

χ k such that G = G 0 ≃ G i ≀ A d i-1 ≀ • • • ≀ A d 0 .
In particular all groups G i have exponential growth. Assume G 0 lies in the class SG, then all the groups G i (which are subgroups of G 0 ) lie in SG. For each integer i define α i to be the minimal ordinal for which G i belongs to SG α i (exists by Fact 8.6). The family {α i } i∈N admits a minimum α i 0 . Now the ordinal α i 0 is not a limit ordinal otherwise G i 0 would belong to SG β for some β < α i 0 . Moreover, the group G i 0 is not a direct limit of a strictly increasing infinite sequence of groups because it is finitely generated. This forces the existence of N and H in SG α i 0 -1 such that the sequence 1 → N → G i 0 → H → 1 is exact. But as G hence G i 0 is residually finite, Proposition 8.1 implies that N = G i 0 +m for some integer m, so that α i 0 +m ≤ α i 0 -1 which contradicts minimality of α i 0 , proving G is not in SG.

d

  and g(v) ∈ S d k the action on the d k children of v. The automorphism g v of the rooted tree T v belongs to the group G(S d k , H k ). The function defined by (5.1) for G(S d k , H k ) will be denoted by ||.|| k .

  denote the ball of radius r associated to the function ν. The next proposition is crucial for our purpose. Proposition 5.4. Consider a spherically homogeneous rooted tree T d of bounded valency 2 ≤ d i ≤ D, a finite subgroup H of H and the function ν constructed above, then the balls B ν (r) ⊂ G(S d 0 , H) grow at most exponentially fast. Namely, there exists a constant C depending only on D and the size of H such that: #B ν (r) ≤ (C) r , for all r sufficiently large.

Figure 4 .

 4 Figure 4. The group Ā. Proposition 7.2. Let d = (d i ) i a bounded sequence of integers 29 ≤ d i ≤ D, the group G 0 = G(A d 0 , A d) belongs to the class χ k where k depends only on D.

  .1) This is clear for the generators in A d i and the generators b in A σ i d have image b = (b ′ , a, 1, . . . , 1) where a belongs to A d i+1 and b ′ to A σ i+1 d by construction. Now remains to prove this injection is onto hence an isomorphism. Given any two elements a 1 , a 2 in A d i+1 there exists b 1 = (b ′ 1 , a 1 , 1, . . . , 1) and b 2 = (b ′ 2 , a 2 , 1, . . . , 1) in A σ i d. Moreover the double transposition σ = (13)(45) belongs to

  d i acts transitively by conjugation on the coordinates, this proves injection (7.1) is onto. Proposition 7.3. Let d = (d i ) i a bounded sequence of integers 5 ≤ d i ≤ D, the group G 0 = G(A d 0 , A d) has exponential growth. Proof. Each group A d contains the double transpositions u = (12)(34) and v = (12)(35). Moreover each of the groups A σ i d ≃ A d i ×A d i+1 ×. . . contains the diagonal elements ū = (u, u, u, . . . ) and v =

  Corollary 7.5 (Theorem 1.1). The groups G(A d 0 , A d) associated to sequences d = (d i ) i of integers 29 ≤ d i ≤ D are (uncountably many pairwise non isomorphic) amenable groups of non uniform exponential growth. Proof. This follows from the Main Theorem 3.1, Wilson's Theorem 7.1, Proposition 7.2 and Proposition 7.3. The bracketted part follows from Corollary 8.2. 7.3. Convergence of the Cayley graphs. This section is devoted to give another proof of some part of Wilson Theorem 7.1. Namely the convergence to 1 of the exponential growth rate of the generating sets {a (n) , b (n) } can be understood as the convergence of the associated Cayley graphs of the group to the Cayley graph of a group H d of intermediate growth introduced in section 6. More precisely, let G = G 0 belong to some class χ k , then by definition of the class there exists a sequence of groups G i in χ k and integers d i ≥ 29 such that G i ≃ G i+1 ≀ A d i . The Theorem 7.1 of Wilson ensures in particular that for each integer i there exists a generating pair of elements a

1 . . . 1 Figure 5 .

 15 Figure 5. Portraits of the elements a (n) 0 and b (n) 0 .

  r 1 , . . . , r k . In particular, G 0 is a quotient of H d hence has subexponential growth. This contradicts Proposition 7.3. 8. Non subexponential amenability 8.1. Description of normal subgroups. The normal subgroups of finite index of groups in a class χ k are completely described by the: Proposition 8.1 (Neumann

  d m-1 . By simplicity, the normal closure of y v is the full alternate group y v A d m-1 = A d m-1 . Moreover the group Aut e (T d 0 ...d m-2 ) acts by conjugation transitively on the coordinates, so that U > y G 0 /Km = A (1...1)

  [g v , [f v , x v ]] =(1, . . . , 1, [η, ξ], 1, . . . , 1), and this for ξ, η in G n+1 arbitrary, which can be rewritten:N ∋ (1, . . . , 1, [η, ξ], 1, . . . , 1) n+1 ∈ G n+1 ≀ Aut e (T d 0 ...dn ),with[η, ξ] in position vs. As this group is perfect, the subgroupN contains 1 × • • • × G (vs) n+1 × • • • × 1. Thetransitivity of the action of Aut e (T d 0 ...dn ) on level n + 1 by conjugation ensures that N contains (G n+1 × • • • × G n+1 ) = K n+1 as required.

  11 12 . . . 1d 1 21 22 . . . v T v . . . d 0 1 d 0 2 . . . d 0 d 1

		∅
	1	2	. . .	d 0
	. . . . . .	v1 v2 . . . vd 2		. . . . . .

1 ), (2.2) where Aut(T d 0 ...d n-1 ) acts by permutation on d 0 . . . d n-1 copies of Aut(T σ n d) the way it acts on the set of leaves ∂T d 0 ...d n-1 (the boundary) of the finite tree T d 0 ...d n-1 . There Figure 1. Spherically homogeneous rooted tree, subtree. is also a projection p n : Aut(T d) → Aut(T d 0 ...d n-1 ), the kernel of which constitutes the stabilizer St n (Aut(T d)) of the nth level. This is a normal subgroup of Aut(T d) isomorphic to the direct product St n

  n where σ n is a random variable in S d 0 and the coordinates Y t n for t ∈ {1, . . . , d 0 } are random variables in G(S d 1 , H 1 ). The point is that (Y t n ) n follows the law of the similarly defined random walk (Y ′ m ) m∈N on G(S d 1 , H 1 ) (which is taking independent equidistributed increments alternatively in S d 1 and H 1 ), but at a slower speed. More precisely: Lemma 4.6 (Similarity of the random walks (Y n ) and (Y ′

m )). Let (Y n ) n∈N the random walk defined above and


F Mathieu

F Larramendy

D Dezest

C Huang

G Lavallee

S Miller

" Eichfeld

W Mansfield

S Trolier-Mckinstry

L Nicu

Reducing parasitic effects of actuation and sensing schemes for !" piezoelectric microelectromechanical resonators #"

The co-integration of piezoelectric actuation and sensing capabilities on !+" microelectromechanical system-based resonators can be a source of electrical cross-talk that, if not !!"

properly taken into account, may dramatically affect the interpretation of the device's output. In this !#" paper, we identify three parasitic electrical effects pertaining to the most commonly used piezoelectric !$" actuation and sensing schemes. To further investigate the impact of such parasitic effects, !%" microcantilevers, bridges and membranes integrating a layer of sol-gel lead zirconate titanate (PZT) !&" were fabricated and electrically characterized. Experimental results on the resonant characteristics !'" were compared with simulations of the studied resonators' equivalent electrical models. Methods for !(" reducing the design-dependent parasitic electrical effects such as mutual capacitances of less than !)" 10fF, electrical wiring or static capacitance mismatches of less than 20% of the integrated !*" piezoelectric films are discussed.

Introduction #$"

Piezoelectric materials enable high performance actuator and sensor systems satisfying #%" dimensional integration constraints from the millimeter to the micrometer scale [1][2][3]. Piezoelectricity #&" is appealing for electromechanical transduction as it allows energy conversion from the mechanical to #'" the electrical domain and vice-versa. Extensive reports are available on the challenges associated with #(" the integration of such materials at the microscale, as well as the direct impact of miniaturization and #)" dimensionality on the electromechanical properties of the resulting devices [4][5][START_REF] Adams | %&[END_REF][START_REF] Nicu | &+[END_REF][START_REF] Koval | [END_REF]. Starting from H. #*" Tilmans work [9] that laid the foundations for developing equivalent electrical circuits of electro-$+" mechanical transducers and their interconnection to the outside world, Studies [10][11][12][13] where lumped $!" element-based modeling is used to predict the behavior of multiphysics microsensors integrating $#" piezoelectric active layers have been carried out. Nonetheless, the provided solutions are most of time $$" application-dependent. Such dependence dramatically constraints the generic aim of the proposed $%" models. In this paper, we propose a simplified electrical modeling of the chip hosting piezoelectric $&" resonators presenting conventional geometries (cantilever, bridge and membrane) and the way in $'" which the "electrical environment" of such integrated resonators may influence their performance.

$("

In the following, a piezoelectric material sandwiched between two metallic electrodes will be $)" called a piezo-cell and a microdevice integrating piezo-cells for actuation and/or sensing purposes as $*" well as the contacting wires will be called a piezo-device. To address the holistic electrical modeling of %+" piezoelectric MEMS, one has to hierarchically answer three questions: %!" (1) The piezo-cell level: is there a generic way to electrically model a piezo-cell taking into %#" account the influence of the associated parasitic "on chip-electrical wiring"? %$"

(2) The piezo-device level: is there an ideal actuation/sensing element topology that minimizes %%" the influence of the parasitic capacitances? %&"

(3) The system level: based upon the answers to (1) and (2), is it possible to devise specific %'" sensing electronics that effectively interface to the piezoelectric devices? %(" The on-chip actuation and sensing schemes studied in this paper along with the use of dedicated %)" electronics developed for this purpose aim to answer these questions. A series of design rules for %*" piezoelectric materials integration is proposed from the standpoint of MEMS-based resonator &+"

development. &!" &#" Device Fabrication &$"
The fabrication process for the MEMS resonators is depicted in Fig. S1 (see Supplementary &%" that is half the full membrane diameter, and an outer ring-like area with a width that is 0.25 of the '#" membrane's diameter.

Information file). A double

'$"

To fabricate these devices, the first step was the growth of a 100-nm-thick thermal silicon dioxide '%" film on the SOI wafer. A metal-complex oxide-metal sandwich structure was then deposited on the '&" thermal SiO 2 . A bottom electrode consisting of a 20 nm titanium adhesion layer and 100nm platinum of ''" Pt was then deposited. A 1.6 m layer of lead zirconate titanate (PZT: Pb(Zr 0.52 Ti 0.48 )O 3 ) was then spin '(" deposited from a sol gel solution on the bottom electrode as described elsewhere [14]. Finally, a top ')" electrode of 100nm of platinum was sputter deposited on the PZT. Subsequent lithography defined the '*" active device structure. The active device pattern was transferred to the top electrode (100nm Pt) and (+" the 1.6 m PZT layer by dry etching using a thick resist as a mask (Shipley SPR220). The dry etch (!" process was run in a Tegal 6540 HRe -capacitively coupled plasma chamber using a mixture of Cl 2 and (#" CF 4 gases and a process pressure of ~ 5 mTorr. The PZT etch was stopped on the bottom platinum ($" electrode. A second set of lithography and plasma etch steps were used to define the bottom electrode (%" patterns. A 400-nm-thick isolation film of silicon dioxide was then sputter deposited to avoid possible (&" hydrogen damage of the PZT piezoelectric layer. Contact lithography and inductively coupled plasma ('" (ICP) oxide etch processes were then used to open contacts to both top and bottom electrodes. The ((" interconnect traces and pads were formed by a liftoff process of an electron beam-evaporated Ti/Au ()" (10/150 nm) metal onto an undercut bi-level resist structure. A separate lithography step was then (*" utilized to define the cantilever device structures; the isolation oxide, device silicon and buried oxide )+" was etched using ICP etch processes (in a PlasmaTherm Versalock cluster tool) to form the beam )!" structures. The cantilever and membrane devices were released utilizing backside thick resist )#" lithography and Bosch etching through the handle wafer. The etching was stopped on the SOI buried )$" oxide layer. During this step, the front-side structures were protected with a thick resist layer. After the )%" handle wafer etch, the protection resist was removed to release both cantilever structures and membrane )&" structures and the device chips were gently extracted from the wafer by applying pressure with tweezers )'" at the separating paths running through the wafer.

Electrical modeling of a generic piezo-cell !++"

The integration of piezoelectric thin films for different MEMS applications is commonly !+!" performed on metalized silicon substrates. Typical doping levels of such substrates (even at low !+#" values, i.e. 10 13 dopant atoms/cm 3 ) render them rather conductive when compared to ceramic !+$" substrates. To electrically isolate piezo-cells from the underlying silicon substrate, one generally !+%" grows silicon dioxide onto the substrate prior to the piezo-cell patterning. This SiO 2 (or ZrO 2 , HfO 2 , !+&" etc.) layer also serves to minimize reactions between Pt metallization and the Si at typical process !+'" temperatures for perovskite piezoelectrics.

!+("

In order to model a simple equivalent electrical circuit, we proceed to a R p -C p base !+)" transformation of the gain-phase measurement and subsequently identify the equivalent R s , C and R !+*" values (where R s is the series resistance of the electrical wiring while C and R are respectively the !!+" capacitance and resistance of the piezo-device including R pz -piezocell's resistance and C pz -!!!" piezocell's capacitance), as depicted in Figure 2(a).

!!#" !!$"
The R p and C p values are respectively given by and whereas the impedance !!%" . Figure 2.(b) compares the measured and modeled data using the extracted responses of a S1 i circular piezo-cell. The graphs include both measured responses (green and blue !##" colored) and corresponding theoretical fit (red and magenta colored).

!!&" parameters for a S1 i circular piezo-cell." !!'" Si SiO2 C1 Cpz C0 Pt P Z T Pt Rs1 Rs2 Rpz C R Rs " !!(" ,-. !!)" !"## !$# !%# !&# !'# # () *+ , -(.-+ , /

!#$"

The R s , C and R parameters are extracted as follows: first, on the R p -C p response, the C p value is !#%" used for the C value, the low frequency R p value is defined as the equivalent resistance R of the piezo-!#&" cell and the R s value is extracted from the high-frequency R p measured response. This strategy is !#'" proved to be physically relevant as the as-extracted R s , R and C values are then implemented in the !#(" corresponding gain-phase response and the resulting simulated response is in excellent agreement with !#)" the measured characteristics.

!#*"

We then proceeded in the same way to extract R s , R and C values for all piezo-cell geometries !$+" on two different chips. Values are listed below (Table 1) together with the theoretical C values for !$!" comparison.

!$#"

Table 1: Values of the series resistance for the electrical wiring (R s ), the resistance of the piezo-layer !$$" (R) and the global capacitance of the piezo-cell and associated electrical wiring (C meas ) extracted from !$%" gain-phase measurements similar to Figure 7

!$("

A qualitative analysis of the Table 1 values reveals that: !$)" 0 R systematically takes high resistance values (over tens of M), which is consistent with the !$*" high resistivities of the patterned sol-gel PZT (R pz ) electrical properties used in this study and !%+" may include the leakage substrate resistive effects; !%!" 0 R s ranges from 15 to 150, which is consistent with the series resistances values of the !%#" metallization measured on the Ti/Pt/Au stacks used to connect on-chip devices, resistive !%$" contact effects and bonding effects; !%%" 0 The measured C values are of the same order of magnitude as the calculated values, but are !%&" not identical. This confirms the impact of the on-chip electrical wiring on the capacitive !%'" behavior of the piezo-cell. The influence of the wiring on the measured capacitance becomes !%(" progressively smaller for larger electrode areas on the PZT. !%)" Actuation and sensing strategies: Minimizing the impact of on-chip, design dependent parasitic !%*"

capacitive effects !&+"

The generic equivalent electrical model of a piezo-cell (R s , R, C) can be subsequently !&!" implemented in piezoelectric transduction schemes [4,[START_REF] Nicu | &+[END_REF]15,16] in order to assess the configuration !&#" that minimizes the impact of parasitic capacitance on the actuation and sensing performance at the !&$" device level. In the present section, we will discuss two widely used actuation and sensing strategies !&%" and comment on the limitations imposed by parasitic effects inherent to silicon chip design. The first !&&" is the "duo-piezo-cell structure" which includes two separated piezo-cells on a single MEMS !&'" resonator, one dedicated to the actuation and the other to the detection. This is often achieved using !&(" clamped-clamped structures, such as bridges or membranes. The second is the "mono-piezo-cell !&)" structure" where the MEMS resonator needs only one piezo-cell. A second identical unreleased piezo-!&*" cell is placed next to the previous one to be used as reference.

!'+"

The "duo-piezo-cell structure" !'!" represented by two RLC equivalent circuits where the gain A corresponds to the electromechanical !)+" transduction coefficient The electrical response is simulated on a 1kHz -10MHz frequency range with !)!" a 100mV excitation amplitude. In addition, the mutual capacitance (C m ) between the two piezo-cells !)#" was considered to assess the impact of this supplementary parasitic element on the global electrical !)$" behavior of the resonator. A large resistance R m (of order 1G) was added to account for surface !)%" leakage currents between the two piezo-cells. Practically, this would describe finite leakages !)&" associated with environmental conditions such as moisture, or surface defects on the chip. For !)'" simplicity, the finite ac field dependence of the permittivity of the PZT films [17] was neglected. The !)(" simulation result for three different values of C m (0, 10fF and 200fF) is presented in Figure 3.(b).

!))"

For the low frequency range (i.e. below the cutoff frequency of the charge amplifier feedback !)*" filter) the output charge amplifier voltage is given by the ratio of the feedback resistance of the charge !*+" amplifier and R m , such as . For frequencies above the cutoff frequency, the influence of !*!" the mutual capacitance can be assessed by observing the modification of the shape of the resonant !*#" frequency peaks. For instance, in Figure 3.(b), in the case of the 200 fF value for C m , the peaks invert. !*$" At these frequencies, the magnitude of the plateau is given by . Thus, these parasitic !*%" effects result from the influence of C m and they cannot be avoided. The main advantage of a duo-!*&" piezo-cell actuation/sensing scheme is the suppression of the effect of the static capacitance of the !*'" sensing piezo-cell. However, the model demonstrates that this comes at a price of considerable impact !*(" of the mutual capacitance on the peak shape. This effect becomes increasingly important as the !*)" dimensions of the resonator shrink, as will be demonstrated in the experimental validation section of !**" this work.

#++"

The "mono-piezo-cell structure" #+!" sensing while a second remote piezo-cell is used as a reference capacitor. Comparable schemes have #+$" been used since Lee et al.'s seminal work [4] proposing the first atomic force microscope, where the #+%" optical detection scheme was replaced by an equivalent piezoelectric one. In this case, the charge #+&" amplifier voltage is given by where and , respectively, account for #+'" the piezoelectric film and the associated on-chip electrical wiring capacitances for each of the piezo-#+(" cells. The gain G allows the piezo-cells to be balanced, so that after tuning G, .

#+)"

In order to assess the global electrical behavior of the mono-piezo-cell configuration, the #+*" equivalent electrical model for the device shown in Figure 4.(a) was implemented using a PSpice #!+" simulator. Two resonant modes of the clamped-free beam (the same as for the previous case) were #!!"

represented by the resonances of two RLC equivalent circuits. In this configuration, the mutual #!#" capacitance between the piezo-cells has no effect on the output. Moreover, this design allows #!$" cancellation of the mutual capacitance due to the distance between the piezo-cells. The piezo-cells' #!%"

Gain = 0.8 Cap = 40pF fo = 500000 Hz Q = 300 Cm2 100f Rm2 10e9 Rp2 Lp2 Cp2 R1 10e9 Rp1 Lp1 Cp1 Rc 10meg C2 {Cap} R2 10e9 C1 {Cap*0.8} {piezo_coef f } {-Gain} U7 AD8065 + 3 - 4 V+ 5 V- 2 OUT 1 #!*" ,-. ##+" ! "!# $!# %!# ! "!# $!# %!# "& "!& "!!& "' "!' ! $# (# )# *# "!# +,-./0/!1*!2
345645/7895,:;/<7= +,-./0/!1>?2 / @A;B4;.CD +,-./0/!1* " ##!" ,/. ###" The simulation was performed for a S 2 cantilever (see Supplementary Information file) with a ##(" static equivalent capacitance of 40pF. The generated charge value from the released piezo-cell is equal ##)" to 3.2pC compared to 4pC generated by the reference piezo-cell because of the 20% difference of ##*" static capacitances. It is clearly seen from Figure 4.(b) that a compensation gain equal to 0.8 perfectly #$+" balances the charges of both piezo-cells. A much cleaner resonant characteristic also results from #$!" appropriate compensation of the static capacitances. A slight detuning of G to 0.795 or 0.805 #$#" immediately unbalances the piezo-cells compensation and subsequently produces residual #$$" corresponding charges of 20fC which increase the off-resonance voltage level to 20mV and hence #$%" dramatically impact the resonance shapes. Thus, for such an actuation/sensing scheme where the #$&" mutual capacitance effects are canceled by design, the compensation of static capacitance is key to #$'" obtain close-to-Lorentzian shaped resonance responses. #$("

Sensing specific electronics set-up and validation #$)"

To experimentally confirm the effects of the mutual capacitance and of capacitive variations as #$*" described in the previous sections, an electronics set-up was designed and constructed as illustrated in #%+" Converter used to drive the microcontroller unit (MCU) and DDS is the Direct Digital Synthesizer.

#%'"

A sinusoidal voltage of constant amplitude (10mV-4V) is applied to the piezo-cells for actuation #%(" by a Direct Digital Synthesizer (AD9832, Analog Device, USA). The same signal may be fed through #%)" a negative gain amplifier and subsequently applied to the piezo-cells used for compensation (in mono-#%*" piezo-cell configuration). The piezo-MEMS output signal is then amplified by a charge amplifier (1 #&+" V/pC) and fed through a wide band-pass filter (10 kHz-2 MHz) to eliminate low frequency and digital #&!" noise. The signal is then multiplied by itself and low-band filtered to obtain a DC signal corresponding #&#"

to its amplitude at a given frequency. This square of the signal is amplified by a settable gain (called #&$" the direct gain). The resulting analog signal is digitized to perform averaging, digital feedback for the #&%" frequency excitation, search for resonance modes in a given frequency band, calculation of the quality #&&" factor, and calculation and driving of the optimal direct gain. Data were sent through a USB #&'" (Universal Serial Bus) to a PC (Personal Computer) with a custom GUI (Graphical User Interface).

#&("

Further actions can be performed subsequently, including: control and setting of the electronic card, #&)" acquisition of wide spectrum response, display of specific resonant modes with associated quality #&*" factors assessment, monitoring of a specific resonant mode, etc.

#'+"

The importance of mutual capacitance effects is amplified when the actuation and sensing #'!" piezo-cells are on the same resonating structure. This configuration is met in the case of the circular #'#" membranes. The first spectrum in Figure 6 represents the effect of the mutual capacitance on the #'$" electrical dynamic response of a membrane (structure S 1 in Figure S3, on the membrane chip). A #'%" 30mV voltage is used to actuate the membrane on its inner electrode (the S 1i piezo-cell) while sensing #'&" is performed using the outer electrode (structure S 1o piezo-cell). It is worth noting that in Fig. 6(a) the #''" voltage response presents inverted resonant peaks. In contrast, the mutual capacitance is eliminated #'(" when a remote piezo-cell (say, S 40 ) on the same chip, is used for sensing. #')" #'*"

! "!! #!! $!! %!! &!!! &!!' "!!' (!!' #!!' )!!' $!!' *!!' %!!' ! &!! "!! (!! %!! &!!! &"!! &#!!
+,-./01/023450,672+83-+9-:;7</7=>?2+@A-#(+" #(!"

#(("

Making use of the same "mono-piezo-cell" configuration, a clamped-free beam bearing a single #()" piezo-cell for both actuation/sensing purposes (e. g. structure S 3 in Figure S3) was excited using a #(*" 40mV voltage. Sensing was done with a piezo-cell reference on a neighboring bridge that was #)+" immobilized using a photoresist drop. Three resonant spectra were recorded (Figure 7) with #)!" compensation gain values set such that the piezo-cells were either perfectly balanced or up to 3%

#)#" different. #)$" #)%" !""# !$ !"" %"" &"" !%"" !&"" !"" %"" '"" ("" !"" %"" &"" !%"" !&""
)*+,-).)"/01)2)&3 456786-9:);<=> )*+,-).)"/01 )*+,-).)"/01)?)%/@3 A8BC8B)DEFB+G6);HI> #)&" #)'" done in order to devise design rules that best exploit the capabilities of the integrated piezoelectric thin #**" films, while avoiding parasitic effects inherent to the integration at the micro-scale. Three main $++" parasitic effects, namely the electrical wiring capacitance, the mutual capacitance between $+!" actuation/sensing piezo-cells and variations in the capacitance of the actuation/sensing piezo-cells, $+#" have been identified, modeled and experimentally investigated. Strategies to minimize their impact on $+$" the resonant behavior of the piezo-MEMS were also discussed.

$+%"

An electrical model of a piezo-cell consisting of piezoelectric thin film sandwiched between two $+&" metallic electrodes was considered in order to systematically assess the role of the associated on-chip $+'" electrical network. Though this kind of consideration is a basic rule for microelectronics designers, it $+(" is often neglected in design of MEMS systems; this, in turn, may induce errors of interpretation of $+)" signals from the piezoelectric sensing piezo-cells. $+*" When two independent piezo-cells are co-hosted on the same microscale resonator, mutual $!+" capacitance between the piezo-cells induces parasitic effects that influence the net piezo-MEMS $!!" electrical behavior. We have proven by simulation that fF-level mutual capacitances may considerably $!#" affect the Lorentzian shape of the resonant peaks. It was shown experimentally that on a $!$" micromembrane geometry bearing both actuation/sensing piezo-cells, use of a remote, motionless $!%" piezo-cell as a reference eliminates this problem.

$!&"

Finally, the compensation of parasitic effects via a reference, unreleased piezo-cell is effective $!'" only if its capacitance value is identical to that of the active piezo-cell. This type of capacitive $!(" compensation scheme can be obtained by finely tuning the gain of the amplifier whose output voltage $!)" drives the reference piezo-cell. $!*" Hence, it is of paramount importance for MEMS designers to have a holistic approach when $#+" dealing with actuation and sensing capabilities integrated at the device level. This is especially true $#!" when the measured electrical charge levels reflect the device motion and when the dimensions of the $##" devices are on the micron scale. This kind of considerations will become progressively more important $#$" with the development of nano-electromechanical systems integrating actuation/sensing capabilities. $#%"

  -side polished silicon-on-insulator (SOI) wafer with a 2 m Si device layer &&" and a 1 m buried oxide layer was used as the starting substrate. The mask set for the wafer hosts chips &'" bearing cantilevers, bridges and membranes. The cantilevers and bridges are 50-m wide and paired, &(" with lengths of 300 m, 500 m or 700 m. The piezoelectric layer covers one quarter of the full length &)" of the cantilevers, beginning at the clamping point to the substrate and one eighth of the bridge length at &*" each of the ends. The membranes are circular with diameters of 50m, 200 m or 450 m. The '+" piezoelectric layer covers two distinct regions on each membrane: an inner full circle with a diameter '!"

Figure 1 :

 1 Figure 1: SEM images of micromachined (a) cantilevers, (b) bridges and (c) membrane devices. *'"

Figure 2 :

 2 Figure 2: (a) Equivalent circuit of a piezo-device; (b) Gain-phase and corresponding R p -C p frequency !#!"

Figure 3 .Figure 3 :

 33 Figure 3.(a) depicts a clamped-clamped resonator bearing two piezo-cells, one of which is used !'#"

Figure 4 .

 4 Figure 4.(a) depicts a clamped-free resonator bearing one piezo-cell used for actuation or #+#"

  capacitances values intentionally differ by 20% in this example in order to assess the role of #!&" fabrication variations on the global electrical behavior of the resonator, where cancels the #!'" and . The simulation result of slight variations of the gain around its #!(" optimal value (G being respectively equal to 0.795, 0.8 and 0.805) is shown in Figure 4.(b): #!)"

Figure 4 :

 4 Figure 4: Schematics and behavioral modeling of a clamped-free beam with a single actuation/sensing ##$"

Figure 5 Figure 5 :

 55 Figure 5. #%!"

Figure 6 :

 6 Figure 6: Experimental responses of a piezoelectrically actuated/sensed membrane using (a) separated #(#"

Figure 7 :

 7 Figure 7: Experimental dynamic responses of a piezoelectrically actuated/sensed cantilever bearing a #)("

Acknowledgements

$#&" This publication was supported by the Pennsylvania State University Materials Research Institute $#'" Nanofabrication Lab and the National Science Foundation Cooperative Agreement No. ECS-0335765. $#(" The French National Agency for Research (program ANR/PNANO 2008, project NEMSPIEZO $#)" ''ANR-08-NANO-015'') is gratefully acknowledged for financial support. $#*" $$+"