FR ÉD ÉRIC MATH ÉUS ¸ois Gautero

POISSON BOUNDARY OF GROUPS ACTING ON R-TREES

Keywords: Subject Classification. 20F65, 60J50, 20F67, 60B15, 20P05 hyperbolic groups, semi-direct products

We give a geometric description of the Poisson boundaries of certain extensions of free and hyperbolic groups. In particular, we get a full description of the Poisson boundaries of free-by-cyclic groups in terms of the boundary of the free subgroup. We rely upon the description of Poisson boundaries by means of a topological compactification as developed by Kaimanovich. All the groups studied here share the property of admitting a sufficiently complicated action on some R-tree.

2000 Mathematics

Introduction

Let G be a finitely generated group and µ a probability measure on G. A bounded function f :

G → R is µ-harmonic if ∀g ∈ G , f (g) = h µ(h)f (gh) .
Denote by H ∞ (G, µ) the Banach space of all bounded µ-harmonic functions on G equipped with the sup norm. There is a simple way to build such functions. Consider a probability G-space (X, λ) such that the measure λ is µ-stationary, which means that

λ = µ * λ def = µ(g) • gλ
To any function F ∈ L ∞ (X, λ) one can assign a function f on G defined, for g ∈ G, by the Poisson formula f (g) = F, gλ = X F (gx)dλ(x). This function f belongs to H ∞ (G, µ) since λ is µ-stationary. In [START_REF] Furstenberg | A Poisson formula for semi-simple Lie groups[END_REF][START_REF] Furstenberg | Random walks and discrete subgroups of Lie groups[END_REF], Furstenberg constructed a measured Gspace Γ with a µ-stationary measure ν such that the Poisson formula states an isometric isomorphism between L ∞ (Γ, ν) and H ∞ (G, µ). The space (Γ, ν) is called the Poisson boundary of the measured group (G, µ). It is well defined up to G-equivariant measurable isomorphism. A natural question is to decide whether a given G-space X endowed with a certain µ-stationary measure λ is actually the Poisson boundary of the pair (G, µ).

Let F be a finitely generated free group, and G a finitely generated group containing F as a normal subgroup. The action of G on F by inner automorphisms extends to an action by homeomorphisms on the geometric boundary ∂F of F . The group G is said to be a cyclic extension of F if the factor G/F is an infinite cyclic group. A probability measure µ on G is non-degenerate if the semigroup generated by its support is G. In this paper, we prove: Theorem 1. Let G be a cyclic extension of a free group F , and µ a non-degenerate probability measure on G with a finite first moment (with respect to some gauge). Then there exists a unique µ-stationary probability measure λ on ∂F and the G-space (∂F, λ) is the Poisson boundary of (G, µ).

The Poisson boundary is trivial for all non-degenerate measures on abelian groups [START_REF] Choquet | Sur l'équation de convolution µ = µ * σ[END_REF] and nilpotent groups [START_REF] Dynkin | Random walk on groups with a finite number of generators[END_REF] : on such groups, there are no non constant bounded harmonic functions. If F is a finitely generated free group then the action mentioned above on ∂F is just the left-action. Let µ be a non-degenerate probability measure on F . Then there exists a unique µ-stationary probability measure λ on ∂F [START_REF] Ledrappier | Some asymptotic properties of random walks on free groups[END_REF]. If µ has a finite first moment, then (∂F, λ) is the Poisson boundary of (F, µ) [START_REF] Dynkin | Random walk on groups with a finite number of generators[END_REF][START_REF] Derriennic | Marche aléatoire sur le groupe libre et frontière de Martin[END_REF][START_REF] Vadim | The Poisson boundary of hyperbolic groups[END_REF].

Assume now that G contains F as a normal subgroup. Then there still exists a unique probability measure λ on ∂F which is µ-stationary for the aforementioned action of G on ∂F , and the G-space (∂F, λ) is a so-called µ-boundary of G, that is a G-equivariant quotient of the Poisson boundary of (G, µ) [START_REF] Vershik | The boundary of the braid group and the Markov-Ivanovsky normal form[END_REF]. So the question is to find conditions on G and µ ensuring that (∂F, λ) is actually the whole Poisson boundary of (G, µ). For instance, if F has finite index in G, then there exists a measure µ 0 on F such that the Poisson boundaries of (G, µ) and (F, µ 0 ) are isomorphic [START_REF] Furstenberg | Random walks and discrete subgroups of Lie groups[END_REF]. Moreover, if µ has a finite first moment, then so has µ 0 [START_REF] Vadim | Poisson boundaries of random walks on discrete solvable groups[END_REF], hence the Poisson boundary of (G, µ) is (∂F, λ).

Theorem 1 above provides an answer when G is a cyclic extension of the free group F . In this case, there is an exact short sequence {1} → F → G → Z → {0}. Such an extension is obviously splitting: there exists an automorphism α of F such that G is isomorphic to the semi-direct product F ⋊ α Z. Of course, the Poisson boundary is already known if α is an inner automorphism of F , because G is then isomorphic to a direct product F × Z. Therefore, since Z is central, its acts trivially on the Poisson boundary of G which can be identified to the Poisson boundary of G/Z ∼ F [START_REF] Kaimanovich | The Poisson boundary of covering Markov operators[END_REF].

There are two types of growth for an automorphism of a finitely generated group: polynomial growth, and exponential growth. It refers to the growth of the iterates of a given element under the automorphism, see Def. 3.1 below. In the case of an automorphism α of a finitely generated free group F , by the pioneering work of Bestvina-Handel [START_REF] Bestvina | Train tracks and automorphisms of free groups[END_REF] (see also [START_REF] Levitt | Counting growth types of automorphisms of free groups[END_REF] for precise and detailed statements about the growth of free group automorphisms) there is a dichotomy:

• either α has a polynomial growth, i.e. |α n (g)| grows polynomially for every g ∈ F ,

• or α has an exponential growth, i.e. there exists g ∈ F such that |α n (g)| grows exponentially. If the automorphism α has exponential growth, then the group G = F ⋊ α Z is hyperbolic relatively to the family H of polynomially growing subgroups [START_REF] Gautero | Hyperbolicity of mapping-torus groups and spaces[END_REF], therefore its Poisson boundary may be identified with its relative hyperbolic boundary ∂ RH (G, H) (see Section 10). But this answer is not completely satisfactory for us since the link between ∂F and the Poisson boundary of G remains unclear.

We list now a few generalizations of Theorem 1. First, we can deal with non-cyclic extension, but we have a restriction on the growth. In the next statement, we denote by Out(F ) = Aut(F )/Inn(F ) the group of outer automorphisms of F where Inn(F ) stands for the group of inner automorphisms.

Theorem 2. Let G be a semi-direct product G = F ⋊ θ P of a free group F with a finitely generated subgroup P over a monomorphism θ : P → Aut(F ) such that θ(P) maps injectively into Out(F ) and consists entirely of polynomially growing automorphisms. Let µ be a non-degenerate probability measure on G with a finite first moment. Then there exists a unique µ-stationary probability measure λ on ∂F and the G-space (∂F, λ) is the Poisson boundary of (G, µ).

If θ(P) is trivial in Out(F ) then G is isomorphic to a direct product F × F ′ where F ′ is a finitely generated free group. Then modifying the proof of Theorem 2 yields: Theorem 3. Let G = F × F ′ be the direct product of two finitely generated free groups. Consider the left-action of G on ∂F given by some fixed monomorphism F ′ ֒→ Inn(F ). Let µ be a non-degenerate probability measure on G with a finite first moment. Then there exists a unique µ-stationary probability measure λ on ∂F and the G-space (∂F, λ) is the Poisson boundary of (G, µ).

Theorems 2 and 3 describe the Poisson boundary of all the groups G = F ⋊ θ P such that θ(P) consists entirely of polynomially growing automorphisms. Below we provide examples where P = Z 2 or P is a free group of rank 2, see Section 9. We do not know how to deal with such groups if θ(P) contains exponentially growing automorphisms.

There is another way to extend Theorem 1 above. Consider a group G which is a cyclic extension of a hyperbolic group Γ. There exists an automorphism α of Γ such that the group G is isomorphic to the semi-direct product Γ ⋊ α Z. Again, the action of G on Γ by inner automorphisms extends to an action on the hyperbolic boundary ∂Γ of Γ. We have: Theorem 4. Let Γ be a torsion free, hyperbolic group with infinitely many ends. Let G = Γ ⋊ α Z be a cyclic extension of Γ. Let µ be a non-degenerate probability measure on G with a finite first moment. Then there exists a unique µ-stationary probability measure λ on ∂Γ and the G-space (∂Γ, λ) is the Poisson boundary of (G, µ).

The same conclusion holds if Γ is the fundamental group π 1 (S) of a compact surface S of genus ≥ 2 and α is an exponentially growing automorphism.

It is well known that automorphisms of the fundamental group of a compact surface S of genus ≥ 2 have either linear or exponential growth. Unfortunately, we are unable to treat the case G = π 1 (S) ⋊ α Z if α has linear (and non-zero) growth.

Beyond the cases of abelian, nilpotent and free groups, there are many other classes of groups for which the Poisson boundary is already known. For instance, if µ is a finitely supported probability measure on a group G with subexponential growth, then its Poisson boundary is trivial [START_REF] Avez | Théorème de Choquet-Deny pour les groupes à croissance non exponentielle[END_REF]. Discrete subgroups of SL(d, R) are treated in [START_REF] Furstenberg | Random walks and discrete subgroups of Lie groups[END_REF] where the Poisson boundary is related to the space of flags. For random walks on Lie groups, the study and the description of the Poisson boundary started in [START_REF] Furstenberg | A Poisson formula for semi-simple Lie groups[END_REF] and was extensively developed in the 70's and the 80's by many authors including Furstenberg [START_REF] Furstenberg | Random walks and discrete subgroups of Lie groups[END_REF][START_REF] Furstenberg | Boundary theory and stochastic processes on homogeneous spaces[END_REF], Azencott [START_REF] Azencott | Espaces de Poisson des groupes localement compacts[END_REF], Raugi and Guivarc'h [START_REF] Raugi | Fonctions harmoniques sur les groupes localement compacts à base dénombrable[END_REF][START_REF] Guivarc | Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence[END_REF][START_REF] Raugi | Un théorème de Choquet-Deny pour les groupes moyennables[END_REF], Ledrappier [START_REF] Ledrappier | Poisson boundaries of discrete groups of matrices[END_REF][START_REF] Ballmann | The Poisson boundary for rank one manifolds and their cocompact lattices[END_REF]. Some analogies between discrete groups and their continuous counterparts are enlightened, see [START_REF] Furstenberg | Random walks and discrete subgroups of Lie groups[END_REF][START_REF] Ledrappier | Poisson boundaries of discrete groups of matrices[END_REF][START_REF] Ballmann | The Poisson boundary for rank one manifolds and their cocompact lattices[END_REF], but some contrast can appear, compare [START_REF] Raugi | Un théorème de Choquet-Deny pour les groupes moyennables[END_REF] with [START_REF] Kaȋmanovich | Random walks on discrete groups: boundary and entropy[END_REF][START_REF] Vadim | Poisson boundaries of random walks on discrete solvable groups[END_REF]. The Poisson boundary of some Fuchsian groups has also been described by Series as being the limit set of the group [START_REF] Series | Martin boundaries of random walks on Fuchsian groups[END_REF]. Kaimanovich and Masur [START_REF] Vadim | The Poisson boundary of the mapping class group[END_REF] proved that the Poisson boundary of the mapping class group is the boundary of Thurston's compactification of the Teichmüller space. Their description also runs for the Poisson boundary of the braid group, see [START_REF] Farb | Superrigidity and mapping class groups[END_REF].

The Poisson boundary is closely related to some other well-known compactifications or boundaries. The Martin boundary is involved in the description of positive harmonic functions (see [START_REF] Ancona | Théorie du potentiel sur les graphes et les variétés[END_REF]). Considered as a measure space with the representing measure of the constant harmonic function 1, it is isomorphic to the Poisson boundary [START_REF] Vadim | Boundaries of invariant Markov operators: the identification problem[END_REF]. If the group G has infinitely many ends (and if µ is non-degenerate) then the Poisson boundary can be identified with the space of ends (with the hitting measure), see [START_REF] Woess | Boundaries of random walks on graphs and groups with infinitely many ends[END_REF][START_REF] Donald | Convergence to ends for random walks on the automorphism group of a tree[END_REF][START_REF] Vadim | The Poisson formula for groups with hyperbolic properties[END_REF]. Furthermore, Kaimanovich obtained an identification of the Poisson boundary for hyperbolic groups with the geometric boundary, see [START_REF] Vadim | The Poisson formula for groups with hyperbolic properties[END_REF]. Concerning the Floyd boundary [START_REF] Floyd | Group completions and limit sets of Kleinian groups[END_REF], Karlsson recently proved that if it is not trivial, then it is isomorphic to the Poisson boundary, see [START_REF] Karlsson | Boundaries and random walks on finitely generated infinite groups[END_REF]. The identification of these various compactifications with the Poisson boundary 1. The method Let µ be a probability on a countable group G, and X be a compact G-space endowed with a µ-stationary probability λ. In [START_REF] Vadim | The Poisson formula for groups with hyperbolic properties[END_REF], Vadim Kaimanovich provided sufficient conditions that should satisfy (X, λ) to be the Poisson boundary of the pair (G, µ) (see Theorem 2.3 below). His results also contain informations on the asymptotic behaviour of the random walk on G governed by µ. The aim of this section is to give a first flavour of the way we construct such a compact G-space X leading to two detailed and somehow technical statements (Theorems 1.1,1.2 and 1.3 below) of which Theorems 1, 2 and 3 will appear as corollaries.

Our approach is inspired by the long line of works about free group automorphims starting at the beginning of the nineties, see [START_REF] Bestvina | Train tracks and automorphisms of free groups[END_REF][START_REF] Bestvina | The Tits alternative for Out(F n ). II. A Kolchin type theorem[END_REF][START_REF] Gaboriau | An index for counting fixed points of automorphisms of free groups[END_REF][START_REF] Gaboriau | The rank of actions on R-trees[END_REF][START_REF] Levitt | Irreducible automorphisms of F n have north-south dynamics on compactified outer space[END_REF][START_REF] Levitt | Automorphisms of free groups have asymptotically periodic dynamics[END_REF][START_REF] Thierry Coulbois | Non-unique ergodicity, observers' topology and the dual algebraic lamination for R-trees[END_REF][START_REF] Thierry Coulbois | R-trees and laminations for free groups. II. The dual lamination of an R-tree[END_REF] to cite only a few of them. Write F d for the rank d free group, d ≥ 2. A common feature to the papers [START_REF] Gaboriau | An index for counting fixed points of automorphisms of free groups[END_REF][START_REF] Gaboriau | The rank of actions on R-trees[END_REF][START_REF] Levitt | Irreducible automorphisms of F n have north-south dynamics on compactified outer space[END_REF][START_REF] Levitt | Automorphisms of free groups have asymptotically periodic dynamics[END_REF][START_REF] Thierry Coulbois | Non-unique ergodicity, observers' topology and the dual algebraic lamination for R-trees[END_REF][START_REF] Thierry Coulbois | R-trees and laminations for free groups. II. The dual lamination of an R-tree[END_REF] is the introduction of a so-called "α-projectively invariant F d -tree", that is a R-tree (a geodesic metric space such that any two points are connected by a unique arc, and this arc is a geodesic for the considered metric) with an isometric action of F d and a homothety H α satisfying the fundamental relation: H α (w.P ) = α(w).H α (P ) (if the homothety H α is an isometry then the R-tree is "α-invariant"). Thanks to this relation, we have in fact an action of the semi-direct product F d ⋊ α Z on the R-tree. Having in mind the problem of compactifying such a semi-direct product, a major drawback of these R-trees is that the usual adjonction of their Gromov-boundary does not provide us with a compact space. This point is however settled by equipping them with the socalled "observers topology": the union of the completion of the R-tree T with its Gromov boundary ∂T , equipped with the observers topology, is denoted by T . This weak topology is thoroughly studied in [START_REF] Thierry Coulbois | Non-unique ergodicity, observers' topology and the dual algebraic lamination for R-trees[END_REF] (see also [START_REF] Favre | Arbres réels et espaces de valuation[END_REF] where it appears in a very different context).

As mentioned in the introduction, we have two different kind of results, depending on the nature of the action of the subgroup of automorphisms involved by the extension: polynomially growing or exponentially growing automorphisms. In Theorems 1.1 and 1.2 stated below, we extracted two particular cases from the various results of the paper (Theorems 6.1, 6.14, 7.2, 7.24, 8.1) to stress this dichotomy.

Theorem 1.1. Let F d be the rank d free group, d ≥ 2. Let P be a finitely generated subgroup of polynomially growing outer automorphisms of F d . Let µ be a probability measure on F d ⋊ P the support of which generates F d ⋊ P as a semi-group. Then there exists a finite index subgroup U in P and a simplicial U-invariant F d -tree T such that, if τ k denotes the (random) sequence of the times at which the path x = {x n } visits F d ⋊ U, then:

• Almost every sub-path {x τ k } of the random walk on (F d ⋊ P, µ) converges to some (random) limit x ∞ ∈ ∂T . • The distribution of x ∞ is a non-atomic µ-stationary measure λ on ∂T ; it is the unique µ-stationary measure on ∂T . • If µ has finite first moment, then the measured space (∂T , λ) is the Poisson boundary of (F d ⋊ P, µ).

Theorem 1.2. Let F d be the rank d free group, d ≥ 2. Let G = F d ⋊ α Z be a cyclic extension of the free group F d over an exponentially growing automorphism α of F d . Let µ be a probability measure on G the support of which generates G as a semi-group. Then there exists an α-projectively invariant F d -tree T such that:

• Almost every path {x n } of the random walk on (G, µ) converges to some limit x ∞ ∈ T (where T is the union of the completion of T with its Gromov boundary ∂T , equipped with the observers topology). • The distribution of x ∞ is a non-atomic µ-stationary measure λ on T ; it is the unique µ-stationary measure on T . • If µ has finite first logarithmic moment and finite entropy, then the measured space ( T , λ) is the Poisson boundary of (G, µ). The measure λ is not concentrated on ∂T .

We describe the Poisson boundaries with more precision in the full statements given farther in the paper. Note that, in the case of an extension by polynomially growing automorphisms which are not trivial in Out(F d ), we need to pass to a finite index subgroup. That is the reason why we require the measure µ to have a finite first moment, and why we obtain the convergence only for a subsequence of a.e. path. See Theorem 2.5 below for more details. On the other hand, no such move to a finite index subgroup is needed in the case of extensions by inner automorphisms, i.e. the case of a direct product. We have:

Theorem 1.3. Let G = F d × F k
be a direct product of two finitely generated free groups and β a monomorphism β : F k ֒→ Inn(F d ). Let µ be a probability measure on G the support of which generates G as a semi-group. Then there exists a β-invariant F d -tree T such that:

• Almost every path {x n } of the random walk on (G, µ) converges to some limit x ∞ ∈ ∂T . • The distribution of x ∞ is a non-atomic µ-stationary measure λ on ∂T ; it is the unique µ-stationary measure on ∂T .

• If µ has finite first logarithmic moment and finite entropy, then the measured space (∂T , λ) is the Poisson boundary of (G, µ).

We also deal with more general hyperbolic groups than the free group. In the case of an exponentially growing automorphism, the reader will notice that the Poisson boundary is (a quotient of) the whole R-tree, and not only the boundary of the tree. At least in the hyperbolic case, this will not seem too surprising for geometric group theorists aware both of the R-tree theory developed for surface and free group automorphisms, and of the existence of the Cannon-Thurston map. Indeed the whole R-tree is homeomorphic to a quotient of ∂F d obtained by identifying the points which are the endpoints of a same leaf of a certain "stable lamination" [START_REF] Thierry Coulbois | Non-unique ergodicity, observers' topology and the dual algebraic lamination for R-trees[END_REF]. The quotient we take amounts to further identifying the points of ∂F d which are the endpoints of a same leaf of a certain "unstable lamination". That this gives the geometric boundary is known in the case of the suspension of a closed hyperbolic surface by a pseudo-Anosov homeomorphism. In the more general setting we work here we know no reference. It could perhaps be alternatively obtained by combining the Cannon-Thurston map defined in [START_REF] Mitra | Relative hyperbolicity, trees of spaces and Cannon-Thurston maps[END_REF] for the relatively hyperbolic setting with the last section of the present paper and a work (yet to be written) of Coulbois-Hilion-Lustig.

As was already written, we work really on the R-trees on which the considered groups act, not on the groups themselves. This leads us to extract the properties that we really need for these actions to provide us with a compactification satisfying all the Kaimanovich properties. In particular the map Q introduced in [START_REF] Levitt | Irreducible automorphisms of F n have north-south dynamics on compactified outer space[END_REF][START_REF] Levitt | Automorphisms of free groups have asymptotically periodic dynamics[END_REF] appears to play a crucial rôle. In fact this map Q also allows us to get the following Corollary 1.4. Let F d be the rank d free group, d ≥ 2. Let G be an extension of the free group F d by a finitely generated subgroup of polynomially growing outer automorphisms. Let µ be a probability measure on G the support of which generates G as a semi-group. Then there exists a finite index subgroup H in G such that, if τ k denotes the (random) sequence of the times at which the path x = {x n } visits H, then:

• There exists a topology on G ∪ ∂F d such that almost every sub-path x τ k converges to some x ∞ ∈ ∂F d . Beware that G∪∂F d is not a compactification of G. We invite the reader to compare the above result with [65, Theorems 1 and 2]. A perhaps more intuitive interpretation of the above result in the case of a random walk on a semi-direct product F d ⋊ α Z is as follows: let F d be the rank d free group together with a basis B and let α be an automorphism of F d . Consider a set of transformations S which consist either of a right-translation by an element in B or of the substitution of an element g by its image α(g) or α -1 (g). Then iterating randomly chosen transformations among the set S amounts to performing a nearest neighbor random walk on F d ⋊ α Z. The above corollary means that the boundary behavior of this random process is entirely described in terms of the Gromov boundary of F d .

There is a difficulty, when dealing with direct products or extensions over polynomially growing automorphisms, which does not appear in the exponentially growing case and is hidden in the statements given here. Indeed, whereas in the exponentially growing case we only need to make act the considered group on a R-tree, in the polynomially growing case we have to make act the group on the product of two (simplicial) R-trees. The reason is that, in order to check the Kaimanovich properties (more precisely the (CP) condition), we need that a single element do not fix more than one or two points, and if it fixes two then it acts as a hyperbolic translation along the axis joining the two points. This is not so difficult in the exponentially growing case because each element fixes at most one point in the (completion of the) tree (exactly one for those acting as strict homotheties), the other ones lying in the Gromov boundary. In the polynomially growing case, some elements fix non-trivial subtrees of the considered R-tree. Collapsing these subtrees is not possible because it might happen that eventually everything gets identified.

The trick is then to make appear, as an intermediate step, a product of trees instead of a single one. Since a semi-direct product structure only depends on the outer-class of the automorphisms, we make the extension considered act in different ways on two copies of the given R-tree. In this way, the fixed points of the action which are an obstruction to some of the Kaimanovich conditions become "small", in some sense, in the ambient space. We mean that there is still a tree to be identified to a single point for some elements but these trees (in ∂( T × T )) are disjoint for two distinct elements. We come back to a single tree by projection on the first factor.

Before concluding this section we would like to comment on possible generalizations of the results exposed here. When considering a group G ⋊ U, the R-trees we work with are only a reminiscence of certain invariant laminations for the action of U on G.

Here the word "invariant lamination" has to be understood in the sense of a geometric lamination or in the more general sense of an algebraic lamination as in [START_REF] Thierry Coulbois | R-trees and laminations for free groups. I. Algebraic laminations[END_REF][START_REF] Thierry Coulbois | R-trees and laminations for free groups. II. The dual lamination of an R-tree[END_REF][START_REF] Thierry Coulbois | R-trees and laminations for free groups. III. Currents and dual R-tree metrics[END_REF]. Even in the restricted geometric setting, such invariant laminations might exist even if the manifolds considered are not suspensions. Thus one can expect to be able to find more general classes of groups than those considered here, like for instance in a first step the fundamental groups of compact 3-manifolds which do not fiber over S 1 but nevertheless admit pseudo-Anosov flows (in the decomposition G ⋊ U we do not need a priori that G and U be finitely generated, but only that G ⋊ U is).

Harmonic functions and random walks on a discrete group

In this section, we recall the construction given by V.Kaimanovich of the Poisson boundary of a countable group endowed with a probability measure, together with a geometric characterization of this boundary. Apart from subsections 2.5 and 2.8, most of the material of this section is taken from [START_REF] Vadim | The Poisson formula for groups with hyperbolic properties[END_REF]. We end with a discussion on the first return measure on a recurrent subgroup.

2.1. Stationary measures and random walks. We write N for the set of nonnegative integers. A (measurable) G-space is any measurable space (X, F) measurably acted upon by a countable group G. If µ and λ are measures respectively on G and X, we denote by µ * λ the measure on X which is the the image of the product measure µ ⊗ λ by the action G × X → X. The measure λ is said to be µ-stationary if one has:

(1) λ = µ * λ = g µ(g)gλ .
Let G be a countable group, and µ a probability measure on G. The (right) random walk on G determined by the measure µ is the Markov chain on G with the transition probabilities p(x, y) = µ(x -1 y) invariant with respect to the left action of the group G on itself. Thus, the position x n of the random walk at time n is obtained from its position x 0 at time 0 by multiplying by independant µ-distributed right increments h i :

(2)

x n = x 0 h 1 h 2 • • • h n .
Denote by G N the space of sample paths x = {x n }, n ∈ N endowed with the σ-algebra A generated by the cylinders {x ∈ G N | x i = g}. The group G acts coordinate-wisely on the space G N . An initial distribution θ on G determines the Markov measure P θ on the path space G N which is the image of the measure θ ⊗ ∞ n=1 µ under the map (2). The one-dimensional distribution of P θ at time n, i.e. the distribution of x n , is θ * µ * n .

In the sequel, we will be mainly interested in random walks starting at the group identity e which correspond to the initial distribution θ = δ e . We denote by P the associated Markov measure. For any initial distribution θ, one easily checks that the probability measure P θ is equal to θ * P and is absolutely continuous with respect to the σ-finite measure P m , where m is the counting measure on G.

The Poisson boundary.

The measure P m on the path space G N is invariant by the time shift T : {x n } → {x n+1 }. The Poisson boundary Γ of the random walk (G, µ) is defined as being the space of ergodic components of the shift T acting on the Lebesgue space (G N , A, P m ) (see [START_REF] Rohlin | On the fundamental ideas of measure theory[END_REF]).

Let us give some details. Denote by ∼ the orbit equivalence relation of the shift T on the path space G N :

(3) x ∼ x ′ ⇐⇒ ∃ n, n ′ ≥ 0, T n x = T n ′ x ′ .
Let A T be the σ-algebra of all the measurable unions of ∼-classes, i.e. the σ-algebra of all T -invariant measurable sets. Denote by A T the completion of A T with respect to the measure P m . Since (G N , A, P m ) is a Lebesgue space, the Rohlin correspondence assigns to the complete σ-algebra A T a measurable partition η of G N called the Poisson partition, which is well defined mod 0. An atom of η is an ergodic component of the shift T , that is, up to a set of P m -measure 0, closed under ∼, A-measurable, and minimal with respect to these propoerties. (Note that the σ-algebra A T is not a priori generated by the atoms of η, see [START_REF] Rohlin | On the fundamental ideas of measure theory[END_REF][START_REF] Babillot | An introduction to Poisson boundaries of Lie groups[END_REF]). The Poisson boundary Γ is the quotient space G N /η. The coordinate-wise action of G on the path space G N commutes with the shift T and therefore projects to an action on Γ. Denote by bnd the canonical map bnd : G N → Γ. The space Γ endowed with the bnd-image of the σ-algebra A T and the measure ν m = bnd(P m ) is a Lebesgue space.

For any initial distribution θ, we set ν θ = bnd(P θ ). The Poisson boundary Γ, which depends only on G and µ, carries all the probability measures ν θ . The measure ν =bnd(P) is called the harmonic measure. One easily checks that ν θ = bnd(θ * P) = θ * ν. It implies that the measure ν is µ-stationary, i.e. ν = µ * ν (whereas the other measures ν θ are not).

Harmonic functions.

As mentioned in the introduction, the space (Γ, ν) enables to retrieve both all the bounded µ-harmonic functions on G and (part of) the asymptotic behaviour of the paths of the random walk. Let us now make this precise.

The Markov operator P = P µ of averaging with respect of the transition probability of the random walk (G, µ) is defined by ( 4)

P µ f (x) = y p(x, y)f (y) = h µ(h)f (xh) . A function f : G → R is called µ-harmonic if P f = f . Denote by H ∞ (G, µ)
the Banach space of bounded µ-harmonic functions on G equipped with the sup-norm.

There is a simple way to build bounded µ-harmonic functions on G. Assume that one is given a probability G-space (X, F, λ) such that the measure λ is µ-stationary. To any function F ∈ L ∞ (X, λ) one can assign a function f on G defined, for g ∈ G, by the Poisson formula f (g) = F, gλ = X F (gx)dλ(x). Since λ = µ * λ, the function f is µ-harmonic.

In the case of the Poisson boundary (Γ, ν), the Poisson formula is indeed an isometric isomorphism from L ∞ (Γ, ν) to H ∞ (G, µ). Actually, one can prove (see [44, theorem 6.1]) that if f is the harmonic function provided by the Poisson formula applied to a function F ∈ L ∞ (Γ, ν), then for P-almost every path x = {x n }, one has F (bnd x) = lim f (x n ). Conversely, if f is any bounded µ-harmonic functions on G, then the sequence of its values along sample paths x = {x n } of the random walk is a martingale (with respect to the increasing filtration of the coordinate σ-algebras in G N ). Therefore, by the Martingale Convergence Theorem, for P-almost every path x = {x n }, there exists a limit F (x) = lim f (x n ). This function F is T -invariant and A T -measurable. Since the Poisson boundary Γ is the quotient of the path space determined by A T , there exists

F ∈ L ∞ (Γ, ν) such that F (bnd x) = lim f (x n ).
2.4. µ-Boundaries. As far as the behaviour of sample paths at infinity is concerned, we first recall the notion of µ-boundary. This notion was first introduced by Furstenberg, see [START_REF] Furstenberg | Random walks and discrete subgroups of Lie groups[END_REF][START_REF] Furstenberg | Boundary theory and stochastic processes on homogeneous spaces[END_REF]. The following definition is due to Kaimanovich. A µ-boundary is a G-equivariant quotient (B, λ) of the Poisson boundary (Γ, ν), i.e. the quotient of Γ with respect to some G-equivariant measurable partition. The Poisson boundary is itself a µ-boundary, and this space is maximal with respect to this property. Particularly, if π is any T -invariant G-equivariant map from the path space (G N , P) to some G-space B, then, by definition of the Poisson boundary, such a map factors through Γ: π : G N → Γ → B and (B, π(P)) is a µ-boundary.

Furstenberg's construction of µ-boundaries runs as follows. Assume that B is a separable compact G-space on which G acts continuously. By compactness, there exists a µ-stationary measure λ on B (see [START_REF] Vadim | The Poisson boundary of the mapping class group[END_REF]). Then the Martingale Convergence Theorem implies that for P-almost every path x = {x n }, the sequence of translations x n λ converges weakly to some measure λ(x). Therefore the map x → λ(x) allows to consider the space of probability measures on B as a µ-boundary. If, in addition, the limit measures λ(x) are Dirac measures, then the space B is itself a µ-boundary (see [START_REF] Furstenberg | Random walks and discrete subgroups of Lie groups[END_REF][START_REF] Furstenberg | Boundary theory and stochastic processes on homogeneous spaces[END_REF]).

For instance, assume that the group G contains a finitely generated free group F as a normal subgroup. Then the action of G on F by inner automorphisms extends to a continuous action on the boundary ∂F of F . Vershik and Malyutin [START_REF] Vershik | The boundary of the braid group and the Markov-Ivanovsky normal form[END_REF] proved that if µ is any nondegenerate measure on G (i.e. the support of µ generates G as a semigroup) then there exists a unique µ-stationary measure λ on the G-space ∂F and (∂F, λ) is a µ-boundary of the pair (G, µ). Their proof heavily relies on the description of some contracting properties of the action of G on ∂. The aim of this paper is to provide various situations for which we can prove that (∂F, λ) is indeed the Poisson boundary of (G, µ).

2.5. Dynkin spaces. The concept of Dynkin space was introduced by Furstenberg in [START_REF] Furstenberg | Poisson boundaries and envelopes of discrete groups[END_REF] in order to prove that a given G-space is a µ-boundary. The purpose of this subsection is to discuss this notion in the case of cyclic extensions of the free group. A Dynkin space for a finitely generated group G is a compact metric G-space X with the following property :

∀ε > 0, ∃ N s.t.∀g ∈ G, |g| ≥ N, ∃U g , V g ⊂ X, diam U g and V g < ε s.t. g(U c
g ) ⊂ V g . One can easily check that the boundary ∂F of a free group F is a Dynkin space for F (for the usual ultra-metric on ∂F ). If G = F ⋊ α Z is a cyclic extension of a free group F by means of some automorphism α of F , then, up to finite index, ∂F is also a Dynkin space for G. For, if ∂α denotes the continuation of α on ∂F , then ∂α has finitely many periodic orbits in ∂F , and the orbit of any element accumulates on a periodic orbit [START_REF] Levitt | Automorphisms of free groups have asymptotically periodic dynamics[END_REF]. Therefore, up to taking some power of α (which amounts to considering a finite index subgroup of G), one may consider that the periodic orbit of α are fixed points, and the orbit of any non-fixed point is attracted to a fixed point.

However, for a G-space X, being a Dynkin space for G is not sufficient to ensure that X is a µ-boundary. Actually, in [START_REF] Furstenberg | Poisson boundaries and envelopes of discrete groups[END_REF], Furstenberg strongly relies on the fact that ∂F is a compactification of the free group F to prove both the fact that ∂F is a µ-boundary and also the convergence of the random walk in F to ∂F . In the case of a cyclic extension G = F ⋊ α Z of F , we do not know a priori how to compactify G with ∂F in such a way to prove that ∂F is a µ-boundary of G, and eventually its Poisson boundary. Indeed, the main achievement of this paper is to construct a compactification G of G containing ∂F as a dense subset of full measure in such a way that G ∪ ∂F has all the properties listed in Corollaries 1.4 and 1.5.

2.6.

Compactifications. The situation we are mainly concerned with is the following. Let G be a compactification of the group G, that is a topological compact space which contains G as an open dense subset and such that the left-action of G on itself extends to a continuous action on G. Assume that P-almost every path x = {x n } converges to a limit x ∞ = lim x n = π(x) ∈ G. Then the map π is obviously T -invariant and Gequivariant, so that the space G equipped with the hitting measure π(P) is a µ-boundary. Moreover, in this case, the Poisson formula yields an isometric embedding of

L ∞ (G, π(P)) into H ∞ (G, µ).
A compactification G is µ-maximal if P-almost every sample path x = {x n } of the random walk (G, µ) converges in this compactification to a (random) limit x ∞ = π(x), and if (G, λ) is isomorphic to the Poisson boundary of (G, µ), where λ = π(P) is the hitting measure .

In [START_REF] Vadim | The Poisson formula for groups with hyperbolic properties[END_REF], Kaimanovich proves a theorem which provides compactifications G of G containing the limit of P-almost every path (theorem 2.4), and another one which is a geometric criterion of maximality of µ-boundaries (theorem 6.5). The combination of these two results yields µ-maximal compactifications of G, see Theorem 2.3 below. This statement will be our central tool.

A compactification G is compatible if the left-action of G on itself extends to an action on G by homeomorphisms. A compactification is separable if, when writing G = G ∪ ∂G, then ∂G is separable (i.e contains a countable dense subset). In the remaining of this section, we shall always assume that G is a compatible and separable compactification of the group G.

We first state a uniqueness criterion. Its proof is mostly inspired from the one of Theorem 2.4 in [START_REF] Vadim | The Poisson formula for groups with hyperbolic properties[END_REF].

Lemma 2.1. Let G = G ∪ ∂G be a compactification of a finitely generated group G satisfying the following proximality property: whenever G ∋ g n → ξ ∈ ∂G, one has g n η → ξ for all but at most one η ∈ ∂G. Let µ be a probability measure on G such that the subgroup gr(µ) generated by its support fixes no finite subset of ∂G. Assume that P-almost every path x = {x n } converges to a (random) limit x ∞ = π(x) ∈ ∂G. Then the hitting measure λ = π(P), which is the distribution of x ∞ , is non-atomic and is the unique µ-stationary probability measure on ∂G.

Proof. Let ν be any µ-stationary measure on ∂G. The hypothesis on gr(µ) ensures that ν is non-atomic (see the proof of Theorem 2.4 in [START_REF] Vadim | The Poisson formula for groups with hyperbolic properties[END_REF] for details). Then the proximality property implies that the sequence of (random) measures x n ν weakly converges to the Dirac measure δ x∞ . Since ν is µ-stationary and µ * n is the distribution of x n , we have

ν = µ * n * ν = g µ * n (g).gν =
x n νdP(x) .

Passing to the limit on n gives

ν = δ x∞ dP(x) = λ .
Remark 2.2. Note that there is no need of compactness in the proof: Lemma 2.1 remains valid if G is not compact. This observation will be used later in the paper.

Kaimanovich's criterion of maximality.

The compactification G satisfies condition (CP) if, for any x ∈ G and for every sequence g n ∈ G converging to a point from ∂G in the compactification G, the sequence g n x converges to the same limit.

The compactification G satisfies condition (CS) if the following holds. The boundary ∂G consists of at least 3 points, and there is a G-equivariant Borel map S assigning to pairs of distinct points (b

1 , b 2 ) from ∂G nonempty subsets (strips) S(b 1 , b 2 ) ⊂ G such that for any 3 pairwise distinct points b i ∈ ∂G, i = 0, 1, 2, there exist neighbourhoods b 0 ∈ O 0 ⊂ G and b i ∈ O i ⊂ ∂G, i = 1, 2 with the property that S(b 1 , b 2 ) ∩ O 0 = ∅ for all b i ∈ O i , i = 1, 2 .
This condition (CS) means that points from ∂G are separated by the strips S(b 1 , b 2 ).

A gauge on a countable group G is any increasing sequence J = (J k ) k≥1 of sets exhausting G. The corresponding gauge function is defined by |g| = |g| J = min{k , g ∈ J k }. The gauge J is finite if all gauge sets are finite. An important class of gauges consists of word gauges, i.e. gauges (J k ) such that J 1 is a set generating G as a semigroup, and

J k = (J 1 ) k is the set of word of length ≤ k in the alphabet J 1 . It is finite if and only if J 1 is finite. A set S ⊂ G grows polynomially with respect to some gauge J if there exist A, B, d such that card [S ∩ J k ] ≤ A + Bk d .
Theorem 2.3. [45, Theorems 2.4 and 6.5] Let G be a finitely generated group. Let G = G ∪ ∂G be a compatible and separable compactification of G satisfying conditions (CP) and (CS). Let µ be a probability measure on G such that the subgroup gr(µ) generated by its support fixes no finite subset of ∂G.

Then P-almost every path x = {x n } converges to a (random) limit x ∞ = π(x) ∈ ∂G. The hitting measure λ = π(P) is non-atomic, the measure space (∂G, λ) is a µ-boundary and λ is the unique µ-stationary probability measure on ∂G.

Moreover, if J is a finite word gauge on G such that the measure µ has a finite first logarithmic moment log |g|µ(g), a finite entropy H(µ) =µ(g) log µ(g), and if each strip S(b 1 , b 2 ) grows polynomially, then the space (∂G, λ) is isomorphic to the Poisson boundary (Γ, ν) of (G, µ) and is therefore µ-maximal.

Note that if a measure µ has a finite first moment |g|µ(g), then the entropy H(µ) is finite, see e.g. [START_REF] Vadim | The Poisson formula for groups with hyperbolic properties[END_REF]Lemma 12.2]. Besides, the finitess of the first (logarithmic) moment does not depend on the choice of the finite word gauge.

2.8. The first return measure. This subsection is devoted to a discussion on the stability of the Poisson boundary when moving to a subroup which is recurrent or normal with finite index. Let G be a finitely generated group, µ a probability measure on G and G 0 a subgroup of G which is a recurrent set for the random walk (G, µ). Define a probability measure µ 0 on G 0 as the distribution of the point where the random walk issued from the identity of G returns for the first time to G 0 . We call µ 0 the first return measure. Furstenberg observed (see [START_REF] Furstenberg | Random walks and discrete subgroups of Lie groups[END_REF]Lemma 4.2]) that the Poisson boundaries of (G, µ) and (G 0 , µ 0 ) are isomorphic.

For instance, if G 0 is a normal subgroup in G such that the random walk (G, µ) projects on the factor group G/G 0 as a recurrent random walk, then the identity in G/G 0 is a recurrent state, therefore G 0 is a recurrent set in G. The case of a normal subgroup of finite index is of special interest: [START_REF] Vadim | Poisson boundaries of random walks on discrete solvable groups[END_REF]Lemma 2.3] Let G be a finitely generated group, µ a probability measure on G and G 0 a normal subgroup of finite index in G. Then G 0 is a recurrent set for the random walk (G, µ) and the Poisson boundaries of (G, µ) and (G 0 , µ 0 ) are isomorphic where µ 0 is the first return measure. Moreover, if µ has a finite first moment (with respect to some finite word gauge) then so has µ 0 .

Lemma 2.4. [27, Lemma 4.2],

Observe that the conclusion of this lemma remains valid if the finite index subgroup

G 0 is not normal, since it contains a subroup G 1 of finite index which is normal in G.
Indeed, what is meant in this lemma is that there is an isomorphism between the spaces of harmonic functions H ∞ (G, µ) and H ∞ (G 0 , µ 0 ). But the Poisson boundaries (Γ, ν) and (Γ 0 , ν 0 ) of (G, µ) and (G 0 , µ 0 ) are distincts objects in nature since Γ 0 is not a priori a G-space -although Γ and Γ 0 are both G 0 -spaces. This is made precise in the following statement, which is a combination of Theorem 2.3 and (the proof of) Lemma 2.4: Theorem 2.5. Let G be a finitely generated group and G 0 a finite index subgroup of G. Let G 0 = G 0 ∪B be a compatible and separable compactification of G 0 satisfying conditions (CP) and (CS) such that the action of G 0 on B extends to a continuous action of G on B. Let µ be a probability measure on G such that the subgroup gr(µ) generated by its support fixes no finite subset of B. Denote by µ 0 the first return measure. Let τ k be the (random) sequence of the times at which the path x = {x n } visits G 0 . Then :

(1) P-almost every sub-path x τ k converges to a (random) limit x ∞ = π(x) ∈ B.

(2) the hitting measure λ = π(P) is non-atomic, the measure space (B, λ) is a µboundary and λ is the unique µ-stationary probability measure on B. (3) if J is a finite word gauge on G such that the measure µ has a finite first moment and each strip grows polynomially, then the space (B, λ) is isomorphic to the Poisson boundary (Γ, ν) of (G, µ).

Proof. Since G 0 has finite index in G, the subgroup gr(µ 0 ) of G 0 generated by the support of µ 0 fixes no finite subset of B. Therefore Theorem 2.3 works for the random walk (G 0 , µ 0 ), which gives Item (1), as well as Items ( 2) and (3) for µ 0 .

Let us prove that λ is the unique µ-stationary probability measure on B. Let ν be any µ-stationary probability measure on B (the compactness of B implies the existence of such a measure). Consider the Poisson formula Π ν :

L ∞ (B, ν) → H ∞ (G, µ). According to the proof of Lemma 4.2 in [27], the restriction Φ : f → f |G 0 maps µ-harmonic functions to µ 0 -harmonic functions. Therefore the composition Φ • Π ν is the Poisson formula L ∞ (B, ν) → H ∞ (G 0 , µ 0 )
so the measure ν is µ 0 -stationary, and ν = λ.

As far as Item ( 3) is concerned, we prove that the Poisson formula Π λ :

L ∞ (B, λ) → H ∞ (G, µ) is an isomorphism. The map Φ : H ∞ (G, µ) → H ∞ (G 0 , µ 0 ) is an isomorphism.
Since µ has a first moment, so has µ 0 . According to Item (3) of Theorem 2. 3,(B,λ) is the Poisson boundary of (G 0 , µ 0 ), therefore the composition Φ • Π λ is an isomorphism, and so is Π λ .

Observe that the way the group G acts on B does not play any role, provided this action extends the action of G 0 . Indeed, the boundary behaviour of the random walk on (G, µ) is governed by (G 0 , µ 0 ). Assume now the following: G 0 is the free group F d , the factor group G/F d is isomorphic to Z and the image random walk on G/F d is recurrent. Then the above construction gives rise to a probability measure µ 0 on F d such that the Poisson boundaries of (G, µ) and (F d , µ 0 ) are isomorphic. The problem is that, in this case, we do not know a priori what the Poisson boundary of (F d , µ 0 ) should be, because the measure µ 0 is potentially too spread out to have a finite entropy, even if the image random walk is the symmetric nearest neighbour random walk on Z. On the other hand, according to Theorem 1, the Poisson boundary of (G, µ) -and that of (

F d , µ 0 ) -is ∂F d .
Besides, if G is, for instance, an extension of the free group F 4 by a Z 2 -subgroup of polynomially growing automorphisms in Aut(F 4 ) (see Subsection 9.2 for such examples) and if µ is any non-degenerate probability measure with finite first moment on G which leads to a recurrent random walk on G/F 4 ∼ Z 2 then the measure µ 0 on F 4 is potentially very spread out, and Corollary 1.4 ensures that the Poisson boundary of (F 4 , µ 0 ) is ∂F 4 .

Generalities about groups, automorphisms and semi-direct products

Let G be a discrete group with generating set S, that we denote by G = S . The Cayley graph of G with respect to S is denoted by Γ S (G). It is equipped with the standard metric which makes each edge isometric to (0, 1). We denote by |γ| S the word-length of an element γ with respect to S, i.e. the minimal number of elements in S ∪ S -1 necessary to write γ. If it is clear, or not important, which generating set is used, we will simply write |γ| for the word-length of γ ∈ G (in particular, when dealing with free groups, |γ| will denote the word-length with some fixed basis).

We denote by Aut(G) the group of automorphisms of G, by Inn(G) the group of inner automorphisms (i.e. automorphisms of the form i g (x) = gxg We say that α has exponential growth if the lengths of the iterates α m (γ) of at least one element γ grow at least exponentially with m → +∞.

The above notions only depend on the outer-class of automorphisms considered. Passing from α to α -1 neither changes the nature of the growth.

Let θ : U → Aut(G) be a monomorphism. We denote by G θ := G ⋊ θ U the semi-direct product of G with U over θ. The semi-direct product only depends on the outer-class of

θ(U): G θ is isomorphic to G θ ′ whenever [θ(U)] = [θ ′ (U)] in Out(G).
For this reason we will also write a semi-direct product

G θ := G ⋊ θ U for θ : U → Out(G).
We will denote by

F n = x 1 , • • • ,
x n the rank n free group. A hyperbolic group is a finitely generated group G = S such that there exists δ ≥ 0 for which the geodesic triangles of Γ S (G) are δ-thin:

For any triple of geodesics

[x, y], [y, z], [x, z] in Γ S (G), [x, z] ⊂ N δ ([x, y] ∪ [y, z])
, where N δ (X) denotes the set of all points at distance smaller than δ from some point in X.

We briefly recall the definition of the Gromov boundary ∂G of a hyperbolic group G [START_REF] Gromov | Hyperbolic groups[END_REF], which allows one to get a geodesic compactification G ∪ ∂G of G. For more details, see [START_REF] Kapovich | Boundaries of hyperbolic groups[END_REF]. The Gromov boundary ∂G of the hyperbolic group G consists of the set of equivalence classes of geodesic rays emanating from the base-point, two rays being equivalent if their Hausdorff distance is finite. A ray emanating from O is the interior of (the image of) an isometric embedding ρ : [0, +∞) → Γ S (G) with ρ(0) = O. Of course the definition of a Gromov boundary naturally extends to any Gromov hyperbolic metric space (a geodesic metric space whose geodesic triangles are δ-thin). In the particular case of a tree T , which will be frequent in this paper, two geodesic rays in T define a same point in ∂T if they eventually coincide. Beware that the Gromov boundary ∂X of a geodesic metric space X does not provide us with a compactification of X if X is not proper (where "proper" here means that the closed balls are compact). For instance, if T is a locally infinite tree, then T ∪ ∂T is not a compactification of T .

About R-trees and group actions on R-trees

The aim of this section is to gather all the vocabulary as well as all the notions and results that we will need about R-trees and group actions on R-trees further in the paper. The reader may choose to skip this chapter, only coming back each time subsequent sections refers to the material developed below. We counsel however to have a quick look at the first part, which motivates the introduction of these R-trees.

4.1.

How R-trees come into play. We take a pedestrian way to show to the reader how R-trees naturally come into play when searching for Poisson boundaries of groups extensions. We consider here the group

F n × Z. Let F n = x 1 , • • • , x n and Z = t .
We assume it is equipped with a probability measure µ and we would like to find a compactification of F n × Z suitable for applying Kaimanovich tools.

The most natural space on which F n × Z acts is of course T × Z, where T is the Cayley-graph of F n with respect to some basis. This is a simplicial, locally finite tree. For compactifying T × Z, if v is a vertex of T we need that t k i v converges to some point O v for any vertex of T . For the (CP) property to be satisfied, we need that the limit-point O v be the same for all v. This point would then be fixed by all elements of F n × Z, so that Theorem 2.3 would not apply. The cause of the problem here holds in the fact that the Z-action on F n has many fixed points . . .

We are now going to take advantage from the fact that a direct product is a particular case of a semi-direct product:

F n ×Z is isomorphic to F n ⋊ α Z with α ∈ Inn(G). We define α ∈ Inn(F n ) by α(x i ) = x 1 x i x -1 1 for i ∈ {1, • • • , n}. The Cayley-graph of F n ⋊ α Z is now the 1-skeleton of t∈Z T ×[t, t+1]/((x, t+1) ∈ T ×[t, t+1] ∼ (f (x), t+1) ∈ T ×[t+1, t+2]),
where f : T → T is a PL-map realizing α, i.e. the image under f of an edge of T with label x i is a PL-path in T with associated edge-path α(x i ). We observe that, contrary to what happened before, there are now two kinds of asymptotic behavior for the α k (w)'s:

for any w = x 1 , lim k→±∞ |α k (w)| |k| = 2 whereas lim k→±∞ α k (x 1 ) |k| = 0.
In other words, for w = x 1 , the length of the α k (w) tend linearly toward infinity with k whereas the lengths of the α k (x 1 ) = x 1 remain constant equal to 1.

Having still in the mind the compactification of the Cayley graph, since some of the orbits separate linearly we scale the metric on T × {i} by the factor 1 |i|+1 . Passing to the limit in the sequence of metric spaces

(T × {0}, |.|) → (T × {1}, |.| 2 ) → • • • → (T × {i}, |.| i + 1 ) → • • •
(see [START_REF] Bestvina | R-trees in topology, geometry, and group theory[END_REF][START_REF] Gillet | Dendrology of groups in low Q-ranks[END_REF][START_REF] Gaboriau | An index for counting fixed points of automorphisms of free groups[END_REF][START_REF] Paulin | Sur les automorphismes extérieurs des groupes hyperboliques[END_REF] for instance) we get a so-called "α-invariant simplicial F n -tree" T . This is a simplicial tree, equipped with an isometric action of F n , and which admits moreover an isometry H which "commutes" with α:

For any P ∈ T , H(wP ) = α(w)H(P ).

There are non trivial vertex F n -stabilizers: these are the conjugates w x 1 w -1 . Thus the tree T is locally infinite. The edge F n -stabilizers are trivial. Although this last property will be found in all the exploited cases, such a tree is only a very particular case of the R-trees below because of its simplicial nature. 4.2. R-trees and the observers topology. A R-tree (T , d) is a geodesic metric space which is 0-hyperbolic for the associated distance-function d. We often simply write T , the distance-function being implicit. The metric completion of an R-tree still is an R-tree. We denote by T the union of the completion of T with its Gromov boundary ∂T . Definition 4.1. [START_REF] Thierry Coulbois | Non-unique ergodicity, observers' topology and the dual algebraic lamination for R-trees[END_REF] Let T be a complete R-tree.

Let P, Q be any two distinct points in T . A direction at P , denoted by D P , is a connected component of T \ {P }. The direction of Q at P , denoted by D P (Q), is the connected component of T \ {P } which contains Q.

A branch-point is any point in T at which there are at least three different directions.

An extremal point is a point P at which there is only one direction, i.e. T \ {P } is connected. The interior tree of T is the tree deprived of its extremal points.

An arc is a subset isometric to an interval of the real line. We denote by [P, Q] the geodesic arc from P to Q (P or Q, or both, may belong to ∂T , in which case [P, Q] denotes the unique infinite, or bi-infinite, geodesic between P and Q).

Among the R-trees we distinguish the simplicial R-trees, defined as the R-trees in which every branch-point v admits a neighborhood N (v) such that each connected component of N (v) \ {v} is homeomorphic to an open interval. In these simplicial trees, it is thus possible to speak of vertices (the branch-points) and of edges (the connected components of the complement of the vertices). The metric of the R-tree defines a length on each edge. If there is a uniform lower-bound on the length of the edges, the vertices form a discrete subset of the simplicial R-tree for the metric topology. Definition 4.2. Let T be a R-tree. The observers'topology on T is the topology which admits as an open neighborhoods basis the set of all directions D P , P ∈ T , and their finite intersections. We denote by T the topological space obtained by equipping T with the observers'topology.

The observers topology is weaker than the metric topology [START_REF] Thierry Coulbois | Non-unique ergodicity, observers' topology and the dual algebraic lamination for R-trees[END_REF]. Any sequence of points (Q n ) turning around a point P of T , meaning that every direction at P contains only finitely many of the Q n 's, converges to P in T . Such a phenomenon is of course only possible in a non locally finite tree. In a locally finite, simplicial tree, the metric and observers topology agree. We summarize below what makes this topology important for us.

Proposition 4.3. [START_REF] Thierry Coulbois | Non-unique ergodicity, observers' topology and the dual algebraic lamination for R-trees[END_REF] Let T be a separable R-tree. Then:

(1) T is compact.

(2) The metric topology and the observers'topology agree on the Gromov boundary ∂T . In particular ∂T ⊂ T is separable. (3) If (P n ) is a sequence of points in T and Q is a fixed point in T , there is a unique point P := lim inf Q n→∞ P n defined by The homothety is strict if λ = 1.

[Q, P ] = ∞ m=0 n≥m [Q, P n ] (4)
By the fixed-point theorem, a strict homothety on a complete R-tree admits exactly one fixed point.

Let T be a R-tree and let H be a homothety of T . Then H induces a permutation on the set of directions of T . Definition 4.5. [START_REF] Liousse | Actions affines sur les arbres réels[END_REF] Let T be a R-tree and let H be a homothety on T . An eigenray of H at a point P is a geodesic ray R starting at P such that H(R) = R.

In particular, if R is an eigenray at P , then H(P ) = P . In [START_REF] Liousse | Actions affines sur les arbres réels[END_REF] it was proven that if a strict homothety on the completion of a R-tree T has its fixed point outside T , then it fixes a unique eigenray starting at this fixed point. In [START_REF] Gaboriau | An index for counting fixed points of automorphisms of free groups[END_REF] (among others) it was noticed that if such a homothety leaves invariant a direction D P (P is thus a fixed point), then it admits a unique eigenray in D P . Definition 4.6. Let G be a discrete group which acts by homeomorphisms on a R-tree (T , d).

The action of G on T is irreducible if there is no finite invariant set in T .

The action of G on T is minimal if there is no proper invariant subtree (where a proper subset of a set is a subset distinct from the set itself).

The action of G on T is an action by homotheties, or an affine action, if there is a morphism λ : G → R + * with d(hx, hy) = λ(h)d(x, y) for any h ∈ G and x, y ∈ T .

A G-tree is a R-tree equipped with an isometric action of G.

We refer to [START_REF] Liousse | Actions affines sur les arbres réels[END_REF] for a specific study of these affine actions. By the very definition, when given an affine action of a discrete group G on a R-tree T , each element is identified to a homothety of T . Thus an affine action of a group G on a R-tree T induces a morphism from G to the permutations on the set of directions of T .

A common way in which affine actions arise is as follows:

Definition 4.7. Let T be a G-tree and let θ : U → Aut(G) be a monomorphism. The G-tree T is a θ(U)-projectively invariant G-tree if θ(U) acts by homotheties on T such that for any g ∈ G, for any u ∈ U and for any point P ∈ T :

H θ(u) (g.P ) = θ(u)(g)H θ(u) (P ) If θ(U) acts by isometries then T is a θ(U)-invariant G-tree.
In the expression "θ(U)-projectively invariant R-tree" we will often substitute the group θ(U) by its generators. In particular, instead of " α -projectively invariant R-tree", we will rather write "α-projectively invariant R-tree".

Remark 4.8. If T is a θ(U)-projectively invariant G-tree, then the semi-direct product G θ := G ⋊ θ U acts by homotheties on T . Conversely, if one has an affine action of G θ on a R-tree T such that the induced action of G ⊳ G θ is an isometric action, then T is a θ(U)-projectively invariant G-tree.

When one has a θ(U)-projectively invariant G-tree T , then T is in fact projectively invariant for any subgroup θ

′ (U) < Aut(G) with [θ(U)] = [θ ′ (U)]. Indeed, let θ ′ (u) ∈ Aut(G) satisfy θ ′ (u)(.) = g 0 θ(u)(.)g -1 0 for some g 0 ∈ G. Then H θ ′ (u) = g 0 H θ(u) satisfies H θ ′ (u) (gP ) = θ ′ (u)(g)H θ ′ (u) (P ). Indeed H θ ′ (u) (gP ) = g 0 H θ(u) (g.P ) = g 0 θ(u)(g)H θ(u) (P ), g 0 θ(u)(g)H θ(u) (P ) = g 0 θ(u)(g)g -1 0 g 0 H θ(u) (P ) = θ ′ (u)(g)g 0 H θ(u) (
P ) and g 0 H θ(u) (P ) = H θ ′ (u) (P ) eventually gives the announced equality. This justifies to state the Definition 4.9. Let T be a G-tree and let θ : U → Out(G) be a monomorphism.

The G-tree T is a θ(U)-projectively invariant G-tree if there is θ :

U → Aut(G) with [ θ(U)] = θ(U) such that T is a θ(U)-projectively invariant G-tree.
4.4. The LL-map Q. The LL-map Q which appears below was introduced in [START_REF] Levitt | Irreducible automorphisms of F n have north-south dynamics on compactified outer space[END_REF][START_REF] Levitt | Automorphisms of free groups have asymptotically periodic dynamics[END_REF] in the setting of free group automorphisms (the two L's of the denomination "LL-map" stand for the name of the authors Levitt-Lustig -see also [START_REF] Thierry Coulbois | Non-unique ergodicity, observers' topology and the dual algebraic lamination for R-trees[END_REF] for more details about this map). Lemma 4.10. Let G be a discrete group acting by homotheties on a R-tree T . Let G = G ∪ ∂G be a compatible compactification of G. Assume that the LL-map

Q : ∂G → T X → lim inf P n→∞ g n P
is well-defined, that is independent from the point P and the sequence (g n ) converging to X in G.

Then for any infinite, not eventually constant sequence (g n ) of elements in G such that (g n P ) tends to some point Q in T , for any R ∈ T , (g n R) tends to Q in T .

Proof. Let R be any point in T and consider the sequence (g n R). By compacity of T (see Proposition 4.3), it has at least one accumulation point. Consider any convergent subsequence (g n k R) and denote by Q ′ its limit. By passing if necessary to a further subsequence we can suppose that (g n k ) tends to some point X ∈ ∂G in G. By the assumption that (g n P ) tends toward Q, the very definition of the map

Q implies Q(X) = Q . By Proposition 4.3, Q ′ = lim inf R n→∞ g n k R and since (g n k ) converges to X, we get Q ′ = Q(X) = Q.
We so proved that Q is the only accumulation-point of (g n R) in T . Since T is compact, this implies that (g n R) tends toward Q in T . The lemma follows.

In the case where G is a hyperbolic group and T is a simplicial G-tree with quasiconvex vertex stabilizers and trivial edge-stabilizers, the existence of the LL-map Q is easy to prove. Observe also that the very existence of the map Q implies the triviality of the arc stabilizers (however, in the cases that we will consider, this triviality is proved in a direct way rather than by appealing to the more complex notion that represents the map Q). Similarly, for the conclusion of Lemma 4.10 to be true, we need the triviality of the arc-stabilizers.

Remark 4.11. Consider a G-tree T with trivial arc-stabilizers. The stabilizers H i of the branch-points are malnormal: for any g ∈ G \ H i , g -1 H i g ∩ H i = {1}. Moreover, any family H of branch-points stabilizers which belong to distinct G-orbits form a malnormal family of subgroups: for any g ∈ G, for any

H i = H j in H, g -1 H i g ∩ H j = {1}.

Some classes of groups with affine actions on R-trees

We gather here various results which provide us with non-trivial classes of groups with affine actions on R-trees. Our results about the Poisson boundary of these groups, developed in the next sections, will rely upon the theorems recalled here. Once again the reader can skip for the moment the exposition of these results and only go back here each time the theorems are referred to. 5.1. Projectively invariant R-trees for single automorphisms. Theorem 5.1. Let G be the fundamental group of a compact hyperbolic surface S and let α ∈ Aut(G) in an outer-class induced by a homeomorphism of S. Assume that α has exponential growth. Then there exists a separable, minimal G-tree T α which is αprojectively invariant and satisfies the following properties:

(1) The action of G α = G ⋊ α Z is irreducible (see Definition 4.6).

(2) The G-action has dense orbits in T α .

(3) There are a finite number of G-orbits of branch-points. The stabilizers of branchpoints are quasiconvex. For any point P in T , the number of Stab G (P )-orbits of directions at P is finite. (4) The LL-map Q : ∂G → T α is well-defined (in particular the arc-stabilizers are trivial for the induced G-action), continuous, G α -equivariant and such that the pre-images of any two distinct points are disjoint compact subsets of ∂G. The pre-image of a point in ∂T is a single point of ∂G.

We refer the reader to [START_REF] Fathi | Travaux de Thurston sur les surfaces[END_REF] for the basis about isotopy-classes of surface homeomorphisms. Since we know no precise reference for the above theorem, we briefly sketch a proof.

Proof. Let us first briefly describe how one gets a R-tree T as in Theorem 5.1. We consider a homeomorphism h inducing [α] ∈ Out(G) as given by the Nielsen-Thurston classification: there is a decomposition of S in subsurfaces S 1 , • • • , S r such that h k i (S i ) = S i and h k i |S i is either pseudo-Anosov or has linear growth. The S i 's for which h k i |S i has linear growth are maximal with respect to the inclusion. Each boundary curve of each S i is fixed by h k i . Since α has exponential growth, there is at least one subsurface S i such that h k i |S i is pseudo-Anosov. Let S 0 be the subsurface for which h k 0 |S 0 is pseudo-Anosov with greatest dilatation factor, denoted by λ 0 . We now consider the universal covering S of S (this is a hyperbolic Poincaré disc D 2 in the closed case, the complement of horoballs in D 2 in the free group case) and a lift h of h. Each connected component of Si , i > 0, each reduction curve and each boundary curve is collapsed to a point. Since the corresponding subgroups are quasi convex and malnormal, the resulting metric space Ŝ is still a Gromov hyperbolic space (of course not proper). The connected components of S0 are equipped with a pair of invariant, transverse, transversely measured singular foliations (F u , µ s ) and (F s , µ u ): h((F u , µ s )) = (F u , 1 λ 0 µ s ) and h((F s , µ u )) = (F s , λ 0 µ u ). We equip Ŝ with the length-metric associated to the transverse measures: |γ| F = γ |dµ u | + |dµ s |. The resulting metric space ŜF is still Gromov hyperbolic. We now consider the limit-metric space

( ŜF × {0}, |.| F ) → ( ŜF × {1}, |.| F λ 0 ) → • • • → ( ŜF × {i}, |.| F λ i 0 ) → • • •
obtained by rescaling the metric |.| F by λ i 0 for i → +∞. From [START_REF] Gaboriau | An index for counting fixed points of automorphisms of free groups[END_REF], the limit-space is a α-projectively invariant G-tree T such that the G-action has trivial arc-stabilizers ( [START_REF] Gaboriau | An index for counting fixed points of automorphisms of free groups[END_REF] deals with free group automorphisms but, as noticed there, the triviality of the arc-stabilizers is proved in the same way when considering a hyperbolic group). Possibly after restricting to a minimal subtree, the G-action has dense orbits, see for instance [START_REF] Paulin | Sur les automorphismes extérieurs des groupes hyperboliques[END_REF]: this is true as soon as the action of α is by strict homotheties and the R-tree is minimal.

Claim: Let L s be the geodesic lamination associated to F s . Each non-boundary leaf of L s is mapped to a point in T which is not a branch-point. Any two points in ŜF on distinct leaves of L s are mapped to distinct points in T . The R-tree T is equivalently obtained by collapsing each leaf of F s to a point. A stabilizers of a branch-point is either conjugate to a subgroup associated to a subsurface S i , i > 0, or is a cyclic subgroup corresponding to a boundary curve of S or to a reduction curve.

Proof: The first two assertions are an easy consequence of the fact that:

• h is contracting by λ 0 along the leaves of L s ,

• given any path p transverse to F s , some iterate of h dilates the length of p by Cλ 0 , for some uniform constant C. The last two assertions of the claim are clear.

Let us check the fourth assertion of the theorem. First we map the boundary of each conjugate of each subgroup associated to a subsurface S i to the branch-point stabilized by this subgroup. Second, since the BBT-property of [START_REF] Gaboriau | An index for counting fixed points of automorphisms of free groups[END_REF] is satisfied, there is a welldefined G-equivariant map from a subset of ∂G onto ∂T which is one-to-one. Thus the LL-map Q is well-defined on points X ∈ ∂G which are limits of sequences (g n ) ⊂ G whose translation-lengths on T tend to infinity. Choose a point P ∈ T and let us now consider a sequence (g n P ) with (g n ) → X ∈ ∂G such that the translation-lengths of the g n 's do not tend toward infinity. Then, it is again a consequence of the BBT property that lim inf n→∞ g n P does not depend on the sequence (g n ) tending toward X. Let us prove that it is independent from P . It suffices to check it for a sequence (g n ) the translation-lengths of which, denoted by (t n ), tend to 0. If (t n ) is eventually constant then for some N all the g k 's with k ∈ N belong to the stabilizer of some branch-point B. Since arc stabilizers are trivial, the g k P turn around B, which is then equal to lim inf n→∞ g n P . The same is true if the g n 's tend to some point in the boundary of the stabilizer of a branch-point. In the remaining case, the g n 's correspond to elements of G which are realized by a sequence of paths (p n ) closer and closer to a given leaf of L s . The terminal points of the p n 's belong to different leaves of L s . Hence this is true for the paths q n associated to a sequence g n q, where q is a fixed path from the base-point to a leaf in L s . Since different leaves of L s correspond to different points of T , the conclusion follows.

The continuity of the map Q is proved as in [START_REF] Thierry Coulbois | Non-unique ergodicity, observers' topology and the dual algebraic lamination for R-trees[END_REF].

Remark 5.2. Assume that α is induced by a pseudo-Anosov homeomorphism h of a closed compact hyperbolic surface S. Let x, y be the endpoints in ∂H 2 of the lift of a stable or unstable leaf of h. Then there are no geodesics in G α from x to y: indeed the exponential contraction of the leaf implies that it can be exhausted by compact intervals I n which are exponentially contracted by λ -t for t → ∞; this forbids the function 2t + λ t |I n | to attain a minimum for |I n | → ∞. In fact, the group G α is hyperbolic and these two points of ∂H 2 get identified to a single one in the Gromov compactification. In this case, Q(x) and Q(y) are the endpoints of a same eigenray, or the endpoints of two eigenrays with a same origin.

Definition 5.3. Let G be a discrete group acting by homotheties on an R-tree T . If R is an eigenray of some homothety H g of T , then closing R means identifying its two endpoints in T .

Proposition 5.4. With the assumptions and notations of Theorem 5.1, let T be the space obtained by closing all the eigenrays in T and let q : T → T be the associated quotient-map. Then:

(1) T is Hausdorff and compact.

(2) The action of G α on T descend, via q, to an irreducible action by homeomorphisms on T . The induced G-action has dense orbits and restricts to an isometric action on the completion of T ⊂ T . The G α -action restricts to an affine action on the completion of T ⊂ T . (3) If X, Y are two distinct points in ∂G which cannot be connected by a bi-infinite

G α -geodesic, then q(Q(X)) = q(Q(Y )).
This result is certainly not a surprise to experts in the field of surface or free group automorphisms. In the case of mapping-tori of pseudo-Anosov surface homeomorphisms, it essentially amounts to the knowledge of the famous Cannon-Thurston map and of what are its fibers. We refer the reader to [START_REF] Cannon | Group invariant Peano curves[END_REF][START_REF] Mahan | Cannon-Thurston maps for hyperbolic group extensions[END_REF] for this point of view. It seems however that the above statement has not been written under this form and under this generality, so that we give a brief proof for completeness.

Proof. The notations are those introduced in the proof of Theorem 5.1. Let l be a leaf of the unstable geodesic lamination L u in ŜF . Either it connects two points in ∂ ŜF , or it connects a point in ∂ ŜF to the collapse of a lift of a reduction curve, a boundary curve or a subsurface S i , i > 0. We now denote by l a leaf as above or the subset of a singular leaf of F u , homeomorphic to R, the endpoints of which are of the above form. Such a subset is a bad subset. Then the endpoints of l are not connected by a G α -geodesic: the argument given in Remark 5.2 in the pseudo-Anosov case applies here.

Conversely, any two points in ŜF ∪ ∂ ŜF which are not the endpoints of a bad subset l as defined above are connected by a G α geodesic. Indeed, they are connected by a path p which can be put transverse to the two foliations so that both its stable and unstable lengths (i.e. the lengths measured by integrating the stable and unstable measures of the foliations) are positive. It follows that sufficiently long subpaths of p are shortened under iterations of h or h-1 until reaching a positive minimal length L. The computations carried on in [START_REF] Gautero | Relative hyperbolization of (one-ended hyperbolic)-by-cyclic groups[END_REF] imply that a G α -geodesic passes in a neighborhood of the path with length L (the size of the neighbrohood depends on L which has been chosen sufficently large enough once and for all).

An eigenray R, with origin O and endpoint w ∈ ∂T satisfies wH(R) = R. Since w acts by isometries, whereas H is a homothety with dilatation factor λ 0 > 1, this implies that the length of any subpath in a geodesic g in ŜF from Q -1 (w) to the boundary of the convex-hull of Q -1 (O) is exponentially contracted under iterations of h-1 . Therefore the endpoints of g cannot be connected by a G α -geodesic. It follows that its endpoints are the endpoints of a bad subset l of ŜF ∪ ∂ ŜF . Conversely it is clear that the endpoints of a bad subset l project, under Q, to the endpoints of an eigenray or to the endpoints of two eigenrays with a same origin. Now the endpoints of the bad subsets form a closed set in the following sense: if (X n , Y n ) ∈ ∂G × ∂G are endpoints of bad subsets tending toward (X, Y ) then X and Y are the endpoints of a bad subset. It follows that the collection of eigenrays satisfy a similar property: if R n is a sequence of eigenrays with origin O n and terminal point w n ∈ ∂T with O n (resp. w n ) tending toward some O (resp. w) in T , then either O and w are the endpoints of an eigenray, or they are the endpoints of two eigenrays with same origin. It readily follows that closing the eigenrays gives a Hausdorff space. The compacity is then easily deduced from the compacity of T .

We so got the first and third items of the proposition. The second item follows from the fact that G α permutes the eigenrays.

There are "generalizations" of these theorem and proposition toward free groups and torsion free hyperbolic groups with infinitely many ends. The paper [START_REF] Gaboriau | An index for counting fixed points of automorphisms of free groups[END_REF] gives the conclusions of Theorem 5.1 in the setting of free group automorphisms and the point (2) of Theorem 5.5 below dealing with polynomially growing automorphisms. The paper [START_REF] Levitt | Automorphisms of free groups have asymptotically periodic dynamics[END_REF][Theorems 10.4 and 10.5] allows us to generalize [START_REF] Gaboriau | An index for counting fixed points of automorphisms of free groups[END_REF] to torsion free hyperbolic groups with infinitely many ends. Let us recall that such a hyperbolic group G is the fundamental group of a graph of groups with trivial edge stabilizers and one-ended hyperbolic groups as vertex stabilizers. The relative length of an element γ of G is the word-length associated to the (infinite) generating set obtained by adding every element of any vertex stabilizer to the given generating set. An automorphism α of a torsion free hyperbolic group with infinitely many ends G has an essential exponential growth if there exists an element γ of G such that the relative length of α j (γ) growths exponentially with j → +∞. Otherwise α has essential polynomial growth. Indeed, any automorphism of a torsion free hyperbolic group G with infinitely many ends has either essential exponential or essential polynomial growth: this is a straightforward consequence of the fact that any free group automorphism has either exponential or polynomial growth, the free group involved here being the quotient of G by the vertex stabilizers mentioned above and their conjugates. Theorem 5.5 ([30, 29, 52, 53, 54, 14]). Let G be a torsion free hyperbolic group with infinitely many ends (in particular G may be a non-abelian free group).

(1) If α is an automorphism with essential exponential growth then the conclusions of Theorem 5.1 and Proposition 5.4 are true. (2) If α is an automorphism with essential polynomial growth then there is a minimal, simplicial α-invariant G-tree T which satisfies all the properties of Theorem 5.1.

Proof of Proposition 5.4 in the setting of Theorem 5.5. We consider the Cayley graph Γ of G with respect to some finite generating set. We consider a tree T as given by Theorem 5.5. Up to conjugacy in G, there are a finite number of stabilizers of branchpoints

H = {H 1 , • • • , H r }.
The family H is quasi convex, and malnormal. We consider the coned-off Cayley graph Γ H (see Section 10). We consider a cellular, piecewise linear map f : Γ H → Γ H which realizes the given automorphism. Since the subgroups H i 's are preserved up to conjugacy, f can be assumed to permutes the vertices of the cones in Γ H . We denote by Γ f H the mapping-telescope of (Γ H , f ), that is the disjoint union of the

K i := Γ H × [0, 1] quotiented by the equivalence relation (x, 1) ∈ K i ∼ (f (x), 0) ∈ K i+1 . A horizontal geodesic in Γ f H is a Γ H -geodesic contained in some stratum Γ H × {j}, j ∈ Z. It is simple if it contains no vertex of any cone in Γ H .
Definition 5.6. A corridor is a union of (possibly infinite or bi-infinite) simple horizontal geodesics, exactly one in each stratum, which connect two orbits of f .

A K-quasi orbit is a sequence of points x 0 , • • • , x j , • • • such that there is a sequence of vertical segments v 1 , • • • , v j+1 , • • • of
lengths at least one satisfying that the initial point of v i is x i-1 and the terminal point of v i lies at horizontal distance at most K from x i . By [START_REF] Gautero | Geodesics in trees of hyperbolic and relatively hyperbolic groups[END_REF], there is a constant K such that, if x is a point in a corridor C, then there is a K-quasi orbit passing through x and contained in C. Definition 5.7. A corridor C is collapsed toward +∞ (resp. toward -∞) if, given any two points x, y in C ∩(Γ H ×{j}), there exist l ∈ N and two K-quasi orbits starting respectively at x and y and ending at the same point in C ∩ (Γ H × {j + l}) (resp. C ∩ (Γ H × {j -l})).

A collapsing corridor is a corridor which collapses either toward +∞ or toward -∞.

The collapsing corridors play the rôle of the stable and unstable leaves in the setting of Theorem 5.1.

Lemma 5.8. The subset of the collapsing corridors is closed in the following sens: if C is a non-collapsing corridor, then there exists L > 0 such that the horizontal L-neighborhood of C contains only non-collapsing corridors.

Corollary 5.9. Let (X n , Y n ) ⊂ ∂Γ H ×∂Γ H be a sequence of points which are the endpoints of a sequence of collapsing corridors. Assume that X n → X and Y n → Y . Then X and Y are the endpoints of a collapsing corridor.

As in the setting of Theorem 5.1, two points in ∂Γ H are connected by a G α -geodesic if and only if they define a non-collapsing corridor. The correspondance between the collapsing corridors and the eigenrays is established as was previously done. The proof of the proposition is then completed in the same way. We leave the reader work out the details.

Subgroups of polynomially growing free group automorphisms.

Theorem 5.10. [START_REF] Bestvina | The Tits alternative for Out(F n ). II. A Kolchin type theorem[END_REF] Let P be a finitely generated group and let θ : P → Out(F n ) be a monomorphism such that θ(P) consists entirely of polynomially growing automorphisms. Then there is a finite-index subgroup U of P, termed unipotent subgroup, and a simplicial θ(U)-invariant F n -tree T with [ θ(U)] = θ(U) which satisfies the following properties:

(1) A vertex of T is fixed by all the isometries H θ(u) , u ∈ U, and this is the unique fixed point of each one. (2) The F n -action has trivial edge stabilizers.

(3) There is exactly one F n -orbit of edges. (4) For each vertex v of T , each Stab Fn (v)-orbit of directions at v is invariant under the U-action.

As was already observed, the map Q is well-defined here since vertex stabilizers are quasiconvex and arc stabilizers are trivial.

Cyclic extensions over exponentially growing automorphisms

The bulk of this section is to prove Theorem 6.1 below about cyclic extensions of surface groups over exponentially growing automorphisms. Then we will briefly show how the same methods apply to similar cyclic extensions of free groups and of torsion free hyperbolic groups with infinitely many ends, see Theorem 6.14.

6.1. The surface case. Theorem 6.1. Let G be the fundamental group of a compact, hyperbolic surface S (with or without boundary). Let α ∈ Aut(G) be an exponentially growing automorphism in an outer-class induced by a homeomorphism of S. Let µ be a probability measure on G α := G ⋊ α Z whose support generates G α as a semi-group.

Then there exists an α-projectively invariant G-tree T with minimal interior such that, if T denotes the space obtained by closing all the eigenrays of T , then:

(1) P-almost every sample path x = {x n } converges to some x ∞ ∈ T .

(2) The hitting measure λ, which is the distribution of x ∞ , is a non-atomic measure on T such that ( T , λ) is a µ-boundary of (G α , µ) and λ is the unique µ-stationary probability measure on T . (3) If the measure µ has finite first logarithmic moment and finite entropy with respect to a word-metric on G α , then the measured space ( T , λ) is the Poisson boundary of (G α , µ).

We consider a R-tree T as given by Theorem 5.1. We denote by t the generator of Z acting on the right as α on G. Beware that t acts on the left as the homothety H -1 α on T .

Choose a point O as base-point and identify G α /Stab Gα (O) with a subset of the points of T by considering the orbit G α .O of the base-point. Of course, in the case where O has a non-trivial stabilizer, infinitely many elements of G α may get identified with a single point. For the clarity and briefness of some arguments we choose as base-point the fixed-point of the homothety H. The following lemma is obvious: Lemma 6.2. If P is any point in T , let N b T (P ) consist of a neighborhood N obs (P ) of P in T together with all the elements gu ∈ G α such that guO ∈ N obs (P ) and |gu| > N for some chosen positive N .

Then G α ∪ T , equipped with the above basis of neighborhoods N b T (.), is a separable, compatible compactification of G α .

The action of G ⊳ G α on T is the given isometric left action. The action of Z < G α on T is given by u.P = H u -1 (P ).

Remark 6.3. If there exists a point O in T with trivial G α -stabilizer we have an identification of G α with the orbit G α .O. In the cases considered in this paper it is always possible to choose such a point O when dealing with exponentially growing automorphisms. This comes from the fact that there are a finite number of orbits of branch-points and of course a countable number of points in a given orbit. Thus there are a countable number of points with non-trivial stabilizer whereas the tree has an uncountable number of points. Thus, carefully choosing the base-point would allow us to get rid of the "size" given by the integer N in the definition of the compactification of Lemma 6.2.

When dealing later with polynomially growing automorphisms, the trees are simplicial and, since edges have trivial G-stabilizers, it suffices to choose the middle of an edge as base-point to get a point with trivial G-stabilizer. However in this simplicial case it might happen that, whatever point is chosen, it has a non-trivial stabilizer in G α . Nevertheless, as the reader shall see, in this case we consider a product of two trees and once again for the given G α -action it is possible to choose a point with trivial G α -stabilizer.

Unfortunately, the compactification above will not satisfy the (CP) condition. We now need to close the eigenrays as in Theorem 6.1. Proposition 6.4. In the compactification of G α by T given in Lemma 6.2, extend the quotient-map q : T → T by the identity on G α .

Define N e T (P ) = q(N b T (q -1 (P ))). Then G α ∪ T , equipped with the basis of neighborhoods N e T (.) is a separable, compatible compactification of G α . The action of G α on T is irreducible.

Proof. By Proposition 5.4, T is Hausdorff and compact. Since the quotient-map q restricts to the identity on T , the neighborhood in G α of a point of T is the same as the neighborhood of this point in T . It is then obvious that Proposition 6.4 gives a compatible compactification of G α by T . The irreducibility of the action comes from Proposition 5.4.

Let us see what happens with respect to the (CP) and (CS) conditions when considering this compactification. Proposition 6.5. The compactification of G α with T given by Proposition 6.4 satisfies Kaimanovich (CP) condition.

Proof. Consider any sequence (w j ) ∈ G which tends to some point P ∈ T , which means that (w j O) tends to P in T . Let v ∈ G ⊳ G α . By Theorem 5.1, the LL-map Q is well-defined. By Lemma 4.10, (w j .(vO)) = (w j vO) tends to the same point P ∈ T . The condition (CP) is thus satisfied for T , and so for T , in this case. If v = t k for some integer k the same conclusion holds trivially from the very definition of the compactification: the elements w and wt k lie in a neighborhood of the same point in the tree since the base-point O has been chosen as the fixed-point of H, that is of the action of t on T .

Let us now consider a sequence t n j with n j → +∞. Of course t n j O tends to O in T so that t n j tends to O in the compactification with T . Let v ∈ G α . If v = t k for some integer k then t n j v = t n j +k tends to O in T so that the (CP) condition still holds. If is G α -equivariant and Borel.

Proof. The G α -equivariance is clear by construction. We only have to check that the map which assigns the elementary sets is Borel. This is straightforward since two distinct points in T × T \ ∆ have distinct image sets. Therefore any set in G α , which is countable, is a countable union of points in T × T \ ∆ e T , hence is Borel. Proposition 6.11. The G α -compactification T , equipped with the collection of strips S(b 1 , b 2 ), satisfies the (CS) condition.

Proof. From Lemma 6.9, no strip is empty. From Lemma 6.10, the assignment of the strips is Borel and G α -equivariant. It only remains to check that any strip S(b 1 , b 2 ) avoids a neighborhood of any third point b 0 ∈ T distinct from both b 1 and b 2 . If no element of G α fixes both b 1 and b 2 , then S(b 1 , b 2 ) consists of a δ-thin pencil of G α -geodesics between two points in a Gromov hyperbolic space. The only accumulation-points of these geodesics are by definition the boundary-points. If some element wt j (j = 0) would fix both b 1 and b 2 , they would be the endpoints of an eigenray. Since eigenrays have been closed in T , if b 1 and b 2 are fixed by a same element of G α this is an element w in G which acts as a hyperbolic isometry whose axis admits b 1 and b 2 as endpoint. It readily follows that these are the only accumulation-points. Proposition 6.12. Let J be a finite word gauge for G α . Then the strips S(b 1 , b 2 ) given by Proposition 6.11 grow polynomially with respect to J .

Proof. The group G α is strongly hyperbolic relative to the mapping-tori of the maximal subgroups where the automorphism α has linear growth (see Section 10, Theorem 10.4this is a polynomial growth in the free group case and an essential polynomial growth in the case of a torsion free hyperbolic group with infinitely many ends). Outside the incompressible tori bounding these submanifolds, the G α -geodesics behave like in a hyperbolic group [START_REF] Farb | Relatively hyperbolic groups[END_REF][START_REF] Osin | Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems[END_REF]. Thus the number of intersection-points of the strip with a ball of region k growths at most linearly with k outside the mapping-tori of the α-polynomial growth subgroups. Inside these mapping-tori, it is easily proved that this same number is bounded above by a polynomial of degree 3. Indeed on the one hand the α-growth of an element is bounded above by a polynomial of degree 2. On the other hand the G α -geodesics remain in a bounded neighborhood of a corridor between the orbits of their entranceand exit-points. Thus the bound is given by the product of a linear map with a degree 2 polynomial (in general -free group case or torsion free hyperbolic group with infinitely many ends case -this is the product of a linear map with a degree d polynomial).

Proof of Theorem 6.1. We consider a G-tree T as given by Theorem 5.1. By Propositions 6.4, 6.11, 6.5, Theorem 2.3 applies and gives the first point of Theorem 6.1. Proposition 6.12 and Theorem 2.3 give the Poisson boundary. Corollary 6.13. With the assumptions and notations of Theorem 6.1:

(1) There is a topology on G α ∪ ∂G such that P-almost every sample path x = {x n } of the random walk converges to some x ∞ ∈ ∂G.

(2) The hitting measure λ (i.e. the distribution of x ∞ ) is a non-atomic measure on ∂G such that (∂G, λ) is a µ-boundary of (G α , µ), and this is the unique µ-stationary measure on ∂G. (3) If µ has finite first logarithmic moment and finite entropy with respect to some finite word-metric on G α , then the measured space (∂G, λ) is the Poisson boundary of (G α , µ).

Proof. The map q • Q : ∂G → T (see Proposition 6.4 for the definition of q and Lemma 4.10 for the definition of the LL-map Q) is continuous, surjective and G α -equivariant. Item (1) of Theorem 6.1 then gives Item (1) of the current corollary. The map q • Q is a continuous projection such that the disjoint sets that get identified to a single point are the disjoint translates of a finite number of disjoint compact subsets of ∂G. We can thus define a hitting measure λ on ∂G such that (q • Q) * λ is the µ-stationary measure on T given by Theorem 6.1 (the pre-image of a point has λ-measure 0). The non-atomicity, µ-stationarity and unicity of λ follow from the non-atomicity, µ-stationarity and the unicity of the measure on T given by Theorem 6.1. We so got Item (2), and Item (3) is a straightforward consequence of Item (3) of Theorem 6.1.

6.2. Generalizations to free and hyperbolic groups. The following result is now given by Theorem 5.5.

Theorem 6.14. Theorem 6.1 and Corollary 6.13 remain true • either if one substitutes a non-abelian free group to the fundamental group of a compact hyperbolic surface, • or more generally if one substitutes a torsion free hyperbolic group with infinitely many ends G to the fundamental group of a compact hyperbolic surface and the automorphism has essential exponential growth.

Observe that, although G has infinitely many ends, the cyclic extension G ⋊ α Z is one-ended.

Cyclic extensions over polynomially growing automorphisms

The core of this section is to prove Theorem 7.2 below. Applications to particular classes of groups follow, see Theorem 7.24. Definition 7.1. Let T be a R-tree. Closing a bi-infinite geodesic in T means identifying its endpoints in ∂T . Theorem 7.2. Let G be a hyperbolic group. Let α ∈ Aut(G) be such that there exists a simplicial α-invariant G-tree T satisfying the following properties:

• The G-action has quasi convex vertex stabilizers, trivial edge stabilizers and only one orbit of edges. • The α-action fixes exactly one vertex O and fixes each Stab G (O)-orbit of directions at O. Let µ be a probability measure on G α := G ⋊ α Z whose support generates G α as a semi-group. Then, if T is obtained from T by closing each geodesic in the G-orbit of some bi-infinite geodesic:

(1) There is a topology on G α ∪ ∂ T such that P-almost every sample path x = {x n } of the random walk converges to some x ∞ ∈ ∂ T . (2) The hitting measure λ is a non-atomic measure on ∂ T such that (∂ T , λ) is a µ-boundary of (G α , µ), and this is the unique µ-stationary measure on ∂ T . (3) If µ has finite first logarithmic moment and finite entropy with respect to some finite word-metric on G α then the measured space (∂ T , λ) is the Poisson boundary of (G α , µ).

An important intermediate step is to first prove Theorem 7.4 below. However we need an additional notion before its statement: Remark 7.5. Of course G Φ , G α and G β are isomorphic groups. We adopt these different notations in the above statement to insist on the different actions: the generator of Z < G α acts as α -1 on T , the generator of Z < G β as β -1 on T and the generator of Z < G Φ as α -1 on the first factor of T × T and as β -1 on the second factor.

We set G = x 1 , • • • , x n . The generator t of Z acts on the left as the isometry H -1 with H the isometry of T satisfying H(wP ) = α(w)H(P ). Let us observe that the simplicial nature of T gives us an easy identification of G with the orbit of some vertex. • w is the element of G associated to the unique geodesic from O to x, • H i is the stabilizer either of O or of O ′ , and each direction at each vertex is associated to a unique element of G.

Equipping T with the observers topology, and after identifying G α with the orbit of O, we get a compatible compactification of G α in a way similar to Lemma 6.2. This compactification however does not necessarily satisfy the (CP) property: indeed it might happen that some isometry vt j , v ∈ G, fixes more than one point. This would imply that, for some w ∈ G, the sequences {(vt j ) k } k=1,••• ,+∞ and {(vt j ) k w} k=1,••• ,+∞ would not have the same limit-point.

By assumption, t fixes each Stab G (x)-orbit of direction at O. Thus, there exists an isometry of G α fixing more than one vertex of T , it has the form vt.

The important observation holds in the following well-known observation:

Lemma 7.7. If both vt and wt k fix more than one vertex in T then there is g ∈ G with (vt) k = g -1 wt k g. If vt and wt k fix the same edge, then

w = vα -1 (v) • • • α 1-k (v).
Proof. Assume that vt and wt both fix at least two vertices. Since there is only one G-orbit of edges, without loss of generality we can assume that vt and a G-conjugate of wt, denoted by g -1 wtg, both fix the same edge E. Then vtg -1 t -1 w -1 g = wα -1 (g -1 )w -1 g fixes E. By the triviality of the edge stabilizers, vtg -1 t -1 w -1 g is trivial. We so get the lemma in the case k = 1, the generalization is straightforward.

Definition 7.8. A singular element for an action of G α on T is an element of G α which fixes at least two vertices of T .

If vt is a singular element, the trick now is to consider another action of G α on T by making t act on T by another automorphism in the same outer-class so that vt acts on this copy of T as a hyperbolic element. Lemma 7.9. With the notations above: there is an (α, β)-action of G Φ on T × T such that any element of G Φ which is a singular element for the action induced on one of the two factors (T × { * } or { * } × T ) acts as a hyperbolic isometry on the other factor and the axis of this hyperbolic isometry either is disjoint from the fixed set of the singular element, or is an axis in this fixed set.

The axis of this hyperbolic isometry is called a singular axis.

Proof. Assume that vt is a singular element. Let T be the tree fixed by vt. We choose a hyperbolic element a of G such that avt acts as an hyperbolic isometry on T , the axis of which is disjoint from T if T = T . We consider the (α, β)-action of G Φ given by: Θ(wt) T × T → T × T (P , Q) → (wH -1 (P ) , wv -1 avH -1 (Q)) If wt is a singular element for the first factor, then by Lemma 7.7, there exists g ∈ G with wt = g -1 vtg. Since vt acts on the second factor as the hyperbolic isometry avt, wt acts on this second factor as g -1 avtg, which is also an hyperbolic isometry.

Assume now that wt acts as a singular element on the second factor. Since vt acts on the second factor like vv -1 avt = avt then a -1 vt is a singular element for the action on the second factor. Thus, by Lemma 7.7, wt acts on the first factor as a conjugate to a -1 vt. Since avt is a hyperbolic isometry of T , so is a -1 vt and so is any of its conjugates.

Of course any (α, β)-action extends to an action on T × T . Definition 7.10. A singular boundary-tree of T × T is a product ∂A × T or T × ∂A where:

• A is a singular axis.

• T is the closure in T ∪∂T of a maximal subtree of T which is fixed by the singular element of singular axis A.

Definition 7.11. We denote by: • T the space obtained from T by closing all the singular axis.

• T 2 the space obtained from T × T by identifying each singular boundary-tree to a point.

Lemma 7.12. The space T 2 is Hausdorff and compact. The (α, β)-action of G Φ on T × T induces an irreducible action on T 2 .

Proof. The tree T is simplicial thus separable. By Proposition 4.3 T is Hausdorff and compact. By construction, there is an axis A of T such that T is obtained from T by identifying two ends in ∂T if and only if there are the ends of an axis gA, g ∈ G. This is easily seen to be a closed property in ∂T × ∂T . It readily follows that T is Hausdorff and compact. By definition of a singular axis, there is an element w of G such that t.A = w.A.

It follows that the whole orbit G.A is invariant under the G Φ -action. Thus the G Φ -action induces an action on T . This action is irreducible because it was on T . It readily follows that T × T is Hausdorff, compact and the (α, β)-action of G Φ is an irreducible action. From Lemma 7.17, to pass to T 2 , one identifies to a point each singular boundary-tree in a single G-orbit. The arguments for completing the proof are then the same as the arguments to pass from T to T .

Although important, the following proposition is however obvious from Lemma 7.12:

Proposition 7.13. Let O ∈ T 2 be the point whose coordinates are the fixed-point of α.

If P is any point in T 2 , let N (P ) consist of a neighborhood N obs (P ) of P in T 2 equipped with the product topology, together with all the elements wt k ∈ G α such that Θ(wt k )(O) ∈ N obs (P ) and |wt k | > N for some chosen positive N .

Then G α ∪ T 2 , equipped with the above basis of neighborhoods N (.), is a separable, compatible compactification of the group G α .

Proposition 7.14. The compactification of G α by G Φ .O ⊂ T 2 satisfies the (CP) condition.

Before proving Proposition 7.14, we study with more details the action of G Φ on T 2 . Even if we will not need the full strength of the two lemmas below, they might be useful to help the reader having a better grasp on what happens here. Definition 7.15. A rectangle in T × T is a product of two geodesics.

A singular rectangle is a rectangle which is the product of a singular axis with a geodesic. A corner of a rectangle R is a point in ∂R ∩ (∂T × ∂T ), where ∂R = R \ R in T × T .

Lemma 7.16. We consider the (α, β)-action of G Φ on T × T given in Lemma 7.9. Let R = g 1 × g 2 be a singular rectangle the stabilizer of which is neither trivial nor cyclic. Then g 1 = g 2 and, up to taking powers, there are a unique v ∈ G and wt ∈ G Φ such that v admits g 1 as hyperbolic axis and g 1 is a singular axis for the action of wt. In particular, the actions of v and wt commute and the stabilizer of R is a Z ⊕ Z-subgroup.

Proof. Since two distinct G-elements cannot share the same axis in T , there is at most one v ∈ G, up to taking powers, admitting both g 1 and g 2 as hyperbolic axis. Since the action of G on T × T is just the diagonal action, if there is one v fixing R then g 1 = g 2 .

Let us now prove that there is at most one element in G Φ , which does not belong to G and admit both g 1 and g 2 as hyperbolic axis. If there were two then, up to taking powers, we can assume that they have the form vt k and wt k . Then vw -1 = vt k t -k w -1 fixes the same axis. This is an element of G and, since T is a α-invariant tree, vw -1 is fixed by the automorphisms vα k (.)v -1 and wα k (.)w -1 . Therefore α k (vw -1 ) = w -1 vα k (vw -1 )v -1 w. Since G is hyperbolic, vw -1 is then either a torsion element, or trivial. Since it fixes a point in ∂T , and edge stabilizers are trivial, it cannot be a torsion element. Thus vw -1 = 1 G . We so got that, up to taking powers, there is at most one vt fixing R.

From the previous two paragraphs, if R is a singular rectangle whose stabilizer is neither trivial nor cyclic, then up to taking powers there are a unique v ∈ G and wt ∈ G Φ fixing R. Since t does not act in the same way on the two factors, if wt fixes R, then one of the two axis is a singular axis for wt. Thus v and wt commute and we get the lemma. Lemma 7.17. We consider the (α, β)-action of G Φ on T × T given in Lemma 7.9.

(1) There is at most one G-orbit of singular rectangles the stabilizers of which are neither trivial nor cyclic. (2) Let R be a rectangle with cyclic stabilizer w 1 t n 1 . For any point P ∈ T × T , the accumulation points of Θ( w 1 t n 1 )(P ) either are two opposite corners of R, or belong to a singular boundary-tree.

Proof. Item (1) comes directly from Lemmas 7.7 (unicity of the singular element) and 7.16. Consider now a rectangle R with cyclic stabilizer in G. If this cyclic stabilizer is not generated by a singular element, we are in the case where the orbit of O accumulates on two opposite corners. If the cyclic stabilizer is generated by a singular element, we are in the second case.

Proof of Proposition 7.14. We first recall that the map Q is well-defined for the actions of G on T (triviality of the edge-stabilizers) so that the action of G alone is not a problem. Consider a sequence v i t n i such that Θ(v i t n i ).O converges to some point P . If the points of the sequence belong to infinitely many rectangles then this is also true for any sequence Θ(v i t n i w).O, w ∈ G. The convergence of Θ(v i t n i ).O to P means that Θ(v i t n i ).O turns around P , and so does Θ(v i t n i w).O. Let us assume that there exists N ≥ 0 such that for all i ≥ N , all the Θ(v i t n i ).O belong to a same rectangle. Then again either the points turn around P and the conclusion for Θ(v i t n i w).O is straightforward. Or the rectangle has a non-trivial stabilizer and the conclusion for Θ(v i t n i w).O is deduced from Lemma 7.17 since each singular boundary-tree has been collapsed to a point. We conclude by noticing that, since t fixes O, the above arguments readily imply the (CP) condition.

last theorem gives the conclusion for G α k and the conclusion for G α follows from Theorem 2.5.

As in the proof of Corollary 6.13, the existence of the LL-map Q : ∂G → T gives:

Corollary 7.25. With the assumptions and notations of Theorem 7.24:

(1) There is a topology on G α ∪ ∂G such that P-almost every sample path admits a subsequence which converges to some x ∞ ∈ ∂G.

(2) The hitting measure λ on ∂G is non-atomic and this is the unique µ-stationary measure λ on ∂G. (3) If µ has a finite first moment, then the measured space (∂G, λ) is the Poisson boundary of (G α , µ).

Remark 7.26. In the case of a direct product there is no need to pass to a finite-index subgroup so that the conditions on the measure may be relaxed to "finite first logarithmic moment and finite entropy" in Theorem 7.24 and Corollary 7.25. Moreover there is no need to pass to a subsequence in the first items of these results.

Extensions by non-cyclic groups

The plan of this section is parallel to the plan of the previous one. As a main goal we have the following Theorem 8.1. Let G be any one of the following two kinds of groups:

• A direct product G × F k where G is a torsion free hyperbolic group with infinitely many ends. • A semi-direct product G ⋊ θ P of a free group G = F n with a finitely generated subgroup P over a monomorphism θ : P → Out(F n ) such that θ(P) consists entirely of polynomially growing outer automorphisms. Let µ be a probability measure on G whose support generates G as a semi-group. Then there exists a finite index subgroup U in P, a simplicial θ(U)-invariant G-tree T , and a set A of bi-infinite geodesics in T whose union forms a proper subtree of T such that, if T denotes the space obtained from T by closing each axis in G.A, then:

(1) There is a topology on G ∪ ∂ T such that P-almost every sample path admits a subsequence which converges to x ∞ ∈ ∂ T . (2) The hitting measure λ is non-atomic and this is the unique µ-stationary measure on ∂ T . (3) If µ has finite first moment, then the measured space (∂ T , λ) is the Poisson boundary of (G, µ).

As in the previous section, we will first focus on an intermediate result involving the product of two copies of a G-invariant tree.

Definition 8.2. Let θ 0 , θ 1 : U → Aut(F n ) be two monomorphisms. Assume that [θ 0 (U)] = [θ 1 (U)] := θ(U) in Out(G). Let T be a θ(U)-invariant G-tree.
The (θ 0 , θ 1 )-action of G = G⋊ θ U on T ×T is the action given by Θ : G → Isom(T ×T ) with:

Θ(wu) T × T → T × T (P , Q) → (wH -1 θ 0 (u) (P ) , wH -1 θ 1 (u) (Q)) Theorem 8.3.
Let G be any one of the following two kinds of groups:

• A direct product G × F k where G is a torsion free hyperbolic group with infinitely many ends. • A semi-direct product G ⋊ θ P of a free group G = F n with a finitely generated subgroup P over a monomorphism θ : P → Out(F n ) such that θ(P) consists entirely of polynomially growing outer automorphisms. Let µ be a probability measure on G whose support generates G as a semi-group. There exist

• two monomorphisms θ 0 , θ 1 : P → Aut(G) with [θ 0 (P)] = [θ 1 (P)] = θ(P) and G = G ⋊ θ P, • a finite index subgroup U in P, • a simplicial θ(U)-invariant G-tree T , • a set A of bi-infinite geodesics in T whose union forms a proper subtree of T such that, if T denotes the space obtained from T by closing each geodesic in G.A and ∂G.O denotes the boundary of the closure of the (θ 0 , θ 1 )-orbit of O in ∂( T × T ) then:

(1) P-almost every sample path x = {x n } admits a subsequence which converges to some x ∞ ∈ ∂G.O. (2) The hitting measure λ is a non-atomic measure and this is the unique µ-stationary probability measure on ∂G.O. (3) If the measure µ has finite first moment with respect to a word-metric on G then the measured space (∂G.O, λ) is the Poisson boundary of (G, µ).

Proof. The proof follows exactly the same scheme as the proof of Theorem 7.4. The only difference lies in the singular elements. Let us first consider the case where G is a direct product G × F 2 (there is no loss of generality in taking F 2 instead of F k ). We first choose a non trivial α ∈ Inn(G) and construct a α-invariant G-tree T as in the previous section. Let t 1 , t 2 be the two generators of F 2 and let x 1 , • • • , x r be the generators of G. We make them act on T as two elliptic isometries fixing a unique vertex v i , v 1 = v 2 and v 1 , v 2 are the two vertices of some edge E: t 1 acts as α whereas t 2 acts as x 1 αx -1 1 for instance. This gives the θ 0 -action. For each t i there is a unique g i ∈ G such that g i t i fixes E. Then we consider another copy of T and make act the t i 's in such a way that g i t i acts as a hyperbolic isometry H i and the subgroup H = H 1 , H 2 is free. This gives the θ 1 -action. Now the subtree T T announced by Theorem 8.3 is the tree of this free subgroup. We consider the collection of bi-infinite geodesics A ′ given by the collection of the axis of the elements of H and the bi-infinite geodesics which are accumulations of such axis. The

set {(X, Y ) ∈ ∂T × ∂T s.t. ∃ A ∈ A ′ with X, Y ∈ ∂A} is a closed subset of ∂T × ∂T .
Thus closing the bi-infinite geodesics in A ′ yields, as in the previous section, a Hausdorff, compact space. We now set A = GA ′ and we close the bi-infinite geodesics in A. The space T we get by closing the bi-infinite geodesics in A is Hausdorff and compact by the same argument as in the previous section. The (θ 0 , θ 1 )-action on T × T is the action made explicit above. All the arguments until the end are then copies of already given arguments.

The adaptation to the case of the extension of a free group by a polynomial growth subgroup is done as follows: Theorem 5.10 gives a finite-index subgroup U for which there exists a θ |U (U)-invariant F n -tree T . Assuming without loss of generality that U is 2-generated, there exist as before g i ∈ G such that g i t i both fix the same edge E since the t i 's fix each Stab G (v)-orbit of directions at v. We leave the reader work out the details for the remaining arguments. We conclude at the end by Theorem 2.5 which allows us to recover the conclusion for the original group G ⋊ θ P, of which G ⋊ θ |U U is a finite-index subgroup.

Theorem 8.1 is deduced from Theorem 8.3 in the same way as Theorem 7.2 was deduced from Theorem 7.4: Lemma 7.22 still holds here.

As was previously done, the existence of the LL-map Q gives the Corollary 8.4. With the assumptions and notations of Theorem 8.1:

(1) There is a topology on G ∪ ∂G such that P-almost every sample path admits a subsequence which converges to x ∞ ∈ ∂G.

(2) The hitting measure λ is non-atomic and this is the unique µ-stationary measure on ∂G.

(3) If µ has a finite first moment, then the measured space (∂G, λ) is the Poisson boundary of (G, µ).

Remark 8.5. In the case where G is a direct product or is a semi-direct product over a unipotent subgroup (see Theorem 5.10) of polynomially growing automorphisms then there is no need to pass to a finite-index subgroup so that the conditions on the measure in item (3) may be relaxed to "finite first logarithmic moment and finite entropy" in Theorem 8.1 and Corollary 8.4. Moreover there is no need to pass to a subsequence in the first items of these results. The morphism θ : U → Aut(F n ) is a monomorphism, i.e. α, β is a free subgroup of rank 2 of Aut(F 3 ) because, for any non-trivial reduced word u in α ±1 , β ±1 , the word u(c) is non-trivial. Hence there are no non-trivial relation between α ±1 and β ±1 .

The subgroup α, β is a subgroup of polynomially growing automorphisms. Indeed a and b have 0-growth under α and β, so under any u ∈ U, whereas c has linear growth. Hence for any w ∈ F 3 the length of u(w) growths at most linearly with respect to |u| a,b,c . 9.2. Extension of a free group by an abelian group of polynomially growing automorphisms. Let α ∈ Aut(F 2 ), F 2 = a, b , be defined by α(a) = a, α(b) = ab. This is a polynomially growing automorphism.

We consider the free group F

Relative hyperbolicity and application to Poisson boundaries

The aim of this section is to give another description of the Poisson boundary of a group G α = G ⋊ α Z when G is either a free group or the fundamental group of a compact hyperbolic surface, and α is an exponentially growing automorphism. There seems to be nothing really new here for geometric group theorists, but only a compilation of folklore, well-known and more recent results. 10.1. Poisson boundary of strongly relatively hyperbolic groups. The Proposition 10.1 below describes the Poisson boundary of a strongly relatively hyperbolic group in terms of its relative hyperbolic boundary. It can be easily deduced from [START_REF] Vadim | The Poisson formula for groups with hyperbolic properties[END_REF]. Beware however that the hyperbolicity of the coned-off Cayley graph (see below) of Farb [START_REF] Farb | Relatively hyperbolic groups[END_REF] is not sufficient: we really need the strong version of the relative hyperbolicity. We briefly recall some basic facts about relative hyperbolicity and otherwise invite the interested reader to consult [START_REF] Farb | Relatively hyperbolic groups[END_REF], [START_REF] Bowditch | Relatively hyperbolic groups[END_REF] for various definitions of relative hyperbolicity. If G is a discrete group with generating set S, and H a is a subgroup of G, the coned-off Cayley graph Γ H S (G) of (G, H) is the graph obtained from by adding to Γ S (G) (the Cayley graph of G with respect to S) a vertex v(gH) for each left H-class and an edge of length 1 2 between v(gH) and all the vertices of Γ S (G) associated to elements in the class gH. The weak relative hyperbolicity of G with respect to H just requires the hyperbolicity of Γ H S (G). The strong relative hyperbolicity requires in addition that Γ H S (G) satisfy the so-called Bounded Coset Penetration property, see [START_REF] Farb | Relatively hyperbolic groups[END_REF]. We will not recall its definition here but just say that this property forbids that two left H-classes remain parallel along arbitrarily long paths in Γ S (G). Combined with the hyperbolicity of Γ H S (G), it follows that any two left H-classes separate exponentially so that, in particular, they define distinct points in the relative hyperbolic boundary ∂ RH (G, H). Of course some care has to be taken in order to get a correct definition of this boundary, since Γ H S (G) is not a proper space (closed balls are not compact) as soon as H is infinite. We refer the reader to [START_REF] Bowditch | Relatively hyperbolic groups[END_REF] or [START_REF] Yaman | A topological characterisation of relatively hyperbolic groups[END_REF] to the definition of this relative hyperbolic boundary. The topology used by Bowditch is close in spirit to the observers topology used before in the current paper to deal with the nonproperness of R-trees. Another construction of this relative hyperbolic boundary can be found in [START_REF] Groves | Dehn filling in relatively hyperbolic groups[END_REF] where the author takes more care in constructing a proper space associated to (G, H) but then gets the relative hyperbolic boundary in an easier way. This last definition is closer in spirit to Gromov approach of relative hyperbolicity as initiated a long time ago in the seminal paper [START_REF] Gromov | Hyperbolic groups[END_REF]. Of course all these notions have a straightforward generalization when substituting a finite family of subgroups H to a single subgroup H. Proposition 10.1. Let G be a finitely generated group which is strongly hyperbolic relatively to a finite family of subgroups H. Let ∂ RH (G, H) be the relative hyperbolic boundary of (G, H). Let µ be a probability measure on G whose support generates G as a semigroup. Then:

(1) P-almost every sample path x = {x n } converges to some x ∞ ∈ ∂ RH (G, H).

(2) The measure λ = π(P) on ∂ RH (G, H) which is the distribution of x ∞ is µstationary and non-atomic. It is the unique µ-stationary measure on ∂ RH (G, H).

The G-space (∂ RH (G, H), λ) is a µ-boundary of (G, µ). 10.2. Applications to cyclic extensions of free and surface groups. We need now some material borrowed from [START_REF] Gautero | Mapping-tori of free group automorphisms are hyperbolic relatively to the polynomially growing subgroups[END_REF]. Let Φ ∈ Out(F n ). A family of Φ-polynomially growing subgroups H = (H 1 , • • • , H r ) is called exhaustive if every element g ∈ F n of polynomial growth is conjugate to an element contained in some of the H i . The family H is called minimal if no H i is a subgroup of any conjugate of some H j with i = j. The following proposition is well-known among the experts of free group automorphisms, see [START_REF] Levitt | Counting growth types of automorphisms of free groups[END_REF] or [START_REF] Gautero | Mapping-tori of free group automorphisms are hyperbolic relatively to the polynomially growing subgroups[END_REF] for a proof. We need to precise a little bit more this notion of "invariance" for a family of subgroups, with respect to the action of an automorphism (and not only an outer automorphism).

For any α ∈ Aut(F n ), a family of subgroups H = (H 1 , . . . , H r ) is called α-invariant up to conjugation if there is a permutation σ of {1, . . . , r} as well as elements h 1 , . . . , h r ∈ G such that α(H k ) = h k H σ(k) h -1 k for each k ∈ {1, . . . , r}. Let H = (H 1 , . . . , H r ) be a finite family of subgroups of G which is α-invariant up to conjugacy. For each H i in H let m i ≥ 1 be the smallest integer such that α m i (H i ) is conjugate in G to H i , and let h i be the conjugator:

α m i (H i ) = h i H i h -1 i
(this conjugator is well-defined because the subgroups in H are malnormal -see Remark 4.11). We define the induced mapping torus subgroup: [START_REF] Azencott | Espaces de Poisson des groupes localement compacts[END_REF]. Let H = (H 1 , . . . , H r ) be a finite family of subgroups of G which is α-invariant up to conjugacy. A family of induced mapping torus subgroups H α = (H α 1 , . . . , H α q ) as above is the mapping torus of H with respect to α if it contains for each conjugacy class in G α of any H α i , for i = 1, . . . , r, precisely one representative. The following theorem is from [START_REF] Gautero | Mapping-tori of free group automorphisms are hyperbolic relatively to the polynomially growing subgroups[END_REF] in the case where G = F n and from [START_REF] Gautero | Geodesics in trees of hyperbolic and relatively hyperbolic groups[END_REF] (see also [START_REF] Gautero | Relative hyperbolization of (one-ended hyperbolic)-by-cyclic groups[END_REF]) in the surface case.

H α i = H i , h -1 i t m i ⊂ G α Definition 10.
Theorem 10.4. The group G ⋊ Φ Z is strongly hyperbolic relative to the mapping-torus of a Φ-characteristic family.

We can now state the theorem of this section, which follows directly from Proposition 10.1 and Theorem 10.4: Theorem 10.5. Let G be the fundamental group of a compact hyperbolic surface or the free group of rank n. Let Φ ∈ Out(G) be an exponentially growing outer automorphism and let α be any automorphism in the class Φ. Let µ be a probability measure on G α = G ⋊ α Z whose support generates G α as a semi-group. Then:

(1) P-almost every sample path x = {x n } converges to some x ∞ ∈ ∂ RH (G α , H α (Φ)).

(2) The hitting measure λ, which is the distribution of x ∞ , is a non-atomic measure on ∂ RH (G α , H α (Φ)) such that (∂ RH (G α , H α (Φ)), λ) is a µ-boundary of (G α , µ) and λ is the unique µ-stationary probability measure on ∂ RH (G α , H α (Φ)). (3) If the measure µ has finite first logarithmic moment and finite entropy with respect to a word-metric on G α , then the measured space (∂ RH (G α , H α (Φ)), λ) is the Poisson boundary of (G α , µ).

When the outer automorphism is hyperbolic, meaning that any conjugacy-class has exponential growth under Φ, the Φ-characteristic family is trivial. This is exactly the case where G α is a hyperbolic group [START_REF] Bestvina | A combination theorem for negatively curved groups[END_REF][START_REF] Gautero | Hyperbolicity of mapping-torus groups and spaces[END_REF][START_REF] Gautero | Geodesics in trees of hyperbolic and relatively hyperbolic groups[END_REF] so that, in this case, Theorem 10.5 tells nothing new with respect to [START_REF] Vadim | The Poisson formula for groups with hyperbolic properties[END_REF].

Frédéric Mathéus, Université de Bretagne-Sud, L.M.A.M., Campus de Tohannic BP 573, 56017 Vannes, France E-mail address: Frederic.Matheus@univ-ubs.fr

  -1 ) and by Out(G) = Aut(G)/Inn(G) the group of outer automorphisms. While Aut(G) acts on elements of G, Out(G) acts on conjugacy-classes of elements. If U < Aut(G) is a subgroup, we denote by [U] < Out(G) its image under the canonical projection. Definition 3.1. An automorphism α of a G has polynomial growth if there is a polynomial function P such that, for any γ in G, for any m ∈ N, |α m (γ)| ≤ P (m)|γ|.

P n holds. 4 . 3 .Definition 4 . 4 .

 4344 If (P n ) is a sequence of point in T which converges to some point P in T then for any Q ∈ T , P = lim inf Q n→∞ Affine actions on R-trees. Let (T , d) be a R-tree. A homothety H : T → T of dilation factor λ ∈ R + * is a homeomorphism of T such that for any x, y ∈ T : d(H(x), H(y)) = λd(x, y).

Definition 7 . 3 .

 73 Let Φ ∈ Out(G) and let T be a Φ-invariant G-tree. Let α, β ∈ Aut(G) with [α] = [β] = Φ. We denote by H α (resp. H β ) the corresponding isometries of T , i.e. for any w ∈ G and P ∈ T , H α (wP ) = α(w)H α (P ) and H β (wP ) = β(w)H β (P ).The (α, β)-action of G Φ := G× Φ Z on T ×T is the action given by Θ :G Φ → Isom(T × T ) with: Q) → (wH -1 α (P ) , wH -1 β (Q)) Theorem 7.4.Let G be a hyperbolic group. Let α ∈ Aut(G) be such that there exists a simplicial α-invariant G-tree T satisfying the following properties:• The G-action has quasi convex vertex stabilizers, trivial edge stabilizers and only one orbit of edges.•The α-action fixes exactly one vertex O and fixes each Stab G (O)-orbit of directions at O. Let µ be a probability measure on G α := G ⋊ α Z. Then there exist β ∈ Aut(G) with [α] = [β] := Φ in Out(G), and a bi-infinite geodesic A such that, if T is obtained from T by closing each geodesic in the G-orbit of A and ∂G Φ .O is the boundary of the closure of the (α, β)-orbit of O in ∂( T × T ) then: (1) P-almost every sample path x = {x n } converges to some x ∞ ∈ ∂G Φ .O. (2) The hitting measure λ, which is the distribution of x ∞ , is a non-atomic measure on ∂G Φ .O such that (∂G Φ .O, λ) is a µ-boundary of (G α , µ) and λ is the unique µ-stationary probability measure on ∂G Φ .O. (3) If the measure µ has finite first logarithmic moment and finite entropy with respect to a word-metric on G α , then the measured space (∂G Φ .O, λ) is the Poisson boundary of (G α , µ).

Lemma 7 . 6 .

 76 Choose the point O given in Theorem 7.4 as base-point and choose an identification of Stab G (O) with the directions at O. If there are two orbits of vertices, choose also a base-point O ′ in the other orbit adjacent to O. Then any vertex x of T is associated to a unique left-class wH i (i = 1, 2) as follows:

9 . Examples 9 . 1 .

 991 Extension of a free group by a free group of polynomially growing automorphisms. Let F 3 = a, b, c and let U := F 2 = t 1 , t 2 . We define θ(t 1 ) := α ∈ Aut(F 3 ) and θ(t 2 ) := β ∈ Aut(F 3 ) by:(1) α(a) = a, α(b) = b and α(c) = ca. (2) β(a) = a, β(b) = b and β(c) = cb.

2k = a 1 , b 1 ,

 11 • • • , a k , b k and the automorphisms α 1 , • • • , α k of F 2k defined by α i (a i ) = a i for any i = 1, • • • , k α i (b i ) = b i a i if i = j α i (b j ) = b j if i = j. The aubgroup α 1 , • • • , α k < Aut(F 2k) is a Z k -subgroup of polynomially growing automorphisms in Aut(F 2k ).

( 3 )

 3 If µ has finite first logarithmic moment and finite entropy then the measured space (∂ RH (G, H), λ) is the Poisson boundary of (G, µ).

Proposition 10 . 2 . [ 51 , 32 ]

 1025132 Every outer automorphism Φ ∈ Out(F n ) possesses a Φcharacteristic family H(Φ), that is a family satisfying the following properties: (a) H(Φ) = (H 1 , . . . , H r ) is a finite, exhaustive, minimal family of finitely generated subgroups H i that are of polynomial growth. (b) The family H(Φ) is uniquely determined, up to permuting the H i or replacing any H i by a conjugate. (c) The family H(Φ) is Φ-invariant (up to conjugacy).

  There exists a topology on G∪∂F d such that almost every path {x n } of the random walk converges to some x ∞ ∈ ∂F d .• The distribution of x ∞ is a non-atomic µ-stationary measure λ on ∂F d ; it is the unique µ-stationary measure on ∂F d . • If µ has finite first logarithmic moment and finite entropy, then the measured space (∂F

• The distribution of x ∞ is a non-atomic µ-stationary measure λ on ∂F d ; it is the unique µ-stationary measure on ∂F d . • If µ has finite first moment, then the measured space (∂F d , λ) is the Poisson boundary of (G, µ). Corollary 1.5. Let F d be the rank d free group, d ≥ 2. Let G be either a cyclic extension F d ⋊ α Z of the free group F d over an exponentially growing automorphism α of F d , or a direct product F d × F k acting on ∂F d via some fixed monomorphism F k ֒→ Inn(F d ). Let µ be a probability measure on G the support of which generates G as a semi-group. Then: • d , λ) is the Poisson boundary of (G, µ).

Acknowledgements: The first author is glad to thank M. Lustig for the numerous, almost uncountable, discussions they had about R-trees and the actions of free groups on R-trees. He is also grateful to the University Blaise Pascal (Clermont-Ferrand II) where he worked during the elaboration of most of this work. Both authors thank Yves Guivarc'h for his permanent encouragement and fruitful comments, and François Dahmani for pointing out Proposition 10.1.

v ∈ G, t k vO admits O and the endpoints of the eigenrays of H as accumulation points. Thus it converges in T to the point O, since the eigenrays have been closed. Therefore the (CP) condition holds in T in this case.

The lemma readily follows.

Remark 6.6. In the last case treated in the proof of Proposition 6.5, we really need to be in T , and not only in T .

We are now going to deal with the (CS) condition. We need a preliminary definition: Definition 6.7. Assume that an identification of Stab G (O) with the directions at the base-point O has been chosen once and for all. Let P ∈ T be a point with non-trivial G-stabilizer H < G. If D P (Q) is the direction of Q at P , the exit-point of H in D P (Q) is the element g ∈ H such that D P (g.O) = D P (Q).

Before giving the definition of the strips, we recall that the mapping-torus group G α of an exponentially growing automorphism of a surface or free group G (or of an essentially exponentially growing automorphism of a torsion free hyperbolic group with infinitely many ends) admits a non-trivial structure of strongly relatively hyperbolic group (see Section 10): the G α -geodesics we evoke below are geodesics for the corresponding socalled "relative" metric. Definition 6.8. We identify G and ∂G with G × {0} and

trivial, where Q is the LL-map defined in Lemma 4.10. Let ∆ be the diagonal of T × T . Let R be a set of representatives of G α -orbits in ( T × T \ ∆)/((x, y) ∼ (y, x)).

is the unique origin of eigenray in q -1 (b i ) otherwise, where q : T → T is the quotient-map which "closes the eigenrays" (see Definition 5.3 and Proposition 5.4). Lemma 6.9. No strip is empty. The exit-point at a vertex b ∈ T in the direction D b (P ) is the element of G associated to this direction by Lemma 7.6. If w is a geodesic in T between P ∈ T and Q ∈ T , the exit-points in w are the exit-points at the vertices v i ∈ w of T in the directions D v i (P ) and

We identify G with G × {0}. Let b 1 , b 2 be any two distinct points in T . The basic strip BS(b 1 , b 2 ) consists of all the exit-points in the unique T -geodesic between b 1 and b 2 . Proof. Observe that everything has been done so that if {(b 1 , c 1 ), (b 2 , c 2 )} is stabilized then the stabilizer is at most a cyclic group generated by w ∈ G ⊳ G Φ , and in this case the strip consists of the exit-points in the T -geodesic which connects the boundary of the axis of this stabilizer. Thus the number of intersections of such a strip with a ball of radius k growths linearly with k. We can thus assume that {(b 1 , c 1 ), (b 2 , c 2 )} is stabilized by no element in G Φ . Then the strip is the union of two basic strips and so is infinite if and only if a basic strip is infinite. Since infinite basic strips are composed of exit-points lined up along a geodesic of T the conclusion follows. Proof of Theorem 7.2. We call singular points the points resulting from the collapsing of the singular boundary-trees. There is a slight abuse of terminology in the lemma below when considering the "projection on the first factor" π : ∂ T 2 \ {singular points} → T . We mean of course the map induced by the projection on the first factor from ∂( T × T ) to T , where

Lemma 7.22. Let π : ∂ T 2 \ {singular points} → T be the projection on the first factor. Then:

(1) For any x ∈ ∂ T lying in the image of π, π -1 (x) ∩ ∂G Φ .O consists of exactly one point.

(2) If V (T ) denotes the set of vertices of T , then there exist either one or two vertices

) and all the translates γ.π -1 (x 0 ) and γ.π -1 (x 1 ),γ ∈ G Φ , are disjoint as soon as the translating element γ is not in the stabilizer. (3) The map π is G Φ -equivariant.

Proof. Items (2) and (3) are clear. Item (1) is a consequence of the fact that the points in ∂T which have neither trivial nor cyclic stabilizer are the endpoints of singular axis. Indeed, if v and wt k both fix P ∈ ∂T then wt k and v commute so that v -1 wt k act as the identity on an axis with terminal point P . Thus P is not in the image of π (we defined π on the complement of the set of singular points).

The set of singular points obviously has λ-measure zero, where λ is the Poisson measure. This is also true for the sets π -1 (x) with x ∈ V (T ). Indeed this follows from Item (2) of Lemma 7.22 and the following Lemma 7.23. [46, Lemma 2.2.2] Let G be a countable group, µ a probability measure on G, and B a G-space endowed with a µ-stationary probability measure ν. Let π : B → C be any G-equivariant quotient map on a G-space C on which G acts with infinite orbits. Then all the fibers π -1 (x) have ν-measure zero.

Proof. Let X = π -1 (x) be a fiber and denote by S the stabilizer of X in G. Consider the function f defined on G/S by f (γ) = ν(gX) where γ = gS ∈ G/S. One has γ∈G/S f (γ) ≤ ν(B) < +∞ therefore f has a maximum value. On the other hand, since ν is µ-stationary, the function f satisfies the following mean-value property :

h∈G µ(h)f (h -1 γ) = f (γ), therefore f must be constant. Since the set G/S is infinite, f must be zero. Thus, by Item (1) of Theorem 7.4 almost every sample path x = {x n } converges to some x ∞ ∈ ∂ T for the measure associated to π * λ. Observe however that there is again a slight abuse here since π * λ is a priori only defined on π(∂ T 2 \ {singular points}), which is only a subset of ∂ T when deprived of the zero measure set V ( T ), but it suffices to extend it by declaring the complement to be of measure zero. From which precedes π * λ is non-atomic and this is the hitting measure for the topology we have on G α ∪ ∂ T .

Observe that ∂ T satisfies the proximality property of Lemma 2.1. By Remark 2.2 and Lemma 2.1 the measure π * λ is thus the unique µ-stationary measure.

By Item (3) of Theorem 7.4 and Lemma 7.22, (∂ T , π * λ) is the Poisson boundary of (G α , µ). Let G be a torsion free hyperbolic group with infinitely many ends. Let G α = G ⋊ α Z be the semi-direct product of G with Z over an essentially polynomially growing automorphism α.

Let µ be a probability measure on G α whose support generates G α as a semi-group. Then there exist a simplicial α-invariant G-tree T and a bi-infinite geodesic A in T such that, if T denotes the space obtained from T by closing each geodesic in G.A, then:

• There is a topology on G α ∪ ∂ T such that P-almost every sample path x = {x n } admits a subsequence which converges to some x ∞ ∈ ∂ T . • The hitting measure λ is non-atomic and it is the unique µ-stationary measure on ∂ T . • If µ has a finite first moment then the measured space (∂ T , λ) is the Poisson boundary of (G α , µ).

Proof. In the case of a direct product of a free group with Z, the construction of Section 4.1 gives a tree T as required by Theorem 7.2. This last theorem gives the conclusion in this case.

In the case of a non-trivial cyclic extension over an essentially polynomially growing automorphism, Theorem 5.5 gives a tree T which satisfies the properties required by Theorem 7.2, at the possible exception of the fact that α fixes each Stab G (x)-orbit of direction. However, the former theorem gives the finiteness of number of Stab G (x)-orbits of directions at each vertex x. Thus, after substituting α by α k , so passing from G α to the finite index subgroup G α k , we get a tree T for G α k as required by Theorem 7.2. This