Nancy Guelman
email: nguelman@fing.edu.uy

Isabelle Liousse
email: liousse@math.univ-lille1.fr

ANY BAUMSLAG-SOLITAR ACTION ON SURFACES WITH A PSEUDO-ANOSOV ELEMENT HAS A FINITE ORBIT

Keywords: Mathematics Subject Classification. Primary: 37C85, 37E30; Secondary: 37B05 Actions on surfaces, Baumslag Solitar group, minimal sets, pseudo-Anosov

We consider f, h homeomorphims generating a faithful BS(1, n)-action on a closed surface S, that is, hf h -1 = f n , for some n ≥ 2. According to [GL], after replacing f by a suitable iterate if necessary, we can assume that there exists a minimal set Λ of the action, included in F ix(f ).

Here, we suppose that f and h are C 1 in neighbourhood of Λ and any point x ∈ Λ admits an h-unstable manifold W u (x). Using Bonatti's techniques, we prove that either there exists an integer N such that W u (x) is included in F ix(f N ) or there is a lower bound for the norm of the differential of h only depending on n and the Riemannian metric on S.

Combining last statement with a result of [AGX], we show that any faithful action of BS(1, n) on S with h a pseudo-Anosov homeomorphism has a finite orbit containing singularities of h ; moreover if f is isotopic to identity it is entirely contained in the singular set of h. As a consequence, there is no faithful C 1 -action of BS(1, n) on the torus with h an Anosov.

Introduction and statements

In 1962, Gilbert Baumslag and Donald Solitar defined the Baumslag-Solitar groups, which play an important role in combinatorial group theory, geometric group theory and dynamical systems. In this paper we will focus on dynamical aspects. Dynamic of Baumslag-Solitar groups in dimension 1, is well understood:

For actions of Baumslag-Solitar groups on the circle, Burslem and Wilkinson ([BW]) gave a classification, up to conjugacy, of real analytic actions. In [START_REF] Guelman | C 1 actions of Baumslag Solitar groups on S 1[END_REF], the authors gave a classification, up to semi-conjugacy of C 1 -actions. Recently Bonatti, Monteverde, Navas and Rivas ( [BMNR]) generalized these results proving a classification, up to conjugacy of C 1 -actions.

The question of existence of global fixed points or finite orbits for group actions on surfaces has been extensively studied and one can consider that the abelian case is now well understood: In 1964, in the context of Lie groups, Lima ([Li]) proved that commuting vectors fields have a common singularity. In 1989, Bonatti ([Bo]) proved that two sufficiently C 1 -close to identity commuting diffeomorphisms of the 2-sphere have a common fixed point. In 2008, Franks, Handel and Parwani ([FHP]) proved that an abelian group of C 1 -diffeomorphisms isotopic to identity of S has a global fixed point, where S is a closed surface of genus at least 2.

It is natural to ask what are the algebraic conditions for the existence of a global fixed point. In [Pl], Plante extended Lima's result for nilpotent Lie groups and exhibited solvable Lie groups acting without global fixed point on surfaces.

In [HW], Hirsch and Weinstein showed that every analytic action of a connected supersoluble Lie group on a non zero Euler characteristic compact surface has a fixed point.

For fixed points of local actions by Lie groups on surfaces, we refer to Hirsch's survey (see [START_REF] Hirsch | Fixed points of Local Actions of Lie groups on on Real and Complex 2-Manifolds[END_REF]) and [START_REF] Hirsch | Fixed points of local actions of nilpotent Lie groups on surfaces[END_REF]).

In [DDF], Druck, Fang and Firmo proved a discrete version of Plante's result on nilpotent groups of diffeomorphisms of the 2-sphere. Very recently, Firmo and Ribón gave topological conditions to insure the existence of finite orbits for nilpotent groups of C 1diffeomorphisms of the 2-torus (see [FR]).

In some sense, solvable Baumslag-Solitar groups, BS(1, n), represent the next discrete step: they are non nilpotent discrete subgroups of the supersoluble Lie group A consisting in homeomorphisms of the real line having the form x → ax+b, a > 0, b ∈ R and therefore are metabelian (2-step solvable).

In this direction, A.McCarthy in [McC], proved that the trivial BS(1, n)-action on a compact manifold does not admit C 1 faithful perturbations.

Before stating our results, we give some definitions and properties.

Let n ∈ N, n ≥ 2, the solvable Baumslag-Solitar group BS(1, n) is defined by

BS(1, n) =< a, b | aba -1 = b n > .
It is well known that BS(1, n) can be represented as the subgroup of A generated by the two affine maps f 0 (x) = x + 1 and h 0 (x) = nx (where f 0 ≡ b and h 0 ≡ a).

In what follows, we will always denote by < f, h > an action of BS(1, n) on a surface, meaning that the homeomorphisms f and h satisfy h

• f • h -1 = f n .
One can easily check the following properties:

• h•f k •h -1 = f nk = (f k ) n for any k ∈ Z, so < f k , h > is also an action of BS(1, n), • h k • f • h -k = f n k for any k ∈ N • h -k • f n k • h k = f for any k ∈ N As consequence we have that: • h -k (F ixf ) ⊂ h -k (F ix(f n k )) = F ix(f ) for any k ∈ N.
In [GL], the authors of this paper exhibited actions of Baumslag-Solitar groups on T 2 without finite orbit (this construction extends on any compact surface) and proved the following:

Theorem. [GL] Let f 0 , h be two homeomorphims, generating a faithful action of BS(1, n) on S. There exists a positive integer N such that f = f N 0 satisfies: • f is isotopic to identity, • f, h generate a faithful action of BS(1, n) on S and • F ixf = ∅ and contains a minimal set Λ for < f, h >, that is also a minimal set for h.

The first point is stated in [GL] for S = T 2 . For closed orientable surface of genus greater than 1, it is a consequence of the facts that the mapping class group of S does not contain distortion element, according to [FLM] and f is a distortion element. A corollary of this Theorem is that there is no minimal actions of BS(1, n) on closed orientable surfaces.

Remark 1. According to last Theorem, we may assume without loss of generality that f is isotopic to identity and F ix(f ) is not empty, noting that if < f N , h > has a finite orbit

F then O f (F ), the f -orbit of F is a finite orbit of < f, h >. Indeed, O f (F ) is finite and h(O f (F )) ⊂ O f (F ) (for x = f k (p) with p ∈ F , one has h(x) = h(f k (p)) = f nk (h(p)) with h(p) ∈ F ).
In [AGX], Alonso, Guelman, Xavier proved that Theorem. [AGX] Let f, h be an action of BS(1, n) on a closed surface S, where f is isotopic to the identity, and h is a (pseudo)-Anosov homeomorphism with stretch factor λ > n. Then f = Id.

We will now fix some definitions, notations and recall some properties that will be used.

Notations. Let S be a closed connected oriented surface embedded in the 3-dimensionnal Euclidean space R 3 , endowed with the usual norm denoted by || ||.

• The distance in S associated to the induced Riemannian metric is denoted by d.

• The injectivity radius of the exponential map associated to d is denoted by ρ.

• The open 2-ball centered at x with radius r with respect to d is denoted by B r (x) or B r for the case that we don't need to specify the center.

Remark.

In what follows, we always assume that radius of balls are strictly less than ρ.

Let F : S → S be a map.

Definition. Let Λ be an F -invariant compact set, we say that

F is C 1 Λ if F is C 1 on a neighbourhood W of Λ in S. Suppose that F is C 1 on an open set W of S.

Notations.

Let x ∈ W , we will denote by:

• DF (x) = D x F : T x S → R 3 the differential at x of F considered as a map F : S → R 3 , • DF (x) = D x F : T x S → T F (x) S the differential at x of F considered as a map F : S → S and • ||D x F || = sup {||D x F.v||, v ∈ T x , ||v|| = 1}. We set S(F, W ) = sup {||D x F ||, x ∈ W }.

Definitions.

The C 1 -norm of F on a subset V of W is defined by:

||F || V = sup x∈V (||F (x)|| + ||D x F ||) .
We say that

F is C 1 -close to identity on V if ||F -Id|| V < . We define V (F ) = {x ∈ S : ||(F -Id)(x)|| + ||D x (F -Id)|| < }. Note that this defi- nition implies that F is C 1 at x, for any x ∈ V (F ).

Properties.

There is a constant

C S ≥ 1 such that for all x, y ∈ B R ⊂ W , with R < ρ, one has (1) ||x -y|| ≤ d(x, y) < C S ||x -y||, (2) d(F (x), F (y)) ≤ S(F, B R )d(x, y).
Theorem 1. Let f, h be two homeomorphisms, generating a faithful action of BS(1, n) on S, Λ be a minimal set of < f, h > included in F ixf and f, h are C 1 Λ . 1. Any point x in Λ is an f -elliptic fixed point, in the sense that eigenvalues of D x f are roots of unit. More precisely, there exists a positive integer N such that the eigenvalue of D x f N is 1, for any x ∈ Λ.

2.

Moreover, for all > 0, there exists δ > 0 and a

C 1 Λ -diffeomorphism f ∈< f, h > such that : • f , h generate a faithful action of BS(1, n) on S and Λ ⊂ F ixf ⊂ F ix(f N ),
where N is given by previous item. • ||f -Id|| B δ (Λ) ≤ , where B δ (Λ) is the union of balls of center in Λ and radius δ (in other words,

B δ (Λ) ⊂ V (f )). More precisely, f is either f N or some n k -root of f N .
Thus, following Mc Carthy ( [McC]), we can adapt Bonatti's tools ( [Bo]) for estimating the norm of the differential of h. More precisely, we prove Theorem 2. Let f , h and Λ as in Theorem 1, let W be a neighbourhood of Λ such that f and h are in C 1 on W . Suppose that any point x ∈ Λ has an h-unstable manifold, W u (x), then either:

(1) there exists N ∈ N such that W u (x) ⊂ F ixf N , for all x ∈ Λ or (2) S(h, W ) ≥ n C S .
Assumption. We refer to [FM], for the definition and properties of pseudo-Anosov homeomorphisms and we always assume that a pseudo-Anosov homeomorphism is a C 1diffeomorphism except at finitely many points: the singularities of the stable and unstable foliations. We recall that if p is a singularity of a foliation, there exists a smooth chart from a neighbourhood of p to the plane that sends leaves to the level sets of a k prong saddle, with k ≥ 3.

Using the result of [AGX], we obtain as a corollary:

Corollary 1. Any faithful action < f, h > of BS(1, n) on S, where f is a C 1 -diffeomorphism isotopic to identity and h is a pseudo-Anosov homeomorphism has a finite orbit. Moreover, this finite orbit is contained in the set of singularities of h and in the set of the fixed points of f .

Using Remark 1 and algebraic properties of BS(1, n), Corollary 1 can be restated as:

Corollary 2. Any Baumslag-Solitar action on a surface with a pseudo-Anosov element has a finite orbit.

Since any pseudo-Anosov homeomorphism of the torus has no singularities, that is, it is an Anosov, then Corollary 3. There is no faithful action < f, h > of BS(1, n) on the torus, where f is a C 1 -diffeomorphism and h is an Anosov.

Proof of Theorem 1.

Proof of item 1. Let f and Λ as in Theorem 1 and x 0 ∈ Λ.

Since Λ is also an h-minimal set, the h-orbit of x 0 is recurrent. Then there exists a subsequence

(n k ) (n k → ∞) such that h -n k (x 0 ) → x 0 . From h n k • f • h -n k = f n n k
, we deduce that:

Dh n k (f (h -n k (x 0 ))) • Df (h -n k (x 0 )) • Dh -n k (x 0 ) = Df n n k (x 0 ).
Moreover, the points x 0 and h -n k (x 0 ) are fixed by f and (Dh

-n k (x 0 )) -1 = Dh n k (h -n k (x 0 )), then: (Dh -n k (x 0 )) -1 • Df (h -n k (x 0 )) • Dh -n k (x 0 ) = (Df (x 0 )) n n k .
So Df (h -n k (x 0 )) and (Df (x 0 )) n n k have same eigenvalues.

Let us denote by: ρ

+ [resp. ρ + k ] the maximum modulus of eigenvalues of Df (x 0 ) [resp. Df (h -n k (x 0 ))] and ρ -[resp. ρ - k ] the minimum modulus of eigenvalues of Df (x 0 ) [resp. Df (h -n k (x 0 ))]. Hence ρ + k = (ρ + ) n n k and ρ - k = (ρ -) n n k . As f is C 1 Λ and h -n k (x 0 ) → x 0 , then Df (h -n k (x 0 )) → Df (x 0 ). Therefore, ρ + k → ρ + and ρ - k → ρ -. Consequently, ρ + = ρ -= 1 and eigenvalues of Df (x 0 ) have modulus 1.
Suppose that Λ is infinite. This means that x 0 is not isolated in F ixf . We are going to prove that 1 is an eigenvalue of Df (x 0 ). By contradiction, suppose that 1 is not an eigenvalue of Df (x 0 ).

Let us introduce local coordinates near x 0 . The map F = f -Id is locally invertible by the Inverse Function Theorem. More precisely, there exist neighborhoods U of x 0 and V of F (x 0 ) = (0, 0) such that F : U → V is a diffeomorphism, hence F -1 (0, 0) = x 0 , which means that x 0 is the unique fixed point of f in U . This contradicts the fact that x 0 is not isolated in F ix(f ).

Consequently, if Λ is infinite, 1 is the unique eigenvalue of Df (x) (since f is isotopic to identity, it is also orientation preserving and -1 can not be an eigenvalue of Df (x)), for any x ∈ Λ.

If Λ is finite, there exists p such that h p (x 0 ) = x 0 . As Df (x 0 ) and (Df (h p (x 0 )) n p = (Df (x 0 )) n p are conjugate, eigenvalues of Df (x 0 ) are roots of unity. Thus, there exists a positive integer N such that eigenvalues of Df N (x 0 ) are 1. As Df N (h -j (x 0 )) and (Df N (x 0 )) n j are conjugate, we get that eigenvalues of Df N (h -j (x 0 )) are 1.

Since Λ = {(h -j (x 0 ), j ∈ N}, we get that eigenvalues of Df N (x) are 1, for any x ∈ Λ.

Proof of item 2. We begin by proving Lemma 2.1. There exists a positive integer N such that:

-either Df N (x) = Id for any x ∈ Λ -or Df N (x) = Id for any x ∈ Λ.

Proof of Lemma 2.1. Let x 0 ∈ Λ, if Df N (x 0 ) = Id. As Df N (h -p (x 0 )) and (Df N (x 0 )) n p are conjugate, we get that Df N (h -p (x 0 )) = Id, for any p ∈ N. Since {h -p (x 0 ), p ∈ N} is dense in Λ, we get that Df N (x) = Id, for any x ∈ Λ.

We can now prove item 2. Fix > 0, N given by item 1. We set f = f N , the action of < f = f N , h > on S is an action of the Baumslag-Solitar group BS(1, n).

Let x 0 ∈ Λ.

Case 1 : D f (x 0 ) = Id. By Lemma 2.1, for all x ∈ Λ, D f (x) = Id and we can find an open ball B r (x) contained in V ( f ). We claim that δ = M in{r x , x ∈ Λ} > 0, where r x is the greater number such that B r (x) ⊂ V ( f ). Indeed, suppose that there exists a sequence x p such that r xp → 0. Without loss of generality, we can suppose that x p → w. By Proposition 2.1, w ∈ Λ and D f (w) = Id. For p sufficiently large x p ∈ B rw 2 (w) and then r p ≥ rw 2 , that is a contradiction.

Hence the δ-neighborhood of Λ is contained in V ( f ) and we conclude by setting f = f . Case 2 : D f (x 0 ) = Id. By Lemma 2.1, D f (x) = Id, for all x ∈ Λ. We can choose in a continuously way an orthonormal basis on each T x S, x ∈ Λ ∩ B, where B is a small open ball centered at x 0 .

According to item 1 and Lemma 2.1, for all x ∈ Λ∩B the matrix of D f (x) (also denoted by D f (x)) is conjugate to 1 1 0 1 by some matrix A x that depends continuously on x.

So norms of A x are uniformly bounded on Λ ∩ B.

Let denote fk

= h -k • f • h k .
As f is isotopic to identity and has fixed points, its conjugate fk is isotopic to identity and has fixed points. One can check that:

(1) fk is a n k -root of f that is

f n k k = (h -k • f • h k ) n k = (h -k • f n k • h k ) = f , (2) < fk , h > generate a faithful action of BS(1, n) on S, that is h • fk • h -1 = h -k+1 • f • h k-1 = h -k • (h • f • h -1 ) • h k = h -k • f n • h k = f n k , (3) F ix( fk ) ⊂ F ix( f ) since h -k (F ix( f )) ⊆ F ix( f ), and (4) Λ ⊂ F ix( fk ), as Λ ⊂ F ix( f ) and h k (Λ) = Λ. Let x ∈ Λ ∩ B, since f and fk commute, the matrices 1 1 0 1 = A x D f (x)A -1 x and A x D fk (x)A -1
x commute. This implies that A x D fk (x)A -1

x is upper-triangular. But D f (x) and D fk (x) are conjugate by Dh k (x), so A x D fk (x)A -1

x has a unique eigenvalue 1, then it is of the form 1 α 0 1 .

Finally 1 1 0 1 = A x D f (x)A -1 x = (A x D fk (x)A -1 x ) n k = 1 α 0 1 n k = 1 αn k 0 1 and then α = 1 n k . Therefore D x ( fk -Id) = A -1 x 0 1 n k 0 0 A x .
Since the norms of the A x are uniformly bounded on Λ ∩ B and we choose orthonormal basis for tangent spaces, we can find an integer k such that sup x∈Λ∩B ||D x ( fk -Id)|| is strictly less than , for any k ≥ k .

Hence for any x 0 ∈ Λ we can find an open ball B r (x 0 ) contained in V ( fk ), therefore, by compactness of Λ there is a positive uniform radius δ such that the δ-neighborhood of Λ is contained in V ( fk ).

We conclude by setting f = fk . This ends the proof of Theorem 1.

3. Proof of Theorem 2.

Flow-like properties.

In this subsection we will prove a local version of Bonatti flow-like properties for C 1closed to identity diffeomorphisms.

Proposition 3.1. Given n ∈ N, for all 0 < η < 1, there exists n η > 0 such that:

||f n (y) -y -n(f (y) -y)|| ≤ η||f (y) -y)||,
for any f ∈ Homeo(S), δ ∈ (0, ρ m ) and y ∈ S satisfying y ∈ B δ (x 0 ) and

B mδ (x 0 ) ⊂ V n η (f ), where x 0 ∈ F ix(f ) and m = (1 + 3C S (n + 1)). Remark 2. Note that B mδ (x 0 ) ⊂ V n η (f ) means that f is C 1 and n η C 1 -close to identity on B mδ (x 0 ).
Proof. Let 0 < η < 1 be given. Proceed using induction on k = 1, ..., n.

The k = 1 case is trivial. Choose 1 η = min ρ m , η 4n , 1 2 . Let k ∈ {1, ..., n -1}, we suppose that ∀η ∈ (0, 1) there exists k η such that for f ∈ Homeo(S), x 0 ∈ F ix(f ) and δ ∈ (0, ρ m ) such that B mδ (x 0 ) ⊂ V k η (f ), we have ||f k (y) -y -k(f (y) -y)|| ≤ η||f (y) -y)||, ∀y ∈ B δ (x 0 ). We have to estimate ||f k+1 (y) -y -(k + 1)(f (y) -y)||. ||f k+1 (y)-y-(k+1)(f (y)-y)|| ≤ ||(f -Id)f k (y)-(f -Id)(y)||+||f k (y)-y-k(f (y)-y)||. We choose, k+1 η ≤ k η 2 , then we get that ||f k (y) -y -k(f (y) -y)|| ≤ η 2 ||f (y) -y)||, for all y ∈ B δ (x 0 ) and B mδ (x 0 ) ⊂ V k+1 η (f ).
To bound the first term, we first show that

f k (y) ∈ B mδ (x 0 ) provided that y ∈ B δ (x 0 ) and B mδ (x 0 ) ⊂ V k+1 η (f ). Indeed d(f k (y), x 0 ) ≤ d(x 0 , y)+d(f k (y), y) ≤ d(x 0 , y)+C S ||f k (y)-y|| ≤ d(x 0 , y)+C S (k+ η 2 )||f (y)-y|| by inductive hypothesis. Moreover, ||f (y) -y|| ≤ ||f (y) -f (x 0 )|| + ||x 0 -y|| ≤ ||f || B δ (x 0 ) d(x 0 , y) + d(x 0 , y) ≤ (||f || B δ (x 0 ) + 1)δ. Since B δ (x 0 ) ⊂ V k+1 η (f ) and k+1 η < 1, we have ||f || B δ (x 0 ) ≤ 2 and therefore we get ||f (y) -y|| ≤ 3δ. Finally d(f k (y), x 0 ) ≤ (1 + 3C S (n + 1))δ, that is f k (y) ∈ B mδ (x 0 ).
The first term can be bounded as follows:

||(f -Id)f k (y)-(f -Id)(y)|| ≤ ||(f -Id)|| B mδ (x 0 ) ||f k (y)-y|| ≤ ||(f -Id)|| B mδ (x 0 ) (k+ η 2 )||f (y)-y|| by inductive hypothesis. Then ||(f -Id)f k (y) -(f -Id)(y)|| ≤ k η 2 (k + η 2 )||f (y) -y||. Moreover k η 2 (k + η 2 ) = k η 2 k + k η 2 η 2 ≤ ηk 4n + η 4
, by the choice of 1 η and the fact that the k η can be chosen to be decreasing in k and in such a way that given k ≤ n, one has k η < k η for 0 < η < η.

Finally, we can bound the first term by η 2 ||f (y) -y|| and therefore one has

||f k+1 (y) -y -(k + 1)(f (y) -y)|| ≤ η||f (y) -y||, for all y ∈ B δ (x 0 ) and B mδ (x 0 ) ⊂ V k+1 η (f ).
Corollary 4. Given n ∈ N, n > 1, there exists > 0 such that for any δ ∈ (0, ρ m ) and f ∈ Homeo(S) such that B mδ (x 0 ) ⊂ V (f ), any f n -fixed point y ∈ B δ (x 0 ) is fixed by f , where x 0 ∈ F ix(f ) and m is defined as in Proposition 3.1.

Proof. By Proposition 3.1 for η = 1, there exists = n 1 > 0 such that:

||f n (y) -y -n(f (y) -y)|| ≤ ||f (y) -y)|| for any f ∈ Homeo(S), x 0 ∈ F ix(f ), y ∈ B δ (x 0 ) and B mδ (x 0 ) ⊂ V (f ).
Then ||f n (y) -y|| ≥ (n -1)||(f (y) -y)||, therefore if y is f n -fixed we have that y is f -fixed.

3.2. Proof of Theorem 2. The key tool of proof of Theorem 2 is the following: Proposition 3.2. Let δ ∈ (0, ρ m ), let h, f, Λ and W as in Theorem 2 verifying that: • for any x, y with d(x, y) < δ, one has d(f (x), f (y)) < ρ 2 , • B δ (Λ) and f (B δ (Λ)) are contained in W ,

• any point of Λ admits an h-unstable manifold. Let 0 < η ≤ 1 such that f -Id B mδ (Λ) < n η , where n η and m are given by Proposition 3.1 ( i. e. B mδ (Λ)

⊂ V n η (f )). If S(h, W ) ≤ n-η
Cs , then there exists r > 0 such that for any x ∈ Λ,

W u loc (x) ∩ B r (x) ⊂ F ix(f ).
Proof. Let x 0 ∈ Λ, there exists p 0 = p 0 (x 0 , δ) such that for all y ∈ W u loc (x 0 ) ∩ B δ (x 0 ), one has d(h -p (x 0 ), h -p (y)) < δ for any p ≥ p 0 . Note that A =

p 0 -1 j=0 h j (B δ (h -j (x 0 )) is a
non-empty open set that contains x 0 , then it intersects W u loc (x) in an open arc containing x 0 .

More precisely, by hyperbolicity, there exists a constant c = c(h) such that Bδ

c (x 0 ) ∩ W u loc (x 0 ) ⊂ h j (W u loc (h -j (x 0 )) ∩ B δ (h -j (x 0 ))), ∀j ∈ N. Then B δ c (x 0 ) ∩ W u loc (x 0 ) ⊂ A.
From now, we will denote r = δ c and W u loc (x 0 ) ∩ B r (x 0 ) by W r (x 0 ). First, we prove that any point y ∈ W r (x 0 ) is f -periodic. Let y ∈ W r (x 0 ), it is easy to check that for all p ∈ N, h -p (y) ∈ B δ (h -p (x 0 )). Indeed, if p ≥ p 0 we have d(h -p (x 0 ), h -p (y)) < δ, by definition of p 0 . And if p < p 0 , by definition of r and A we have that y ∈ h p (B δ (h -p (x 0 ))).

If f (y) = y, in particular y is f -periodic. If f (y) = y , there exists some l = l(y) such that h -l (y) is close enough to h -l (x 0 ) ∈ F ixf to ensure that f (h -l (y)) -h -l (y) < f (y) -y . Therefore there exists some p

= p(y) ≤ l, such that f (h -(p+1) (y)) -h -(p+1) (y) < f (h -p (y)) -h -p (y) .
Indeed, by contradiction, let us denote y j := h -j (y), for j ∈ N, we would have :

||f (y) -y|| ≤ ||f (y 1 ) -y 1 || ≤ ... ≤ ||f (y l ) -y l || < ||f (y) -y||. Notice that as h -(p+1) (y) ∈ B δ (h -(p+1) (x 0 )), we have that d(y p+1 , f (y p+1 )) ≤ d(y p+1 , h -(p+1) (x 0 )) + d(f (y p+1 ), h -(p+1) (x 0 )) ≤ ≤ δ + d(f (y p+1 ), f (h -(p+1) (x 0 ))) ≤ ρ. Since y p+1 and f (y p+1 ) are in W , one has d(h(f (y p+1 )), h(y p+1 )) ≤ S(h, W ).d(f (y p+1 ), y p+1 ).
We adapt McCarthy's argument:

f n (y p ) -y p = hf h -1 (y p ) -y p ≤ d(hf h -1 (y p ), y p ) = d(hf (y p+1 ), h(y p+1 )) ≤ S(h, W ).d(f (y p+1 ), y p+1 ) ≤ S(h, W )C S f (y p+1 ) -y p+1 < S(h, W )C S f (y p ) -y p . Since, y p ∈ B δ (h -p (x 0 )) and B mδ (h -p (x 0 )) ⊂ V n η (f ), we can apply Proposition 3.1 to obtain f n (y p ) -y p ≥ (n -η) f (y p ) -y p and consequently (n -η) f (y p ) -y p < S(h, W )C S f (y p ) -y p .
Therefore, either (n -η) < S(h, W )C S that contradicts the hypothesis or f (y p ) -y p = 0 which implies that y p = h -p (y) is f -fixed, then y is f n p -fixed. Hence we have proven that any point y ∈ W r (x 0 ) is f -periodic. Now, we will prove that any point y ∈ W r (x 0 ) is f -fixed.

By contradiction, let y ∈ W r (x 0 ) \ F ixf and p 0 be the smallest integer such that y is f n p 0 -fixed point. Since F ix(f n p ) = h p (F ix(f )), we have that p 0 is the smallest integer such that h -p 0 (y) is in F ix(f ).

By definition of p 0 we have that h -(p 0 -1) (y) is in F ix(f n ) \ F ix(f ). And by definition of r and A, h -(p 0 -1) (y) ∈ B δ (h -(p 0 -1) (x 0 )). This is impossible, according to Corollary 4.

End of the proof of Theorem 2.

Let η > 0 and n η given by Proposition 3.1. Suppose that S(h, W ) ≤ n-η C S . By Theorem 1, there exists δ > 0 and f η = f n η such that ||f η -Id|| B mδ (Λ) < n η and f η , h generate a faithful action of BS(1, n) on S and Λ ⊂ F ix(f η ).

By Proposition 3.2, there exists r such that for all x ∈ Λ, W r (x) ⊂ F ix(f η ).

Let x 0 ∈ Λ, we are going to prove that W u (x 0 ) ⊂ F ixf η . By contradiction, suppose that W u (x 0 ) is not contained in F ix(f η ). Therefore, there exists a segment [a, b] in W u (x 0 ) and c ∈ (a, b) such that [a, c] ⊂ F ix(f η ) and (c, b) is disjoint of F ix(f η ).

Let p be an integer such that h -p ((c, b)) ∈ B r (h -p (x 0 )) ∩ W u (h -p (x 0 )) ⊂ F ix(f η ). Hence (c, b) ⊂ F ix(f n p η ). We have proven that there exists a segment [a, b] ) is a f η -invariant continuum, one can check that main Theorem of [Wea] applies. Then all but a finite number of points in C have the same least period. So all but a finite number of points in [a, b] are fixed by f η , this is a contradiction.

Finally, we have proved that given η > 0, either S(h, W ) > n-η C S or W u (x 0 ) ⊂ F ix(f η ) ⊂ F ix(f N ).

Therefore, if W u (x 0 ) is not included in F ix(f N ) then for any η > 0, S(h, W ) > n-η C S . Let η converges to 0, then if W u (x 0 ) is not included in F ix(f N ) then S(h, W ) ≥ n C S .

  in W u (x 0 ) and c ∈ (a, b) such that [a, c] ⊂ F ix(f η ) and (c, b) ⊂ F ix(f n p η ) \ F ix(f η ). The set C = n p -1 k=0 f k η ([a, b]

Proof of Corollaries 1 and 2.

In this section, we prove that any Baumslag-Solitar action with a pseudo-Anosov element has a finite orbit.

Proof of Corollary 1. Let < f, h > be a faithful action of BS(1, n) on S. We suppose that f is a C 1 diffeomorphism isotopic to identity and h is pseudo-Anosov homeomorphism. According to Theorem [GL], f admits fixed points.

Let k ∈ N, f and h k generate a faithful action of BS(1, n k ) and by Theorem [GL], there is a minimal subset Λ k of this action that is included in F ix(f ). As h k is a pseudo-Anosov homeomorphism of S, any point admits an h k -unstable manifold.

Note that for all k ∈ N, h k is C 1 on S except at Σ the set of singularities of h. Suppose that the compact set F ix(f ) is disjoint from Σ, then there exists a neighbourhood W of F ix(f ) and of Λ k , for any k ∈ N such that f and [FM]).

Hence, we can apply Theorem 2 to f and h k , and we obtain that either

(1) there exists

In the first case, as h k is pseudo-Anosov homeomorphism , there exists x ∈ Λ k such that the h k -unstable manifold of x is dense in S and therefore F ix(f N ) = S. Hence f N = Id and the action is not faithful.

Consequently, it holds that S(h k , W ) ≥ n k C S , for all k ∈ N. Finally, This implies that λ ≥ n.

According to [AGX], λ ≤ n and therefore λ = n ∈ N but this is not possible. We conclude that F ix(f ) ∩ Σ = ∅, as Σ is a finite h-invariant set, we deduce that the action has a finite orbit contained in F ix(f ) ∩ Σ.

Proof of Corollary 2. Indeed, if an element g of < f, h >= BS(1, n) acts on S as a pseudo-Anosov then it is the image of a strict homothety of BS(1, n) (since the homotopy class of the image of a translation has finite order according to [GL]).

Eventually changing g by its inverse and conjugating by a translation element, we can suppose that g is the image of x → n p x, for some positive integer p, that is g = h p . Therefore < f, g > BS(1, n p ), by Corollary 1, F ixf N (N given by Theorem [GL]) contains a singular periodic point of g and therefore a singular periodic point of h. Hence < f N , h > has a finite orbit and according to the Remark 1, < f, h > also has a finite orbit.


Nancy Guelman

Isabelle Liousse

INTERVAL EXCHANGE TRANSFORMATIONS GROUPS FREE ACTIONS AND DYNAMICS OF VIRTUALLY ABELIAN GROUPS

Keywords: 2020 Mathematics Subject Classification. 37E05, 57S30, 37C85, 20K35, 20F50 Interval exchange transformations, virtually abelian groups, free actions, dynamical classification

In this paper, we study groups acting freely by IETs. We first note that a finitely generated group admits a free IET action if and only if it is virtually abelian. Then, we classify the free actions of non virtually cyclic groups showing that they are "conjugate" to actions in some specific subgroups G n , namely G n ≃ (G 2 ) n ⋊ S n where G 2 is the group of circular rotations seen as exchanges of 2 intervals and S n is the group of permutations of {1, ..., n} acting by permuting the copies of G 2 .

We also study non free actions of virtually abelian groups and we obtain the same conclusion for any such group that contains a conjugate to a product of restricted rotations with disjoint supports and without periodic points. As a consequence, we get that the group generated by f ∈ G n periodic point free and g / ∈ G n is not virtually nilpotent. Moreover, we exhibit examples of finitely generated non virtually nilpotent subgroups of IETs, some of them are metabelian and others are not virtually solvable.

Introduction.

Definition 1.1. An interval exchange transformation (IET) is a bijective map f : [0, 1) → [0, 1) defined by a finite partition of the unit interval into half-open subintervals and a reordering of these intervals by translations. If a partition has cardinality r, we say that f is an r-IET. We denote by G the group consisting of all IETs.

Circular rotations identify with 2-IETs and an IET g whose support is a subinterval I = [a, b) is a restricted rotation if the orientation preserving affine map from I to [0, 1) conjugates g |I to a 2-IET. It is denoted by R α,I where α ∈ [0,b-a) b-a∼0 is represented by g(a) -a.

An affine interval exchange transformation (AIET) is a bijective map f : [0, 1) → [0, 1) defined by a finite partition of the unit interval into half-open subintervals such that the restriction to each of these intervals is an orientation preserving affine map.

An AIET that is a homeomorphism of [0, 1) is called PL-homeomorphism.

A group is said to act freely if the only element acting with fixed points is the trivial element, in particular a free action is faithful.

By the classification of finitely generated abelian groups, it is plain that any such group G admits a free IET action by rotations. But, Hölder's Theorem (see e.g. Theorem 2.2.32 of [START_REF] Navas | Groups of circle diffeomorphisms[END_REF]) stating that groups that act freely by circle homeomorphisms are abelian is no longer true for free IET actions.

Indeed, it is easy to see that any finite group F has a free IET action by taking a partition in #F subintervals of equal length labelled with the elements of F and the action by left multiplication in the group F . However, some infinite non abelian groups can also admit free IET actions: we claim that the following IETs a and b generate a free and minimal action of the Klein bottle group BS(1, -1) := ⟨ a, b | bab -1 = a -1 ⟩ provided that 1, α, β 1 and β 2 are rationally linearly independent numbers. 0

1 2 1 1 2 1 α 1 -α 1 2 -α 1 2 + α IET a 0 1 2 1 1 2 + β1 β2 IET b Figure 1
Note that this example is a subgroup of some very specific groups of G, namely the groups G n described in the following Definition 1.2. Let n be a positive integer and S n = [0, 1 n ]/0= 1 n be the circle of length 1 n . We define G n as the set of IETs on [0, 1) that preserve the partition [0, 1) = [0,

1 n ) ∪ [ 1 n , 2 n )... ∪ [ n-1 n , 1
) and whose restrictions to the intervals

I i = [ i-1 n , i n ), i ∈ {1, • • • , n},
are IETs with only one interior discontinuity.

For g ∈ G n , we define σ g as the element of S n given by σ g (i) = j if g(I i ) = I j .

It follows that, for x ∈ I i , one has g(x) = R α i ,I i (x) + σ g (i) -i n .

We define the rotation vector of g by α g = (α 1 , ..., α n ) = (α 1 (g), ..., α n (g)) ∈ S n n and we denote g = (α g , σ g ).

The group G 1 is the group of circular rotations R α (α ∈ S 1 ), regarded as 2-IETs.

Examples 1. The IET a and b, represented above by their graphs, are expressed as:

a, b ∈ G 2 ,
α a = (α, -α) σ a = Id and α b = (β 1 , β 2 ) σ b is the transposition (1, 2) .

Remark 1.3. A straightforward consequence of this definition is that G n is a group and composing two elements f and g of G n , we get

α f •g = σ g (α f ) + α g σ f •g = σ f • σ g
, where σ g acts on the vector α f by permuting its coordinates. Therefore, the map (α, σ) : G n → (S n ) n ⋊ S n is an isomorphism. In particular, the group G n is virtually abelian. Note that G n can be seen as the isometry group of the disjoint union of n circles of the same length 1/n and adopting this point of view, it follows from Dahmani, Fujiwara and Guirardel (Proposition 1.2 of [START_REF] Dahmani | Solvable groups of interval exchange transformations[END_REF]) that any finitely generated virtually abelian group is isomorphic to a subgroup of some G n .

In Section 2, we will specify the related construction of [START_REF] Dahmani | Solvable groups of interval exchange transformations[END_REF] in order to provide free actions, this leads to Proposition 1.4. (The Dahmani-Fujiwara-Guirardel Proposition 1.2 revisited) A finitely generated group admits a free IET action if and only if it is virtually abelian.

The main result of this paper establishes that this revisited construction is "basically" the only one that provides free IET actions of non virtually cyclic groups. It is related to the dynamical Hölder's Theorem: "free actions on the circle are topologically semiconjugate to actions of rotations groups" (see e.g. comments following Theorem 2.2.32 in [START_REF] Navas | Groups of circle diffeomorphisms[END_REF]), which suggests that it makes sense to give a dynamical classification of free actions by elements of G. We will do this classification up to conjugacy by maps that are not required to be in G, however, as noted by the referees, our conclusion can be reformulated in terms of G-conjugation (see Theorem 1' and its preliminary comments).

Actually our conjugating maps are given by Definition 1.5.

An element f ∈ G is PL • G-conjugate to F ∈ G n , if f is conjugate to F through a map P • E, where E ∈ G and P is a PL-homeomorphism such that P -1 is affine on [ k-1 n , k n ) for any k ∈ {1, ..., n}. Theorem 1.
The image in G of any free action of a finitely generated non virtually cyclic group is PL • G-conjugate to a subgroup of some G n .

Roughly speaking, being PL•G-conjugate to a subgroup of some G n can be reformulated as being G-conjugate to a subgroup of products of restricted rotations on finitely many disjoint sub-intervals modulo a permutation of them. More precisely, Theorem 1'. Any free IET-action of a finitely generated non virtually cyclic group is G-conjugate to a subgroup of the semi-direct product G(P ) := R(P ) ⋊ S(P ) < G for some partition P of [0, 1) in half-open intervals, where R(P ) denotes the group of all products of restricted rotations supported in intervals of P and S(P ) the subgroup of the symmetric group on P consisting of elements that only exchange intervals of the same size.

The first step in the proof of Theorem 1 is Proposition 1.6. Any free action of Z 2 in G consists of periodic point free maps having an iterate that is conjugate in G to a product of restricted rotations with disjoint supports. In particular its image contains a conjugate to a product of restricted rotations with disjoint supports and without periodic points.

It is not plain that if all elements of a finitely generated subgroup of G have an iterate that is conjugate in G to a product of restricted rotations with disjoint supports, then they are simultaneously conjugate to their models. However, dealing with virtually abelian groups, the situation turns out to be much more rigid and this conclusion can be proved only requiring the existence of one conjugate in G to a product of restricted rotations with disjoint supports and without periodic points.

The proof of Theorem 1 does not strictly require actions to be free, but rather the presence of specific elements that occur when considering sufficiently large free actions. Moreover, the finite generation condition is involved only when applying Proposition 1.4.

More precisely, the second step is Theorem 2. Let G be a virtually abelian subgroup of G. If G contains an element that is conjugate in G to a product of restricted rotations with disjoint supports and without periodic points, then G is PL • G-conjugate to a subgroup of some G n .

Some extra dynamical assumptions provide stronger rigidity results:

Definition 1.7.

• An IET f is totally minimal if for all p ∈ Z * , the map f p is minimal. More generally:

• An action by elements of G of a group G is totally minimal if any finite index subgroup of G acts minimally on [0, 1).

Remark. According to [START_REF] Keane | Interval exchange transformations[END_REF], an IET that satisfies the Keane I.D.O.C condition is totally minimal.

Corollary 1. Let G be virtually abelian group containing a copy of Z 2 . Then any faithful totally minimal action of G in G is conjugate in G to an action by rotations.

By Schreier Lemma and the classification of finitely generated abelian groups, any finitely generated virtually abelian group is either virtually cyclic or it contains a copy of Z 2 . Combining this with the theorem of [START_REF] Dahmani | Solvable groups of interval exchange transformations[END_REF] stating that virtually polycyclic IET subgroups are virtually abelian, we get Corollary 2. Let G be a finitely generated subgroup of G that contains a totally minimal IET non conjugate to a rotation then G is either virtually cyclic or not virtually polycyclic.

Using Theorem 2 and its ingredients of proof, we can exhibit explicit examples of non virtually polycyclic subgroups of G. Other examples were known as the lamplighter groups considered in [START_REF] Dahmani | Solvable groups of interval exchange transformations[END_REF], however the situations, we present here, are easy to achieve. Corollary 3. The group G generated by the following f and g is not virtually polycyclic.

(1) f is an irrational rotation and g is not a rotation.

(2) f ∈ G n is periodic point free and g / ∈ G n .

As a consequence of Theorem 2 of [START_REF] Dahmani | Solvable groups of interval exchange transformations[END_REF], for finitely generated IET subgroups, the properties of being virtually abelian, virtually nilpotent or virtually polycyclic are equivalent. This justifies why the groups involved in Corollaries 2 and 3 are described in the abstract as non virtually nilpotent.

Item (1) is the particular case n = 1 of Item (2) and it is then a corollary of Gromov's polynomial growth theorem [START_REF] Gromov | Groups of polynomial growth and expanding maps[END_REF] and Proposition 5.9 of [START_REF] Juschenko | Extensive amenability and an application to interval exchanges[END_REF] that claims that a finitely generated subgroup of IET that contains an irrational rotation is either a subgroup of G 1 or has exponential growth.

In addition, in the last section, we provide examples of finitely generated non virtually polycyclic subgroups of G that have extra properties as being metabelian or non virtually solvable. In particular, these subgroups are generated by rotations and torsion elements as the groups considered by Boshernitzan in [START_REF] Boshernitzan | Subgroup of interval exchanges generated by torsion elements and rotations[END_REF].

This criterium applies to the group ⟨r, s⟩ constructed in Theorem 8.1 of [START_REF] Dahmani | Free groups of interval exchange transformations are rare[END_REF] by taking s ∈ G 3 (obviously r / ∈ G 3 ) and also to the lamplighter groups A ≀ Z of Proposition 4.1 of [START_REF] Dahmani | Solvable groups of interval exchange transformations[END_REF], since the infinite order generator belongs to G n and the support of the finite order elements can be chosen in such a way that these maps do not belong to G n .

Another motivation for studying the dynamics of the virtually abelian subgroups of G comes from the Klein bottle group BS(1, -1) and its connections with reversible maps: O'Farrell and Short ( [START_REF] Anthony | Reversibility in dynamics and group theory[END_REF]) have pointed out the importance of reversibility in dynamical systems and group theory and they have raised the following question: given a group G, are all reversible elements of G reversible by an involution? In [START_REF] Guelman | Reversible maps and products of involutions in groups of iets[END_REF], we study reversible IETs and free actions of BS(1, -1) play a key role for O'Farrell and Short question. In particular, Theorem 1 applies to such actions (since the subgroup ⟨ a, b 2 ⟩ is isomorphic to Z 2 and of index 2) leading to Corollary 4. Any free action of BS(1, -1) by elements of G is P L • G-conjugate to a free action of BS(1, -1) by elements of some G n .

Acknowledgements. We deeply thank the referees for their valuable comments and very specific suggestions, in particular for an enlightening reformulation of Theorem 1 and an alternative proof of Corollary 2.

We acknowledge the support received from the I.F.U.M.I, the projects MathAmSud (GDG 18-MATH-08), Project CSIC 883174, Labex CEMPI (ANR-11-LABX-0007-01), ANR Gromeov (ANR-19-CE40-0007) and the Universities of Lille and de la República.

Algebraic description of finitely generated groups acting freely by

elements of G: Proof of Proposition 1.4

Proof. Since translations commute, the orbit of any point under a finitely generated subgroup H of G has polynomial growth. Therefore, if H acts freely then it has polynomial growth so it is virtually nilpotent by Gromov [START_REF] Gromov | Groups of polynomial growth and expanding maps[END_REF]. According to Novak [START_REF] Christopher | Discontinuity-growth of interval-exchange maps[END_REF], IET groups do not contain distorted copy of Z so, by standard arguments, H is virtually abelian (this can also be deduced from Theorem 2 of [DFG20] since finitely generated nilpotent groups are polycyclic).

Let G be a finitely generated virtually abelian group. By the classification of finitely generated abelian groups, there exists H ¡ G isomorphic to some Z m such that F = G/H is finite. In addition, H can be represented as a group of circle rotations A seen as 2-IETs of [0, 1). Namely,

A = ⟨R α 1 , ..., R αm ⟩ and 1, α 1 , • • • , α m are Q linearly independent.
According to Krasner-Kaloujnine (see [START_REF] Krasner | Produit complet des groupes de permutations et problème d'extension de groupes[END_REF]) G embeds in H F ⋊ F , where F acts on H F by permuting coordinates. Thus, it suffices to show that A F ⋊ F acts freely by elements of some G n .

To any (a s ) s∈F ∈ A F and σ ∈ F , we associate a map

E((a s ), σ) : [0, 1) × F → [0, 1) × F (x, s) → (a s (x), σs) Noticing that E((a s ), σ) • E((a ′ s ), σ ′ ) = E((a σ ′ s • a ′ s ), σσ ′ ), we get that E induces an injective morphism from A F ⋊ F to the group MG 2 ([0, 1) × F ) := {E((a s ), σ), (a s ) ∈ (G 2 ) F , σ ∈ F }.
In addition, its image acts freely on [0, 1) × F because 1, α 1 , ..., α m are Q linearly independent. Finally, MG 2 ([0, 1)×F ) embeds in G #F by subdividing [0, 1) in #F intervals of same length. □

Dynamical properties of IETs and centralizers

3.1. Dynamical properties of IETs. Definition 3.1.

• The break point set of an IET f is obtained by adding the initial point 0 to the discontinuity set of f , it is denoted by BP(f ). The set consisting of the f -orbits of points in BP(f

) is denoted by BP ∞ (f ). The translation set of g is {g(x) -x, x ∈ [0, 1)}. • The break point set of a finitely generated subgroup G of G is BP(G) := g∈G BP(g) and its translation set is {g(x) -x, g ∈ G, x ∈ [0, 1)}.
We let the reader check the following

Properties 3.2. Let f , g in G and n ∈ N. (a) BP(f • g) ⊆ BP(g) ∪ g -1 (BP(f )), (b) BP(f -1 ) = f (BP(f )) and (c) BP(f n ) ⊆ BP(f ) ∪ f -1 (BP(f )) ∪ ... ∪ f -n+1 (BP(f )).
Remark 3.3. By these properties, #BP(f n ) is bounded above by #BP(f )×|n|. However, it is possible for #BP(f n ) to be bounded independently of n, for IETs of finite order but also for minimal IETs as irrational rotations and then also for all elements of G n , in this case we say that f is of bounded break point type. Definition 3.4. Let f ∈ G.

• Let p ∈ N * , we denote by Per p (f ) the set of points x such that, O f (x), the f -orbit of x has cardinality p. The fix point set of f is Fix(f ) = Per 1 (f ).

• Let V be a subset of [0, 1), we say that V is of type M if it is a non empty finite union of intervals each of the form [b, c), with b, c in BP ∞ (f ). A type M and f -invariant set that is minimal for the inclusion among non empty, type M and f -invariant subsets of [0, 1) is called an f -component.

The well-known decomposition into minimal and periodic components of an IET was first given for measured surface flows by Mayer in 1943 ([May43]) and restated for IETs by Keane ([Kea75]) and Arnoux ([Arn81]). Here, we detail Arnoux formulation (see [START_REF] Arnoux | échanges d'intervalles et flots sur les surfaces[END_REF] Proposition p. 20):

Theorem. The interval [0, 1) can be decomposed as [0, 1) = P 1 ∪ ...P l ∪ M 1 ... ∪ M m , where

• P i is an f -periodic component: P i is the f -orbit of an interval [b, c) with b, c in BP ∞ (f ) and all iterates f k of f are continuous on [b, c). In particular points in P i are periodic of the same period.

• M j is an f -minimal component: for any x ∈ M j , the orbit O f (x) is dense in M j .
For convenience, the f -component containing x ∈ [0, 1) will be denoted M f (x).

According to Remark 3.3, it makes sense to give the following Definition 3.5. We define the growth rate of the number of discontinuities for the iterates of an IET f on the f -orbit through a given point x by

N x (f ) = lim n→+∞ #{BP(f n ) ∩ O f (x)} n . Properties 3.6. Let f , g in G and x ∈ [0, 1). (1) If BP (f ) ∩ O f (x) = ∅ then N x (f ) = 0. (2) N g(x) (g • f • g -1 ) = N x (f ).
Proof.

Item (1) follows from Item (c) of Properties 3.2.

Item (2). Let N ∈ N, we write A ⊂ N B if A ⊂ B ∪ F where #F ≤ N . Let C = 2#BP(g) and f g = g • f • g -1 , we have O fg (g(x)) = g(O f (x)).
It follows from Item (a) of Properties 3.2 that:

• BP(f n g ) ⊆ BP(g -1 ) ∪ g(BP(f n )) ∪ g • f -n (BP(g)) ⊂ C g(BP(f n )) and • BP(f n ) = BP(g -1 • f n g • g) ⊆ BP(g) ∪ g -1 (BP(f n g )) ∪ g -1 • f -n g (BP(g -1 )) ⊂ C g -1 (BP(f n g )). Therefore BP(f n g ) ∩ O fg (g(x)) ⊂ C g BP(f n ) ∩ g -1 (O fg (g(x))) = g BP(f n ) ∩ O f (x)
and

BP(f n ) ∩ O f (x) ⊂ C g -1 (BP(f n g ) ∩ g(O f (x)) = g -1 (BP(f n g ) ∩ O fg (g(x))). Finally, we get # (BP(f n ) ∩ O f (x)) -C ≤ # BP(f n g ) ∩ O fg (g(x)) ≤ # (BP(f n ) ∩ O f (x)) + C.
We conclude by dividing by n and taking the limit. □

In addition, N x (f ) provides a characterization of the bounded break point type. More precisely, from [START_REF] Christopher | Discontinuity-growth of interval-exchange maps[END_REF], we deduce Theorem 3.7. Adapted version of Theorems 1.1 and 1.2 of [START_REF] Christopher | Discontinuity-growth of interval-exchange maps[END_REF].

Let f ∈ G, the following assertions are equivalent

(1) For any x ∈ [0, 1), we have N x (f ) = 0.

(2) The map f is of bounded break point type.

(3) There exists a positive integer p such that f p is conjugate in G to a product of restricted rotations of pairwise disjoint supports. (4) There exist two positive integers p and n, an IET E and a PL-homeomorphism

P : [0, 1) → [0, 1) such that Φ = (P • E) • f p • (P • E) -1 ∈ G n , σ Φ =
Id and the map P -1 is affine on each

I i = [ i-1 n , i n ). Proof. Since BP(f n ) = a∈BP(f ) BP(f n ) ∩ O f (a), the implication (1) =⇒(2) is a direct conse- quence of Theorem 1.1 of [Nov09] stating that either #BP(f n ) is bounded or #BP (f n ) has linear growth.
The implication (2) =⇒(3) is exactly Theorem 1.2 of [START_REF] Christopher | Discontinuity-growth of interval-exchange maps[END_REF].

We prove (3) =⇒(4). Let E be an IET that conjugates f p to F , a product of restricted rotations of pairwise disjoint supports. We decompose I as a disjoint union of consecutive half open intervals J i , i = 1, ..., n, where J i is either the support of a restricted rotation in F or a connected component of Fix(F ). We define P as the PL-homeomorphism such that P (J i ) = I i and

P |J i is affine. It is easily seen that Φ = P • F • P -1 ∈ G n and σ Φ = Id.
It remains to prove (4) =⇒(1). As Φ ∈ G n , it holds that N x (Φ) = 0 for any x ∈ [0, 1), then N x (P -1 • Φ • P ) = 0 since P is a homeomorphism. Therefore Item (2) of Properties 3.6 implies that N x (f p ) = 0 for any x ∈ [0, 1). It follows that N x (f ) = 0 for any x ∈ [0, 1).

Indeed, let x ∈ [0, 1) and n ∈ N * , we have O f (x) = p-1 j=0 O f p (f j (x)) and then # BP(f pn ) ∩ O f (x) pn ≤ 1 p p-1 j=0 # BP(f pn ) ∩ O f p (f j (x)) n
Taking, the limit when n goes to infinity, we get that

N x (f ) ≤ 1 p p-1 j=0 N f j (x) (f p ).
Finally, by Item (2) of Properties 3.6, it holds that N x (f p ) = N f j (x) (f p ) for any integer j and then N x (f ) ≤ N x (f p ) = 0. □ 3.2. Centralizers.

Generalities.

Let G be a group and f ∈ G. It is well-known that the centralizer of f in G, C(f

) := {h ∈ G : h • f = f • h}, is a subgroup of G.
Lemma 3.8. Let f, h be bijections of a set X and h ∈ C(f ), then:

• For any x ∈ X, h(O f (x)) = O f (h(x)).
• For any p ∈ N * , the set of f -periodic points with exact period p, Per p (f ), is hinvariant.

Proof.

• The first item is a direct consequence of the fact that for any x ∈ X, one has

h(f n (x)) = f n (h(x)). • Let p ∈ N * , x ∈ Per p (f ) means that #O f (x) = p. By the first item, we get that #O f (h(x)) = p and therefore h(x) ∈ Per p (f ). □ Lemma 3.9. Let f, h ∈ G, h ∈ C(f ) and x be a non f -periodic point. Then h(x) is not f -periodic and h(M f (x)) = M f (h(x)
). In addition, there exists a positive integer r such that h r (M f (x)) = M f (x).

Proof. As x is not f -periodic, the orbit O f (x) is dense in M f (x). By Lemma 3.8, the point h(x) is not f -periodic and h(O f (x)) = O f (h(x)), taking closure we get h(M f (x)) = M f (h(x)) even if h is not continuous.

Indeed, the dynamics of the group G generated by f and h can be lifted, by a Denjoy blow-up, to that of a homeomorphisms group of X = [0, 1) ⊔ BP - ∞ (G), where BP - ∞ (G) is an abstract copy of BP ∞ (G). To any x ∈ BP ∞ (G) correspond two points of X: x + ∈ [0, 1) and x -∈ BP - ∞ (G). The topology on X is given by the total order defined on X by taking the usual order on BP - ∞ (G) and [0, 1) and setting that for x -∈ BP - ∞ (G) and y ∈ [0, 1): x -< y ⇐⇒ x + ≤ y. For more details see e.g [START_REF] Arnoux | échanges d'intervalles et flots sur les surfaces[END_REF] or [Cor20] §3.2.

In addition, the finiteness of the number of f -minimal components implies that there exist two distinct integers s and t such that

h s (M f (x)) = h t (M f (x)) therefore h t-s (M f (x)) = M f (x). □ 3.2.2.
Centralizer for IETs of bounded break point type.

We begin by recalling Properties 3.10. ([Nov09], Lemma 5.1 and its proof.)

(1) Two irrational rotations R α and R β , with α ̸ = β mod 1, are not conjugate in G.

(2) The centralizer in G of an irrational rotation is the rotation group G 2 .

This extends to

Lemma 3.11. Let F ∈ G n without periodic points then C(F ) ⊂ G n .

Proof. Let F ∈ G n without periodic points, noting that C(F ) ⊂ C(F n ) holds for all n ∈ Z and after replacing F by some iterate F p , it may be assumed that σ F = Id and then the intervals

I i = [ i-1 n , i n )
are the minimal components of F , since F has no periodic points.

Let H ∈ C(F ). By Lemma 3.9, there exists a permutation γ ∈ S n such that H(I i ) = I γ(i) and H conjugates the irrational rotation F |I i to the irrational rotation F |I γ(i) . Thus, Item (1) of Properties 3.10 implies that α γ(i) (F ) = α i (F ).

Next, we consider the map K of G n defined by α(K) = 0 and σ K = γ. It is easily seen that K ∈ C(F ), therefore K -1 • H is an IET that commutes with F and fixes each I i . By Item (2) of Properties 3.10, one has (

K -1 • H) |I i = R β i for any i meaning that K -1 • H ∈ G n , so does H. □
More generally, we have Proposition 3.12. Let f be a periodic-point free IET that is conjugate to F ∈ G n by a PL-homeomorphism P such that P -1 |I i is an orientation preserving affine map. Then the map P conjugates C(f ) to a subgroup of G n .

Proof. As in the proof of Lemma 3.11, the map f has no periodic points then after replacing f by some iterate, we can suppose that the intervals J i := P -1 (I i ) are the minimal components of f . Let g ∈ C(f ), according to Lemma 3.9, there exists a permutation γ ∈ S n such that g(J i ) = J γ(i) . In particular J i and J γ(i) have same lengths and then the restrictions of P to J i and to J γ(i) are affine with same slopes.

This implies that the a priori AIET G = P • g • P -1 , which sends P (J i ) = I i to P (J γ(i) ) = I γ(i) for any i, is actually an IET. Finally G is an IET that commutes with F (i.e. G ∈ C(F )), therefore, by Lemma 3.11, G belongs to G n . □

Free actions of Z 2 by IET

The aim of this section is to prove Proposition 1.6. By Theorem 3.7, it can be reformulated as: "Any free action of Z 2 in G consists of IETs of bounded break point type."

This proof, inspired by the description given by Minakawa for the free actions of Z 2 by circle PL-homeomorphisms (see [START_REF] Minakawa | Classification of exotic circles of PL + (S 1 )[END_REF]), uses the properties of the rates N x (f ).

As the action ⟨f, h⟩ of Z 2 is free, for every x ∈ I and k, q distinct integers, one has:

O f (h k (x)) ∩ O f (h q (x)) = ∅.
Therefore, as #BP (f ) is finite, there exists a positive integer N 0 such that:

∀n ≥ N 0 , O f (h n (x)) ∩ BP (f ) = ∅.
Using Item (1) of Properties 3.6, one get N h n (x) (f ) = 0 for n ≥ N 0 and, by Item (2) of Properties 3.6,

N h n (x) (h n • f • h -n ) = N x (f ).
By commutativity, it holds that

N h n (x) (h n • f • h -n ) = N h n (x) (f ).
Combining these properties, we get N x (f ) = 0 for any x ∈ I. Finally, by Theorem 3.7, f has bounded break point type.

Proof of Theorem 2.

After a possible conjugation by an IET, it suffices to prove Proposition 5.1. Let G be a non trivial virtually abelian subgroup of G. Suppose that there exists f 1 ∈ G which is a product of restricted rotations with disjoint supports and without periodic points then G is conjugate to a subgroup of some G n by the canonical PL-homeomorphism that conjugates f 1 to an element of G n .

Proof of Proposition 5.1.

Let G < G and H ¡ G be an abelian subgroup of G with finite index, p.

Claim 5.2. To any f ∈ G we can associate a positive integer p f in order that the maps f p f are pairwise commuting.

Indeed, let f ∈ G, since the set G/H is finite, there exist integers 0 ≤ s < t such that f s and f t belong to a same class modulo H, that is f t-s ∈ H. So taking p f = t -s leads to the required conclusion.

Consider P the PL-homeomorphism associated to f 1 as involved for proving (3) =⇒(4) of Theorem 3.7. In particular, the f 1 -components are the intervals J i = P -1 (I i ) for i = 1, ..., n ; they are minimal and

P |J i has constant slope for any i ∈ {1, • • • , n}. Let f ∈ G and ϕ = f • f 1 • f -1 . By Claim 5.2, there exist p f 1 , p ϕ ∈ N * such that f p f 1 1
and ϕ p ϕ commute. Thus, Proposition 3.12 applies to ϕ p ϕ ∈ C(f p f 1 1 ) and implies that ϕ p ϕ is conjugate by P to an element of G n , namely

P • ϕ p ϕ • P -1 ∈ G n .
Moreover, as ϕ p ϕ is periodic points free, eventually replacing p ϕ by a multiple, we may assume that the components of ϕ p ϕ are the intervals J i and they are minimal.

Therefore, as f conjugates ϕ p ϕ and f p ϕ 1 , the map f permutes the J i . This implies that the a priori AIET P • f • P -1 is an IET. Indeed, for any i = 1, ..., n, it holds that

• the map P -1 sends I i to J i with constant slope |J i | |I i | ,
• the IET f sends J i to some J k of same length, with constant slope 1 and

• the map P sends J k to I k with constant slope |I k | |J k | = |I i | |J i | . Now, it remains to prove that P • f • P -1 ∈ G n .
Lemma 5.3. Let n be a positive integer and T ∈ G n be a periodic point free IET. If

S ∈ G is such that T ′ = ST S -1 ∈ G n then S ∈ G n .
Proof.

• If n = 1 then T and T ′ are irrational rotations and T ′ S = ST . Identifying [0, 1) with S 1 , the maps T and T ′ become continuous and computing break point sets, we get

BP(S) = BP(T ′ S) = BP(ST ) = T -1 (BP(S)).
Thus BP(S) is a finite T -invariant set so the minimality of T implies that it is empty and then S is a rotation.

• If n ≥ 2 then eventually changing T for some iterate, we can suppose that σ T and σ ST S -1 are trivial.

• If S preserves each I j , we apply the case n = 1 to the restrictions to I j of T and S. We conclude that S| I j is a rotation and then S ∈ G n .

• If S does not preserve each I j . As S conjugates two periodic point free maps in G n there exists a permutation γ such that S(I j ) = I γ(j) . We consider the element K ∈ G n defined by σ K = γ and α K = 0. We have K -1 S • T • S -1 K ∈ G n and K -1 S preserves each I j . According to the previous case, K -1 S ∈ G n and therefore S ∈ G n . □

We conclude by applying Lemma 5.3 to

T = P • f p ϕ 1 • P -1 ∈ G n and S = P • f • P -1 .
6. Proofs of Theorem 1 and corollaries 1 and 2.

Proof of Theorem 1. Let G be a finitely generated group acting freely by IET and Γ < G be the image of a copy of Z 2 contained in G. According to Proposition 1.4, G is virtually abelian. It is plain that Γ also acts freely, then Proposition 1.6 implies that Γ contains IETs that are conjugate in G to products of restricted rotations with disjoint supports and without periodic points. Therefore, G satisfies the hypothesis of Theorem 2 and then its conclusion.

Proof of Corollary 1. Let G be a virtually abelian subgroup of G containing a copy ⟨a, b⟩ of Z 2 and whose action is totally minimal.

Let K be a finite index abelian normal subgroup of G. Claim 5.2 implies that there exist integers p and q such that a p and b q generate a copy Z of Z 2 in K. We claim that that Z acts freely.

Indeed, we argue by contradiction supposing that there exist h ∈ Z \ {Id} and x ∈ I such that x ∈ Fix(h). As K is abelian, Lemma 3.8 implies that Fix(h) is a non empty K-invariant set. By totally minimality of G, all K-orbits are dense, hence Fix(h) = [0, 1) and then h = Id, a contradiction.

Therefore, by Proposition 1.6, G contains maps which are conjugate in G to products of restricted rotations with disjoint supports and without periodic points. Let f 1 be such an element of G, Theorem 2 applies to G and we conclude that there exists a map H = P • E that conjugates G to a subgroup of some G n , with P a PL-homeomorphisms, E ∈ G and H of constant slope on each minimal component of f 1 .

Moreover, as the conjugation by H preserves the totally minimality (according to classical arguments for topological conjugacy and extra arguments given in the proof of Lemma 3.9 and its footnote), this subgroup is also totally minimal.

We claim that no subgroup of

G n , n ≥ 2, is totally minimal. Indeed, if G < G n then G 0 = {g ∈ G | σ g = Id} is a finite index subgroup of G that preserves any I i , i = 1, ..., n.
Thus, for G 0 to be minimal it is necessary that n = 1. Finally n = 1, so the map f 1 is minimal and then H has constant slope 1. In conclusion, G is conjugate in G to a subgroup of G 1 .

Proof of Corollary 2. Let G be a finitely generated subgroup of G and f ∈ G be a totally minimal IET that is not conjugate to a rotation.

We first note that the action of G is also totally minimal, since any finite index subgroup of G contains some non trivial power of f . We next argue by contradiction, supposing that G is virtually polycyclic and not virtually cyclic. By [START_REF] Dahmani | Solvable groups of interval exchange transformations[END_REF], G is virtually abelian and not virtually cyclic then it contains a copy of Z 2 . Therefore, applying Corollary 1, we get that f is conjugate in G to a rotation, a contradiction.

Corollary 2 can also be proven more directly: arguing by contradiction and using [DFG20], we may assume that up to a finite index subgroup, G is abelian and it contains a totally minimal IET f non conjugate to a rotation. By [START_REF] Christopher | Discontinuity-growth of interval-exchange maps[END_REF], the centraliser of f is cyclic and then G is cyclic, a contradiction. 7. Non virtually abelian subgroups of G.

Proof of Corollary 3.

Item (1) is the special case n = 1 in Item (2), so we only prove Item (2): We argue by contradiction supposing that G is virtually abelian. Hence, according to Claim 5.2, there exist integers p and q such that f p ∈ G n and gf q g -1 commute. Therefore gf q g -1 ∈ G n by Lemma 3.11 and then Lemma 5.3 leads to g ∈ G n , a contradiction.

Related examples involving arithmetic conditions.

Notations. Let s ∈ N * and α 1 , ..., α s be s Q-independent irrational numbers. We denote by ⟨1, α 1 , ..., α s ⟩ Q the Q-vector space generated by 1 and the α i 's. We set

• G Q = {g ∈ G | g(x) -x ∈ Q, ∀x ∈ I} and • G α 1 ,...,αs = {g ∈ G | g(x) -x ∈ ⟨1, α 1 , ..., α s ⟩ Q , ∀x ∈ I}.
Corollary 5. Let p, r ∈ N * , r < p and α 1 , ..., α p be p Q-independent irrational numbers.

If f ∈ G α 1 ,...,αr is totally minimal not conjugate to a rotation and g ∈ G α r+1 ,...,αp then the group generated by f and g is not virtually polycyclic.

Proof. According to Corollary 2, G is either virtually cyclic or not virtually polycyclic. Suppose, contrary to our claim that G is virtually cyclic. Hence, there exists h ∈ G such that Z = ⟨h⟩ has finite index in G and then f m = h n and g q = h t for some positive integers m, n, q and t. Therefore f mt = h nt = g qn ∈ G α 1 ,,...,αr ∩ G α r+1 ,...,αp = G Q that only contains finite order maps 1 , in particular f is not totally minimal, a contradiction. Finally, G is not virtually polycyclic. □

In the following two subsections, we exhibit examples of finitely generated non virtually polycyclic subgroups of G that have extra properties. In the first one, we provide metabelian examples and in the second one, non virtually solvable groups.

Metabelian examples.

Let α ∈ [0, 1 3 ) \ Q and G be the subgroup of G generated by R α and the map g ∈ G 3 described by Figure 2.

Proposition 7.1. The group G is metabelian and non virtually polycyclic. More precisely its commutators subgroup [G, G] is a non finitely generated abelian group whose non trivial elements have order 3.

Remark. This metabelian group shares many properties with lamplighter groups

A ≀ Z, but its abelianization G [G,G] = Z × Z 2Z
whereas the one of a lamplighter group is Z.

0 1 3 2 3 1 1 3 2 3 1 + 2 3 +0 -2 3 g 0 1 -α 1 α 1 +α +α-1 R α 0 1 3 -α 1 3 2 3 α 1 3 1 3 +α 2 3 2 3 +α 1 + 2 3 +α +α-1 3 +α -2 3 +α R α • g 0 α 1 3 2 3 1 [R α , g] Figure 2
Note that, since g has order 2, any f ∈ G can be written as

f = R ps α g R p s-1 α g • • • R p 1 α g R p 0 α
, where p j ∈ Z. Lemma 7.2. For any f ∈ G, there exists a unique ℓ(f ) ∈ Z such that for all x ∈ [0, 1),

f (x) = x + ℓ(f ) α + p f (x) 3
, where p f (x) ∈ Z.

Proof. We argue by induction on the length L S (f ) of f ∈ G as a word in S = {g, R α }. Indeed, if L S (f ) = 1 then either f = g or f = R ±α and the property holds.

Let n ∈ N * , suppose that G-elements of length less or equal than n have the required property. Consider f ∈ G with L S (f ) = n + 1, then either f = R ±α f 0 or f = gf 0 , where L S (f 0 ) = n. So, for any x ∈ [0, 1), we have either

f (x) = f 0 (x) ± α + ϵ or f (x) = f 0 (x) + ϵ 2 3
, where ϵ ∈ {-1, 0, 1}.

Therefore, either

f (x) = x + (ℓ(f 0 ) ± 1) α + p f 0 (x) + 3ϵ 3 or f (x) = x + ℓ(f 0 ) α + p f 0 (x) + 2ϵ 3 .
In both cases, f has the required form and ℓ(f ) is unique since α is irrational. □ Lemma 7.3.

(1) The map ℓ : G → Z is a morphism.

(2) Any f ∈ G can be written as (1) We obviously have ℓ(Id) = 0 and ℓ(R α ) = 1. Given f, h ∈ G and x ∈ [0, 1), the computation of hf (x) leads to

f = P f R ℓ(f ) α = R ℓ(f ) α Q f with P f , Q f ∈ ker ℓ. (
hf (x) = x + (ℓ(f ) + ℓ(h)) α + p f (x) + p h (f (x)) 3 Therefore ℓ(hf ) = ℓ(f ) + ℓ(h) and p hf (x) = p f (x) + p h (f (x))
In particular, ℓ is a morphism from G to (Z, +).

(

2) Let f ∈ G, one has ℓ f R -ℓ(f ) α = ℓ(f ) + ℓ(R -ℓ(f ) α ) = ℓ(f ) -ℓ(f ) = 0. Then P f = f R -ℓ(f ) α ∈ ker ℓ and f = P f R ℓ(f ) α . Similarly, we get Q f = R -ℓ(f ) α f ∈ ker ℓ and f = R ℓ(f ) α Q f .
(3) Let x ∈ [0, 1), the ker ℓ-orbit of x is the set {f (x) | f ∈ ker ℓ} and it is included in S = x + p 3 , p ∈ Z ∩ [0, 1) which has cardinality 3 since S ⊂ [0, 1) and the distance between two distinct elements of S is 1 3 or 2 3 . (4) Since ℓ is a morphism to an abelian group, its kernel contains [G, G]. In addition, let f ∈ G per of order m ∈ N * and x ∈ [0, 1). One has

f m (x) = x + mℓ(f ) α + p f m (x) 3 = x
By irrationality of α, we get ℓ(f ) = 0 that is f ∈ ker ℓ.

(

) Let β ∈ [0, 1 3 ), we compute the map [R β , g] = R β g R -β g. [0, β) g -→ + 2 3 [ 2 3 , 2 3 + β) R -β -→ -β [ 2 3 -β, 5
-→ +0 [ 2 3 -β, 2 3 ) R β -→ β [ 2 3 , 2 3 + β) [β, 2 3 ) g
→ + 2 3 [ 2 3 + β, 1) R -β -→ -β [ 2 3 , 1 -β) g -→ -2 3 [0, 1 3 -β) R β -→ β [β, 1 3 ) [ 1 3 , 1 3 + β) g -→ 0 [ 1 3 , 1 3 + β) R -β -→ -β [ 1 3 -β, 1 3 ) g -→ + 2 3 [1 -β, 1) R β -→ β-1 [0, β) [ 1 3 + β, 1 3 ) g -
3 )

g -→ 0 [ 1 3 + β, 2 3 ) R -β -→ -β [ 1 3 , 2 3 -β) g -→ 0 [ 1 3 , 2 3 -β) R β -→ β [ 1 3 + β, 2
3 ) Proof.

[ 2 3 , 2 3 + β) g -→ -2 3 [0, β) R -β -→ 1-β [1 -β, 1) g -→ -2 3 [ 1 3 -β, 1 3 ) R β -→ β [ 1 3 , 1 3 + β) [ 2 3 + β, 1) g -→ -2 3 [β, 1 3 ) R -β -→ -β [0, 1 3 -β) g -→ + 2 3 [ 2 3 , 1 -β) (
(1) It is plain that the maps R n α gR -n α , n ∈ Z belong to ker ℓ. We previously noted that any f ∈ G can be decomposed as

f = R ps α gR p s-1 α g • • • R p 1 α gR p 0 α
, where p j ∈ Z, and then we can write:

f = R ps α gR -ps α R α ps+p s-1 gR p s-2 α • • • R p 1 α gR p 0 α
Iterating this process, we get:

f = R ps α gR -ps α R ps+p s-1 α gR -(ps+p s-1 ) α • • • R ps+p s-1 +•••p 1 α gR -(ps+p s-1 +•••p 1 ) α R s j=0 p j α
Moreover, f ∈ ker ℓ if and only if s j=0 p j = 0, thus any f ∈ ker ℓ is the product of elements of the form R p α gR -p α .

(2) We first note that for all p, q ∈ Z, we have

( * ) R p α gR -p α R q α gR -q α = R p α gR -p α g gR q α gR -q α = [R p α , g] [R q α , g] -1
. Next, we claim that any element of [G, G] can be written as a product of an even number of R n α gR -n α so as a product of [R p α , g] and [R q α , g] -1 by ( * ). Indeed, it suffices to prove this property for a commutator f = [a, b] with a, b ∈ G. By Lemma 7.3 (2), f can be written as

f = P a R ℓ(ab) α Q b Q -1 a R -ℓ(ab) α P -1 b = P a R ℓ(ab) α Q b R -ℓ(ab) α R ℓ(ab) α Q -1 a R -ℓ(ab) α P -1 b
According to Item (1), P a , P b , Q a and Q b can be written as a product of R n α gR -n α . Moreover, for any

c ∈ G, Q c = R -ℓ(c) α P c R -ℓ(c) α so the maps P c , P -1 c , Q c , Q -1
c and their conjugates by R m α (m ∈ Z) can be decomposed as products of the same number of R n α gR -n α . Finally, f can be written as a product of an even number of R n α gR -n α .

(

[R n α , g] = [g, R n α gR -n α ]. Indeed, noting that if a 2 = b 2 = Id then [a, b] = (ab) 2 and that g and R n α gR -n α have order 2, we get [g, R n α gR -n α ] = (gR n α gR -n α ) 2 = [g, R n α ] 2 = [R n α , g] -2 . In addition, by Lemma 7.3 (5), the map [R n α , g] = [R nα , g] has order 3 then [g, R n α gR -n α ] = [R n α , g]. □
Lemma 7.5.

(1) Let f ∈ ker ℓ and J ⊂ [0, 1 3 ) such that J, J + 1 3 and J + 2 3 are continuity intervals of f then there exists σ = σ f,J ∈ S 3 , the group of permutations of the set {0, 1, 2} such that f (J + i 3 ) = J + σ(i) 3 .

(2) Let f, h ∈ ker ℓ and J ⊂ [0, 1 3 ) such that J, J + 1 3 and J + 2 3 are continuity intervals of f and h then J, J + 1 3 and J + 2 3 are continuity intervals of f h and σ f h,J = σ f,J • σ h,J Proposition 7.6. The group G is elementary amenable, non virtually solvable and its commutator subgroup [G, G] is a locally finite torsion group3 , moreover [G, G] is perfect but it is not simple. This proposition will be proved after Lemma 7.13. Lemma 7.7. Let f ∈ G. Then there exists a unique ℓ(f ) ∈ Z such that for all x ∈ [0, 1) one has

f (x) = x + ℓ(f )α + p f (x) n
, where p f (x) ∈ Z.

Proof. As in Lemma 7.2, we argue by induction on the length L S (f ) of f ∈ G as a word in S = {A n , R α }. The change being in the case f = gf 0 of the induction step. Here, we have f = af 0 with a ∈ A n and the involved translation ϵ 2 3 is replaced by pa(x) n . □ Lemma 7.8.

(1) The map ℓ : G → Z is a morphism.

(2) Any f ∈ G can be written as

f = P f R ℓ(f ) α = R ℓ(f ) α Q f where P f , Q f ∈ ker ℓ.
The proof is analogous to that of Lemma 7.4.

Lemma 7.9.

(1) The group ker ℓ is generated by Proof.

{R k α aR -k α , k ∈ Z, a ∈ A n }. (
(1) We previously noted that any f ∈ G can be written as

f = R ps α a s R p s-1 α a s-1 • • • R p 1 α a 1 R p 0 α
, where p j ∈ Z, and as in the proof of Lemma 7.4 we get:

f = R ps α a s R -ps α R ps+p s-1 α a s-1 R -(ps+p s-1 ) α • • • R ps+p s-1 +•••p 1 α a 1 R -(ps+p s-1 +•••p 1 ) α R s j=0 p j α
Moreover, as ℓ is a morphism f ∈ ker ℓ if and only if belongs to [ker ℓ, ker ℓ] and then f belongs to [ker ℓ, ker ℓ], which establishes that ker ℓ is perfect.

In addition, as ℓ is a morphism to an abelian group, [G, G] < ker ℓ and the other inclusion holds since ker ℓ is perfect.

(3) Let x ∈ [0, p n ) and f ∈ ker ℓ. Lemma 7.7 gives f (x) = x + p f (x) n for some p f (x) ∈ Z and f (x) ∈ [0, 1) forces p f (x) ∈ {0, 1, • • • , n -1}. Moreover, by choosing a ∈ A n with a(1) = p, we obtain t a (x) = x + p-1 n . Therefore the A n -orbit of x is x + p n , p = 0, • • • , n -1 . Finally, ker ℓ is contained in G Q so in G per and the other inclusion is a consequence of α / ∈ Q. □ Definition 7.10. Let f ∈ ker ℓ, x ∈ [0, 1 n ) and β ∈ (0, 1 n ) such that f is continuous on the intervals J p (x, β) = [x + p-1 n , x + p-1 n + β) for all p = 1, • • • , n. As the translations of f belong to { p n , p ∈ Z}, the map f permutes the J p (x, β) and then there exists ω ∈ S n such that

f (J p (x, β)) = J ω(p) (x, β) = [x + ω(p) -1 n , x + ω(p) -1 n + β)
It is easy to see that ω does not depend on the choice of suitable β and it is denoted by ω(f, x) and called the local permutation of f at x.

Proposition 7.11. Let x ∈ [0, 1 n ). The map ω x : ker ℓ → S n f → ω(f, x) is a morphism and its image is A n .

Proof. Let x ∈ [0, 1 n ), it is obvious that ω(Id, x) = Id. Let f, h ∈ ker ℓ and β ∈ (0, 1 n ) small enough so that f and h are continuous on the intervals J p (x, β). Hence, for any p ∈ {1, • • • , n}, we have h • f (J p (x, β)) = h J ω(f,x)(p) (x, β) = J ω(h,x)ω(f,x)(p) (x, β), therefore ω(h • f, x) = ω(h, x) ω(f, x).

Since ker ℓ is perfect the image of ω x is contained in [S n , S n ] = A n . In addition, it is easy to check that the IET t associated to τ ∈ A n satisfies ω(t, x) = τ and then we get ω x (ker ℓ) = A n . □ Proposition 7.12. For any β ∈ (0, 1 n ), the set S β of maps in ker ℓ having support in n p=1 p -1 n , p -1 n + β is a normal subgroup of ker ℓ and its image by ω 0 is A n .

Proof. The normality of S β is direct consequence of the facts that the translation set of ker ℓ is { p n , p ∈ Z ∩ [-(n -1), n -1]} and the support of h • f • h -1 is the image by h of the support of f . Consequently, the image by ω 0 of S β is a normal subgroup of A n which is simple, therefore ω 0 (S β ) is either trivial or equal to A n . Hence, the proof of Proposition 7.12 reduces to prove that ω 0 (S β ) is not trivial. This is provided, taking β ∈ (0, 1 n ) ∩ { p n + kα, p, k ∈ Z} so that R β ∈ G, by the following Lemma 7.13. Let β ∈ (0, 1 n ), τ ∈ A n and t ∈ ker ℓ be its associated IET. Then the map C β,t = [R β , t] = R β tR -β t -1 has support in

  3) The ker ℓ-orbit of any x ∈ [0, 1) consists in at most 3 points. (4) The group ker ℓ contains [G, G] and ker ℓ = G per = {f ∈ G : ∃m ∈ N * , f m = Id}.(5) For any β ∈ [0, 1), the map [R β , g] has order 3. 2 Proof.

  3) The group [G, G] coincides with [ker ℓ, ker ℓ].

  2) The group ker ℓ is perfect and [G, G] = ker ℓ. (3) The ker ℓ-orbit of x ∈ [0, p n ) is {x + p n , p = 0, • • • , n -1} and ker ℓ = G per .

s

  j=0 p j = 0, thus any f ∈ ker ℓ is the product of elements of the form R k α aR -k α .(2) Let f ∈ ker ℓ, by the previous point, f can be written asf = j R n j α a j R -n j αwhere n j ∈ Z and a j ∈ A nAs A n is perfect and A n < ker ℓ, any a j belongs to [ker ℓ, ker ℓ]. Finally, [ker ℓ, ker ℓ] being normal in G, any R n j α a j R -n j α

  n + β and its local permutation at 0 is ω(C β,t , 0) = [σ, τ ], where σ is the n-cycle (1, 2, • • • , n) ∈ S n .

Let f ∈ G Q and q be the least common multiple of the denominators of the translations of f , then O f (x) ⊂ {x + p q , p ∈ Z} ∩ [0, 1) is finite.

This will be used for maps in G, that is for β = nα mod 1.

i.e its finitely generated subgroups are finite

and its restriction is periodic of exact period 3. Therefore, [R β , g] has order 3.

The case β ∈ ( 2 3 , 1) can be deduced from the previous case by noting that β ′ = 1 -β ∈ (0, 1 3 ) and R β = R -1 β ′ . Finally, for the case β ∈ [ 1 3 , 2 3 ], let us write β = 1 3 + β 0 with β 0 ∈ [0, 1 3 ]. A direct computation similar to that in the case β ∈ [0, 1

3 ) shows that [R β , g] is continuous on the intervals

(1) The group ker ℓ is generated by

Proof.

(1) Let f ∈ ker ℓ, since J ⊂ [0, 1 3 ) is a continuity interval of f , the map p f is constant on J equal to some p ∈ Z and f

3 ) ∩ [0, 1). Therefore p ∈ {0, 1, 2} and f (J) is either J, J + 1 3 or J + 2 3 . Analogously, for i = 1, 2, we get that f (J + i

3 ) is either J, J + 1 3 or J + 2 3 . This defines the required σ ∈ S 3 .

(2) Let f, h ∈ ker ℓ and J ⊂ [0, 1 3 ) such that f and h are continuous on J, J + 1 3 , J + 2 3 . For any i ∈ {0, 1, 2}, we have that h is continuous on J + i 3 and

By Item (1) of Corollary 3, the group G is not virtually polycyclic.

We claim that all maps in [G, G] \ {Id} commute and have order 3.

\ {Id} be a non trivial commutator. By Lemma 7.4 (3), we can suppose that f, h ∈ ker ℓ. Let us decompose [0, 1) as a union of triples of intervals J, J

Noting that the non trivial elements of A 3 are 3-cycles, we get that σ [f,h],J is either trivial or has order 3 and the same holds for C |J by its continuity. As C ̸ = Id, there is some J such that C |J ̸ = Id and then C has order 3.

As A 3 is abelian, two commutators in [G, G] have commuting associate permutations on their common continuity intervals, so they commute.

Finally, we get the claim since [G, G] is generated by the commutators and the product of two elements of order 3 that commute is either of order 3 or trivial.

In addition, [G, G] is not finitely generated. This is due to the fact that any finitely generated abelian torsion group has to be finite but the [G, G]-elements [R n α , g], n ∈ Z are pairwise distinct, since the proof of Item (5) of Lemma 7.3 indicates that they have pairwise distinct break point sets.

Non virtually solvable examples.

Let n ∈ N * , n ≥ 5, we make the alternating group A n act on [0, 1) as a subgroup of G n so that the IET associated to a ∈ A n , t a , sends the interval

n ) and has a trivial rotation vector. In particular, for any x ∈ [0, 1) there exists an integer p a (x) such that t a (x) = x + pa(x) n . Let G be the subgroup of G generated by A n and R α , α ∈ [0, 1 n ) \ Q. Note that any f ∈ G can be written as

, where a j ∈ A n and p j ∈ Z.

Proof. We explicitely compute the map C β,t = [R β , t] = R β tR -β t -1 . We first note that the break point set of C β,t is contained in { p-1 n , p-1 n + β; p ∈ {1, ..., n}} and we have p -1

In addition, its local permutation at 0 is given by ω(p) -1 = τ (τ -1 (p) -1) that is

then R β and t are not commuting and C β,t is not trivial.

Proof of Proposition 7.6. We first prove the properties related to [G, G]. Lemma 7.9 implies that [G, G] = ker ℓ is a perfect torsion group and, by Proposition 7.12, it is not simple because of its non trivial normal subgroups S β . We claim that the finitely generated subgroups of [G, G] = ker ℓ are finite. Indeed, let H = ⟨m i ⟩ < ker ℓ be a finitely generated subgroup of ker ℓ. By Properties 3.2, the set BP(H) is contained in the H-orbit of the finite set ∪BP(m i ) so it is finite since all ker ℓ-orbits are finite. The group G is elementary amenable since G/[G, G] ≃ Z is abelian and [G, G] = ker ℓ is the direct union of its finitely generated subgroups that are finite.

By Item (1) of Corollary 3, G is not virtually polycyclic. Moreover, we are going to prove that no finite index subgroup of ker ℓ is solvable. Indeed, let R be a finite index subgroup of ker ℓ. Since a subgroup of finite index always contains a normal subgroup also of finite index, we may suppose that R is normal in G. Therefore for any x ∈ [0, 1 n ), the group ω x (R) is either trivial or equal to A n . As ker ℓ is infinite, R is not trivial and so there exists x ∈ [0, 1 n ) such that ω x (R) is not trivial and then equal to A n . Finally, since A n is simple, ω x maps any iterated commutators subgroup of R surjectively to A n . In particular, R is not solvable and therefore ker ℓ and then G are not virtually solvable.

Remark 7.14. Due to the SAF-invariant (see [START_REF] Boshernitzan | Subgroup of interval exchanges generated by torsion elements and rotations[END_REF]), the groups constructed in Sections 7.3 and 7.4, with rationally independent α are not conjugated in G.

Moreover, similar arguments show that Proposition 7.6 holds for the groups generated by R α 1 , ..., R αm and A n for n ≥ 5. In particular, these groups are not linear since by Schur Theorem ( [START_REF] Schur | Uber gruppen periodischer substitutionen[END_REF]) any linear torsion group is virtually abelian.

Conflict of interest

The authors have no conflict of interest to declare that are relevant to this article.


Isabelle Liousse

Non strongly relative hyperbolicity of some groups of Piecewise Linear homeomorphisms of the interval or the circle

We prove that some groups of PL homeomorphisms of the interval or the circle, included the classical Thompson groups T and F , are not strongly relatively hyperbolic.

1 Introduction.

Originally, the notion of relatively hyperbolic groups was proposed by M. Gromov in [G] and developed by B. Bowditch [B], B. Farb [F], and Osin [O]. There is now an interesting and rich literature on the subject. In this paper, we refer to Osin's paper ( [O]), for the definitions.

Definition of Thompson-Stein groups.

Let's r, s be two positive integers and 1 < n 1 < ... < n s be s integers. We denote by m the lcm(n 1 , ..., n s ).

The Thompson-Stein group F r,(n i ) [resp. T r,(n i ) ] is the group consisting of piecewise linear homeomorphisms of the interval [0, r] [resp. the circle [0,r] 0=r of lenght r] with finitely many madic break points, that is of the form N m q for q, N ∈ N and derivatives of the form n p 1 1 ×.....×n ps s for p i ∈ Z. We identify F r,(n i ) to the subgroup of T r,(n i ) consisting of g ∈ T r,(n i ) such that g(0) = 0.

The groups F r,(n i ) are torsion free and M. Stein, in [S], proved that F r,(n i ) and T r,(n i ) are finitely presented. In this paper, we show that :

Theorem. Let r, s be two positive integers and 1 < n 1 < ... < n s be s integers. The Thompson-Stein groups F r,(n i ) and T r,(n i ) are not strongly relatively hyperbolic.

This fact for the classical Thompson group F consisting of 2-adic PL homeomorphisms of the interval has been already proved, among many things, by Anderson, Aramayona and Shackleton in [AAS]. As in [AAS], our proof is based on Osin's result stating the hyperbolicity of the centralizers of hyperbolic elements, in strongly relatively hyperbolic groups.

2 Basic properties on Thompson-Stein groups.

The Bieri-Strebel criterion. Let Λ ⊂ R + * be a multiplicative subgroup, and A ⊂ R be an additive subgroup, invariant by multiplication by elements of Λ.

1)-There exists a PL homeomorphism with slopes in Λ and breaks in A that takes the interval [a, c] into the interval [a , c ] if and only if c -a and c -a are equal modulo (1 -Λ)A, where (1

-Λ)A = { i (1 -λ i )a i , λ i ∈ Λ, a i ∈ A}. (for a proof, see [S] appendix).
2)-Suppose that Λ = < n i > is the multiplicative group generated by the n i 's and

A = Z[ 1 m ] is the set of m-adic numbers, with m = lcm(n i ). Then (1 -Λ)A = d Z[ 1 m ]
, where d = gcd(n i -1).

Proof of 1. See [BS] or [S] appendix.

Proof of 2. That is (1 -Λ)A = dA.

Step 1 : (1 -Λ)A ⊂ dA. As (1 -Λ)A is a submodule, it suffices to prove that (1 -λ)a ∈ dA, with a ∈ A and λ = n s 1 1 ...n sp p ∈ Λ. By reducing to the same denominator, (1

-λ)a = (1 -n a 1 1 ...n ap p )a = (n b 1 1 ...n bp p -n c 1 1 ...n cp p
)a with b i , c i positive integers, a ∈ A. By replacing the n i by k i d + 1 and developping, we get (1 -λ)a = (dN )a with N ∈ N.

Step 2 : dA ⊂ (1 -Λ)A. Let a ∈ A, by Bezout identity, there exist u i ∈ Z such that : d = u 1 (n 1 -1)+....+u p (n p -1) and hence da = (n i -1)(u i a) ∈ (1-Λ)A.

Proposition 1. Let r, s be positive integers and 1 < n 1 < ... < n s be s integers.

1-Given a non degenerated interval I ⊂ [0, r], there exists an infinite order element of F r,(n i ) with (open) support contained in I.

2-The group F r,(n i ) is generated by B 0,1 : the set of g ∈ F r,(n i ) such that either Dg(0) = 1 or Dg(1) = 1. 3-Let m ≥ 2 be an integer. Given a m-adic number a, there exists f ∈ F r,m and k ∈ {0, ..., m -1} such that f ( k m ) = a. 3'-Let m ≥ 2 be an integer. The group T r,m contains an element t = t(r, m) such that t is of infinite order and t k (0) = k m , for all k ∈ Z, in particular 0 is a t-periodic point of period rm. 4-The group T r,(n i ) is generated by F r,(n i ) and t(r, m) with m = lcm(n 1 , ..., n s ). Proof of proposition 1. 1-Recall that the (open) support of a map f is the set of the non fixed points of f , it's an open set. Since f is PL it's a finite union of open intervals. Let ω, b be two points in A = Z[ 1 m ] such that [ω, ω + b] ⊂ I. Such points exist since A is a dense subset of [0, r]. Let's set ω 0 = ω ω 1 = ω + b n 1 +1 and ω 1 = ω + n 1 b n 1 +1 ω 2 = ω + b
We define a PL homeomorphism f I with break points ω 0 , ω 1 and ω 2 , with derivatives power of n 1 and with support ]ω 0 , ω 2 [, by setting :

f I : [0, ω 0 ] → [0, ω 0 ] is defined by f I (x) = x f I : [ω 0 , ω 1 ] → [ω 0 , ω 1 ] is defined by f I (x) = n 1 (x -ω 0 ) + ω 0 f I : [ω 1 , ω 2 ] → [ω 1 , ω] is defined by f I (x) = n -1 1 (x -ω 2 ) + ω 2 f I : [ω 2 , r] → [ω 2 , r] is defined by f I (x) = x Moreover, the point ω is a repulsive fixed point, so f I is of infinite order. 0 r r ¢ ¢ ¢ | ω 0 | ω 1 | ω 2 - ω 0 - ω 1 - ω 2 Figure 1. 2-Let k be in {1, ..., s} and a ∈ A = Z[ 1 m ] such that (n k + 1)a < r (such a point exists since A is dense in [0, r]). Let's denote by V k the PL-homeomorphism defined as follows : 0 r r ¢ ¢ ¢ ¢ ¢ | a | (n k +1)a - n k a - (n k +1)a n k n -1 k 1 V k (x) = n k x if x ∈ [0, a] V k (x) = 1 n k (x -a) + n k a if x ∈ [a, (n k + 1)a] V k (x) = x if x ∈ [(n k + 1)a, r] Figure 2.
Let f be in F r,(n i ) , by definition the derivative Df (0) can be written as :

Df (0) = n e 1 1 ....n es s , where e i ∈ Z. Now, the homeomorphism f := f • (V e 1 1 • .... • V es s ) -1 satisfies D f (0) = 1. Therefore f (0) ∈ B 0,1 and f = f • (V e 1 1 • .... • V es s ) is a product of elements of B 0,1 . 3-Using the Bieri-Strebel criterion, two points a 1 , a 2 in Z[ 1 m ] belong to the same F r,m -orbit if and only if 2 -a 1 ) = N (m-1) m p
, where p ∈ N and N is a positive integer non multiple of m. Since a 1 and a 2 are in Z[ 1 m ], there exist p 1 , p 2 positive integers and N 1 , N 2 positive integers non multiple of m such that

a 1 = N 1 m p 1 and a 2 = N 2 m p 2 . Then a 1 , a 2 belong to the same F r,m -orbit if and only if ( N 1 m p 1 -N 2 m p 2 ) = N (m-1) m p
. Since N , N 1 and N 2 are non multiple of m, we have p ≤ max(p 1 , p 2 ) and eventually changing a 1 by a 2 we can suppose that p 1 ≤ p 2 . Finally, a 1 , a 2 belong to the same F r,m -orbit if and only if there is an integer P ≥ 0 such that m P N 1 ≡ N 2 mod (m -1). But since m P ≡ 1 mod (m -1), we can conclude that two points a 1 , a 2 in Z[ 1 m ] belong to the same F r,m -orbit if and only if N 1 ≡ N 2 mod (m -1). This implies that there are exactly m orbits of m-adic points under the action of F r,m : the orbits of 0, 1 m , ..., m-1 m . 3'-The rotation by 1 m has the required properties except the fact that it has finite order, we slighty modified it by the following way :

0 | 1 m 3 | 1 m 2 + 1 m 3 | 1 m - 1 m - 1 m + 1 m 2 + 1 m 3 - 1 m + 1 m 2 - 2 m ¢ ¢ ¢ ¢ ¢ t(x) = mx + 1 m si x ∈ [0, 1 m 3 ] t(x) = 1 m (x -1 m 3 ) + 1 m + 1 m 2 si x ∈ [ 1 m 3 , 1 m 2 + 1 m 3 ] t(x) = x + 1 m si x ∈ [ 1 m 2 + 1 m 3 , r] Figure 3.
We still have t k (0) = k m , for all k ∈ Z. But Dt rm (0) = m, therefore 0 is a repulsive periodic point and t has infinite order.

4-Let

g ∈ T r,(n i ) \ F r,(n i ) , g(0) is a point of Z[ 1 m ] \ {0}
, where m = lcm(n i ). By the previous point 3, there exists an element f ∈ F r,(n i ) (in fact f ∈ F r,m a subgroup of F r,(n i ) ) such that f (g(0)) = k m for some k ∈ {1, ..., m -1}. So, using the point 3', f (g(0)) = t k (0) for some k ∈ {1, ..., m -1}. Finally,

t -k • f • g(0) = 0, hence t -k • f • g ∈ F r,(n i )
and g is in the group generated by t and F r,(n i ) . This ends the proof of the proposition 1.

3 Relatively hyperbolic groups, Osin's results.

In this paper we don't really need the definition of the strong relative hyperbolicity. We need the following definition and the fact that there exists a finite collection (H i ) i∈I of subgroups of G, for which the conclusions of the two following results hold.

Definition. An element g ∈ G is said to be parabolic if there is some i ∈ I and γ ∈ G such that g ∈ H γ i , where

H γ i := γH i γ -1 is called a parabolic subgroup of G.
An non parabolic element is said to be hyperbolic.

Proposition 2.36. Let G be a group and (H i ) i∈I a finite collection of subgroups of G. Suppose that G strongly hyperbolic relative to (H i ). Then the following conditions hold :

1)-The intersection

H γ i ∩ H δ j is finite whenever γ = δ 2)-The intersection H γ i ∩ H i is finite for any γ / ∈ H i .
3)-Let r be an element of infinite order in G, if there is an integer n such that r n ∈ H γ i then r ∈ H γ i . Theorem 4.19. Let G be a finitely generated group strongly hyperbolic relative to a collection (H i ) and g be an hyperbolic element of G. Then the centralizer C(g) of g is hyperbolic.

4 Proof of the theorem.

Recall that, two homeomorphisms having disjoint support commute, hence, using proposition 1.1, one can easily produce free abelian subgroup of rank greater than 2 in F r,(n i ) . Therefore, F r,n i and T r,n i are not hyperbolic groups. Moreover, the centralizer of an element of F r,n i with non full support contains a subgroup that is isomorphic to

F r,n i : Consider [a, a ] ⊂ [0, r] such that a, a in Z[ 1 m ], a -a ≡ r(mod d) and f (x) = x for all x ∈ [a, a ]
. By the Bieri-Strebel criterion, there exists a PL homeomorphism H with breaks in Z[ 1 m ] and slopes of the form m s for s ∈ Z that maps [a, a ] into [0, r]. Therefore H -1 F r,n i H is a group of PL homeomorphisms of [a, a ] with breaks in Z[ 1 m ] and slopes of the form m s , it can be injected in F r,n i by prolonging every element of it by the identity on [0, a] and [a , r].

Suppose that F r,n i [resp T r,n i ] is strongly hyperbolic relatively to a non trivial (i.e containing at least 2 elements) finite family (H i ).

Proposition 2. 1)-Let f ∈ B 0,1 \ {Id} then f is parabolic and its centralizer C(f ) is contained in a parabolic subgroup.

2)-Let f and g be two homeomorphisms in B 0,1 \ {Id} then f and g belong to the same H γ i , in other words B 0,1 ⊂ H γ i . Proof of proposition 2.

1)-Let f ∈ B 0,1 \ {Id}, suppose that Df (0) = 1 (the other case is analogous) then, there exists a non degenerated interval [0, a] such that f (x) = x on [0, a]. The centralizer of f contains a copy F r,(n i ) : it's not hyperbolic. Therefore, by Osin's results, f is parabolic : f belong to a H γ i . Now, if g ∈ C(f ) then the intersection of gH γ i g -1 = H gγ i with H γ i contains the cyclic group generated by f , since f = Id and F r,(n i ) has no torsion, this intersection is infinite and Osin's proposition 2.36 implies that g ∈ H γ i . 2)-Let f, g ∈ B 0,1 . We claim (we'll prove this after we have concluded the proof of 2) that it's possible to find two infinite order elements f ∈ C(f ) and ḡ ∈ C(g) that commute.

Hence, by the point 1) f, C(f ), f , C( f ) are contained in a same parabolic subgroup and g, C(g), ḡ, C( f ), g are contained in a same parabolic subgroup. Since F r,(n i ) has no torsion, Osin's proposition 2.36 implies that parabolic subgroups coincide or have trivial intersection. But ḡ ∈ C( f ) so they coincide, hence f and g belong to a same parabolic subgroup.

Proof of the claim. Case 2' : Df (1) = 1 and Dg(0) = 1 is analogous to the case 2.

Proof of the theorem for F r,(n i ) . By proposition 2, the set B 0,1 is contained in a parabolic subgroup, since it generates F r,(n i ) , the whole group F r,(n i ) is contained in this parabolic subgroup, this contradicts the hypothesis.

Proof of the theorem for T r,(n i ) . By the previous point, the subgroup F r,(n i ) of T r,(n i ) is contained in a parabolic subgroup H γ i . Consider the element t of infinite order defined in proposition 1 part 3'. Since t rm fix 0, it belongs to F r,m so it belongs to H γ i . By Osin's result, it follows that t also belongs to H γ i . Finally, by proposition 1 part 4, t and F r,(n i ) generate T r,(n i ) so T r,(n i ) is contained in H γ i a contradiction.

Case 1 :

 1 Df (0) = Dg(0) = 1. There exists a non degenerated interval [0, a] such that f (x) = g(x) = x for all x ∈ [0, a]. Then f = ḡ ∈ F r,(n i ) with support in [0, a] (such a map exists by proposition 1.1 or Bieri-Strebel criterion) satisfy the claim. Case 1' : Df (1) = Dg(1) = 1 is analogous to the case 1. Case 2 : Df (0) = 1 and Dg(1) = 1. There exist two non degenerated intervals [0, a] and [b, r] such that f (x) = x for all x ∈ [0, a] and g(x) = x for all x ∈ [b, r]. Taking f ∈ F r,(n i ) with support in [0, a] ∩ [0, b] and ḡ ∈ F r,(n i ) with support in [b, r] ∩ [a, r], then f and ḡ satisfy the claim.