Alexandre Martin
email: alexandre.martin@univie.ac.at

Jacek Świątkowski
email: jacek.swiatkowski@math.uni.wroc.pl

Infinitely-ended hyperbolic groups with homeomorphic Gromov boundaries

Keywords:

We show that the Gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its factors. We generalize this result to graphs of hyperbolic groups over finite subgroups. Finally, we give a necessary and sufficient condition for the Gromov boundaries of any two hyperbolic groups to be homeomorphic (in terms of the topology of the boundaries of factors in terminal splittings over finite subgroups).

Introduction

It is well known that the Gromov boundary of the free product G 1 * G 2 of two infinite hyperbolic groups is some combination of the Gromov boundaries of its factors. However, the nature of this combination seems to be not clarified. In particular, it is not clear whether the topology of ∂(G 1 * G 2 ) is uniquely determined by the topology of ∂G 1 and ∂G 2 . In this paper we show, among other things, that the answer to the above question is positive.

In [START_REF] Papasoglu | Quasi-isometries between groups with infinitely many ends[END_REF], P. Papasoglu and K. Whyte show that, up to quasi-isometry, such a free product is uniquely determined by the quasi-isometry types of its factors. Since there are hyperbolic groups whose boundaries are homeomorphic, but the groups themselves are not quasiisometric, this result does not help to answer the question above. For instance, there exist two non-quasi-isometric groups such that each of them is quasi-isometric to a hyperbolic building with the Menger curve as its Gromov boundary [START_REF] Bourdon | Immeubles hyperboliques, dimension conforme et rigidité de Mostow[END_REF].

Several descriptions of the boundary ∂(G 1 * G 2 ) appear more or less explicitely in the literature (see e.g. [START_REF] Tirel | Z-structures on products[END_REF], [START_REF] Woess | A description of the Martin boundary for nearest neighbour random walks on free products[END_REF]). However, they depend a priori on the groups G i , not only on the topology of their Gromov boundaries, and hence the question requires a more careful analysis.

We now present the main results of the paper, which generalize the above mentioned result concerning the free product. Recall that the fundamental group π 1 G of a graph of groups G with all vertex groups hyperbolic and all edge groups finite is itself hyperbolic.

Theorem 1.1. For i = 1, 2, let G i be graphs of groups with all vertex groups hyperbolic and all edge groups finite, and suppose that the groups π 1 G i have infinitely many ends. Denote by h(G i ) the set of homeomorphism types of Gromov boundaries of those vertex groups in G i which are nonelementary hyperbolic (i.e. are not finite and not virtually cyclic). If h(G 1 ) = h(G 2 ) then the Gromov boundaries ∂(π 1 (G i )) are homeomorphic.

Recall that a hyperbolic group is 1-ended if its Gromov boundary is a nonempty connected space. Theorem 1.1 has the following partial converse. Theorem 1.2. Under assumptions and notation as in Theorem 1.1, suppose additionally that all the vertex groups of both G i are either finite or 1-ended. If the Gromov boundaries ∂(π 1 (G i )) are homeomorphic then the sets h(G i ) of homeomorphism types of boundaries of 1-ended vertex groups in G i are equal.

Recall that, by a result of M. Dunwoody [START_REF] Dunwoody | The accessibility of finitely presented groups[END_REF], each finitely presented group G has a terminal splitting over finite subgroups, i.e. it is isomorphic to the fundamental group π 1 (G) of a graph of groups G whose vertex groups are 1-ended or finite and whose edge groups are finite. If G is hyperbolic, all vertex groups in any such splitting are also hyperbolic. The next corollary is a direct consequence of Theorems 1.1 and 1.2.

Corollary 1.3. For i = 1, 2, let G i be terminal splittings over finite subgroups of hyperbolic groups G i with infinitely many ends. Then the Gromov boundaries ∂G i are homeomorphic iff we have h(G 1 ) = h(G 2 ) (i.e. the sets of homeomorphism types of boundaries of 1-ended vertex groups in G i coincide).

The paper is organized as follows. In Sections 2-4 we deal with the case of the free product G = G 1 * G 2 of two groups. More precisely, in Section 2 we describe some topological space δΓ = δΓ(G 1 , G 2 ), arranged out of copies of the Gromov boundaries ∂G 1 and ∂G 2 , and equipped with a natural action of G. In Section 3 we show that δΓ is G-equivariantly homeomorphic to the Gromov boundary ∂G. We do not claim priority for the content of those two sections, as similar ideas seem to be known to various researchers (see Remark 3.2 and comments at the beginning of Subsection 2.2). Section 4 is the core of this article: given two free products A 1 * A 2 and B 1 * B 2 such that A i and B i have homeomorphic Gromov boundaries, we specify an isomorphism between the associated Bass-Serre trees, which allows us to prove Theorem 1.1 in the case of a free product (see Theorem 4.1). In Section 5 we deduce full Theorem 1.1 from the special case considered in Section 4, and from the result of Papasoglu and Whyte [START_REF] Papasoglu | Quasi-isometries between groups with infinitely many ends[END_REF]. Finally, in Section 6 we prove Theorem 1.2.

2 Geometric model for a free product.

In this section, we associate to the free product of two hyperbolic groups a graph quasiisometric to that free product and describe its Gromov boundary. The constructions and results presented in this section extend in a straightforward way to free products of finitely many hyperbolic groups.

Tree of spaces associated to a free product.

Bass-Serre tree of the splitting. Let G = A * B be the free product of two hyperbolic groups A and B. Let τ be a copy of the unit interval, and denote by v A and v B its vertices. We define a tree T = T (A, B), called the Bass-Serre tree of the splitting, as G × τ divided by the equivalence relation ∼ induced by the equivalences

(g 1 , v A ) ∼ (g 2 , v A ) if g -1 2 g 1 ∈ A, (g 1 , v B ) ∼ (g 2 , v B ) if g -1 2 g 1 ∈ B.
This tree comes with an action of G. More precisely, if [g, x] denotes the equivalence class of (g, x) ∈ G × τ , we set g • [g , x] = [gg , x]. A strict fundamental domain for this action is any edge [g, τ ] = {[g, x] : x ∈ τ }. Edges have trivial pointwise stabilizers, while vertex stabilizers are the conjugates in G of the factors A and B. For more information about Bass-Serre trees we refer the reader to [START_REF] Serre | Arbres, amalgames, SL 2[END_REF].

Tree of Cayley graphs. To get a convenient geometric model for G we "blow up" the vertices of T , by replacing them (in an equivariant way) with copies of the Cayley graphs of the associated vertex stabilizers. Let ΓA, ΓB be the Cayley graphs of A and B, with respect to some chosen finite sets of generators, with distinguished vertices as basepoints. Define an auxiliary graph X as the union of ΓA, ΓB and τ , where the vertex v A of τ is identified with the basepoint of ΓA and v B with the basepoint of ΓB. Define a graph Γ = Γ(A, B) as G × X divided by the equivalence relation induced by

(g 1 , x 1 ) ∼ (g 2 , x 2 ) if x 1 , x 2 ∈ ΓA, g -1 2 g 1 ∈ A and g -1 2 g 1 x 1 = x 2 , (g 1 , x 1 ) ∼ (g 2 , x 2 ) if x 1 , x 2 ∈ ΓB, g -1 2 g 1 ∈ B and g -1 2 g 1 x 1 = x 2 .
The canonical map X → τ obtained by collapsing each Cayley graph on its basepoint extends to an equivariant continuous map Γ → T , which equips Γ with the structure of a tree of spaces (compare [START_REF] Scott | Topological methods in group theory[END_REF]), and in particular of a tree-graded space as considered in [START_REF] Druţu | Tree-graded spaces and asymptotic cones of groups[END_REF]. We denote by ΓG v the preimage of a vertex v of T under this map. This notation agrees with the fact that ΓG v is a subgraph of Γ isomorphic to the Cayley graph of the stabilizing subgroup G v of G in its action on T (which is an appropriate conjugate of either A or B in G). Thus, the structure of a tree of spaces over T for Γ consists of the subgraphs ΓG v corresponding to the vertices of T , and of edges connecting them. These connecting edges are in a natural bijective correspondence with the edges of T , and we call them lifts of the corresponding edges of T under the above projection map Γ → T .

Compactification and boundary of Γ.

We now define a compactification Γ of Γ, using Gromov boundaries ∂A, ∂B of the groups A and B as ingredients, and its boundary δΓ = Γ \ Γ. This boundary consists of two disjoint sets, corresponding to the two ways of "approaching infinity" in the tree of spaces Γ. Similar descriptions, in slightly different contexts and expressed in different terms (e.g. by an explicit metric or by description of convergent sequences), can be found in [START_REF] Tirel | Z-structures on products[END_REF] and [START_REF] Woess | A description of the Martin boundary for nearest neighbour random walks on free products[END_REF]. Our description is inspired by a construction given in [START_REF] Dahmani | Combination of convergence groups[END_REF], which does not apply directly to our case.

Boundaries of stabilizers.

Let δ Stab Γ be the set G× ∂A∪∂B divided by the equivalence relation induced by

(g 1 , ξ 1 ) ∼ (g 2 , ξ 2 ) if ξ 1 , ξ 2 ∈ ∂A, g -1 2 g 1 ∈ A and g -1 2 g 1 ξ 1 = ξ 2 , (g 1 , ξ 1 ) ∼ (g 2 , ξ 2 ) if ξ 1 , ξ 2 ∈ ∂B, g -1 2 g 1 ∈ B and g -1 2 g 1 ξ 1 = ξ 2 . We denote by [g, ξ] the equivalence class of an element (g, ξ). The set δ Stab Γ comes with a natural action of G given by g • [g, ξ] = [g g, ξ]
. It also comes with a natural projection to the set V(T ) of vertices of the Bass-Serre tree T . We denote by ∂G v the preimage of a vertex v of T under this projection. This notation is consistent with the fact that ∂G v can be identified with the Gromov boundary of the stabilizing subgroup G v of G in its action on T .

Boundary of the Bass-Serre tree. Denote by ∂T the set of ends of the tree T , i.e. the set of infinite geodesic rays in T divided by the equivalence relation obtained by identifying rays when they coincide except at some bounded intial parts. Clearly, ∂T comes with the action of G induced from the action on T .

We define the boundary of Γ, δΓ := δ Stab Γ ∂T , and the compactification of Γ, Γ = Γ δΓ. This set comes with the action of G (described separately on the parts Γ and δΓ), and with the natural map p : Γ → T ∪ ∂T . The preimage of a vertex v of T is ΓG v ∪ ∂G v , which we identify (at this moment only set theoretically) with the compactification ΓG v of the Cayley graph of the corresponding stabilizing subgroup by means of its Gromov boundary.

Topology of the compactification Γ. For a point x ∈ Γ ⊂ Γ, we set a basis of open neighbourhoods of x in Γ to be also a basis of open neighbourhoods of x in Γ. We now define a basis of open neighbourhoods for points of δΓ ⊂ Γ. Fix a vertex v 0 in the Bass-Serre tree T .

• Let ξ ∈ δ Stab Γ and let v be the vertex of T such that ξ ∈ ∂G v . Let U be a neighbourhood of ξ in ΓG v . Define V U to be the set of all elements z ∈ Γ with projection p(z) = v, and such that the geodesic in T from v to p(z) starts with an edge e that lifts through p to an edge of Γ which is glued to ΓG v at a point of U . We then set

V U (ξ) = U ∪ V U .
As a basis of neighbourhoods of ξ in Γ we take a collection of sets V U (ξ) as above, where U runs through some basis of open neighbourhoods of ξ in ΓG v .

• Let η ∈ ∂T and let n ≥ 1 be an integer. Let T n (η) be the subtree of T that consists of those elements x ∈ T for which the geodesic from v 0 to x has the same first n edges as the geodesic ray [v 0 , η). Denote by T n (η) ⊂ T ∪ ∂T the union of T n (η) and the set of those eneds of T which are represented by rays contained in T n (η). Put also u n (η) to be the vertex at distance n from v 0 on the same geodesic ray [v 0 , η). We then set

V n (η) = p -1 T n (η) \ {u n (η)} .
As a basis of open neighbourhoods of η in Γ we take the collection of sets V n (η) for all integer n ≥ 1.

We skip a straightforward verification that the above collections of sets satisfy the axioms for basis of open neighbourhoods.

3 Identification of δΓ with the Gromov boundary ∂G.

Proposition 3.1. The space δΓ, equipped with the topology induced from the above described topology on Γ, is a compact metric space which is G-equivariantly homeomorphic to the Gromov boundary ∂G of G.

Here again, it should be noted that Proposition 3.1 and its proof below extend without any change to the case of free products of finitely many hyperbolic groups.

Remark 3.2. (1) Proposition 3.1 can be proved by repeating the arguments given by Francois Dahmani in Sections 2 and 3 of [START_REF] Dahmani | Combination of convergence groups[END_REF]. More precisely, using his arguments one can show that the natural action of G on δΓ is a uniform convergence action. (This does not follow directly from the results of Dahmani, as he considers graphs of groups with infinite edge groups only.) By a theorem of Bowditch [START_REF] Bowditch | A topological characterisation of hyperbolic groups[END_REF], it follows that δΓ is then Gequivariantly homeomorphic to the Gromov boundary of G. A convincing account of the fact that Dahmani's arguments yield Proposition 3.1 would be rather long, even though no essential adaptations are required.

(2) Proposition 3.1 (or at least its main assertion not dealing with G-equivariance) is a reformulation (and a restriction to hyperbolic groups) of a result by Wolfgang Woess (see [START_REF] Woess | A description of the Martin boundary for nearest neighbour random walks on free products[END_REF] or Section 26.B in [START_REF] Woess | Random walks on infinite graphs and groups[END_REF]). Woess studies an object called the Martin boundary of a graph (or group), which is known to coincide with the Gromov boundary if the graph (or the group) is hyperbolic, see e.g. Section 27 of [START_REF] Woess | Random walks on infinite graphs and groups[END_REF]. He describes the Martin boundary of the free product of two groups, in terms of Martin boundaries of the factors. His description uses a slightly different language than ours, and the topology is introduced in terms of convergent sequences rather than neighbourhoods of points. To justify precisely our statement of Proposition 3.1 by referring to the result of Woess, one needs to provide a translation of his setting to ours, and this cannot be done in one sentence.

(3) Proposition 3.1 follows also from a much more general theorem of the first author, namely Corollary 9.19 in [START_REF] Martin | Non-positively curved complexes of groups and boundaries[END_REF].

For the reader's convenience, we include here a separate short proof concerning the case we need in this paper.

We now pass to the proof of Proposition 3.1. Recall that an edge e of T canonically lifts to an edge e of Γ such that the projection Γ → T maps e onto e. We call e the lift of e. For an edge e = [v, v ] of T , we call the points e ∩ ΓG v and e ∩ ΓG v the attaching points of e.

Observe that we can obtain a Cayley graph for G by collapsing to points lifts of all edges of T in Γ. Moreover, the corresponding quotient map is G-equivariant. In particular, Γ is G-equivariantly quasi-isometric to G, so Γ is a hyperbolic space and the Gromov boundaries ∂Γ and ∂G are G-equivariantly homeomorphic. It is thus sufficient to construct a G-equivariant homeomorphism ∂Γ → δΓ.

Gromov boundary ∂Γ. We now describe the Gromov boundary ∂Γ. As a set, it consists of equivalence classes of geodesic rays in Γ (started at arbitrary vertices), where two geodesic rays are equivalent if they remain at finite distance from one another. Such geodesic rays ρ are easily seen to have one of the following two forms: (r 1 ) ρ is the concatenation of an infinite sequence of polygonal paths of the form

(3.1.1) ρ = [u 1 , w 1 ]τ 1 [u 2 , w 2 ]τ 2 . . .
where each [u i , w i ] is a geodesic path in a single subgraph ΓG v i in Γ ([u 1 , w 1 ] may be reduced to a trivial path), and where each τ i is the lift of an edge e i of T ; in particular, the concatenation e 1 e 2 . . . defines a geodesic ray in T ;

(r 2 ) ρ is the concatenation of a finite sequence of polygonal paths of the form

(3.1.2) ρ = [u 1 , w 1 ]τ 1 . . . [u k-1 , w k-1 ]τ k-1 ρ k
for k ≥ 0, where the paths [u i , w i ] and the edges τ i are as in (r 1 ), and where ρ k is a geodesic ray in ΓG v k based at the attaching point of τ k-1 .

An easy observation is that two geodesic rays ρ, ρ in Γ are equivalent if and only if:

(e 1 ) either they both have form (3.1.1) and their corresponding infinite sequence of edges (τ i ) i≥1 and (τ i ) i≥1 eventually coincide (i.e. they coincide after deleting some finite initial parts), or (e 2 ) they both have form (3.1.2), and in this case their terminal geodesic rays ρ k , ρ k belong to the same subgraph ΓG v k = ΓG v k and are equivalent in it.

Defining the map h : ∂Γ → δΓ. We now construct a bijection h : ∂Γ → δΓ as follows: Let z ∈ ∂Γ be an equivalence class as in (r 1 ). With the notations of (r 1 ), it follows from (e 1 ) that we can associate to z the end η of T induced by the corresponding class of rays e 1 e 2 . . . in T . We thus set h(z) := η ∈ ∂T .

Let z ∈ ∂Γ be an equivalence class as in (r 2 ). With the notations of (r 2 ), it follows from (e 2 ) that we can associate to z the point ξ of the Gromov boundary ∂G v k represented by the equivalence class of geodesic rays ρ k . We thus set h(z) := ξ ∈ δ Stab Γ.

The resulting map h : ∂Γ → δΓ is easily seen to be a G-equivariant bijection. Since ∂Γ = ∂G is compact, to get Proposition 3.1 it is enough to prove that δΓ is Hausdorff and h is continuous.

Lemma 3.3. The space δΓ is Hausdorff.

Proof. Since the points in δΓ are of two different natures, there are three cases to consider.

Case 1: Let η, η be two distinct points of ∂T . Let n be the length of the maximal common subsegment in the geodesic rays [v 0 , η) and [v 0 , η ). It follows from the definition of the topology of δΓ that the associated neighbourhoods V n+1 (η) and V n+1 (η ) are disjoint.

Case 2: Let η ∈ ∂T and ξ ∈ δ Stab Γ. Let v be the vertex of T such that ξ ∈ ∂G v . The intersection of the geodesic rays [v 0 , η) ∩ [v, η) is a geodesic ray of the form [w, η) for some vertex w of T . Let n be the combinatorial distance between v 0 and w in T . Let e be the first edge on the geodesic ray [v, η) and e its lift to Γ. Let U be a neighbourhood of ξ in ΓG v that misses the attaching point of e. The definition of the topology of δΓ then implies that the neighbourhoods V U (ξ) and V n+1 (η) are disjoint.

Case 3: Let ξ, ξ be two distinct points of δ Stab Γ. Let v, v be the vertices of T such that ξ ∈ ∂G v and ξ ∈ ∂G v .

If v = v , then as ΓG v is Hausdorff, we can choose disjoint neighbourhoods U, U of ξ, ξ in EG v , which yields disjoint neighbourhoods V U (ξ) and V U (ξ ) in δΓ.

If v = v , let e (resp. e ) be the edge of the geodesic segment [v, v ] which contains v (resp. v ). Let e, e be their lifts to Γ. We choose a neighbourhood U (resp. U ) of ξ (resp. ξ ) in ΓG v (resp. ΓG v ) which misses the attaching point of e (resp. e ). The definition of the topology of δΓ then implies that the neighbourhoods V U (ξ) and V U (ξ ) are disjoint.

The map h is continuous. As ∂Γ is metrisable, it is enough to prove that h is sequentially continuous. To do this, we fix a vertex u 0 ∈ ΓG v 0 (where v 0 is our chosen base vertex of T ), and we view it as a basepoint of Γ. Points z ∈ ∂Γ are then represented by (the equivalence classes of) geodesic rays started at u 0 and convergence of sequences in ∂Γ is characterized by

(3.1.3) z n → z iff d(u 0 , [z, z n ]) → ∞,
where d is the standard geodesic metric on Γ and [z, z n ] is any geodesic in Γ connecting the corresponding pair of boundary points (see e.g. Remark 3.17 [START_REF] Dunwoody | The accessibility of finitely presented groups[END_REF] and Exercise 3.18(3) on p. 433 in [START_REF] Bridson | Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften[END_REF]).

The proof of the continuity of h is split in two parts, Corollary 3.5 and Lemma 3.6, corresponding to the two parts ∂T and δ Stab Γ of δΓ. We start by a useful lemma used in both Corollary 3.5 and Lemma 3.6, in the statement of which we refer to the natural map ∂Γ → V(T ) ∪ ∂T resulting from conditions (e 1 ) and (e 2 ), which we again call a projection. Lemma 3.4. Let z n : n ∈ N and z be points of δΓ, x n : n ∈ N and x their projections in V(T ) ∪ ∂T . Suppose that the intersection

[v 0 , x n ] ∩ [v 0 , x] is a fixed (i.e. independent of n) geodesic segment in T that is strictly contained in both [v 0 , x n ] and [v 0 , x]. Then the sequence of distances (d(u 0 , [z, z n ])) is bounded.
Proof. The intersection of geodesics [v 0 , x n ] and [v 0 , x] can be written as the concatenation e 1 . . . e m of a finite sequence of edges of T . Then, in accordance with (3.1.1) or (3.1.2), a geodesic ray in the equivalence class z started at u 0 has a form

[u 0 , w 1 ] e 1 [u 2 , w 2 ] . . . e m [u m+1 , w m+1 ]τ m+1 . . .
where e i is the lift of e i , and where the appearance of τ m+1 follows from the assumption that [v 0 , x n ] ∩ [v 0 , x] is strictly contained in both intersected geodesics. Since Γ has a structure of tree of spaces, for any n every geodesic [z, z n ] passes through the vertex w m+1 . Thus we have d(u 0 , [z, z n ]) ≤ d(u 0 , w m+1 ), hence the lemma. Corollary 3.5. Let η ∈ ∂Γ be a point such that h(η) ∈ ∂T (i.e. geodesic rays representing η have form (2.1.1)), and let (z n ) be a sequence converging to η in ∂Γ. Then h(z n ) converges to h(η) in δΓ.

Proof. Let x n ∈ V(T )∪∂T be the projections of z n . Since by (3.1.3) we have d(u 0 , [η, z n ]) → ∞, Lemma 3.4 implies that x n converges to η in T ∪ ∂T . By definition of the topology of Γ, this implies that h(z n ) converges to h(η) in δΓ.

Lemma 3.6. Let ξ ∈ ∂Γ be a point such that h(ξ) ∈ δ Stab Γ (i.e. geodesic rays representing ξ have form (3.1.2)), and let (z n ) be a sequence of elements of ∂Γ converging to ξ. Then h(z n ) converges to h(ξ) in δΓ.

Proof. Let v be the vertex of T such that h(ξ) ∈ ∂G v (i.e. v is the image of ξ through the projection ∂Γ → V(T ) ∪ ∂T ). Denote by

[u 0 , w 1 ]τ 1 . . . [u k , w k ]τ k ρ k+1
a geodesic ray in the equivalence class ξ, where ρ k+1 is a geodesic ray in ΓG v based at the attaching point of τ k in ΓG v , which we denote u k+1 . Let x n ∈ V(T ) ∪ ∂T be the projections of z n . By Lemma 3.4, we have that for all sufficiently large n the geodesic [v 0 , x n ] contains [v 0 , v]. For such large enough n, put y n := z n if x n = v. If x n = v, the equivalence class z n consists of geodesic rays of form

[u 0 , w 1 ]τ 1 . . . [u k , w k ]τ k [u k+1 , w n k+1 ]τ k+1 . . . ,
where the vertex w n k+1 is uniquely determined by z n (i.e. it is common for all geodesic rays as above). For such n put y n := w n k+1 . It follows from the structure of Γ that for all n as above we have

d(u 0 , [ξ, z n ]) = d(u 0 , u k+1 ) + d(u k+1 , [ξ, y n ]),
and thus, by (3.1.3), d(u k+1 , [ξ, y n ]) → ∞. Let us extend the map h : ∂Γ → δΓ to a map Γ ∪ ∂Γ → Γ ∪ δΓ, still denoted h, by setting h to be the identity map on Γ. For every n, we thus have h(y n ) ∈ ΓG v ∪ ∂G v (seen as a subset of Γ ∪ δΓ). Since ΓG v is geodesically convex in Γ, the latter convergence implies that h(y n ) → h(ξ) in ΓG v ∪ ∂G v = ΓG v with its natural topology. From the description of neighbourhoods of h(ξ) in the topology of δΓ, it follows that for every neighbourhood U of h(ξ) in δΓ, all but finitely many h(z n ) belong to U , which implies that h(z n ) → h(ξ) in δΓ.

It follows from Corollary 3.5 and Lemma 3.6 that h is continuous, which completes the proof of Proposition 3.1.

Step 1. Consider the smallest k for which b(g k ) has not yet been defined. Choose some integer l such that d H h l , f (π(g k )) < 1/k and such that h l was not yet chosen as image of any g i (such an element exists because H is dense in H ∪ ∂H). Set b(g k ) = h l .

Step 2. Consider the smallest k for which h k was not yet chosen as the image of any g, and choose any g ∈ G \ {1} such that d H h k , f (π(g)) < d H h k , ∂H + 1/k and such that b has not yet been defined on g (such an element g exists since π(G) is dense in ∂G, hence

f (π(G)) is dense in ∂H). Set b(g) = h k .
Then b is obviously a bijection. A straightforward verification shows that b satisfies property ( * ).

As a consequence, viewing each group as the vertex set of its Cayley graph, and the group unit as the base vertex of this graph, we get the following.

Corollary 4.3. There exists a bijection α : A 1 → A 2 (respectively β : B 1 → B 2 ) such that α(1) = 1 (respectively β(1) = 1), and the following property holds:

Let ξ ∈ ∂A 2 (respectively ξ ∈ ∂B 2 ) and let U 2 be an open neighbourhood of ξ in ΓA 2 (respectively, in ΓB 2 ). Then there exists a neighbourhood

U 1 of h -1 (ξ) in ΓA 1 (respectively ΓB 1 ) such that for every element a ∈ A 1 ∩ U 1 (respectively b ∈ B 1 ∩ U 1 ) we have α(a) ∈ U 2 (respectively β(b) ∈ U 2 ).
Proof. We prove only the part concerning the existence of α (the corresponding part for β clearly follows by the same argument). Let α : A 1 → A 2 be any bijection as in Lemma 4.2, i.e. such that ᾱ := α ∪ h A : A 1 ∪ ∂A 1 → A 2 ∪ ∂A 2 is a homeomorphism. We will show that this α is as required. Viewing A i ∪ ∂A i as subspaces in ΓA i , we get that the preimage ᾱ-1 (U 2 ) is an open neighbourhood of h -1 (ξ) in A 1 ∪ ∂A 1 . Since the complement of this set, K = (A 1 ∪∂A 1 )\ ᾱ-1 (U 2 ), is compact, and since h -1 (ξ) ∈ ΓA 1 \K, there is a neighbourhood U 1 of h -1 (ξ) in ΓA 1 disjoint from K, and consequently such that U 1 ⊂ ᾱ-1 (U 2 ). One verifies directly that this U 1 is as required.

Given bijections α, β as above, we can now define a specific isomorphism ι : T 1 → T 2 as follows. For i = 1, 2, let τ i = [u i , v i ] be the edge of T i such that u i , v i are stabilized by A i , B i respectively. Recall that each element g ∈ H 1 can be expressed uniquely, in reduced form, as g = a 1 b 1 . . . a n b n , with a j ∈ A 1 \ {1}, b j ∈ B 1 \ {1}, allowing also that a 1 = 1 and that b n = 1. It is not hard to realize that there is an isomorphism T 1 → T 2 which maps any edge a

1 b 1 . . . a n b n • τ 1 to the edge α(a 1 )β(b 1 ) . . . α(a n )β(b n ) • τ 2 ,
and it is obviously unique. We denote this isomorphism by ι, and we observe that ι(v 1 ) = v 2 .

Construction of the homeomorphism

∂(A 1 * B 1 ) → ∂(A 2 * B 2 ).
To show that the Gromov boundaries ∂H 1 , ∂H 2 are homeomorphic, it is sufficient to describe a homeomorphism δΓ 1 → δΓ 2 . To do this, we need the notion of a reduced representative of an element [g, ξ] ∈ δ Stab Γ 1 . Note that the set of all representatives of [g, ξ] has the form (ga, a -1 ξ) : a ∈ A 1 when ξ ∈ ∂A 1 , and the form (gb, b -1 ξ) : b ∈ B 1 when ξ ∈ ∂B 1 . In any case we choose for the reduced representative this pair (g , ξ ) in which the reduced form of g is the simplest one among the elements of the corresponding coset. More precisely, when ξ ∈ ∂A 1 , we choose the unique ga = a 1 b 1 . . . a n b n for which n is the smallest possible (in which case we have b n = 1), and when ξ ∈ ∂B 1 , we choose the unique gb = a 1 b 1 . . . a n b n for which b n = 1.

Let α : A 1 → A 2 , β : B 1 → B 2 , τ 1 , v 1 and ι : T 1 → T 2 be as in the previous subsection. We now define a map F : δΓ 1 → δΓ 2 as follows.

• Let [g, ξ] be an element of δ Stab Γ 1 , such that (g, ξ) is its reduced representative. Write the reduced expression g = a • Let η be an element of ∂T . We can represent it as an infinite word η = a 1 b 1 . . ., such that for each n the subword consisting of its first n letters corresponds to the n-th edge of the geodesic from v 1 to η (via the correspondence g → g • τ 1 ). We set F (a 1 b 1 . . .) = α(a 1 )β(b 1 ) . . .

where the infinite word on the right is similarly interpreted as a geoedesic ray in T 2 started at v 2 . Note that this amounts to defining the restriction of F to ∂T 1 as the map ∂T 1 → ∂T 2 induced by the isomorphism ι :

T 1 → T 2 .
Compatibility of the above described map F with the isomorphism ι manifests also by the following: for any vertex v of the tree T 1 the subset ∂(H 1 ) v ⊂ δΓ 1 is mapped by F bijectively to the subset ∂(H 2 ) ι(v) ⊂ δΓ 2 .

Proposition 4.4. The map F is a homeomorphism.

Proof. The map F is clearly bijective, so we need to prove that it is continuous.

Let ξ ∈ δ Stab Γ 1 and let v be the vertex of T 1 such that ξ ∈ ∂(H 1 ) v . Let U 2 be a neighbourhood of F (ξ) in Γ(H 2 ) ι(v) . Let U 1 be a neighbourhood of ξ in Γ(H 1 ) v as in Corollary 4.3 (where we identify canonically Γ(H 1 ) v with ΓA 1 or ΓB 1 , and Γ(H 2 ) ι(v) with ΓA 2 or ΓB 2 ). It is straightforward to check that F V U 1 (ξ) ∩ δΓ 1 ⊂ V U 2 (F (ξ)) ∩ δΓ 2 , and hence F is continuous at ξ.

Let η ∈ ∂T 1 . The isomorphism ι : T 1 → T 2 extends to a map ι : T 1 ∪ ∂T 1 → T 2 ∪ ∂T 2 so that ι| ∂T 1 coincides with F | ∂T 1 . Let n ≥ 1 be an integer, and consider the subtree (T 2 ) n F (η) ⊂ T 2 , defined with respect to the base vertex v 2 . Then ι -1 (T 2 ) n F (η) = (T 1 ) n (η), where the latter subtree is defined with respect to the base vertex v 1 . It is then straightforward to check that F V n (η) ∩ δΓ 1 = V n F (η) ∩ δΓ 2 , and hence F is continuous at η. This completes the proof of Theorem 4.1.

1 b 1

 1 . . . a n b n . We setF ([a 1 b 1 . . . a n b n , ξ]) = [α(a 1 )β(b 1 ) . . . α(a n )β(b n ), h(ξ)].

This section is devoted to the proof of Proposition 3.1 below. This proposition is a reformulation or a slight modification of already known results, but it cannot be easily justified by simply referring to the literature, for the reasons explained in Remark 3.2. Thus, for completeness and for the reader's convenience, we provide here a direct proof.

{1, . . . , k i }. Since a homeomorphism ∂G 1 = ∂π 1 (G 1 ) → ∂π 1 (G 2 ) = ∂G 2 clearly maps components homeomorphically onto components, we get h(G 1 ) = h(G 2 ), which concludes the proof.

The first author is supported by the European Research Council (ERC) grant of G. Arzhantseva, grant agreement n o 259527.

† The second author is partially supported by the Polish Ministry of Science and Higher Education (MNiSW), Grant N201 541738, and by the Polish National Science Centre (NCN), Grant 2012/06/ST1/00259.

4 Homeomorphism type of the boundary of a free product.

This section is devoted to the proof of the following special case of Theorem 1.1.

Theorem 4.1. For i = 1, 2, let H i = A i * B i be the free product of infinite hyperbolic groups, and suppose that we have homeomorphisms ∂A 1 ∼ = ∂A 2 and ∂B 1 ∼ = ∂B 2 between Gromov boundaries. Then the boundaries ∂H i are homeomorphic.

Before starting the proof, which occupies the next two subsections, we fix some notation. Let h A : ∂A 1 ∼ -→ ∂A 2 and h B : ∂B 1 ∼ -→ ∂B 2 be some fixed homeomorphisms of boundaries resulting from the assumptions of the theorem. We denote by h :

In what follows, we use the same notation as in the previous two sections. We denote by T 1 (respectively T 2 ) the Bass-Serre tree associated to the product A 1 * B 1 (respectively A 2 * B 2 ). We also denote by Γ 1 = Γ(A 1 , B 1 ) and Γ 2 = Γ(A 2 , B 2 ) the corresponding trees of Cayley graphs, as described in Subsection 2.1.

Specifying an isomorphism between Bass-Serre trees.

Since we deal with infinite finitely generated factor groups, the Bass-Serre tree of each of the splittings H i = A i * B i is abstractly isomorphic to the unique (up to simplicial isomorphism) simplicial tree with infinite countable valence at every vertex. Thus the associated Bass-Serre trees of the two splittings are abstractly isomorphic. In this subsection, we specify an isomorphism between those trees with some additional properties. We will use this isomorphism to describe a homeomorphism between the two boundaries ∂H i .

Recall that given a hyperbolic group G, the natural associated topology turns G ∪ ∂G (where ∂G is the Gromov boundary) into a compact metric space. Lemma 4.2. Let G, H be two infinite hyperbolic groups, and let f : ∂G → ∂H be a homeomorphism between their Gromov boundaries. Then there is a bijection b :

To choose b as above, put b(1) = 1, order G \ {1} and H \ {1} into sequences (g k ), (h k ), and iterate the following two steps alternately:

5 The proof of Theorem 1.1.

The proof is based on Proposition 3.1 and the following result of P. Papasoglu and K. Whyte (see [START_REF] Papasoglu | Quasi-isometries between groups with infinitely many ends[END_REF]).

Theorem 5.1 (Papasoglu-Whyte). Let G, H be finitely generated groups with infinitely many ends and let G, H be their graph of groups decompositions with all edge groups finite. If G, H have the same set of quasi-isometry types of infinite vertex groups (without multiplicities) then G and H are quasi-isometric.

Theorem 5.1 implies in particular that if G is any group distinct from Z 2 then the groups G * Z ∼ = G * 1 and G * G are quasi-isometric. In view of this, Theorem 5.1 has also the following consequence.

Corollary 5.2. Under assumptions and notation of Theorem 5.1, if groups G 1 , . . . , G m represent all quasi-isometry types of those infinite vertex groups of G which are not virtually cyclic then

Proof of Theorem 1.1. Recall that, by the assumption, the groups π 1 (G i ) have infinitely many ends. Recall also that quasi-isometric hyperbolic groups have homeomorphic Gromov boundaries, and that a group is quasi-isometric to a virtually cyclic group iff it is virtually cyclic. Thus, if the case m = 0 in Corollary 5.2 applies to G 1 , it also applies to G 2 , and the theorem follows in this case from the corollary. In the remaining cases, not changing the quasi-isometry types, we replace the groups π 1 (G i ) with the corresponding free products of infinite vertex groups, accordingly with Corollary 5.2. By applying Theorem 5.1 again, we get that for some integer p ≥ 2 and for i = 1, 2 the groups π 1 (G i ) are quasi-isometric to some free products of infinite groups H i = G i,1 * • • • * G i,p such that for j = 1, . . . , p the groups G 1,j , G 2,j have homeomorphic Gromov boundaries. The theorem then follows by induction from Theorem 4.1.

6 Hyperbolic groups with homeomorphic boundaries.

In this section, after a few preparations dealing with detecting the connected components of the Gromov boundary of a free product of hyperbolic groups (Propositions 6.3 and 6.4), we prove Theorem 1.2.

Let G = A 1 * • • • * A k be the free product of finitely many hyperbolic groups. We represent G as the fundamental group of a graph of groups G over a finite graph Q with vertex set {v 1 , . . . , v k }, such that the vertex group at v j is A j and all the edge groups are trivial. Using the constructions of Sections 2.1 and 2.2, we associate to G a graph Γ having the structure of a tree of spaces over a Bass-Serre tree T of G, as well as the boundary δΓ = δ Stab Γ ∪ ∂T which is G-equivariantly homeomorphic to the Gromov boundary of G.

We start by stating two basic topological properties of δΓ, whose straightforward proofs we omit. Lemma 6.1. For any g ∈ G and any j ∈ {1, . . . , k}, the map f = f g,j : ∂A j → δΓ given by f

Next lemma requires some preparatory notation. Given an oriented edge e of T , consider the two subtrees obtained by deleting from T the interior of e, and denote by T + this subtree which contains the terminal vertex of e, and by T -the remaining subtree. Consequently, denote by ∂T + and ∂T -the corresponding sets of ends of the subtrees, which we naturally view as subsets of ∂T . Lemma 6.2. For any oriented edge e of T the subsets W e + := p -1 (T + ∪ ∂T + ) and W e -:= p -1 (T -∪ ∂T -) form an open-closed partition of δΓ.

We will call the pairs of subsets of δΓ as in the above lemma the natural halfspaces in δΓ. The above two lemmas have the following useful consequences. Proposition 6.3. A point η ∈ ∂T ⊂ δΓ is its own connected component in δΓ.

Proof. A straigtforward observation shows that any point ξ ∈ δΓ distinct from η can be separated from η by a pair of natural halfspaces. In view of Lemma 6.2, this clearly implies the assertion. Proposition 6.4. Let v be a vertex of T which is projected to v j by the natural projection T → Q, and suppose that the group A j is 1-ended. Then the subset ∂G v = p -1 (v) is a connected component of δΓ.

Proof. Note that the subset ∂G v coincides then with the image of the map f g,j : ∂A j → δΓ from Lemma 6.1, for some appropriately chosen g. Since A j is 1-ended, ∂A j is connected, and since f g,j is continuous, its image ∂G v is also connected. On the other hand, any point ξ ∈ δΓ \ ∂G v can be easily separated from ∂G v by a pair of natural halfspaces (which are open and closed due to Lemma 6.2), which completes the proof.

Proof of Theorem 1.2. By Theorem 5.1, without changing the quasi-isometry type of π 1 (G i ) and the 1-ended vertex groups of G i , we can assume that G i represent free products G i = A i,1 * • • • * A i,k i of 1-ended groups A i,j . By Propositions 6.3 and 6.4 (accompanied with the extension of Proposition 3.1 mentioned after its statement), the connected components of ∂G i are single points and subspaces homeomorphic to boundaries ∂A i,j for all j ∈


André Kabakouala
email: andre.kabakouala@lmpt.univ-tours.fr

A Remark on the Stability of Peakons for the Degasperis-Procesi Equation

Introduction

In this paper, we consider the Degasperis-Procesi equation (DP)

u t -u txx + 4uu x = 3u x u xx + uu xxx , (t, x) ∈ R + × R, (1.1)
with u(0) = u 0 ∈ L 2 (R) and (1 -∂ 2 x )u 0 ∈ M + (R). The DP equation is completely integrable (see [START_REF] Degasperis | A new integrable equation with peakon solutions[END_REF]) and has been proved to be physically relevant for water waves (see [START_REF] Constantin | The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations[END_REF]). It possesses, among others, the following conservation laws

E(u) = R yv = R 4v 2 + 5v 2 x + v 2 xx , F (u) = R u 3 = R -v 3 xx + 12vv 2 xx -48v 2 v xx + 64v 3 , (1.2)
where y = (1 -∂ 2 x )u and v = (4 -∂ 2 x ) -1 u. One can notice that the conservation law E(•) is equivalent to

• 2 L 2 (R)
. Indeed, using integration by parts (we assume that u(±∞) = v(±∞) = v x (±∞) = 0), it holds

u 2 L 2 (R) = R u 2 = R (4v -v xx ) 2 = R 16v 2 + 8v 2 x + v 2 xx ∼ E(u). (1.3)
In the sequel we will denote u H = E(u).

(1.4)

Applying (1 -∂ 2 x ) -1 (•) to (1.1), we obtain

u t + 1 2 ∂ x u 2 + 3 2 (1 -∂ 2 x ) -1 ∂ x u 2 = 0, (t, x) ∈ R + × R. (1.5)
In this form, the DP equation admits explicit solitary waves called peakons (see [START_REF] Degasperis | A new integrable equation with peakon solutions[END_REF]) that are defined by

u(t, x) = ϕ c (x -ct) = cϕ(x -ct) = ce -|x-ct| , c ∈ R * , (t, x) ∈ R + × R. (1.6) 1
Our goal is to simplify the proof given in [START_REF] Lin | Stability of peakons for the Degasperis-Procesi equation[END_REF] of the stability of a single peakon for the DP equation. Recall that the proof of the stability for the Camassa-Holm equation (CH) in [START_REF] Constantin | Stability of peakons[END_REF] follows from two integral relations between two conservation laws of CH, max R u and functions related to u. In [START_REF] Lin | Stability of peakons for the Degasperis-Procesi equation[END_REF] the proof is more complicated, since all the local maxima and minima of v = (4 -∂ 2

x ) -1 u are involved in the relations. In this paper we present a simplification of this proof, where only the maximum of v is involved in the relations. Our proof is thus closer to the proof for CH in [START_REF] Constantin | Stability of peakons[END_REF]. The main idea is the following: since u is L 2 -close to the peakon ϕ c (• -ξ), for some ξ ∈ R, and (1 -∂ 2

x )u ∈ M + (R), it is easy to check that u is actually C 0 -close to the peakon, and thus v is C 2 -close to the smooth-peakon:

ρ c (x -ξ) = (4 -∂ 2 x ) -1 ϕ c (x -ξ) = c 3 e -|x-ξ| - c 6 e -2|x-ξ| , x ∈ R. (1.7)
First, since ρ c , ρ ′ c and ρ ′′ c are very small with respect to the amplitude c outside of the interval Θ 0 = [-6.7, 6.7], we can restrict ourself to study v on Θ ξ = [ξ -6.

Preliminaries

In this section, we briefly recall the global well-posedness results for the Cauchy problem of the DP equation (see [START_REF] Escher | Global weak solutions and blow-up structure for the Degasperis-Procesi equation[END_REF] and [START_REF] Liu | Global existence and blow-up phenomena for the Degasperis-Procesi equation[END_REF]), and its consequences. For I a finite or infinite time interval of R + , we denote by X (I) the function space 1

X (I) = u ∈ C I; H 1 (R) ∩ L ∞ I; W 1,1 (R) , u x ∈ L ∞ (I; BV (R)) . (2.1)
Theorem 2.1 (Global Weak Solution; See [START_REF] Escher | Global weak solutions and blow-up structure for the Degasperis-Procesi equation[END_REF] and [START_REF] Liu | Global existence and blow-up phenomena for the Degasperis-Procesi equation[END_REF]). Assume that u 0 ∈ L 2 (R) with y 0 = (1 -∂ 2 x )u 0 ∈ M + (R). Then the DP equation has a unique global weak solution u ∈ X (R + ) such that

y(t, •) = (1 -∂ 2 x )u(t, •) ∈ M + (R), ∀t ∈ R + (2.2) and |u x (t, x)| ≤ u(t, x), ∀(t, x) ∈ R + × R. ( 2
(R) into L ∞ (R) lead to u L ∞ (R) ≤ 1 √ 2 u H 1 (R) ≤ u L 2 (R) .
(2.4)

Stability of peakons

In this section, we present our simplification of the proof of stability of peakons for the DP equation.

y 0 = (1 -∂ 2 x )u 0 ∈ M + (R) (3.1)
and

u 0 -ϕ c H ≤ ε 2 , with 0 < ε < ε 0 , (3.2) then u(t, •) -ϕ c (• -ξ(t)) H ≤ C √ ε, ∀t ∈ [0, T [, (3.3)
where ξ(t) ∈ R is the only point where the function v(t,

•) = (4 -∂ 2 x ) -1 u(t, •) attains its maximum.
We first recall that E(u) ∼ E(ϕ c ) and

F (u) ∼ F (ϕ c ) in R, if u ∼ ϕ c in L 2 (R)
, with y ∈ M + (R) (see for instance [START_REF] Lin | Stability of peakons for the Degasperis-Procesi equation[END_REF] or [START_REF] Kabakouala | Stability in the energy space of the sum of N peakons for the Degasperis-Procesi equation[END_REF]).

Lemma 3.1 (Control of Distances Between Energies; See [START_REF] Kabakouala | Stability in the energy space of the sum of N peakons for the Degasperis-Procesi equation[END_REF]).

Let u ∈ L 2 (R) with y = (1 -∂ 2 x )u ∈ M + (R). If u -ϕ c H ≤ ε 2 , then |E(u) -E(ϕ c )| ≤ O(ε 2 ) (3.4) and |F (u) -F (ϕ c )| ≤ O(ε 2 ), (3.5)
where O(•) only depends on the speed c.

To prove Theorem 3.1, by the conservation of E(•), F (•) and the continuity of the map

t → u(t) from [0, T [ to H (since H ≃ L 2 ), it suffices to prove that for any function u ∈ L 2 (R) satisfying y = (1 -∂ 2 x )u ∈ M + (R), (3.4) and (3.5), if inf z∈R u -ϕ c (• -z) H ≤ ε 1/4 , (3.6)
then u -ϕ c (• -ξ) H ≤ C √ ε, (3.7)
where ξ ∈ R is the only point of maximum of v.

Let us present some important properties of smooth-peakons, defined in (1.7), which will play a crucial role in the proof of Theorem 3.1. The smooth-peakon ρ c belongs to H 3 (R) ֒→ C 2 (R) (by the Sobolev embedding) since ϕ c belongs to H 1 (R) (defined in (1.6)). It is a positive even function, which admits a single maximum c/6 at point 0, and decays at infinity to 0 (see Fig. 1a). Its derivative ρ ′ c belongs to H 2 (R) ֒→ C 1 (R), it is an odd function, which vanishes only at the origin and has negative values on [0, +∞[. It admits a a single minimum -c/12 at point ln2 and tends at infinity to 0 (see Fig. 1b). Its second derivative ρ ′′ c belongs to H 1 (R) ֒→ C 0 (R), it is an even function, which vanishes at ±ln2, takes positive values on ] -∞, -ln2[∪]ln2, +∞[ and negative values on [-ln2, ln2]. It admits a single minimum -c/3 at point 0 and two maxima c/24 at points ±ln4, and decays at infinity to 0 (see Fig. 1c).

Next, we will need the following estimates.

Lemma 3.2 (C 0 , C 1 and C 2 Approximations). Let u ∈ L 2 (R) with y = (1 -∂ 2 x )u ∈ M + (R). If u -ϕ c H ≤ ε 1/4 , then u -ϕ c C 0 (R) + v -ρ c C 2 (R) ≤ O(ε 1/8 ) (3.8) and v -ρ c C 1 (R) ≤ O(ε 1/4
is equivalent to v H 2 (R) , since v H 2 (R) ≤ u H ≤ 5 v H 2 (R) . Then, assumption u is H-close to ϕ c implies that v is H 2 -close to ρ c . Now, using the Sobolev embedding of H 2 (R) into C 1 (R), we deduce (3.9).
For the first estimate, note that the assumption y = (1 (2.3)). Then, applying triangular inequality, and using that |ϕ ′ c | = ϕ c on R and (2.4), we have

-∂ 2 x )u ≥ 0 implies that u = (1 -∂ 2 x ) -1 y ≥ 0 and satisfies |u x | ≤ u on R (see
u -ϕ c H 1 (R) ≤ u H 1 (R) + ϕ c H 1 (R) ≤ 2 u L 2 (R) + 2 ϕ c L 2 (R) ≤ 2 u -ϕ c L 2 (R) + 4 ϕ c L 2 (R) ≤ O(ε 1/4 ) + O(1),
where ϕ c L 2 (R) = c. Therefore, applying the Gagliardo-Nirenberg inequality and using that u-ϕ c H ≤ ε 1/4 (with H ≃ L 2 ), we obtain

u -ϕ c C 0 (R) ≤ u -ϕ c 1/2 L 2 (R) u -ϕ c 1/2 H 1 (R) ≤ O(ε 1/8 ) O(ε 1/8 ) + O(1) ≤ O(ε 1/8 ) .
Finally to estimate the second term of the left-hand side of (3.8), we first notice that the continuity of (4

-∂ 2 x ) -1 (•) from H s (R) to H s+2 (R) and the above estimates ensure that v -ρ c H 3 = O(1) and v -ρ c H 2 = O(ε 1/4
) . These last estimates combined with the Gagliardo-Nirenberg inequality yield the result as above.

The following lemma specifies the distance to minimize for stability. Lemma 3.3 (Quadratic Identity; See [START_REF] Lin | Stability of peakons for the Degasperis-Procesi equation[END_REF]). For any u ∈ L 2 (R) and ξ ∈ R, it holds

E(u) -E(ϕ c ) = u -ϕ c (• -ξ) 2 H + 4c v(ξ) - c 6 , (3.10)
where v = (4 -∂ 2 x ) -1 u and ρ c (0) = c/6. Proof. We follow the idea of Constantin and Strauss with the CH equation (see [START_REF] Constantin | Stability of peakons[END_REF], Lemma 1). We compute

E(u -ϕ c (• -ξ)) = E(u) + E(ϕ c ) -2 (1 -∂ 2 x )ϕ c (• -ξ), (4 -∂ 2 x ) -1 u H -1 ,H 1 = E(u) + E(ϕ c ) -2 (1 -∂ 2 x )ϕ c (• -ξ), v H -1 ,H 1 , (3.11)
where •, • H -1 ,H 1 denotes the duality bracket H -1 (R), H 1 (R). Now, using the definition of ϕ ′ c (• -ξ) and integration by parts, we have

(1 -∂ 2 x )ϕ c (• -ξ), v H -1 ,H 1 = R vϕ c (• -ξ) + R v x ϕ ′ c (• -ξ) = R vϕ c (• -ξ) + ξ -∞ v x ϕ c (• -ξ) - +∞ ξ v x ϕ c (• -ξ) = 2cv(ξ). (3.12)
Recalling that the energy of peakons is given by

E(ϕ c ) = (1 -∂ 2 x )ϕ c , (4 -∂ 2 x ) -1 ϕ c H -1 ,H 1 = R ρ c ϕ c + R ρ ′ c ϕ ′ c = R ρ c ϕ c + 0 -∞ ρ ′ c ϕ c - +∞ 0 ρ ′ c ϕ c = 2cρ c (0) = c 2 3 , (3.13)
we obtain the lemma.

Now we will study carefully the local extrema of

v = (4 -∂ 2 x ) -1 u. Let u ∈ L 2 (R) with y = (1 -∂ 2 x )u ∈ M + (R)
, and assume that (3.6) holds for some z ∈ R. We consider the interval in which the mass of smooth-peakons is concentrated, and the interval in which the mass of second derivative of smoothpeakons is strictly negative. In the sequel of this paper, the notation α ≃ β means that 0.9 × β ≤ α ≤ 1.1 × β. We set, for any z ∈ R, One can clearly see that V 0 is a subset of Θ 0 (since 20/(20 -√ 399) > √ 2). We chose the values ±6.7 such that ρ c (±6.7) ≃ c/2400 ≃ 4.1 × 10 -4 c as in [START_REF] Kabakouala | Stability in the energy space of the sum of N peakons for the Degasperis-Procesi equation[END_REF]. Also, we have ρ ′ c (-6.7) = -ρ ′ c (6.7) ≃ 4.1 × 10 -4 c and ρ ′′ c (±6.7) ≃ 4.1 × 10 -4 c. Then ρ c , ρ ′ c and ρ ′′ c very small with respect to the amplitude c on R \ Θ 0 . We claim the following result.

Θ z = [z -6.

Lemma 3.4 (Uniqueness of the Local Maximum

). Let u ∈ L 2 (R) with y = (1 -∂ 2 x )u ∈ M + (R)
, that satisfies (3.6) for some z ∈ R. There exists ε 0 > 0 only depending on the speed c, such that if 0 < ε < ε 0 , then the function v = (4 -∂ 2

x ) -1 u admits a unique local extremum on Θ z . This extremum is a maximum, and it holds

v(x) ≤ c 300 , ∀x ∈ R \ Θ z , (3.16) u(x) ≤ c 300 , ∀x ∈ R \ Θ z . (3.17)
Proof.

The key is to study the impact of the assumption y ∈ M + (R) on v. First, let us show that |v x | ≤ 2v on R. We recall that from the assumption y ≥ 0, we have u ≥ 0 and v ≥ 0 on R. According to the definition of v, we have for all x ∈ R,

v(x) = e -2x 4 x -∞ e 2x ′ u(x ′ )dx ′ + e 2x 4 +∞ x e -2x ′ u(x ′ )dx ′ and v x (x) = - e -2x 2 x -∞ e 2x ′ u(x ′ )dx ′ + e 2x 2 +∞ x e -2x ′ u(x ′ )dx ′ , which yields |v x (x)| ≤ 2v(x), ∀x ∈ R. (3.18)
Second, let us show that u ≤ 6v on R. Using the Fourier transform, one can check that

(1 -∂ 2 x ) -1 (4 -∂ 2 x ) -1 (•) = F -1 1 3(1 + ω 2 ) - 1 3(4 + ω 2 ) (•) = 1 3 (1 -∂ 2 x ) -1 (•) - 1 3 (4 -∂ 2 x ) -1 (•), (3.19)
and one can rewrite v as

v = (4 -∂ 2 x ) -1 (1 -∂ 2 x ) -1 y = 1 3 (1 -∂ 2 x ) -1 y - 1 3 (4 -∂ 2 x ) -1 y. (3.20)
Then for all x ∈ R,

u(x) -6v(x) = -(1 -∂ 2 x ) -1 y(x) + 2(4 -∂ 2 x ) -1 y(x) = - 1 2 R e -|x-x ′ | y(x ′ )dx ′ + 1 2 R e -2|x-x ′ | y(x ′ )dx ′ ≤ 0, (3.21) since e -2|•| ≤ e -|•| on R.
We are now ready to prove the uniqueness of local maxima in Θ z . Let first study the sign of v xx on V z . One can easy check that for all x ∈ V 0 ,

ρ ′′ c (x) ≤ √ 2 -2 6 c. (3.22)
Then, combining (3.8) and (3.22), taking 0 < ε < ε 0 with ε 0 ≪ 1, we have for all x ∈ V z ,

v xx (x) ≤ √ 2 -2 6 c + O(ε 1/4 ) ≤ √ 2 -2 600 c < 0,
which implies that v x is strictly decreasing on V z . Let us study the sign of v x on Θ z \ V z . One can easily check that

ρ ′ c -ln √ 2 = √ 2 -1 6 c and ρ ′ c ln √ 2 = - √ 2 -1 6 c, (3.23)
and that ρ ′ c (x) ≥ 10 -4 c for all x ∈ [-6.7, -ln √ 2]. Then using (3.9) and taking 0 < ε < ε 0 with ε 0 ≪ 1, we have v x (x) ≥ 4 × 10 -5 c > 0 for all x ∈ [z -6.7, z -ln √ 2]. Proceeding in the same way, we obtain v x (x) ≤ -4 × 10 -5 c < 0 for all x ∈ [z + ln √ 2, z + 6.7]. Since v x is strictly decreasing on V z and changes sign, v x vanishes once on V z and thus on Θ z . Hence, v admits a single local extremum on Θ z , which is a maximum since v xx < 0 on V z . Now, using that ρ c is increasing on R -, (3.9) and taking 0 < ε < ε 0 with ε 0 ≪ 1, it holds for all x ∈] -∞, z 6.7[,

v(x) = ρ c (x -z) + O(ε 1/4 ) ≤ c 2400 + O(ε 1/4 ) ≤ c 300 .
Proceeding in the same way for x ∈]z + 6.7, +∞[, we obtain (3.16). Combining (3.8), (3.21) and proceeding as for the estimate (3.16), we get (3.17). Note that ϕ c (±6.7) ≃ 1.2 × 10 -3 c. This completes the proof of the lemma.

Under the assumptions of Lemma 3.4, v has got a unique point of global maximum on R. In the sequel of this section, we will denote by ξ this point of global maximum and we set M = v(ξ) = max x∈R v(x). The next two lemmas can be directly deduced from the similar lemmas established in [START_REF] Lin | Stability of peakons for the Degasperis-Procesi equation[END_REF] (see also [START_REF] Kabakouala | Stability in the energy space of the sum of N peakons for the Degasperis-Procesi equation[END_REF]). Lemma 3.5 (Connection Between E(•) and M 2 ; See [START_REF] Lin | Stability of peakons for the Degasperis-Procesi equation[END_REF]). Let u ∈ L 2 (R) and v = (4 -∂ 2

x ) -1 u ∈ H 2 (R). Define the function g by

g(x) = 2v + v xx -3v x , x < ξ, 2v + v xx + 3v x , x > ξ. (3.24) Then it holds R g 2 (x)dx = E(u) -12M 2 . (3.25) Lemma 3.6 (Connection Between F (•) and M 3 ; See [7]). Let u ∈ L 2 (R) and v = (4 -∂ 2 x ) -1 u ∈ H 2 (R). Define the function h by h(x) = -v xx -6v x + 16v, x < ξ, -v xx + 6v x + 16v, x > ξ. (3.26) Then it holds R h(x)g 2 (x)dx = F (u) -144M 3 . (3.27)
Sketch of proof. The proof of Lemmas 3.5-3.6 follows by direct computation, using integration by parts, with v x (ξ) = 0 and v(±∞) = v x (±∞) = v xx (±∞) = 0. See [START_REF] Lin | Stability of peakons for the Degasperis-Procesi equation[END_REF] (also [START_REF] Kabakouala | Stability in the energy space of the sum of N peakons for the Degasperis-Procesi equation[END_REF]) to undersand the technique.

We can now connect the conservation laws.

Lemma 3.7 (Connection Between E(•) and F (•)). Let u ∈ L 2 (R), with y = (1 -∂ 2 x )u ∈ M + (R), that satisfies (3.6) for some z ∈ R. There exists ε 0 > 0 only depending on the speed c, such that if 0 < ε < ε 0 , then it holds M 3 - 1 4 E(u)M + 1 72 F (u) ≤ 0. (3.28)
Proof. The key is to show that h ≤ 18M on R. Note that by (3.9) we know that 18M ≥ c/4 and that Lemma 3.4 ensures that ξ ∈ Θ z for ε 0 small enough. Let us set λ = z -6.7, µ = z + 6.7, and rewrite the function h as

h(x) =          -v xx -6v x + 16v, x < λ, u -6v x + 12v, λ < x < ξ, u + 6v x + 12v, ξ < x < µ, -v xx + 6v x + 16v, x > µ .
If x ∈ R \ Θ z , using that v xx = 4v -u, (3.16) and (3.17), it holds

h ≤ |v xx | + 6|v x | + 16v ≤ u + 32v ≤ c 9 ≤ 18M.
If λ < x < ξ, then v x ≥ 0, and using that u ≤ 6v on R, we have h = u -6v x + 12v ≤ 18v.

If ξ < x < µ, then v x ≤ 0, and similarly using that u ≤ 6v on R, we get

h = u + 6v x + 12v ≤ 18v.
Therefore, it holds

F (u) -144M 3 = R h(x)g 2 (x)dx ≤ h L ∞ (R) R g 2 (x)dx ≤ 18M (E(u) -12M 2 ),
and we obtain the lemma.

Proof of Theorem 3.1. We argue as El Dika and Molinet in [START_REF] El | Stability of multipeakons[END_REF]. As noticed after the statement of the theorem, it suffices to prove (3.7) assuming that u ∈ L 2 (R) satisfies Using that E(ϕ c ) = c 2 /3 and F (ϕ c ) = 2c 3 /3, our inequality becomes

M - c 6 2 M + c 3 ≤ O(ε 2 ).
Next, substituting M by c/6 -δ and using that [M + c/3] -1 < 3/c, we obtain where C > 0 only depends on the speed c. This completes the proof of the stability of peakons.

δ 2 ≤ O(ε 2 ) ⇒ δ ≤ O(ε). ( 3

Theorem 3 . 1 (

 31 Stability of Peakons). Let u ∈ X ([0, T [), with 0 < T ≤ +∞, be a solution of the DP equation and ϕ c be the peakon defined in (1.6), traveling to the right at the speed c > 0. There exist C > 0 and ε 0 > 0 only depending on the speed c, such that if

Figure 1 :

 1 Figure 1: Variation of the smooth-peakon with the amplitude 1/6 at initial time.

  h(x) ≤ 18 max x∈R v(x) = 18M, ∀x ∈ R. (3.29) Now, combining (3.25), (3.27) and (3.29), we get

  (3.1), (3.2) and(3.6). We recall that M = v(ξ) = max x∈R v(x) and we set δ = c/6 -M . We first remark that if δ ≤ 0, combining (3.4) and (3.10), it holdsu -ϕ c (• -ξ) H ≤ |E(u 0 ) -E(ϕ c )| 1/2 ≤ O(ε),that yields the desired result. Now suppose that δ > 0, that is the maximum of the function v is less than the maximum of ρ c . Combining (3.4), (3.5) and (3.28), we getM 3 -1 4 E(ϕ c )M + 1 72 F (ϕ c ) ≤ O(ε 2 ).

  .30) Finally, combining (3.4), (3.10) and (3.30), we infer thatu -ϕ c (• -ξ) H ≤ C √ ε,

  [START_REF] Lin | Stability of peakons for the Degasperis-Procesi equation[END_REF], ξ + 6.7]. Now we observe that ρ ′′ This forces v x to change sign only one time on Θ ξ , and thus v has only one local extremum (which is a maximum) on Θ ξ . This fact will considerably simplify the proof of the stability.

	c has 2] and √ c strictly positive on [-6.7, -ln 2], with ρ ′ √ 2, ln √ strictly negative values in the interval V 0 = [-ln ρ ′ c strictly negative on [ln √ 2, 6.7].

W 1,1 (R) is the space of L 1 (R) functions with derivatives in L 1 (R) and BV (R) is the space of function with bounded variation.

Acknowledgements. The author would like to thank his PhD advisor Luc Molinet for his help and his careful reading of this manuscript.