Amenable actions of amalgamated free products of free groups over a cyclic subgroup and generic property

BLAISE PASCAL

Soyoung Moon

Amenable actions of amalgamated free products of free groups over a cyclic subgroup and generic property Volume 18, n^o2 (2011), p. 211-229.

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Amenable actions of amalgamated free products of free groups over a cyclic subgroup and generic

property

Soyoung Moon

Abstract

We show that the amalgamated free products of two free groups over a cyclic subgroup admit amenable, faithful and transitive actions on infinite countable sets.

This work generalizes the results on such actions for doubles of free group on two generators.

Actions moyennables de produits amalgamés de groupes libres sur un sous-groupe cyclique et propriétés génériques

Résumé

On montre que les produits amalgamés de groupes libres sur un sous-groupe cyclique admettent des actions moyennables, fidèles et transitives sur un ensemble dénombrable infini. Ce travail généralise le résultat concernant de telles actions pour les produits amalgamés de groupes libres sur deux générateurs.

1. Introduction

An action of a countable group G on a setX is amenable if there exists a sequence {A_n}_n≥1 of finite non-empty subsets ofX such that for every g∈G, one has

n→∞lim

|A_nMg·A_n|

|A_n| = 0.

Such a sequence is called a Følner sequence for the action of G on X.

Thanks to a result of Følner [3], this definition is equivalent to the existence of a G-invariant mean on subsets ofX.

Keywords:Amenable actions, amalgamated free products, Baire’s property.

Math. classification:43A07, 20E06, 57M07.

Definition 1.1. We say that a countable group G is in the class A if it admits an amenable, faithful and transitive action on an infinite countable set.

The question of understanding which groups are contained in A was raised by von Neumann and recently studied in a few papers ([1], [4], [5], [6]). In this note we add the following:

Theorem 1.2. Let n, m≥1. Let G=Fm+1∗_ZFn+1 be an amalgamated free product of two free groups over a cyclic subgroup such that the image of the generator of Z is cyclically reduced in both free groups. Then any finite index subgroup of Gis in A.

The methods used in this work are analogous to those used in [6] to obtain Theorem 1.2 in case of m=n= 1. The role of the generic permu- tationαin [6] is now played by an-tuple of permutations (α1, ..., αn) and, for a cyclically reduced word c=c(α₁, ..., α_n), we now prove genericity of the set of such n-tuples for which the permutation c has infinitely many orbits of size k ∈ N^∗, and all orbits finite. This new result allows us to apply the method of [6] in our new setting.

ForXan infinite countable set, recall thatSym(X) with the topology of pointwise convergence is a Baire space, i.e. every intersection of countably many dense open subsets is dense in Sym(X). So for every n ≥ 1, the product space (Sym(X))ⁿ is a Baire space. A subset of a Baire space is calledmeagreif it is a union of countably many closed subsets with empty interior; and generic ordense G_δ if its complement is meagre.

Remark 1.3. The amalgamated products appearing in Theorem 1.2 are known in combinatorial group theory as “cyclically pinched one-relator groups” (see [2]). These are exactly the groups admitting a presentation of the formG=ha₁, . . . , a_n, b₁, . . . , b_m|c=diwhere 16=c=c(a₁, . . . , a_n) is a cyclically reduced non-primitive word (not part of a basis) in the free group Fn =ha₁, . . . , a_ni, and 1 6=d=d(b₁, . . . , b_m) is a cyclically reduced non- primitive word in the free group Fm =hb₁, . . . , bmi. The most important examples of such groups are the surface groups, i.e. the fundamental group of a compact surface. The fundamental group of the closed orientable sur- face of genus g has the presentation ha₁, b1, . . . , ag, bg|[a₁, b1]· · ·[ag, bg] = 1i. By letting c = [a1, b1]· · ·[ag−1, bg−1] and d = [ag, bg]⁻¹, the group decomposes as the free product of the free group F2(g−1) on a₁, b₁, . . .,

ag−1, bg−1 and the free group F2 on ag, bg amalgamated over the cyclic subgroup generated by c in F2(g−1) and d in F2, hence it is a cyclically pinched one-relator group.

Acknowledgment. This work is supported by the grant 20_118014/1 of the Swiss National Science Foundation (SNSF).

2. Graph extensions

A graph Gconsists of the set of verticesV(G) and the set of edgesE(G), and two applicationsE(G)→E(G);e7→e¯such that ¯¯e=eand ¯e6=e, and E(G)→V(G)×V(G); e7→(i(e), t(e)) such that i(e) =t(¯e). An element e ∈ E(G) is a directed edge of G and ¯e is the inverse edge of e. For all e∈E(G), i(e) is theinitial vertex of eand t(e) is the terminal vertex of e.

Let S be a set. A labeling of a graph G = (V(G), E(G)) on the set S^±1 =S∪S⁻¹ is an application

l:E(G)→S^±1;e7→l(e)

such thatl(¯e) =l(e)⁻¹. Alabeled graph G= (V(G), E(G), S, l) is a graph with a labeling l on the setS^±1. A labeled graph is well-labeled if for any edges e,e⁰ ∈E(G),i(e) =i(e⁰) and l(e) =l(e⁰) implies thate=e⁰.

A wordw=w_m· · ·w₁on{α^±1_n ,α^±1_n−1,. . .,α^±1₁ , β^±1}is calledreducedif w_k+1 6=w⁻¹_k ,∀1≤k≤m−1. A wordw=wm· · ·w1 on{α^±1_n ,α_n−1^±1 ,. . ., α^±1₁ ,β^±1}is calledweakly cyclically reducedifwis reduced andwm 6=w⁻¹₁ ; this definition allows w_m and w₁ to be equal. Given a reduced word, we define two finite graphs labeled on {α^±1_n ,α^±1_n−1,. . .,α^±1₁ , β^±1}as follows:

Definition 2.1. Letw=w_m· · ·w₁be a reduced word on{α^±1_k ,α_k−1^±1 ,. . ., α^±1₁ , β^±1}. The path of w (Figure 2.1) is a finite labeled graph P(w, v0) labeled on {α^±1_k ,. . .,α^±1₁ ,β}consisting ofm+ 1 vertices and mdirected edges {e₁,. . .,em}such that

• i(ej+1) =t(ej),∀1≤j≤m−1;

• v0 =i(e1)6=t(em);

• l(ej) =wj,∀1≤j ≤m.

e₁ e₂ v₀

e_i e_i+1 e_m

Figure 2.1. A path

e₁

e₂ e_m

e_m-1 v₀

Figure 2.2. A cycle

The pointv0 is thestartpoint and the pointt(em) is theendpoint of the path P(w, v₀). The two points are the extreme points of the path.

Definition 2.2. Letw=wm· · ·w1be a reduced word on{α^±1_k ,α_k−1^±1 ,. . ., α^±1₁ ,β^±1}. Thecycle of w (Figure 2.2) is a finite labeled graphC(w, v₀) labeled on {α^±1_k , . . ., α^±1₁ , β} consisting of m vertices and m directed edges {e₁,. . .,em}such that

• i(e_j+1) =t(e_j),∀1≤j≤m−1;

• v0 =i(e1) =t(em);

• l(ej) =wj,∀1≤j ≤m.

The point v₀ is the startpoint of the cycleC(w, v₀).

Notice that sincewis a reduced word, the graphP(w, v0) is well-labeled.

If wis weakly cyclically reduced, then C(w, v₀) is also well-labeled.

Conversely, if P ={e₁,e₂,. . .,e_n} is a well-labeled path with i(e₁) = v0, labeled by l(ei) = gi, ∀i, then there exists a unique reduced word w =g_n· · ·g₁ such that P(w, v₀) is P. If C = {e₁, e₂, . . ., e_n} is a well- labeled cycle with t(en) =i(e1) =v0, labeled by l(ei) =gi,∀i, then there exists a unique weakly cyclically reduced word w₁ = g_n· · ·g₁ such that C(w, v0) isC.

Let X be an infinite countable set. Let β be a transitive permutation of X. Thepre-graph G₀ is a labeled graph consisting of the set of vertices V(G0) = X and the set of directed edges all labeled by β such that every vertex has exactly one entering edge and one outgoing edge, and t(e) =β(i(e)). One can imagine G0 as the Cayley graph of Z with 1 as a generator.

Definition 2.3. Anextension ofG0 is a well-labeled graphG labeled by {α^±1_k ,α^±1_k−1,. . .,α^±1₁ ,β^±1}, containingG₀, withV(G) =V(G₀) =X. We will denote it by G0 ⊂G.

In order to have a transitive action with some additional properties of the hα_k, . . . , α1, βi-action on X, we shall extend inductively G0 on 1 ≤ i ≤k by adding finitely many directed edges labeled by α_i on G₀ where the edges labeled by β are already prescribed. In order that the added edges represent an action on X, we put the edges in such a way that the extended graph is well-labeled, and moreover we put an additional edge labeled by αi on every endpoint of the extended edges by αi; more precisely, if we have added nedges labeled by α_i betweenx₀,x₁, . . ., x_n successively, we put anαi-edge from xntox0 to have a cycle consisting of n+ 1 edges, which corresponds to a α_i-orbit of sizen+ 1. On the points where no αi-edges are involved, we can put anyαi-edge in a way that the extended graph is well-labeled and every point has an entering edge and an outgoing edge labeled byα_i (for example we can put a loop labeled by αi, corresponding to the fixed points). In the end, the graph represents an hα_k,. . .,α₁,βi-action onX, i.e.G will be a Schreier graph.

Definition 2.4. Let G,G⁰ be graphs labeled on a set S^±1. A homomor- phism f : G → G⁰ is a map sending vertices to vertices, edges to edges, such that

• f(i(e)) =i(f(e)) andf(t(e)) =t(f(e));

• l(e) =l(f(e)), for all e∈E(G).

If there exists an injective homomorphism f :G→G⁰, we say that f is an embedding, and Gembeds inG⁰.

Lemma 2.5. Let k ≥ 1. Let w_k = w_k(α_k, αk−1, . . . , α₁, β) be a reduced word on {α^±1_k , α^±1_k−1, . . ., α₁^±1, β^±1}. For every finite subset F of G₀,

there is an extension G of G0 on which the path P(w_k, v0) embeds in G, the image of P(w_k, v₀) inGdoes not intersect withF, andG\G₀ is finite.

Proof. Let us show this by induction onk. Ifk= 1, it follows from Propo- sition 6 in [6]. Indeed, in the proof of Proposition 6 in [6], we start by choosing any element z₀ ∈X to construct a path. Since the setX is infi- nite and there is no assumption on the starting pointz0 of the path, there are infinitely many choices for z₀.

For the proof of the induction step, consider the case w_k=α^a_k^2mw_k−1^2m−1α^a_k^2m−2· · ·α^a_k⁴w_k−1³ α^a_k²w¹_k−1.

with wⁱ_k−1 = w_k−1ⁱ (αk−1, . . . , α₁, β) a reduced word on {α^±1_k−1, . . ., α^±1₁ , β^±1}, for alli. To simplify the notation, we assume thata_j is positive,∀j.

LetF ⊂X be a finite subset ofX. By hypothesis of induction, there is an extensionG₁ofG₀and an embeddingf¹such thatf¹:P(w_k−1¹ , v₀),→ G₁ and the image of P(w¹_k−1, v₀) in G₁ does not intersect with F. Let

f¹(v₀) =f¹ i(P(w_k−1¹ , v₀))=:z₀ and

f¹ t(P(w¹_k−1, v0))=:z1.

Inductively on each 2≤i≤m, we apply the following algorithm:

Algorithm

(1) Take an extensionG2i−2 ofG₀ such that

• P(w²ⁱ⁻¹_k−1 , v2i−2) embeds inG2i−2such that the image does not intersect withF;

• G2i−2∩G2i−3=G0 (this is possible since there are infinitely many extensionsG⁰_2i−2 of G₀ by hypothesis of induction and G2i−3\G0 is finite).

(2) Letf²ⁱ⁻¹:P(w_k−1²ⁱ⁻¹, v2i−2),→G2i−2∪G2i−3 =:G⁰_2i−1 with

• f²ⁱ⁻¹ i(P(w²ⁱ⁻¹_k−1, v2i−2))=f²ⁱ⁻¹(v2i−2) =:z2i−2;

• f²ⁱ⁻¹ t(P(w_k−1²ⁱ⁻¹, v2i−2))=:z2i−1.

(3) Choosea2i−2−1 points{p^(a₁ ²ⁱ⁻²⁾,. . .,p^(a_a2i−2²ⁱ⁻²₋₁⁾ }outside of the finite set of all points appeared until now, and put the directed edges labeled by αk from

• z2i−3 top^(a₁²ⁱ⁻²⁾;

• p^(a_j ²ⁱ⁻²⁾ top^(a_j+1²ⁱ⁻²⁾,∀1≤j≤a2i−2−2;

• p^(a_a2i−2²ⁱ⁻²₋₁⁾ toz2i−2,

and let G2i−1 := G⁰_2i−1∪ {the additional α_k-edges between z2i−3

and z2i−2}.

In the ends, we choose new a2m points {p^(a₁ ^2m), . . ., p^(aa2m^2m)} and put the directed edges labeled byα_k fromz2m−1 top^(a₁ ^2m), and fromp^(a_j ^2m) to p^(a_j+1^2m),∀1≤j≤a_2m−1.

By construction, the resulting graphG2m−1∪P(α^a2m, z2m−1) =:Gis an extension ofG₀satisfyingP(w_k, v₀),→Gsuch that the image ofP(w_k, v₀)

does not intersect with F.

Lemma 2.6. Letw=w(αn, . . . , α1, β)be a weakly cyclically reduced word on {α^±1_n , . . ., α^±1₁ , β^±1} such that αi appears in the word w for some i (i.e. w /∈ hβi). For every finite subset F of G₀, there exists an extension Gn+1 of G0 such that the cycle C(w, v0) embeds in Gn+1 and the image of C(w, v₀) in G₀ does not intersect withF.

Proof. Let us consider the case

w=α^a_i^2mw2m−1α^a_i^2m−2· · ·α^a_i⁴w3α^a_i²w1

written as the normal form of hα_n, . . . , α_i+1, αi−1, . . . , α₁, βi ∗ hα_ii.

Since w⁰ = w2m−1α^a_i^2m−2· · ·α^a_i⁴w3α_i^a2w1 is reduced, by Lemma 2.5, there is an extension G⁰_n+1 of G₀ and a homomorphism f : P(w⁰, v₀) → G⁰_n+1such thatf(P(w⁰, v₀)) is a path inG⁰_n+1outside ofF. Letf(v₀) =:z₀ be the startpoint of f(P(w⁰, v0)) andf(w⁰(z0)) =:z2m−1 be the endpoint of f(P(w⁰, v₀)). To simplify the notation, we assume that a_j is positive,

∀j.

Choose a_2m −1 new points {p_am, . . ., p_a2m−1} and put the directed edges labeled by αi from

• z2m−1 top1;

• pj topj+1,∀1≤j ≤a2m−2;

• pa2m−1 toz0.

e1

e2

i(e₁) = i(e₂)

e₁~ e₂

Figure 2.3

P₁ P₂ P₃

Figure 2.4

By construction, the resulting graphGn+1:=G⁰_n+1∪P(α^a2m, z2m−1) is an extension of G₀ andC(w, v₀) embeds inG_n+1 outside ofF. Let c =c(αn, . . . , α1, β) be a weakly cyclically reduced word on {α^±1_n , . . .,α^±1₁ ,β^±1}such thatc /∈ hβi and letw=w(α_n,αn−1,. . .,α₁,β) be a reduced word on {α^±1_n ,. . .,α^±1₁ , β^±1}such that w /∈ hci. Let C(c, v0) be the cycle ofcwith startpoint atv₀, and letP(w, v₀) be the path ofwwith the same startpointv₀asC(c, v₀) such that every vertex ofP(w, v₀) (other thanv0) is distinct from every vertex inC(c, v0). Letwv0 be the endpoint ofP(w, v₀). LetC(c, wv₀) be the cycle ofcwith startpoint atwv₀such that every vertex of C(c, wv0) (other than wv0) is distinct from every vertex in P(w, v₀)∪C(c, v₀). Let us denote by Q₀(c, w) the union of C(c, v₀), P(w, v0) and C(c, wv0). Let Q(c, w) be the well-labeled graph obtained from Q₀(c, w) by identifying the successive edges with the same initial vertex and the same label. That is, Qis the quotient graph Q₀/[e₁ ∼e₂] where e1 ∼e2 ifi(e1) =i(e2) and l(e1) =l(e2) (see Figure 2.3).

Notice that the well-labeled graph Q(c, w) can have one, two or three cycles, and in each type of Q(c, w), the quotient map Q0(c, w)Q(c, w) restricted to C(c, v₀) and to C(c, wv₀) is injective (each one separately).

Lemma 2.7. There is an extensionG_n+1 of G₀ such that Q(c, w)embeds in G_n+1.

Proof. By Lemma 2.5 and 2.6, it is enough to show that every cycle in Q contains edges labeled by α^±1_i for some i. For the cases where Q has one or two cycles, it is clear since the cycles in Q represent C(c, v0) and C(c, wv₀), andc /∈ hβi. In the case whereQ(c, w) has three cycles,Q(c, w) has three paths P1, P2 and P3 such that P1 ∩P2 ∩P3 are exactly two extreme points of P_i’s, and P₁ ∪P₂, P₂∪P₃ and P₁∪P₃ are the three cycles in Q(c, w) (see Figure 2.4). So we need to prove that, if one of the three paths has edges labeled only on {β^±1}, then the other two paths both contains edges labeled by α^±1_i for some i. For this, it is enough to prove:

Claim. If the reduced wordc=γλ is conjugate to the reduced wordγλ⁰ via a reduced wordw, whereγ ∈ hα_n,αn−1,. . .,βi \ hβiandλ∈ hβi, then wc = cw. Furthermore, the word c can not be conjugate to the reduced word γ⁻¹λ⁰ withλ⁰ ∈ hβi.

Let us see how we can conclude Lemma 2.7 using the Claim. First of all, notice thatcdoes not commute withwsince we are treating the case where Qhas three cycles. More precisely, in a free group, two elements commute if and only if they are both powers of the same word. So if cw=wc, then c = γ^k and w =γ^l with k 6=l, whereγ is a non-trivial word, so that Q has one cycle. Suppose that P₁ consists of edges labeled only on {β^±1}.

One of the cycles among P1∪P2, P2∪P3 and P1∪P3 consists of edges labeled by the letters ofcup to cyclic permutation, let us sayP₁∪P₂ (i.e.

ifc=c1· · ·cm, given any startpointv0inP1∪P₂, the directed edges of the cycleC(c, v₀) are labeled on a cyclic permutation of the sequence{c_m,. . ., c₁}). Another cycle amongP₂∪P₃ andP₁∪P₃ consists of edges labeled by the letters of the reduced form of w⁻¹cw up to cyclic permutation. Since c /∈ hβi, the path P₂ has edges labeled by α^±1_i for some i. Now, if the cycle representing w⁻¹cw is P₁∪P₃, then the path P₃ has edges labeled by α^±1_i since w⁻¹cw /∈ hβi and P1 has only edges labeled on {β^±1}(this is because two words in the free group F define conjugate elements of F if and only if their cyclic reduction in F are cyclic permutations of one another). Suppose now that the cycle representing w⁻¹cw is P2∪P3 and P₃ has edges labeled only on{β^±1}. Then, cwould be the form γλup to cyclic permutation where γ ∈ hα_n, αn−1, . . ., βi \ hβi (representing P2) and λ ∈ hβi (representing P₁); and w⁻¹cw would be the form γ^±1λ⁰ up to cyclic permutation where λ⁰ ∈Fn(representing P3); but the Claim tells

us that this is not possible, thereforeP3 contains edges labeled by α^±1_i for somei.

Now we prove the Claim. Let c = γλ and w⁻¹cw = γλ⁰ such that γ ∈ hα_n,αn−1,. . .,βi \ hβiandλ,λ⁰∈ hβi. Without loss of generality, we can suppose that γ =γmλm−1· · ·λ1γ1, with γi ∈ hα_n,αn−1, . . ., βi \ hβi and λi∈ hβi. Sinceγλand γλ⁰ are conjugate in a free group, there exists 1≤k≤m such that

γkλk−1· · ·λ1γ1λγmλm−1· · ·γk+1λk =γλ⁰ =γmλm−1· · ·λ1γ1λ⁰. By identification of each letter, one deduces that λ⁰ =λ_k =λ_j, for every j multiple of k in Z/mZ, and λ = λm−k. In particular, λ = λ⁰ so that c = γλ = γλ⁰ = w⁻¹cw and thus cw = wc. For the seconde statement, suppose by contradiction that there exists w such that w⁻¹cw = γ⁻¹λ⁰. Then by the similar identification as above we deduce that λ⁻¹ =λ⁰, so w⁻¹cwwould be a cyclic permutation ofc⁻¹, which is clearly not possible.

3. Construction of generic actions of free groups

LetXbe an infinite countable set. We identifyX=Z. Letβbe a transitive permutation of X (which is identified to the translationx7→x+ 1).

Let c be a non trivial weakly cyclically reduced word on {α^±1_n , α^±1_n−1, . . .,α^±1₁ ,β^±1}such that the sumS_c(β) of the exponents ofβ in the word c is zero. Thus necessarily ccontainsαi for somei.

Let us denote byS⁺_c(β) the sum of positive exponents ofβ in the word c; by denotingS_c⁻(β) the sum of negative exponents ofβ in the wordc, we have 0 = Sc(β) =S_c⁺(β) +S_c⁻(β) (for example, ifc =α1β⁻¹α2β⁻¹α²_nβ², then S_c⁺(β) = 2). Ifcdoes not contain β, we set S_c⁺(β) = 0.

Let{A_m}_m≥1 be a sequence of pairwise disjoint intervals ofXsuch that

|A_m| ≥ m+ 2S_c⁺(β), ∀m ≥1. Clearly this sequence is a pairwise disjoint Følner sequence for β.

Proposition 3.1. Let c be a weakly cyclically reduced word as above.

There existsα = (α₁, . . . , α_n)∈(Sym(X))ⁿ such thathα_n,αn−1, . . ., α₁, βi is free of rank n+ 1, and

(1) the action ofhα_n,αn−1,. . .,α1,βionX is transitive and faithful;

(2) for all non trivial word w on {α^±1_n , α^±1_n−1, . . ., α^±1₁ , β^±1} with w /∈ hci, there exist infinitely many x ∈ X such that cx = x, cwx=wx and wx6=x;

(3) there exists a pairwise disjoint Følner sequence {A_k}_k≥1 for hα_n, αn−1, . . ., α₁, βi which is fixed by c, and |A_k|=k, ∀k≥1;

(4) for all k≥1, there are infinitely many hci-orbits of sizek;

(5) every hci-orbit is finite;

(6) for every finite index subgroup H of hα_n, αn−1, . . ., α1, βi, the H-action on X is transitive.

With the notion of the permutation type, the conditions (4) and (5) mean that the word c has the permutation type (∞,∞,. . ., ; 0).

Proof. For the proof, we are going to exhibit six generic subsets of (Sym(X))ⁿ that will do the job.

We start by claiming that the set U₁ =

nα= (α₁, . . . , αn)∈(Sym(X))ⁿ∀_k∈Z\{0},∃x∈X such that c^kx6=x^o is generic in (Sym(X))ⁿ. Indeed, for every k ∈ Z\ {0}, let V_k = {α ∈ (Sym(X))ⁿ|∀x ∈ X, c^kx = x}. The set V_k is closed since if {γ_m}_m≥1 is a sequence in V_k converging to γ, then c^k(γm) converges to c^k(γ). To see the interior of V_k is empty, let α ∈ V_k and let F ⊂X be a finite subset.

There is an extensionG_n+1 ofG₀ such that P(c^k(α⁰), v₀) embeds inG_n+1 outside of F by Lemma 2.5. So in particular there is x∈X\F such that c^k(α⁰)x6=x, so α⁰ ∈ V/ _k. By definingα⁰|_F =α|_F, we have shown that U₁ is generic in (Sym(X))ⁿ.

Let us show that the set

U₂=ⁿα= (α₁, . . . , αn)∈(Sym(X))ⁿ∀w6= 1∈ hα_n, . . . , α₁, βi \ hci, there exist infinitely many x∈Xsuch that

cx=x,cwx=wxand wx6=x^o is generic in (Sym(X))ⁿ.

Indeed, for every non trivial word w in hα_n, . . . , α1, βi \ hci, let V_w = {α∈(Sym(X))ⁿ|there exists a finite subset K ⊂X such that (Fix(c)∩ w⁻¹Fix(c) ∩supp(w)) ⊂ K} = ^S_Kfinite⊂X{α ∈ (Sym(X))ⁿ|(Fix(c) ∩ w⁻¹Fix(c)∩supp(w))⊂K}. We shall show that the setV_w is meagre. It is an easy exercise to show that the set

V_w,K ={α∈(Sym(X))ⁿ|(Fix(c)∩w⁻¹Fix(c)∩supp(w))⊂K}

is closed. To show that the interior of V_w,K is empty, let α ∈ V_w,K, and F ⊂ X be a finite subset. We need to prove that for some α⁰ defined as α⁰|_F =α|_F, we can extend the definition ofα⁰ outside of the finite subset such that α⁰ ∈ V/ _w,K. By Lemma 2.7, we can take an extension Gn+1 of G₀ such thatQ(c(α⁰), w) embeds inG_n+1 outside ofF∪α(F)∪K, which proves the genericity ofU₂.

Now let us show that the set

U₃= { α= (α1, . . . , αn)∈(Sym(X))ⁿ|∃{A_mk}_k≥1 a subsequence of {A_m}_m≥1 such that A_mk ⊂Fix(α_i),∀k≥1,∀1≤i≤n}

is generic in (Sym(X))ⁿ.

Indeed, the set U₃ can be written as U₃ = ^T_N≥1{α = (α1, . . . , αn) ∈ (Sym(X))ⁿ|∃k≥N such thatAk ⊂Fix(αi),∀i}. We claim that for every N ≥1, the set V_N ={α∈(Sym(X))ⁿ|∀k≥N,A_k(∩_iFix(αi)}is closed and of empty interior. It is closed sinceV_N =^T_k≥N{α ∈(Sym(X))ⁿ|A_k(

∩_iFix(α_i)}and the set{α∈(Sym(X))ⁿ|A_k (∩_iFix(α_i)}is clearly closed.

For the emptiness of its interior, let α ∈ V_N and let F ⊂ X be a finite subset. Let k ≥ N such that A_k∩(F ∪α(F)) = ∅. We can then take α⁰ ∈(Sym(X))ⁿ fixingAk and satisfying α⁰|_F =α|_F.

For (4), we show that the set

U₄= { α= (α1, . . . , αn)∈(Sym(X))ⁿ|for everym, there exist infinitely manyhci-orbits of size m}

is generic in (Sym(X))ⁿ.

For all m ≥1, let V_m ={α ∈(Sym(X))ⁿ| there exists a finite subset K ⊂Xsuch that everyhci-orbit of sizemis contained inK}=^S_Kfinite⊂X V_m,K, where

V_m,K ={α∈(Sym(X))ⁿ|if|hci ·x|=m, thenhci ·x⊂K}.

· V_m,K is of empty interior. LetF ⊂Xbe a finite subset. Letα∈ V_m,K. Takex /∈(F∪α(F))∪K. Sinceccontainsα_i for somei, we can construct a cyclec^m(α⁰) outside ofF∪α(F)∪Ksuch thatα⁰|_F =α|_F (Lemma 2.6), so that the orbit of x underα⁰ is of sizem and not contained inK.

· V_m,K is closed. Let{γ_l}_l≥1 ⊂ V_m,K converging toγ ∈(Sym(X))ⁿ. Let x∈Xsuch that|hc(γ)i·x|=m. Sinceγ_lconverges toγ,c(γ_l) converges to c(γ). Sincehc(γ)i·xis finite, there existsl₀such thathc(γ)i·x=hc(γ_l)i·x,

∀l ≥ l0. Since γ_l ∈ V_m,K and m = |hc(γ)i ·x| = |hc(γ_l)i ·x|, we have hc(γ_l)i ·x⊂K,∀l≥l₀. Therefore hc(γ)i ·x⊂K, so that γ ∈ V_m,K.

About (5), we prove that the set

U₅ ={α= (α₁, . . . , α_n)∈(Sym(X))ⁿ|∀x∈X,hci ·x is finite} is generic in (Sym(X))ⁿ.

For all x∈X, let V_x ={α ∈(Sym(X))ⁿ|hci ·x is infinite }. It is clear that the setV_xis closed. To see that the interior ofV_x is empty, letF ⊂X be a finite subset and letα∈ V_x. We shall show that there existsα⁰ ∈ V/ _x such that α|_F =α⁰|_F. Denotec=c(α) and c⁰ =c(α⁰). We choose p >>1 large enough so that

B(c^−p−1x,|c|)∪B(c^p+1x,|c|)∩(F ∪α(F)) =∅;

(F∪α(F))⊂B(x,|c^p|),

where |c|is the length ofc and B(x, r) is the ball centered onx with the radius r.

We construct a path ofc⁰outside ofB(x,|c^p|) starting fromc^p+1xwhich ends on c^−p−1x, i.e. c⁰(c^p+1x) =c^−p−1x. This is possible since c⁰ contains αi for some i(Lemma 2.5). On the points inB(x,|c^p+1|), we define

α⁰|_B(x,|cp+1|)=α|_B(x,|cp+1|). In particular, α⁰|_F =α|_F, and |hc⁰i ·x|is finite.

Finally for (6), let

U₆= { α= (αn, . . . , α1)∈(Sym(X))ⁿ|for every finite index subgroup H of hα₁, βi, the H-action on X is transitive}.

By Proposition 4 in [6], the set W = {α₁ ∈ Sym(X)| for every finite index subgroupHofhα₁, βi, theH-action onXis transitive}is generic in Sym(X). ThusU₆is generic in (Sym(X))ⁿsinceU₆=W ×(Sym(X))ⁿ⁻¹.

Now let α = (α1, . . . , αn) ∈ ∩⁶_i=1U_i. It remains us to prove (3) and (6) in the Proposition. To simplify the notation, let A_m := A_mk be the subsequence of Am fixed by αi,∀1≤i≤n(genericity of U₃).

Without loss of generality, let c = w1β^b1w2β^b2· · ·wlβ^bl, where wj are reduced words on {α^±1_n ,. . ., α^±1₁ }, ∀1 ≤ j ≤ l. Recall that {A_m}m≥1 is a sequence of pairwise disjoint intervals such that |A_m| ≥ m+ 2S⁺_c(β).

If c does not containβ, then we can take the subinterval A⁰_m of A_m such that |A⁰_m|=m for the Følner sequence which is fixed by c. If not, for all m > S_c⁺(β), let

Em=β^b1(Am)∩β^b2+b1(Am)∩ · · ·

∩β^{bl−1+bl−2+···+b1}(Am)∩β^{bl+bl−1+···+b1}(Am).

Notice that β^{bl+bl−1+···+b1}(A_m) =A_m. We claim that the setE_m is not empty. Indeed, for every 1≤i≤l, the set

β^{bi+bi−1+···+b1}(A_m)∩β^{bp+bp−1+···+b1}(A_m)

is not empty, ∀1 ≤ p ≤ i−1 since |b_i +bi−1 +· · ·+bp+1| ≤ S_c⁺(β) <

|A_m|. Moreover, a family of intervals which meet pairwise, has non-empty intersection so that E_m 6=∅.

In addition, let us show that c fixes the elements of Em. Let x ∈ E_m and let 1 ≤ p ≤ l −1. There exists al−p+1 ∈ A_m such that x = β^{bl−p+bl−p−1+···+b1}(al−p+1). Then

β^{bl−p+1+···+bl−1+bl}(x) = β^{bl+bl−1+···+bl−p+1}(x)

= β^{bl+bl−1+···+bl−p+1}·β^{bl−p+bl−p−1+···+b1}(al−p+1)

= al−p+1 ∈A_m.

Sincew_jfixes every element inA_m, and the elementβ^{bl−p+1+···+bl−1+bl}(x) is in Am for every 1≤p≤l−1, the word cfixes x,∀x∈Em. Clearly the set Em is a Følner sequence forhα_n,αn−1,. . .,α₁,βi.

Furthermore, we have

A_m∩β^Sc+(β)A_m∩β^S−c(β)A_m⊆E_m, and

|A_m∩β^Sc+(β)A_m∩β^S−c(β)A_m|=|A_m| −2S_c⁺(β)≥m.

So |E_m| ≥m, and upon replacingE_m by a subintervalE_m⁰ ofE_m such that |E_m⁰ | = m, we can suppose that |E_m| = m, ∀m ≥ 1. Thus the

sequence {E_m}_m≥1 is a Følner sequence satisfying the condition in (3) in the Proposition 3.1.

Furthermore, if H is a finite index subgroup of hα_n, . . . , α1, βi, then Q=H∩hα₁, βiis a finite index subgroup ofhα₁, βi, so by the genericity of U₆theQ-action is transitive and therefore theH-action onXis transitive.

4. Construction of Fⁿ⁺¹∗_ZF^m+1-actions, n, m≥1

Let X be an infinite countable set. LetG=hα_n,αn−1, . . ., α1, βi y X be the group action constructed as in Proposition 3.1 with the pairwise disjoint Følner sequence {A_k}_k≥1. Let H =hα_m, αm−1, . . ., α1, βi y X be the group action constructed as in Proposition 3.1 with the pairwise disjoint Følner sequence{B_k}k≥1 and letdbe a weakly cyclically reduced word on {α_m,αm−1,. . .,α1,β}that does the role ofcin Proposition 3.1.

Let Z = {σ ∈ Sym(X)|σc =dσ}. By virtue of the points (4) and (5) of Proposition 3.1, the set Z is not empty. Let

H^σ =σ⁻¹Hσ=hσ⁻¹αmσ, σ⁻¹αm−1σ, . . . , σ⁻¹α1σ, σ⁻¹βσi.

Forσ∈Z, consider the amalgamated free productG∗_hc=diH^σofGand H^σ along hc=di. The action of G∗_hc=diH^σ onX is given byg·x=gx, and h·x=σ⁻¹hσx,∀g∈Gand ∀h∈H.

Notice that the set Z is closed inSym(X). In particular, Z is a Baire space.

Proposition 4.1. The set

O₁ ={σ ∈Z | the action of G∗_hc=diH^σ on X is faithful } is generic in Z.

Proof. For every non trivial word w∈ G∗_hc=diH^σ, let us show that the set

V_w ={σ ∈Z|∀x∈X, w^σx=x}

is closed and of empty interior. It is obvious that the set V_w is closed.

To prove that the set V_w is of empty interior, let us treat the case where w = agnhn· · ·g1h1 with a ∈ hci, gi ∈ G\ hci, and hi ∈ H \ hdi, n ≥ 1. The corresponding element of Sym(X) given by the action is w^σ = agnσ⁻¹hnσ· · ·g1σ⁻¹h1σ. Let σ ∈ V_w. Let F ⊂X be a finite subset. We

shall show that there exists σ⁰ ∈ Z \ V_w such that σ⁰|_F = σ|_F. For all g∈G\ hciand h∈H\ hdi, let

bg={x∈X |cx=x, cgx=gxand gx6=x }, bh={x∈X|dx=x, dhx=hxand hx6=x }.

By (2) of Proposition 3.1, these sets are infinite.

Choose any x0 ∈ Fix(c)\(F ∪σ(F)). By induction on 1 ≤i ≤n, we choose x4i−3 ∈hb_i such that x4i−3, h_ix4i−3 ∈/ (F ∪σ(F)) are new points.

This is possible since hb_i is infinite. Then we define

σ⁰(x4i−4) :=x4i−3 and σ⁰(σ⁻¹(x4i−3)) :=σ(x4i−4).

We set x4i−2 := h_ix4i−3, which is different from x4i−3 and which is fixed by d, by definition of h^b_i. We choose x4i−1 ∈g_bi such that x4i−1,g_ix4i−1 ∈/ (F∪σ(F)) are again new points. This is again possible sinceg_bi is infinite.

Then we define

σ⁰(x4i−1) :=x4i−2 and σ⁰(σ⁻¹(x4i−2)) :=σ(x4i−1).

We finally set x4i := gix4i−1. Then every point x on which σ⁰ is defined verifies σ⁰c(x) =dσ⁰(x). Indeed,

• σ⁰c(x4i−4) = σ⁰(x4i−4) = x4i−3 = d(x4i−3) = dσ⁰(x4i−4) since x4i−4 ∈Fix(c) and x4i−3∈Fix(d);

• σ⁰c(σ⁻¹(x4i−3)) = σ⁰(σ⁻¹(x4i−3)) = σ(x4i−4) = dσ(x4i−4) = dσ⁰(σ⁻¹(x4i−3)) since σ⁻¹(x4i−3) ∈Fix(c) and σ(x4i−4) ∈Fix(d) because σ∈Z;

• σ⁰c(x4i−1) = σ⁰(x4i−1) = x4i−2 = d(x4i−2) = dσ⁰(x4i−1) since x4i−2 ∈Fix(d) andx4i−1 ∈Fix(c);

• σ⁰c(σ⁻¹(x4i−2)) = σ⁰(σ⁻¹(x4i−2)) = σ(x4i−1) = dσ(x4i−1) = dσ⁰(σ⁻¹(x4i−2)) since σ⁻¹(x4i−2) ∈Fix(c) and σ(x4i−1) ∈Fix(d) because σ∈Z.

By construction, the 4n points defined by the subwords on the right of w^σ0 are all distinct. In particular, w^σ0x₀ = x_4n 6= x₀. If w = h ∈ H \ {Id}, choose x₀ ∈ Fix(c)\(F ∪σ(F)), x₁ ∈ ˆh\(F ∪σ(F)∪ {x₀}), x2 ∈Fix(c)\(F∪σ(F)∪ {x₀, x1}) and define σ⁰(x0) =x1, σ⁰(x2) =hx1, σ⁰(σ⁻¹(x1)) =σ(x0), σ⁰(σ⁻¹(hx1)) =σ(x2) so that w^σ0x0 =x2 6=x0. At