Alexandre Martin

Non-positively curved complexes of groups and boundaries

Z-compactication.

The existence of EZ-structures, and their generalisation for groups with torsion, is known for groups that admit a classifying space for proper actions with a suciently nice geometry. For a group G acting properly and cocompactly on a CAT(0) space X, the compactication of X obtained by adding the visual boundary ∂X yields an EZ-structure for G. In the case of a torsion-free hyperbolic group G, a classifying space is given by an appropriate Rips complex (see for instance CoornaertDelzantPapadopoulos [START_REF] Coornaert | Géométrie et théorie des groupes[END_REF]). Bestvina and Mess [START_REF] Bestvina | The boundary of negatively curved groups[END_REF] proved that such a space can be compactied by adding the Gromov boundary of G to get an EZ-structure for G. This result was further generalised in [START_REF] Meintrup | A model for the universal space for proper actions of a hyperbolic group[END_REF] to the case of hyperbolic groups with torsion, where they show that such a compactication yields an EZ-structure in the sense of CarlssonPedersen [START_REF] Carlsson | Controlled algebra and the Novikov conjectures for Kand L-theory[END_REF]. The existence of such an EZ-structure is also known for systolic groups by work of Osajda and Przytycki [START_REF] Osajda | Boundaries of systolic groups[END_REF].

In this article, we adress the following combination problem: Given a group G acting cocompactly by simplicial isometries on a simplicial CAT(0) complex X, are there natural conditions under which it is possible to build an EZ-structure for G, assuming that the stabilisers of simplices all admit such a structure? There are already some special cases for which such a combination theorem is known to hold. For instance, Tirel [START_REF] Tirel | Z-structures on products[END_REF] explained how to build a Z-boundary for free and direct 1 products of groups admitting Z-boundaries. Furthermore, Dahmani [START_REF] Dahmani | Classifying spaces and boundaries for relatively hyperbolic groups[END_REF] built an EZstructure for a torsion-free group that is hyperbolic relative to a group admitting an EZstructure.

This article deals with acylindrical actions on CAT(0) spaces. Recall that an action is called acylindrical if the diameter of sets with innite pointwise stabiliser is uniformly bounded above 1 .This is a rst step towards developing geometric tools to study groups through their cocompact actions on non-positively curved complexes of arbitrary dimension. This is particularly relevant for groups that do not split and lack the rich geometry of groups that are non-positively curved in a broad sense (hyperbolic, CAT(0), systolic). An example of such a phenomenon is the case of the mapping class group of a non-exceptional surface, acting on its curve complex. The action is acylindrical by a result of Bowditch [START_REF] Bowditch | Tight geodesics in the curve complex[END_REF] and the curve complex is hyperbolic by a celebrated result of Masur and Minsky [START_REF] Masur | Geometry of the complex of curves. I. Hyperbolicity[END_REF]. However, it is known that the mapping class group is not relatively hyperbolic by work of Behrstock, Druµu and Mosher [START_REF] Behrstock | Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity[END_REF].

In this article, we consider a non-positively curved, complex of groups G(Y) = (G σ , ψ a , g a,b ) over a nite simplicial complex Y endowed with a M κ -structure, κ ≤ 0, in the sense of Bridson [START_REF] Bridson | Geodesics and curvature in metric simplicial complexes[END_REF], such that the stabiliser of every simplex σ of Y admits an EZ-structure (EG σ , ∂G σ ). We further assume that these structures dene an EZ-complex of space compatible with G(Y) (see Denition 2.2 for a precise denition), that is, there are embeddings φ σ,σ : EG σ → EG σ , for all σ ⊂ σ , that are equivariant with respect to the local maps of G(Y), and such that the induced diagram of embeddings is commutative up to multiplication by twisting elements of G(Y).

Combination Theorem for boundaries of groups. Let G(Y) be a non-positively curved complex of groups over a nite simplicial complex Y endowed with a M κ -structure, κ ≤ 0. Let G be the fundamental group of G(Y) and X be a universal covering 2 of G(Y). Suppose that the following global condition holds:

(i) The action of G on X is acylindrical.

Further assume that there is an EZ-complex of spaces compatible with G(Y) that satises each of the following local conditions:

(ii) the limit set property: For every pair of simplices σ ⊂ σ of Y , the embedding EG σ → EG σ realises an equivariant homeomorphism from ∂G σ to the limit set ΛG σ ⊂ ∂G σ . Furthermore, for every simplex σ of Y , and every pair of subgroups H 1 , H 2 in the family 1 The original denition of acylindricity by Sela [START_REF] Sela | Acylindrical accessibility for groups[END_REF] considers nontrivial stabilisers instead of innite ones. Here we use a more general notion of acylindricity introduced by Delzant [START_REF] Delzant | Sur l'accessibilité acylindrique des groupes de présentation nie[END_REF] that is more suitable for proper actions. 2 The simplicial complex X naturally inherits a Mκ-structure from that of Y , which makes it a complete geodesic metric space by work of Bridson [START_REF] Bridson | Geodesics and curvature in metric simplicial complexes[END_REF]; the CAT(0) property follows from the Cartan-Hadamard Theorem.

F σ = n i=1 g i G σ i g -1 i | g 1 , . . . , g n ∈ G σ , σ 1 , . . . , σ n ⊂ st(σ), n ∈ N , we have ΛH 1 ∩ ΛH 2 = Λ(H 1 ∩ H 2 ) ⊂ ∂G σ ,
(iii) the convergence property: for every pair of simplices σ ⊂ σ in Y and every sequence (g n ) of G σ whose projection is injective in G σ /G σ , there exists a subsequence such that (g ϕ(n) EG σ ) uniformly converges to a point in EG σ , (iv) the nite height property: for every pair of simplices σ ⊂ σ of Y , G σ has nite height in G σ , that is, there exist an upper bound on the number of distinct cosets

γ 1 G σ , . . . , γ n G σ ∈ G σ /G σ such that the intersection γ 1 G σ γ -1 1 ∩ . . . ∩ γ n G σ γ -1 n is innite.
Then G admits an EZ-structure (EG, ∂G) in the sense of FarrellLafont.

Furthermore, the following properties hold:

(ii ) For every simplex σ of Y , the map EG σ → EG realises an equivariant embedding from ∂G σ to ΛG σ ⊂ ∂G. Moreover, for every pair H 1 , H 2 of subgroups in the family F =

n i=1 g i G σ i g -1 i | g 1 , . . . , g n ∈ G, σ 1 , . . . , σ n ∈ S(Y ), n ∈ N , we have ΛH 1 ∩ ΛH 2 = Λ(H 1 ∩ H 2 ) ⊂ ∂G.
(iii ) For every simplex σ of Y , the embedding EG σ → EG satises the convergence property.

(iv ) For every simplex σ of Y , the local group G σ has nite height in G.

As an application of the previous construction, we prove a higher dimensional combination theorem for hyperbolic groups, in the case of acylindrical complexes of groups of arbitrary dimension.

Combination Theorem for hyperbolic groups. Let G(Y) be a strictly developable nonpositively curved simple complex of groups over a nite simplicial complex Y endowed with a M κ -structure, κ ≤ 0. Let G be the fundamental group of G(Y) and X be a universal covering of G(Y). Assume that:

• The universal covering X is hyperbolic 3 ,

• The local groups are hyperbolic and all the local maps are quasiconvex embeddings,

• The action of G on X is acylindrical.

Then G is hyperbolic. Furthermore, the local groups embed in G as quasiconvex subgroups.

Note that a complex of groups over a simply connected simplicial complex is developable if and only if it is strictly developable. Hence one might try to create new hyperbolic groups as fundamental groups of non-positively curved complexes of hyperbolic groups over a simply-connected nite complex (see BridsonHaeiger [9, Theorem II.12.28]). 3 For instance, when κ < 0.

Such a result is already known for acylindrical graphs of groups: the hyperbolicity is a direct consequence of the much more general combination theorem of Bestvina and Feighn [START_REF] Bestvina | A combination theorem for negatively curved groups[END_REF], while the quasiconvexity of vertex stabilisers follows from a result of Kapovich [START_REF] Kapovich | The combination theorem and quasiconvexity[END_REF]. Mj and Sadar [START_REF] Mj | A Combination Theorem for Metric Bundles[END_REF] have, using a dierent approach, a combination theorem that deals with the case where all the local groups are the same.

Our construction follows the strategy of Dahmani [START_REF] Dahmani | Combination of convergence groups[END_REF], who applied this idea to amalgamate Bowditch boundaries of relatively hyperbolic groups in the case of acylindrical graphs of groups. The proofs in our case are signicantly more involved as the topology of X can be much more complicated than that of a tree. Generalising an argument of Dahmani, we prove that G is a uniform convergence group on ∂G (see Section 6 for denitions), which implies the hyperbolicity of G by a celebrated result of Bowditch [START_REF] Bowditch | A topological characterisation of hyperbolic groups[END_REF] and Tukia [START_REF] Tukia | Conical limit points and uniform convergence groups[END_REF].

The article is organised as follows. In Section 1, we study complexes of spaces over a simplicial complex. These spaces are direct generalisations of graphs of spaces studied by Scott and Wall in the context of BassSerre theory [START_REF] Scott | Topological methods in group theory[END_REF]. In Section 2, we give conditions under which it is possible to build a classifying space for proper actions of the fundamental group of a complex of groups as a complex of spaces over its universal covering. We also dene the boundary ∂G of G and the compactication EG of EG as sets. In Section 3, we investigate geometric properties of geodesic in CAT(0) M κ -complexes. In Section 4, we study the geometry of some subcomplexes of X, called domains, which are used to dene ∂G. Section 5 is devoted to the proof of some geometric results that are used throughout the paper. We dene a topology on EG in Section 6 and we prove that it makes EG a compact metrisable space in Section 7. The proof of the Combination Theorem for boundaries of groups is completed in Section 8, where the properties of ∂G are investigated. Finally, Section 9 is devoted to the dynamics of G on its boundary and to the proof of the Combination Theorem for hyperbolic groups.

Notation. Throughout this paper, X is a simplicial complex. For a point x of X, we denote by σ x the unique simplex containing x in its interior. For a simplex σ of X, we denote by st(σ) the open star of σ, that is, the reunion of the open simplices whose closure contain σ. We also denote st(σ) its closed star, that is, the reunion of the closed simplices containing σ. We denote by S(X) the set of simplices of X, and by V (X) the set of its vertices. For a simplex σ of X and a constant r > 0, we denote by B(σ, r) (resp. B(σ, r)) the open (resp. closed) metric r-ball around σ.

All the types of classifying spaces we will consider in this paper are classifying spaces for proper actions (see Section 2 for denitions). Consequently, we will simply speak of classifying spaces rather than classifying spaces for proper actions. Moreover, although the notation EG is well spread in the literature to mean a classifying space for proper actions of a discrete group G, we will simply use the notation EG so as to avoid the somehow unaesthetic notation EG when speaking of an EZ-compactication of a classifying space for proper actions of G.

Acknowledgements. I am grateful to Thomas Delzant for proposing this problem to me, as well as for his help and advice during this work. I would also like to thank Frédéric Haglund for many helpful conversations. Finally, I would like to thank the referee for his/her very careful reading of this article, as well as for many useful remarks and corrections.

1

Complexes of spaces and their topology.

In this section, we study a class of spaces with a projection to a given simplicial complex X, called complexes of spaces over X, which are high-dimensional analogues of graphs of spaces studied in the context of actions on trees (see ScottWall [START_REF] Scott | Topological methods in group theory[END_REF]). This notion of complexes of spaces is close to the one studied by Corson [START_REF] Corson | Complexes of groups[END_REF] and Haeiger [START_REF] Haeiger | Extension of complexes of groups[END_REF].

1.1 A few geometric facts about M κ -simplical complexes.

Since the present article deals with nonproper actions of a group, the simplicial complex on which it acts is generally non locally nite. In [START_REF] Bridson | Geodesics and curvature in metric simplicial complexes[END_REF], Bridson dened a class of spaces that is suitable for a geometric approach.

Denition 1.1 (M κ -simplicial complexes with κ ≤ 0 [START_REF] Bridson | Geodesics and curvature in metric simplicial complexes[END_REF]). Let κ ≤ 0. A simplicial complex X is called a M κ -simplicial complex if it satises the following two conditions:

• Each simplex of X is modeled after a geodesic simplex in some M n κ , where M n κ is the simply-connected n-Riemannian manifold of constant curvature κ.

• If σ and σ are two simplices of X sharing a common face τ , the identity map from τ ⊂ σ to τ ⊂ σ is an isometry.

To such a M κ -complex X is associated a canonical simplicial metric.

Theorem 1.2 (Bridson [8]). If X is a M κ -simplicial complex, κ ≤ 0, with nitely many isometry types of simplices, the simplicial metric is complete and geodesic.

From now on, every simplicial complex will implicitely be given the structure of a M κ -complex, κ ≤ 0, with nitely many isometry types of simplices.

1.2 Complexes of spaces.

Denition 1.3. A complex of spaces C(X ) over X consists of the following data:

• for every simplex σ of X, a pointed CW-complex C σ , called a bre,

• for every pair of simplices σ ⊂ σ , an embedding φ σ ,σ : C σ → C σ , called a gluing map, such that for every σ ⊂ σ ⊂ σ , we have φ σ,σ = φ σ,σ • φ σ ,σ .

Denition 1.4 (realisation of a complex of spaces). Let C(X ) be a complex of spaces over X. The realisation of C(X ) is the quotient space

|C(X )| = σ∈S(X) σ × C σ / where (i σ,σ (x), s) (x, φ σ,σ (s)) for x ∈ σ ⊂ σ and s ∈ C σ , where i σ,σ : σ → σ is the natural inclusion. The class in |C(X )| of a point (x, s) will be denoted [x, s].
Denition 1.5. A complex of spaces C(X ) will be called locally nite if for every simplex σ of X and every point x ∈ C σ , there exists an open set U of C σ containing x and such that there are only nitely many simplices σ in the open star of σ satisfying U ∩ Im(φ σ,σ ) = ∅.

Proposition 1.6. Let C(X ) be a locally nite complex of spaces. Then |C(X )| admits a natural locally nite CW-complex structure, for which the σ×C σ embed as subcomplexes.

1.3 Topology of complexes of spaces with contractible bres.

Denition 1.7 (quotient complex of spaces). Let C(X ) be a complex of spaces over X and Y ⊂ X a subcomplex. We denote C Y (X ) the complex of spaces over X dened as follows: Lemma 1.8. Let C(X ) be a locally nite complex of spaces over X with contractible bres, and let Y be a nite subcomplex of X.

• (C Y ) σ = C σ if σ Y , (C Y ) σ is the basepoint of C σ otherwise, • φ Y σ,σ is the composition (C Y ) σ → C σ φ σ,σ ---→ C σ (C Y ) σ .
Then p Y : |C(X )| → |C Y (X )| is a homotopy equivalence.
Proof. It amounts to proving the result for Y consisting of a single closed simplex σ. We have the following commutative diagram:

|C(X )| p Y / / |C Y (X )| |C(X )|/ (σ × C σ ) = / / |C Y (X )|/ (σ × ) .
The vertical arrows are homotopy equivalences, since we are quotienting by contractible subcomplexes, hence the result.

Theorem 1.9 (Dowker [17]). The (continuous) identity map X → X from X with its CW topology to X with its simplicial metric is a homotopy equivalence.

Proposition 1.10. Let C(X ) be a locally nite complex of space over X with contractible bres. If X has nitely many types of simplices and is contractible, then |C(X )| is contractible.

Proof. By the previous theorem, it is enough to show that the projection p : |C(X )| → X induces isomorphisms on homotopy groups, when X is endowed with its CW topology. For that topology, a continuous map from a compact space to X has its image contained in a nite subcomplex, to which Lemma 1.8 applies.

2

Construction of EG and ∂G for developable complexes of groups.

In this section, given a developable simple complex of groups G(Y) over a nite simplicial complex Y , we build a classifying space for its fundamental group.

Notation: We choose once and for all a non-positively curved complex of groups G(Y) over a nite simplicial complex endowed with a M κ -structure, κ ≤ 0 (where Y is the small category without loops whose vertices correspond to simplices of Y and whose oriented edges come from inclusion of simplices of Y ). Recall that a complex of groups consists of the data (G σ , ψ a , g a,b ) of local groups (G σ ), local maps (ψ a ) and twisting elements (g a,b ).

For the background on complexes of groups, we refer the reader to BridsonHaeiger [START_REF] Bridson | Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften[END_REF].

We x a maximal tree T in the 1-skeleton of the rst barycentric subdivision of Y , which allows us to dene the fundamental group G = π 1 (G(Y), T ) and the canonical morphism ι T : G(Y) → G given by the collection of injections G σ → G [9, p.553]. Finally, we dene X as the universal covering of G(Y) associated to ι T . The simplicial complex X naturally inherits a M κ -structure with nitely many isometry types of simplices from that of Y and the simplicial metric d on X makes it a complete geodesic metric space by work of Bridson [8]. This space is CAT(0) by the curvature assumption on G(Y) [9, p.562].

Construction of EG and

∂ Stab G.
Denition 2.1 ((conite and nite dimensional) classifying space for proper actions). Let Γ be a countable discrete group. A conite and nite dimensional classifying space for proper actions of Γ (or briey a classifying space for Γ) is a contractible CW-complex EΓ with a proper cocompact and cellular action of Γ, and such that:

• for every nite subgroup H of Γ, the xed point set EΓ H is nonempty and contractible,

• every innite subgroup H of Γ has an empty xed point set.

Denition 2.2. A complex of spaces D(Y) compatible with the complex of groups G(Y) consists of the following:

• For every vertex σ of Y, a space D σ that is a model of classifying space for proper actions EG σ of the local group G σ ,

• For every edge a of Y with initial vertex i(a) and terminal vertex t(a), an embedding φ a : EG i(a) → EG t(a) which is G i(a) -equivariant, that is, for every g ∈ G i(a) and every x ∈ EG i(a) , we have φ a (g.x) = ψ a (g).φ a (x),

and such that for every pair (a, b) of composable edges of Y, we have:

g a,b • φ ab = φ a φ b ,
Note that a complex of spaces compatible with the complex of groups G(Y) is not a complex of spaces over Y when the twisting elements g a,b are not trivial. Nonetheless, this data is used to build a complex of spaces over X, as explained in the following denition. Denition 2.3. We dene the space

Cl D(Y) = G × σ∈V (Y) (σ × EG σ ) / where (g, i σ,σ (x), s) gι T [σσ ] -1 , x, φ (σσ ) (s) if [σσ ] ∈ Edges(Y ), s ∈ EG σ , x ∈ σ , g ∈ G, (gg , x, s) (g, x, g s) if x ∈ σ, s ∈ EG σ , g ∈ G σ , g ∈ G. The canonical projection G × σ∈V (Y) (σ × EG σ ) → G × σ∈V (Y) σ yields a map p : Cl D(Y) → X.
The action of G on G × σ∈V (Y) (σ × EG σ ) on the rst factor by left multiplication yields an action of G on Cl D(Y) , making the projection p :

Cl D(Y) → X a G-equivariant map.
Note that Cl D(Y) can be seen as a complex of spaces over X, the bre of a simplex [g, σ] being the classifying space EG σ . Indeed, for en edge [g, a] of the rst barycentric subdivision of X, the gluing map φ [gι T (a) -1 ,i(a)],[g,t(a)] : EG i(a) → EG t(a) is dened as φ i(a),t(a) .

For every simplex σ of X, we denote by EG σ the bre over σ of that complex of space. For simplices σ, σ of X such that σ ⊂ σ, we denote by φ σ ,σ : EG σ → EG σ the associated gluing map.

Theorem 2.4. The space Cl D(Y) is a classifying space for proper actions of G.

From now on, we denote by EG this classifying space.

Proof. Local niteness: Let σ be a simplex of X and U be an open set of EG σ that is relatively compact. It is enough to prove that for any injective sequence (σ n ) of simplices of X containing σ there are only nitely many n such that the image of φ σ,σn meets U . By cocompactness of the action, we can assume that all the σ n are in the same G-orbit, and let σ be a simplex in that orbit. Since the action of G σ on EG σ is proper, it follows that for every compact subset K of EG σ , only nitely many distinct cosets gEG σ in EG σ can meet K, hence the result. Proper action: As Cl D(Y) is a locally nite CW-complex, hence a locally compact space, it is enough to show that every nite subcomplex intersects only nitely many of its Gtranslates.

Let us rst show that for every cell τ of Cl D(Y) , there are only nitely many g ∈ G such that gτ = τ . Indeed, let g ∈ G such that gτ = τ . The canonical projection Cl D(Y) → X is G-equivariant and sends a cell of Cl D(Y) on a simplex of X, thus g also stabilises the simplex p(τ ) ⊂ X. Since G acts without inversion on X, g pointwise stabilises the vertices of p(τ ). Let s be such a vertex. Then g ∈ G s and, by construction of Cl D(Y) , the restriction to G s of the action of G on Cl D(Y) is just the action of G s on EG s . Thus, by denition of a classifying space for proper actions, this implies that there are only nitely many possibilites for g. Fixed sets: Let H be a nite subgroup of G. As G acts without inversion on the CAT(0) complex X, the subset X H is a nonempty convex subcomplex of X. Furthermore, for every simplex σ of X We now turn to the construction of a boundary of G. As introduced by Farrell Lafont [START_REF] Farrell | EZ-structures and topological applications[END_REF], the denition of an EZ-structure only applies to torsion-free groups. Here we use a notion of Z-structure suitable for groups with torsion, which was introduced by Dranishnikov [START_REF] Dranishnikov | On Bestvina-Mess formula[END_REF]. Denition 2.5 (Z -structures, EZ-structures). Let Γ be a discrete group. A Z-structure for Γ is a pair (Y, Z) of spaces such that:

• Y is a Euclidean retract, that is, a compact, contractible and locally contractible space with nite covering dimension,

• Y \ Z is a classifying space for proper actions of Γ,

• Z is a Z-set in Y , that is, Z is a closed subpace of Y such that for every open set U of Y , the inclusion U \ Z → U is a homotopy equivalence,
• Compact sets fade at innity, that is, for every compact set K of Y \ Z, every point z ∈ Z and every neighbourhood U of z in Z, there exists a subneighbourhood V ⊂ U with the property that if a Γ-translate of K intersects V , then it is contained in U .

The pair (Y, Z) is called an EZ-structure if in addition we have:

• The action of Γ on Y \ Z continuously extends to Y . Denition 2.6. We say that a complex of spaces D(Y) compatible with a complex of groups G(Y) extends to an EZ-complex of spaces if it satises the following extra conditions:

• Each bre D σ = EG σ is endowed with an EZ-structure (EG σ , ∂G σ ),

• The equivariant embeddings (φ a ) extend to equivariant embeddings φ a : EG i(a) → EG t(a) , such that for every pair (a, b) of composable edges of Y, we have:

g a,b • φ ab = φ a φ b .
Denition 2.7. We dene the space

Ω(Y) = G × σ∈V (Y) ({σ} × ∂G σ ) / where gg , (x, s) g, (x, g s) if x ∈ σ, s ∈ EG σ , g ∈ G σ , g ∈ G.
It should be noted here that {σ} denotes a point labeled by σ and not the simplex itself.

The set Ω(Y) comes with a natural projection to the set of simplices of X. If σ is a simplex of X, we denote by ∂G σ the preimage of {σ} under that projection. We now dene

∂ Stab G = Ω(Y)/ ∼
where ∼ is the equivalence relation generated by the following identications:

g, {σ} , ξ ∼ gF ([σσ ]) -1 , {σ }, φ [σσ ] (ξ) if g ∈ G, [σσ ] ∈ Edges(Y ) and ξ ∈ ∂G σ .
The action of G on G × σ∈V (Y) ({σ} × ∂G σ ) by left multiplication on the rst factor yields an action of G on Ω(Y) and on ∂ Stab G.

Denition 2.8. We dene the spaces ∂G = ∂ Stab G ∪ ∂X and EG = EG ∪ ∂G.

Our aim is to endow EG with a topology that makes (EG, ∂G) an EZ-structure for G.

Notation: Since the φ σ,σ are embeddings, we will identify φ σ,σ (EG σ ) with EG σ . For instance, if U is an open subset of EG σ we will simply write we have EG σ ⊂ U in EG σ instead of we have φ σ,σ (EG σ ) ⊂ U in EG σ .

From now on, we assume that there is a complex of spaces D(Y) that extends to an EZ-complex of spaces compatible with the complex of groups G(Y).

2.2 Further properties of EZ-complexes of spaces.

In this paragraph, we dene additional properties of EZ-complexes of spaces, which will enable us to study the properties of the equivalence relation ∼ previously dened.

2.2.1 The limit set property.

Recall that for a discrete group Γ together with an EZ-structure (EΓ, ∂Γ) and a subgroup H, the limit set ΛH of H in ∂Γ is the set Hx ∩ ∂Γ, where x is an arbitrary point of EΓ.

Denition 2.9 (Limit set property for an EZ-complex of spaces). We say that the EZcomplex of spaces D(Y) compatible with the complex of groups G(Y) saties the limit set property if the following conditions are satised:

• For every pair of simplices σ ⊂ σ of Y , the map φ σ,σ is an equivariant homeomorphism from ∂G σ to the limit set ΛG σ ⊂ ∂G σ .

• For every simplex σ of Y , and every pair of subgroups H 1 , H 2 in the family

F σ = n i=1 g i G σ i g -1 i | g 1 , . . . , g n ∈ G σ , σ 1 , . . . , σ n ⊂ st(σ), n ∈ N , we have ΛH 1 ∩ ΛH 2 = Λ(H 1 ∩ H 2 ).
Remark. (i) Let Γ be a hyperbolic group, and H a subgroup. Then H is quasiconvex if and only if its limit set in ∂Γ is equivariantly homeomorphic to ∂H, by a result of Bowditch [START_REF] Bowditch | Convergence groups and conguration spaces[END_REF].

(ii) Let Γ be a hyperbolic group and ∂Γ its Gromov boundary. Let H 1 and H 2 be two quasiconvex subgroups of Γ. Then ΛH 1 ∩ ΛH 2 = Λ(H 1 ∩ H 2 ) by a result of Gromov [START_REF] Gromov | Asymptotic invariants of innite groups[END_REF].

The nite height property.

Recall that, for Γ a discrete group and H a subgroup, the height of H is the supremum of the set of integers n ∈ N such that there exist distinct cosets γ 1 H, . . . , γ n H ∈ G/H such that the intersection γ 1 Hγ -1 1 ∩ . . . ∩ γ n Hγ -1 n is innite. If such a supremum is innite, we say that H is of innite height in Γ. Otherwise, H is said to be of nite height in Γ. A quasiconvex subgroup of a hyperbolic group is of nite height, by a result of GitikMitra RipsSageev [START_REF] Gitik | Widths of subgroups[END_REF].

Denition 2.10 (nite height property). We say that the EZ-complex of spaces D(Y)

compatible with the complex of groups G(Y) saties the nite height property if for every pair of simplices σ ⊂ σ of Y , G σ is of nite height in G σ .

3

Geodesics in M κ -complexes.

In this section, we study the geometry of the set of geodesics of an M κ -complex. Recall that X is assumed to be a M κ -complex, κ ≤ 0, with nitely many isometry types of simplices.

3.1 The niteness lemma.

Denition 3.1. For subsets K, L of X, we dene Geod(K, L) as the set of points lying on a geodesic segment from a point of K to a point of L.

Denition 3.2 (Simplicial neighbourhood). Let K be a subcomplex of X. The subcomplex spanned by the closed simplices that meet K is called the closed simplicial neighbourhood of K, and denoted N(K ). The union of the open simplices whose closure meets K is called the open simplicial neighbourhood of K, and denoted N(K ).

We recall the following proposition of Bridson, which follows from the Claim contained in the proof of Theorem 1.11 of [START_REF] Bridson | Geodesics and curvature in metric simplicial complexes[END_REF].

Proposition 3.3 (containment lemma, Bridson [START_REF] Bridson | Geodesics and curvature in metric simplicial complexes[END_REF]). For every n there exists a constant k such that for every nite subcomplex K ⊂ X spanned by at most n simplices, any geodesic path contained in the open simplicial neighbourhood of K meets at most k simplices.

We also recall this useful related result, which follows from Theorem 1.11 of [START_REF] Bridson | Geodesics and curvature in metric simplicial complexes[END_REF].

Corollary 3.4 (Bridson [8]). For every n there exists a constant k such that every geodesic segment of length at most n meets at most k simplices.

Lemma 3.5 (Finiteness lemma). Let X be as before. For subcomplexes K, K ⊂ X, Geod(K, K ) meets only nitely many open simplices.

Proof. It is enough to prove the result when K and K consist of two closed simplices σ and σ . For every x ∈ σ and every x ∈ σ , we consider the sequence of open simplices σ 1 , . . . , σ n met by the geodesic segment [x, x ] and set C x,x = σ ∪ σ 1 ∪ . . . ∪ σ n ∪ σ . Note that by Corollary 3.4 there is a uniform k bound on the number of simplices contained in C x,x . Since there is only nitely many isometry types of simplices in X, there is, up to simplicial isometry xing pointwise σ and σ , nitely many subcomplexes of the form C x,x .

Following Bridson, we call such an equivalence class of subcomplexes a model (see the proof of I.7.57 in BridsonHaeiger [START_REF] Bridson | Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften[END_REF]).

We now claim that for every x, y ∈ σ and every x , y ∈ σ such that C x,x and C y,y are in the same model, we have C x,x = C y,y . Indeed, choose a simplicial isometry φ : C x,x → C y,y that xes pointwise σ and σ . Then φ sends the geodesic segment [x, x ] ⊂ C x,x to a simplicial path of the same length between φ(x) = x and φ(x ) = x . As X is CAT(0), geodesic segments are unique, hence φ pointwise xes [x, x ]. We thus have [x, x ] = φ [x, x ] ⊂ C y,y , hence C x,x ⊂ C y,y . The same reasoning applied to the geodesic segment [y, y ] yields C y,y ⊂ C x,x , hence C x,x = C y,y .

We have

Geod(σ, σ ) ⊂ x∈σ,x ∈σ C x,x
and the previous discussion shows that this is a nite union, which concludes the proof.

3.2 Paths of simplices of uniformly bounded length. Denition 3.6. A path of simplices is a sequence of open simplices σ 1 , . . . , σ n such that σ i ⊂ σ i+1 or σ i+1 ⊂ σ i for every i = 1, . . . , n -1. Equivalently, it is a nite path in the rst barycentric subdivion of X. The integer n is called the length of the path of simplices.

Up to rescaling the metric, we also make the following assumption:

From now on, we will assume that the distance from any simplex to the boundary of its (closed) simplicial neighbourhood is at least 1.

Here we prove the following lemma: Lemma 3.7 (Short paths of simplices). For every n ≥ 1, there exists m ≥ 1 such that the following holds: Let K be a convex subcomplex of X and K a connected subcomplex of X, both containing at most n simplices. Let x, y ∈ K and x , y ∈ K and assume that there exists a path in K between x and y that does not meet K. Let τ, τ be two simplices of N(K ) \ K such that the geodesic segment [x, x ] (resp. [y, y ]) meets the interior of τ (resp. τ ). Then there exists a path of simplices in N(K ) \ K of length at most m between τ and τ . Denition 3.8 (I.7.8 of BridsonHaeiger [START_REF] Bridson | Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften[END_REF]). For x ∈ X, let

η(x) = inf{d(x, σ)| σ ⊂ st(σ x ), x / ∈ σ}.
The constant is such that for every y ∈ B(x, η(x)), we have σ x ⊂ σ y .

The following lemma is a controlled version of Lemma I.7.54 in [START_REF] Bridson | Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften[END_REF].

Lemma 3.9. There exist constants η 0 > ε 0 > 0 such that:

• for every simplex σ of X, the 2η 0 -neighbourhood of σ is contained in the open simplicial neighbourhood of σ;

• for every point x ∈ X, there exists y ∈ B(x, η 0 ) such that B(x, ε 0 ) ⊂ B(y, η(y)).

Proof. For a simplex σ of X, let

η(σ) = inf{d(σ, τ )| τ ⊂ N(σ), σ ∩ τ = ∅}.
The above set of distances is nite since X has only nitely many isometry types of simplices, thus η(σ) > 0. For the same reason, we can dene η 0 = 1/2 • min η(σ) > 0, where the minimum is taken over all the simplices of X. Now that η 0 is dened, we construct constants η 1 , . . . , η D by induction, where D is the maximal dimension of a simplex of X, as well as subets T 0 , . . . ,

T D of X, such that each T k is an open neighbourhood of the k-skeleton X (k) of X. Let T 0 = v∈V (X) B(v, η 0 ),
where η 0 is as above. Suppose that η 0 , . . . , η k and T 0 , . . . , T k are dened. For each simplex σ ⊂ X of dimension k + 1, the function η (as dened in 3.8) is continuous on the compact set σ \ T k and does not vanish, hence is bounded below by a constant η k+1 (σ) > 0. As X has nitely many isometry types of simplices, we dene η k+1 = 1/2 • min η k+1 (σ) > 0, where the minimum is taken over all simplices of dimension k + 1. We can further assume that η k+1 ≤ η k . Let

T k+1 = T k ∪ σ⊂X, dim σ=k+1 x∈σ\T k B(x, η k+1 ) . Finally, let ε 0 = η D . We have T 0 ⊂ . . . ⊂ T D = X. Let x ∈ X. There exists a unique k such that x ∈ T k \ T k-1 . For such a k, there exists y ∈ X (k) \ T k-1 with d(x, y) ≤ η k (in particular d(x, y) ≤ η 0 ). As ε 0 ≤ η k , we get B(x, ε 0 ) ⊂ B(x, η k ) ⊂ B(y, 2η k ) ⊂ B(y, η k (σ y )) ⊂ B(y, η(y)),
which concludes the proof.

Proof of Lemma 3.7. First notice that since X has only nitely many isometry types of simplices, there exists a constant l, which depends only on n and X, points x = x 0 , . . . , x l = y in K and x = x 0 , . . . , x l = y in K such that for every k, d(x k , x k+1 ) < ε 0 , d(x k , x k+1 ) < ε 0 , x k , x k+1 belong to the same simplex of K, and x k , x k+1 belong to the same simplex of K . For every k

= 1 . . . , l -1, let τ k be a simplex of N(K ) \ K whose interior meets [x k , x k ].
In order to prove Lemma 3.7, it is thus enough to consider the case where d(x, y) < ε 0 , d(x , y ) < ε 0 , x, y belong to the same simplex σ of K, and x , y belong to the same simplex σ of K . We treat separately two cases.

Case 1: Suppose that the geodesic segments [x, x ] and [y, y ] are both contained in the open η 0 -neighbourhood of K. Recall that by denition of η 0 , this implies that they are contained in the open simplicial neighbourhood of K. The geodesic segment [x, x ] yields a geodesic segment, contained in N(K ) \ K by convexity of K, between a point in the interior of τ and x . By Proposition 3.3, there exists a constant m 1 (which depends only on X and n) such that there exists a path of simplices in N(K ) \ K of length at most m 1 between τ and σ . Reasoning similarly for [y, y ], we get a path of simplices in N(K ) \ K of length at most m 1 between τ and σ . We thus get a path of simplices in N(K ) \ K of length at most 2m 1 between τ and τ .

Case 2: Suppose that the geodesic segment [x, x ] is not contained in the η 0 -neighbourhood of K. We then choose a point u on that geodesic segment which belongs to B(K,

2η 0 ) \ B(K, η 0 ) (such a subset is contained in N(K ) by denition of η 0 ). By Lemma 3.9, we can choose z ∈ X \ K such that B(u, ε 0 ) ⊂ B(z, η(z)). Since d(x, y) < ε 0 and d(x , y ) < ε 0 , the CAT(0) geometry of X implies that [y, y ] meets the ball B(u, ε 0 ) ⊂ B(z, η(z)) at a point v.
By denition of η(z), we thus have σ z ⊂ σ u and σ z ⊂ σ v , which yields the path of simplices σ u , σ z , σ v in N(K ) \ K between σ u and σ v . Now the geodesic segment [x, x ] (resp. [y, y ]) yields a path of simplices in N(K ) \ K (by convexity of K) of length at most m 1 between τ and σ u (resp. between τ and σ v ). We thus get a path of simplices in N(K ) \ K of length at most 2m 1 + 1 between τ and τ .

4

The geometry of the action.

In this section, we gather a few geometric tools that will be used to construct a topology on EG = EG∪∂G. From now on, we assume that the EZ-complex of spaces D(Y) compatible with G(Y) satises the limit set property 2.9 and the nite height property 2.10. We further assume that the action of G on X is acylindrical and we x an acylindricity constant A > 0, that is, a constant such that every subcomplex of X of diameter at least A has a nite pointwise stabiliser.

Domains and their geometry.

In this section, we study the topological properties of the identications made to build the boundary of G. Recall that Ω(Y) is dened in 2.7 as the disjoint union of the ∂G v 's (v ∈ V (X)) and that ∂ Stab G is a quotient of Ω(Y) dened by making identications along edges of X. We start by proving the following proposition: Proposition 4.4. Let v be a vertex of X. Then the projection π :

∂G v → ∂G is injective. Denition 4.5. Let ξ ∈ ∂ Stab G. A ξ-path is the data {(v i ) 0≤i≤n , (ξ i ) 0≤i≤n , (x i ) 1≤i≤n } of: • a sequence v 0 , . . . , v n of adjacent vertices of X,
• a sequence ξ 0 , . . . , ξ n of elements of Ω(Y), such that ξ i ∈ ∂G v i for every i, and such that each ξ i is in the equivalence class ξ,

• a sequence x 1 , . . . , x n of elements of Ω(Y), such that x i ∈ ∂G [v i-1 ,v i ] for every i, and such that φ v i-1 ,[v i-1 ,v i ] (x i ) = ξ i-1 (resp φ v i ,[v i-1 ,v i ] (x i ) = ξ i ).
To lighten notations, a ξ-path will sometimes just be denoted [v 0 , . . . , v n ] ξ . The path in the 1-skeleton of X induced by a ξ-path is called the support of [v 0 , . . . , v n ] ξ , and denoted [v 0 , . . . , v n ]. If v 0 = v n , a ξ-path will rather be called a ξ-loop.

Lemma 4.6. Let v 0 , . . . , v n be vertices of X, H = ∩ 0≤i≤n G v i , and K be a connected subcomplex of X pointwise xed by H. Suppose that H is innite, and

let ξ ∈ ∂ Stab G such that, in G v 0 , we have ξ ∈ ΛH ⊂ ∂G v 0 . Then ξ ∈ ΛH ⊂ ∂G σ for every simplex σ of K, hence K ⊂ D(ξ).
Proof. As K is connected, it is enough to prove that for every path of simplices σ 0 = v 0 , . . . , σ d contained in K, we have ξ ∈ ∂H ⊂ ∂G σ d . Now this follows from an easy induction on the number of simplices contained in such a path.

Lemma 4.7. Let ξ ∈ ∂ Stab G, [v 0 , . . . , v n ] ξ a ξ-path and H = ∩ 0≤i≤n G v i . Then • H is innite,
• ξ ∈ ΛH ⊂ ∂G v i for every i = 0, . . . , n.

Proof. We show the result by induction on n ≥ 1. The result is immediate for n = 1 by denition of ∼. Suppose the result true up to rank n and let ξ ∈

∂ Stab G together with a ξ-path [v 0 , . . . , v n+1 ] ξ . By restriction, we get a ξ-path [v 0 . . . , v n ] ξ for which the result is true by the induction hypothesis. Thus ξ ∈ Λ(∩ 0≤i≤n G v i ) ⊂ ∂G vn . But since ξ is also in ∂G [vn,v n+1 ] = ΛG [vn,v n+1 ] by assumption, we get ξ ∈ Λ 0≤i≤n G v i ∩ ΛG [vn,v n+1 ] = Λ 0≤i≤n+1 G v i ⊂ ∂G vn ,
the previous equality following from the limit set property 2.9. Now, by Lemma 4.6, we get

ξ ∈ Λ(∩ 0≤i≤n+1 G v i ) ⊂ ∂G v i for every i = 0, . . . , n + 1, which concludes the induction.
Proof of Proposition 4.4. Let ξ, ξ be two elements of Ω(Y) in the image of ∂G v , that are equivalent for the equivalence relation ∼. Then there exists a ξ-loop (H d ): For every ξ ∈ ∂ Stab G and every ξ-loop {(v i ) 0≤i≤n , (ξ i ) 0≤i≤n , (x i ) 1≤i≤n } admitting a hull containing at most d triangles, we have ξ 0 = ξ n .

{(v i ) 0≤i≤n , (ξ i ) 0≤i≤n , (x i ) 1≤i≤n } with ξ 0 = ξ, ξ n = ξ . It is enough to prove the result when the support [v 0 , . . . , v n ] of that ξ-loop is injective. Let Y be
If d = 1, then n = 2, and the hull considered is just a single triangle σ. Since H ⊂ G σ because H stabilises σ pointwise, we can choose x ∈ ∂G σ such that φ v 1 ,σ (x) = ξ 1 . From the commutativity of the diagram of embeddings for a simplex, it follows that φ

[v 0 ,v 1 ],σ (x) = x 1 and φ [v 1 ,v 2 ],σ (x) = x 2 . Hence ξ 0 = φ v 0 ,[v 0 ,v 1 ] (x 1 ) = φ v 0 ,σ (x) = φ v 0 ,[v 2 ,v 0 ] (x 2 ) = ξ 2 .
Suppose the result true up to rank d, and let ξ ∈ ∂ Stab G, together with a ξ-loop

{(v i ) 0≤i≤n , (ξ i ) 0≤i≤n , (x i ) 1≤i≤n } admitting a hull F containing at most d + 1 triangles. Choose any triangle σ of F containing the segment [v 1 , v 2 ]. As σ is stabilised by H, we can nd x ∈ ∂G σ such that φ v 1 ,σ (x) = ξ 1 .
There are now two possible cases, depending of the nature of σ:

• If another side of σ is contained in the support of the ξ-loop, for example [v 2 , v 3 ], we set x = φ [v 1 ,v 3 ],σ (x).
Now the commutativity of the diagram of embeddings for σ yields the following new ξ-loop

{(v 0 , v 1 , v 3 , v 4 , . . . , v n ), (ξ 0 , ξ 1 , ξ 3 , . . . , ξ n ), (x 1 , x , x 4 , . . . , x n )}.
A hull for that new loop is given by the closure of F \ σ, thus containing at most d triangles, and we are done by induction.

• If no other side of σ is contained in the support of the ξ-loop, we set a to be the remaining vertex of σ, α = φ a,σ (x),

x 2 = φ [v 1 ,a],σ (x) and x 2 = φ [a,v 2 ],σ (x).
The commutativity of the diagram of embeddings for σ yields the following new ξloop:

{(v 0 , v 1 , a, v 2 , . . . , v n ), (ξ 0 , ξ 1 , α, ξ 2 , . . . , ξ n ), (x 1 , x 2 , x 2 , x 3 , . . . , x n )}.
A hull for that new loop is given by the closure of F \ σ, thus containing at most d triangles, and we are done by induction.

Proof of Proposition 4.2. Convexity : Let x, x be two points of D(ξ). Let v (resp. v ) be a vertex of σ x (resp. σ x ). We can thus nd a ξ-path Finiteness : Let σ be a simplex of D(ξ) and σ 1 , σ 2 , . . . be a (possibly empty) sequence of simplices containing strictly σ and contained in D(ξ). It follows from the proof of Proposition 4.4 that ξ ∈ ∂G σ i ⊂ ∂G σ for every i. Thus, the limit set property 2.9, the nite height property 2.10 and the cocompactness of the action imply that there can be only nitely many such simplices. Thus D(ξ) locally nite. To prove that it is also bounded, consider x, x two points of D(ξ). By Lemma 4.7 the stabiliser of {x, x } is innite. Thus the domain of ξ has a diameter bounded above by the acylindricity constant. The complex D(ξ) is locally nite and bounded, hence nite. Moreover, it is clear from the above argument that the bound can be chosen uniform on ξ.

{(v i ) 0≤i≤n , (ξ i ) 0≤i≤n , (x i ) 1≤i≤n } with v 0 = v and v n = v . As ξ ∈ ∂G σx and ξ ∈ ∂G σ x ,

Nestings and Families.

Denition 4.8 (the convergence property). We say that an EZ-complex of spaces compatible with G(Y) satises the convergence property if, for every pair of simplices σ ⊂ σ in Y and every injective sequence (g n G σ ) of cosets of G σ /G σ , there exists a subsequence such that (g ϕ(n) EG σ ) uniformly converges to a point in EG σ .

From now on, besides the limit set property 2.9, the nite height property 2.10 and the acylindricity assumption, we assume that the EZ-complex of spaces D(Y) satises the convergence property 4.8. Denition 4.9. Let ξ ∈ ∂ Stab G, v a vertex of D(ξ), and U a neighbourhood of ξ in EG v .

We say that a subneighbourhood V ⊂ U containing ξ is nested in U if its closure is contained in U and for every simplex σ of st(v) not contained in D(ξ), we have Proof. We show this by contradiction. Consider a countable basis (V n ) n of neighbourhoods of ξ in EG v , and suppose that for every n, there exists a simplex σ n ∈ st(v) \D(ξ) such that EG σn ∩V n = ∅ and EG σn U . Up to a subsequence, we can assume that (σ n ) n is injective. By cocompactness of the action, we can also assume that all the σ n cover a unique simplex σ of Y . Now the convergence property 4.8 implies that there should exist a subsequence σ λ(n) such that EG σ λ(n) uniformly converges to a point in EG v , a contradiction. Since, in ∂G, boundaries of stabilisers of vertices are glued together along boundaries of stabilisers of edges, we will construct neighbourhoods in EG of a point ξ ∈ ∂ Stab G using neighbourhoods of the representatives of ξ in the various EG v , where v runs over the vertices of the domain of ξ.

EG σ ∩ V = ∅ ⇒ EG σ ⊂ U.

Denition 4.11 (ξ-family

). Let ξ ∈ ∂ Stab G. A collection U of open sets {U v , v ∈ V (ξ)}
is called a ξ-family if for every pair of vertices v, v of X that are joined by an edge e and every x ∈ EG e , φ v,e (x)

∈ U v ⇔ φ v ,e (x) ∈ U v .
Proposition 4.12. Let ξ ∈ ∂ Stab G. For every vertex v of D(ξ), let U v be a neighbourhood of ξ in EG v . Then there exists a ξ-family U such that U v ⊂ U v for every vertex v of D(ξ).

Proof. For every simplex σ of D(ξ), we construct open sets U σ by induction on dim(σ), starting with simplices of maximal dimension, that we denote d.

If dim(σ) = d, we set U σ = v∈σ φ -1 v,σ (U v ).
Assume the U σ constructed for simplices of dimension at least k ≤ d, and let σ 0 be of dimension k -1. If no simplex of dimension ≥ k contains σ 0 , set

U σ 0 = v∈σ φ -1 v,σ 0 (U v ).
Otherwise, since the φ σ,σ are embeddings,

σ 0 ⊂σ⊂D(ξ) dim(σ)=k φ σ 0 ,σ (U σ ) is open in dim(σ)=k σ 0 ⊂σ⊂D(ξ) φ σ 0 ,σ (EG σ ).
We

U if for every vertex v of D(ξ), U v is nested in U v . Furthermore we say that U is n-nested in U if there exist ξ-families U = U [0] ⊂ . . . ⊂ U [n] = U with U [i] nested in U [i+1]
for every i = 0, . . . , n -1.

A geometric toolbox.

We now prove some results which will be our main tools in studying EG and ∂G. Since the proofs in this section rely heavily on the geometry of X, we start with a few denitions.

Denition 5.1. Let ξ ∈ ∂ Stab G, x ∈ X, η ∈ ∂X and ε ∈ (0, 1).

Let d be the simplicial metric on X, and choose a basepoint v 0 ∈ X. We denote by [v 0 , x] the unique geodesic segment from v 0 to x, and by γ x : [0, d(v 0 , x)] → X its parametrisation. We denote by [v 0 , η) the unique geodesic ray from v 0 to η, and by γ η : [0, ∞) → X its parametrisation.

We denote by D ε (ξ) the open ε-neighbourhood of D(ξ). We say that a geodesic in X parametrised by γ goes through (resp. enters ) D ε (ξ) if there exist t 0 such that γ(t 0 ) ∈ D ε (ξ) and t 1 > t 0 such that γ(t 1 ) / ∈ D ε (ξ) (resp. if there exists t 0 such that γ(t 0 ) ∈ D ε (ξ)) .

If the geodesic [v 0 , x] goes through D ε (ξ), we dene an exit simplex σ ξ,ε (x) as the rst simplex touched by

[v 0 , x] after leaving D ε (ξ). If x ∈ D ε (ξ), we set σ ξ,ε (x) = σ x .
Note that, by the assumption on the distance from a simplex to the boundary of its closed simplicial neighbourhood, we always have

D ε (ξ) ⊂ N(D(ξ)). Denition 5.2. Let ξ ∈ ∂ Stab G , U a ξ-family and ε ∈ (0, 1). We dene Cone U ,ε (ξ) (resp.
Cone U ,ε (ξ)) as the set of points x of X \ D(ξ) such that the geodesic [v 0 , x] goes through (resp. enters) D ε (ξ) and such that for some vertex v of D(ξ) (hence for every by Denition 4.11) contained in the exit simplex σ ξ,ε (x), we have, in EG v : a path of open simplices contained in N D(ξ) \ D(ξ). Suppose that U is n-nested in U (Denition 4.13), and that σ 1 ⊂ N U D(ξ) . Then for every k ∈ {1, . . . , n} and every vertex

EG σ ξ,ε (x) ⊂ U v .
v of D(ξ) contained in σ k , we have EG σ k ⊂ U v in EG v .
Suppose now that σ n ⊃ σ n+1 , and let v be a vertex of D(ξ

) contained in σ n+1 . Since v is also in σ n , EG σn ⊂ U [n] v d in EG σn , so we have EG σ n+1 ∩ U [n] vn = ∅, which in turn implies EG σ n+1 ⊂ U [n+1] v since U [n] is nested in U [n+1]
. Now by the denition of ξ-families 4.11, the same result holds for every vertex v of D(ξ) contained in σ n+1 . Let K be a convex subcomplex of X and K a connected subcomplex of X, both containing at most max(n, d max ) simplices. Let x, y ∈ K and x , y ∈ K and assume that there exists a path in K between x and y that does not meet K. Let τ, τ be two simplices of N(K ) \ K such that the geodesic segment [x, x ] (resp. [y, y ]) meets the interior of τ (resp. τ ). Then there exists a path of simplices in N(K ) \ K of length at most m between τ and τ .

Let ξ ∈ ∂ Stab G, U a ξ-family. A ξ-family that is m-nested in U is said to be n-rened in U. For n the number of simplices of D(ξ), we denote by m ξ such a choice of m. Lemma 5.6. Let ξ ∈ ∂ Stab G. There exists a ξ-family V ξ such that for every vertex v of D(ξ) and every simplex σ of st(v) \ D(ξ) ∩ Geod v 0 , D(ξ) , we have (V ξ ) v ∩ EG σ = ∅.

Proof. Let σ a simplex of N(D(ξ)) \ D(ξ) whose interior meets Geod(v 0 , D(ξ)). Let v be a vertex of D(ξ) ∩ σ. Let U v be a neighbourhood of ξ in EG v that is disjoint from EG σ . For every other vertex w of D(ξ), set U w = EG w . By Proposition 4.12, we choose a ξ-family V ξ that is (d ξ + 1)-rened in the collection of open sets {U w , w ∈ V (ξ)}. The result now follows from Denition 5.5. Lemma 5.7. Let ξ ∈ ∂ Stab G. Let U be a ξ-family that is m ξ -nested in V ξ (recall that V ξ is assumed to satisfy Lemma 5.6). Let x ∈ X \ D(ξ) be such that there exists a simplex σ ⊂ N(D(ξ)) \D(ξ) that meets Geod x, D(ξ) and such that for some (hence any) vertex

v of σ ∩ D(ξ) we have EG σ ⊂ U v . Then x / ∈ Geod(v 0 , D(ξ)). x D(ξ) σ σ v 0 z z Figure 2
Since U is m ξ -nested in V ξ , it follows from the inclusion EG σ ⊂ U v and Lemma 3.7 that EG σ ⊂ (V ξ ) v , contradicting the denition of V ξ .

The next lemma gives a useful criterion that ensures that a given path is a global geodesic.

Lemma 5.8 (geodesic reattachment lemma). Let ξ ∈ ∂ Stab G, V a ξ-family satisfying Lemma 5.6, U a ξ-family which is (m ξ + d ξ )-nested in V, and x ∈ X \ D(ξ). Suppose that there exists a simplex σ ⊂ N(D(ξ)) \ D(ξ) that meets Geod x, D(ξ) such that for some (hence any) vertex v of σ ∩ D(ξ) we have EG σ ⊂ U v . Then [v 0 , x] meets D(ξ) and x ∈ Cone V,ε (ξ) for every ε ∈ (0, 1).

In such a case, the geodesic from v 0 to x meets D(ξ), and is the concatenation of a geodesic segment in Geod(v 0 , D(ξ)) and a geodesic in Geod(D(ξ), x).

Proof. Let K = Geod(v 0 , D(ξ)) ∪ Geod(D(ξ), x) and let [v 0 , x] K be the geodesic from v 0 to x in K (which meets nitely many simplices by Lemma 3.5). Our aim is to prove that

[v 0 , x] K = [v 0 , x]. By Lemma 5.7, x / ∈ Geod(v 0 , D(ξ)). As D(ξ) is convex by Proposition 4.2, let v 1 , v 2 ∈ D(ξ) be such that [v 0 , x] K = [v 0 , v 1 ] ∪ [v 1 , v 2 ] ∪ [v 2 , x] and such that [v 0 , v 1 ) and (v 2 , x] do not meet D(ξ). Let ε ∈ (0, 1). Let a ∈ [v 0 , v 1 ] be such that d(a, v 1 ) = ε. If x / ∈ D ε (ξ) let b ∈ [v 2 , x] be such that d(v 2 , b) = ε. Otherwise, let b = x. Since X is CAT(0), it is enough to prove that [v 0 , x] K is a local geodesic at every point. We already have that [v 0 , v 1 ] ∪ [v 1 , v 2 ] and [v 1 , v 2 ] ∪ [v 2 , x] are geodesics, so it is sucient to prove the result when v 1 = v 2 . We thus have [v 0 , x] K = [v 0 , v 1 ] ∪ [v 1 , x], with [v 0 , v 1 ] ⊂ Geod(v 0 , D(ξ)) and [v 1 , x] ⊂ Geod(D(ξ), x). Assume by contradiction that [v 0 , x] K is not a local geodesic at v 1 . Then the geodesic segment [a, b] does not meet D(ξ).
This geodesic segment yields a path of simplices between σ a and σ b of length at most d ξ in N(D(ξ)) \ D(ξ). Furthermore, there is a path of simplices between σ and σ b of length at most m ξ in N(D(ξ)) \ D(ξ) by Denition 5.5. Thus, there is a path of simplices between σ and σ a of length at most

m ξ + d ξ in N(D(ξ)) \ D(ξ). But since EG b ⊂ U v and U is (m ξ + d ξ )
-nested in V, the crossing lemma 5.4 implies EG a ⊂ V v , which contradicts the fact that V satises Lemma 5.6. Thus [v 0 , x] K = [v 0 , x] and σ b = σ ξ,ε (x). It follows from the above discussion that for some (hence every) vertex v of σ ξ,ε (x

) we have EG σ ξ,ε (x) ⊂ V v , hence x ∈ Cone V,ε (ξ).
From now on, every ξ-family will be assumed to be contained in a ξ-family U ξ satisfying Lemma 5.8.

As a consequence, we get the following: Corollary 5.9. Let ξ ∈ ∂ Stab G, U a ξ-family and ε ∈ (0, 1). Then for every x ∈ Cone U ,ε (ξ), the geodesic segment [v 0 , x] meets D(ξ).

Proof. By Lemma 5.7 applied to x and σ ξ,ε (x), we get x / ∈ Geod(v 0 , D(ξ)). Let y be a point of σ ξ,ε (x) ∩ [v 0 , x] ∩ D ε (ξ). It follows from the geodesic reattachment lemma 5.8 applied to y and σ ξ,ε (x) that [v 0 , y], hence [v 0 , x], meets D(ξ).

The renement lemma.

Lemma 5.10 (renement lemma). Let ξ ∈ ∂ Stab G, U a ξ-family and n ≥ 1. Let U be a ξ-family which is n-rened in U. Then the following holds:

For every ε ∈ (0, 1) and every path of simplices σ 1 , . . . , σ n in X \ D(ξ) such that there exists a point x 1 ∈ σ 1 such that [v 0 , x 1 ] enters D ε (ξ) and σ ξ,ε (x 1 ) ⊂ N U D(ξ) , we have σ 1 , . . . , σ n ⊂ Cone U ,ε (ξ).

Proof. Let us prove that for every x ∈ ∪ 1≤i≤n σ i , the geodesic segment [v 0 , x] meets D(ξ). Let x 1 ∈ σ 1 such that σ ξ,ε (x 1 ) ⊂ N U (D(ξ)). Note that Corollary 5.9 implies that [v 0 , x 1 ] meets D(ξ). Let v be a vertex of D(ξ) ∩ σ ξ,ε (x 1 ).

Let x ∈ ∪ 1≤i≤n σ i and σ be a simplex of N(D(ξ)) \ D(ξ) touched by [v, x] after leaving D(ξ). Let also w be a vertex of σ ∩D(ξ). We can apply Lemma 3.7 to the geodesic segments [v, x] and (a portion of ) [v 0 , x 1 ], and to simplices σ and σ ξ,ε (x 1 ). Since EG σ ξ,ε (x 1 ) ⊂ U v and U is n-rened in U, we get EG σ ⊂ U w . Thus the geodesic reattachment lemma 5.8 implies that [v 0 , x] meets D(ξ).

Let x ∈ ∪ 1≤i≤n σ i and let w be a vertex of σ ξ,ε (x) ∩ D(ξ). We apply apply once again Lemma 3.7, this time to portions of the geodesic segments [v 0 , x] and [v 0 , x 1 ], and to simplices σ ξ,ε (x) and σ ξ,ε (x 1 ).

Now since U is n-rened in U and EG σ ξ,ε (x 1 ) ⊂ U v , we get EG σ ξ,ε (x) ⊂ U w , hence x ∈ Cone U ,ε (ξ). v 0 x 1 x D(ξ) σ σ ξ,ε (x 1 )
v w Proof. Let T = dist(v 0 , x), and let γ x : [0, T ] → X be the parametrisation of the geodesic segment [v 0 , x]. Let t 0 > 0 such that [v 0 , x] leaves D ε (ξ) at time t 0 . Since D(ξ) is convex by Proposition 4.2, the map z → dist(z, D(ξ)) is convex. Thus, there exists r > 0 such that

γ x [t 0 -r, t 0 ) ⊂ D ε (ξ), γ x [t 0 -r, t 0 ] ⊂ st (σ ξ,ε (x)) .
We also choose τ > 0 such that for every y -, y

+ in the τ -neighbourhood of γ x [t 0 -r, t 0 ] , the geodesic segment [y -, y + ] is contained in st(σ ξ,ε (x)). Let k = ε -dist(γ x (t 0 -r), D(ξ)) > 0. We set δ 1 = 1/10 • min(k, τ, r). If x ∈ D ε (ξ), set δ = δ 1 . If x / ∈ D ε (ξ)
, we can assume without loss of generality that δ 1 < 1/10 • (T -t 0 ). By convexity of the distance, we have d γ x (t 0 + δ 1 ), D(ξ) > ε, and we set δ = 1/2

• min δ 1 , d(γ x (t 0 + δ 1 ), D ε (ξ)) > 0.
Let y ∈ B(x, δ) \ D ε (ξ), and let γ y be its parametrisation. Since δ ≤ r, we have d(v 0 , y) ≥ t 0 -r. Now, γ x and γ y parametrise geodesics starting at v 0 and such that d(x, y) < δ, so since X is a CAT(0)-space, we get d(γ x (t 0 -r), γ y (t 0 -r)) ≤ 2δ ≤ τ . The inequality 10δ ≤ k now implies

d γ y (t 0 -r), D(ξ) ≤ d γ x (t 0 -r), D(ξ) + d γ x (t 0 -r), γ y (t 0 -r) ≤ (ε -10δ) + 2δ < ε, so γ y (t 0 -r) ∈ D ε (ξ). Since y / ∈ D ε (ξ)
, it follows that the geodesic segment [v 0 , y] goes through D ε (ξ) and leaves it for some t 1 > t 0 -r.

Moreover, after leaving D ε (ξ) the geodesic [v 0 , y] meets the τ -ball centred at γ x (t 0 ) for some

t 2 ≥ t 1 . Indeed, this is clear if x ∈ D ε (ξ) since d(x, y) < δ ≤ τ . If x / ∈ D ε (ξ), then [v 0 , y] meets the 2δ-ball centred at γ x (t 0 + δ 1 ), which is contained in (X \ D ε (ξ)) ∩ B(γ x (t 0 ), 2δ 1 ) by denition of δ . Hence, [v 0 , y] meets B(γ x (t 0 ), τ ) \ D ε (ξ) for some t 2 ≥ t 1 .
We thus have d(γ x (t 0 -r), γ y (t 0 -r)) ≤ τ and d(γ

x (t 0 ), γ y (t 2 )) ≤ τ . By denition of τ , it follows that γ y [t 0 -r, t 2 ] ⊂ st (σ ξ,ε (x)) , which implies σ ξ,ε (y) ⊂ st σ ξ,ε (x) .
The star lemma 5.11 immediately implies the following:

Corollary 5.12. Let ξ ∈ ∂ Stab G, U a ξ-family and ε ∈ (0, 1). Then the sets Cone U ,ε (ξ)

and Cone U ,ε (ξ) are open in X.

6

The topology of EG.

In this section, we dene a topology on EG and study its rst properties.

6.1 Denition of the topology. We now turn to the case of points of the boundary of X. Recall that since X is a simplicial CAT(0) space with countably many simplices, the bordication X = X ∪ ∂X has a natural metrisable topology, though not necessarily compact if X is not locally nite. For every η ∈ ∂X, a basis of neighbourhoods of η in that bordication is given by the family of V r,δ (η) = x ∈ X d(v 0 , x) > r and γ x (r) ∈ B(γ η (r), δ) , r, δ > 0.

Remark. For r, δ > 0, η ∈ ∂X and if γ is the parametrisation of a geodesic such that there exists T ≥ 0 with γ(T ) ∈ V r,δ (η), then γ(t) ∈ V r,δ (η) for every t ≥ T . We denote by O X (η) this basis of neighbourhoods of η in X. Endowed with that topology, X is a secound countable metrisable space (see BridsonHaeiger [START_REF] Bridson | Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften[END_REF]).

Note that the topology of X satises the following properties: Lemma 6.2. Let η ∈ ∂X. Then there exists a basis of neighbourhoods (U n ) of η in X such that U n and U n \ ∂X are contractible for every n ≥ 0.

Proof. For r, δ > 0, let U r,δ (η) = V r,δ (η) ∪ B(γ η (r), δ). This denes a basis of neighbourhoods of η in X. As U r,δ (η) \ ∂X can be retracted by strong deformation along geodesics starting at v 0 onto B(γ η (r), δ), it is contractible. Furthermore, as U r,δ (η) can be retracted by strong deformation onto U r,δ (η) \ ∂X, the same holds for U r,δ (η).

Lemma 6.3. Let η ∈ ∂X, U a neighbourhood of η in X and k ≥ 0. Then there exists a neighbourhood U of η in X that is contained in U and such that d(U ∩ X, X \ U ) > k.

Proof. The denition of the topology of X implies the following: if (x n ) and (y n ) are two sequences of X such that d(x n , y n ) is bounded, then (x n ) converges to a point of ∂X if and only if (y n ) converges to the same point. Reasoning by contradiction thus implies the lemma.

Denition 6.4. Let η ∈ ∂X, and let U be a neighbourhood of η in X. We set

V U (η) = p -1 (U ∩ X) ∪ (U ∩ ∂X) ∪ {ξ ∈ ∂ Stab G|D(ξ) ⊂ U } .
When U runs over the basis O X (η) of neighbourhoods of η in X, the above formula denes a collection of neighbourhoods for η in EG, denoted O EG (η).

We nally dene open neighbourhoods for points in ∂ Stab G.

Denition 6.5. Let ξ ∈ ∂ Stab G, U ⊂ U ξ be a ξ-family, and ε ∈ (0, 1). A neighbourhood V U ,ε (ξ) is dened in four steps as follows:

• Let W U ,ε (ξ) be the set of points x ∈ EG whose projection x ∈ X belongs to D ε (ξ) and is such that for some (hence every) vertex v of D(ξ) ∩ σ x , we have φ v,σx ( x) ∈ U v .

• Let W 1 be the set of points of EG whose projection in X belongs to Cone U ,ε (ξ).

• Let W 2 be the set of points of ∂X that belong to Cone U ,ε (ξ).

• Let W 3 be the set of points ξ ∈ ∂ Stab G such that D(ξ ) \ D(ξ) ⊂ Cone U ,ε (ξ) and for every vertex v of D(ξ) ∩ D(ξ ) we have ξ ∈ U v . Now set V U ,ε (ξ) = W U ,ε (ξ) ∪ W 1 ∪ W 2 ∪ W 3 .
This collection of neighbourhoods of ξ in EG is denoted O EG (ξ). Note that these neighbourhoods depend on the chosen basepoint v 0 . If we need to specify the basepoint used to dene the various sets Cone U ,ε (ξ), V U ,ε (ξ), we will indicate it in superscript. In that case, we will speak of the topology (of EG) centred at a given point.

Note that for ξ-families U ⊂ U and ε < ε, we do not necessarily have the inclusion V U ,ε (ξ) ⊂ V U ,ε (ξ) since these two neighbourhoods are dened by looking at the way geodesics leave two (a priori non related) dierent neighbourhoods of the domain D(ξ).

However, the crossing lemma 5.4 immediately implies the following: Lemma 6.6. Let ξ ∈ ∂ Stab G, U, U two ξ-families, and 0

< ε < ε. If U is d ξ -nested in U, then V U ,ε (ξ) ⊂ V U ,ε (ξ).
Denition 6.7. We dene a topology on EG by taking the topology generated by the elements of O EG (x), for every x ∈ EG. We denote by O EG the set of elements of O EG (x) when x runs over EG. Thus, any an open set in EG is a union of nite intersections of elements of O EG .

We will show in the next subsection that O EG is actually a basis for the topology of EG.

A basis of neighbourhoods.

Here we prove that the set of neighbourhoods we just dened is a basis for the topology of EG. In order to do that, we need the following:

Filtration Lemma. Let z, z ∈ EG and U ∈ O EG (z) an open neighbourhood of z. If z ∈ U , then there exists an open neighbourhood of z , U ∈ O EG (z ), such that U ⊂ U .
Since points of EG are of three dierent natures (EG, ∂X, and ∂ Stab G), the proof breaks into six distinct cases. We rst introduce a notation that will be useful to treat similar cases at once. Denition 6.8. We extend the projection p : EG → X to a map p from EG to the set of subsets of X in the following way:

• For x ∈ EG, we dene p(z) to be the singleton {p( x)}.

• For η ∈ ∂X, we dene p(η) to be the singleton {η}.

• For ξ ∈ ∂ Stab G, we set p(ξ) = D(ξ).

• Finally, for K ⊂ EG, we set p(K) = z∈K p(z).

U of η in X, such that V U (η ) ⊂ V U (η). Proof. Since O X is a basis of neighbourhoods for X, there exists a neighbourhood U ∈ O X (η ) such that U ⊂ U . Now one clearly has η ∈ V U (η ) ⊂ V U (η). Lemma 6.11. Let x ∈ EG, η ∈ ∂X and U an open neighbourhood of η in X. If x ∈ V U (η), then there exists an open neighbourhood U of x in EG, U ∈ O EG ( x), such that U ⊂ V U (η). Proof. It is enough to choose an arbitratry open neighbourhood U of x contained in p -1 (U ∩ X). Lemma 6.12. Let ξ ∈ ∂ Stab G, η ∈ ∂X and U ∈ O X (η) an open neighbourhood of η in X. If ξ ∈ V U (η), then there exist ε ∈ (0, 1) and a ξ-family U such that V U ,ε (ξ) ⊂ V U (η).
Proof. The subcomplex D(ξ) ⊂ U is nite, hence compact, so choose ε ∈ (0, 1) such that D ε (ξ) ⊂ U . Let U be any ξ-family. For every x ∈ Cone U ,ε (ξ), the geodesic segment [v 0 , x] meets D(ξ) by Corollary 5.9. As D(ξ) is contained in U , the same holds for x. It then follows that V U ξ ,ε (ξ) ⊂ V U (η). Lemma 6.13. Let η ∈ ∂X, ξ ∈ ∂ Stab G, U a ξ-family and ε ∈ (0, 1). If η ∈ V U ,ε (ξ), then there exists an open neighbourhood U of η in X such that V U (η) ⊂ V U ,ε (ξ).

Proof. Let γ η : [0, ∞) → X be a parametrisation of the geodesic ray [v 0 , η). The subcomplex D(ξ) being nite by Proposition 4.2, choose R > 0 such that D(ξ) ⊂ B(v 0 , R), and let

x = γ η (R + 1). Since η ∈ V U ,ε (ξ), we have x ∈ Cone U ,ε (ξ), which is open in X by Corollary 5.12. Let δ > 0 such that B(x, δ) ⊂ Cone U ,ε (ξ). Now if we set U = V R+1,δ (η) ∈ O X (η), it follows that V U (η) ⊂ V U ,ε (ξ). Lemma 6.14. Let x ∈ EG, ξ ∈ ∂ Stab G, U a ξ-family and ε ∈ (0, 1). If x ∈ V U ,ε (ξ), then there exists a U ∈ O EG ( x) such that U ⊂ V U ,ε (ξ). Proof. It is enough to prove that V U ,ε (ξ) ∩ EG is open in EG. First, since the maps φ σ,σ are embeddings, it is clear that W U ,ε (ξ) is open in EG. Let y ∈ V U ,ε (ξ) ∩ EG with y = p( y) / ∈ D ε (ξ).
The star lemma 5.11 yields a δ > 0 such that for every z ∈ B(y, δ)\D ε (ξ), the geodesic segment [v 0 , z] goes through D ε (ξ) and σ ξ,ε (z) ⊂ st(σ ξ,ε (y)). We can further assume that B(y, δ) ⊂ st(σ y ). It now follows immediately from the construction of V U ,ε (ξ) that p -1 (B(y, δ)) is an open neighbourhood of x contained in V U ,ε (ξ), which concludes the proof. Lemma 6.15. Let ξ, ξ ∈ ∂ Stab G, U a ξ-family and ε ∈ (0, 1). If ξ ∈ V U ,ε (ξ), then there exists a ξ -family U and ε ∈ (0, 1) such that V U ,ε (ξ ) ⊂ V U ,ε (ξ).

By Lemma 5.11, let δ ∈ (0, ε) be such that for all y ∈ D δ (ξ ) \ D ε (ξ), the geodesic segment [v 0 , y] goes through D ε (ξ) and is such that σ ξ,ε (y) ⊂ st (σ ξ,ε (x)), for some x ∈ D(ξ ). We now dene a ξ -family using the following lemma. Lemma 6.16. There exist nested ξ -families U [d ξ ] ⊃ . . . ⊃ U [0] = U such that the following holds: Let x be a point of Cone U ,δ (ξ ) such that the geodesic from v 0 to x leaves D δ (ξ ) at a point which is still inside D ε (ξ). Let σ 1 = σ ξ ,δ (x), . . . , σ n = σ ξ,ε (x) (n ≤ d ξ ) be the path of simplices met by the geodesic segment [v 0 , x] inside D ε (ξ) after leaving D δ (ξ ) (cf Figure 4).

D ε (ξ) v 0 D δ (ξ ) σ ξ ,δ (x)
σ ξ,ε (x)

We then have the following, for every 1 ≤ k ≤ n:

(i) The simplex σ k is contained in v ∈V (ξ)∩V (ξ ) st(v ) but not in v∈V (ξ)\V (ξ ) st(v). (ii) For every vertex v of σ k contained in D(ξ ), the inclusion EG σ k ⊂ U [k] v holds in EG v . Proof. If v is a vertex of D(ξ) ∩ D(ξ ), then for every vertex v of st(v ) ∩ (D(ξ) \ D(ξ )), choose a neighbourhood W v,v of ξ in EG v missing EG [v,v ]
, and set

W v =   v∈st(v )∩(V (ξ)\V (ξ )) W v,v   ∩ U v . If v is a vertex not in D(ξ), set W v = EG v . We now dene U to be a ξ -family that is d ξ -nested in the family of W v , v ∈ D(ξ ), that is, U is a ξ -family such that there exists a sequence of nested ξ -families U [d ξ ] ⊃ . . . ⊃ U [0] = U satisfying W v ⊃ U [d ξ ] v ⊃ . . . ⊃ U [0] v = U v for every vertex v of D(ξ ).
We now prove (i) and (ii) by induction on k. Since the geodesic segment [v 0 , x] leaves D δ (ξ ) while inside D ε (ξ), we have σ 1 = σ ξ ,δ (x) ⊂ v ∈V (ξ)∩V (ξ ) st(v ). To prove (i) for k = 1, we reason by contradiction. Suppose there exists a vertex v of D(ξ) ∩ D(ξ ) and a vertex v of D(ξ) \ D(ξ

) such that σ 1 ⊂ st [v, v ] , then we have EG σ 1 ⊂ EG [v,v ] in EG v . But the former set is contained in U v since x ∈ V U ,δ (ξ )
, and the latter is disjoint from U v by construction of U , which is absurd.

Suppose the result has been proved up to rank k. If σ k+1 ⊂ σ k , the result is straightforward, so we suppose that σ k ⊂ σ k+1 . We prove (i) by contradiction. Suppose there exists a vertex v of D(ξ) ∩ D(ξ ) and a vertex v of D(ξ) \ D(ξ

) such that σ k+1 ⊂ st [v, v ] .
Then by the induction hypothesis, we have EG

[v,v ] ∩ U [k] v = ∅ in EG v , hence EG [v,v ] ⊂ U [k+1] v ⊂ W v since U [k] is nested in U [k+1]
,and the last inclusion contradicts the denition of U .

We now prove (ii

). Let v k a vertex of D(ξ) ∩ D(ξ ) contained in σ k (hence in σ k+1 ). Thus we have EG σ k+1 ⊂ EG σ k ⊂ U [k] v k ⊂ U [k+1] v k in EG v k . Now let v be another vertex of D(ξ ) ∩ D(ξ) contained in σ k+1 (if any). We thus have EG [v k ,v ] ∩ U [k] v k = ∅ in EG v k , so EG [v k ,v ] ⊂ U [k+1] v k in EG v k . But by Proposition 4.12, this implies EG [v k ,v ] ⊂ U [k+1] v , which proves (ii). Proof of Lemma 6.15. Let us show now that V U ,δ (ξ ) ⊂ V U ,ε (ξ). Let z ∈ V U ,δ (ξ ) and x ∈ p(z). The geodesic [v 0 , x] meets D δ (ξ ), hence D ε (ξ). To prove that z ∈ V U ,ε (ξ), it is now enough to prove that x ∈ Cone U ,ε (ξ). If x ∈ W U ,δ (ξ ) ∩ D ε (ξ), it follows from the denition of U (dened in 6.16) that z ∈ W U ,ε (ξ).
If the geodesic segment [v 0 , x] meets D δ (ξ ) outside D ε (ξ), it follows from the denition of δ that there exists x ∈ D(ξ ) \ D(ξ) such that σ ξ,ε (x) ⊂ st (σ ξ,ε (x )). But since x ∈ Cone U ,ε (ξ), the same holds for x.

Thus the only case left to consider is when the geodesic segment [v 0 , x] leaves D δ (ξ ) while still being inside D ε (ξ). But by the previous lemma, we get that for every vertex v of σ ξ,ε (x

) contained in D(ξ), EG σ ξ,ε (x) ⊂ U [n] v ⊂ U v in EG v , which now implies x ∈ Cone U ,ε (ξ).
This concludes the proof. Theorem 6.17. O EG is a basis for the topology of EG, which makes it a second countable space. For this topology, EG embeds as a dense open subset.

Proof. To prove that O EG is a basis for the topology of EG, it is enough to show that for every open sets U 1 , U 2 of EG and every z ∈

U 1 ∩ U 2 , there exists an open neighbourhood W ∈ O EG such that z ∈ W ⊂ U 1 ∩ U 2 .
If z ∈ EG: By the results from the previous paragraph, there exists

V 1 , V 2 ∈ O EG (z) such that V 1 ⊂ U 1 and V 2 ⊂ U 2 . Then take W to be any element of O EG (z) = O EG (z) contained in V 1 ∩ V 2 . If z = η ∈ ∂X: By the results from the previous paragraph, let O 1 , O 2 ∈ O X (η) such that V O 1 (η) ⊂ U 1 and V O 2 (η) ⊂ U 2 . Choosing a neighbourhood W ∈ V X (η) contained in O 1 ∩ O 2 , it follows that V W (η) ⊂ U 1 ∩ U 2 . If z = ξ ∈ ∂ Stab G: By the results from the previous paragraph, let V U 1 ,ε 1 (ξ), V U 2 ,ε 2 (ξ) such that V U 1 ,ε 1 (ξ) ⊂ U 1 and V U 2 ,ε 2 (ξ) ⊂ U 2 . Let U be a ξ-family which is d ξ -nested in {(U 1 ) v ∩ (U 2 ) v , v ∈ V (ξ)}, and let ε = min(ε 1 , ε 2 ). It follows from Lemma 6.6 that V U ,ε (ξ) ⊂ V U 1 ,ε 1 (ξ) ∩ V U 2 ,ε 2 (ξ) ⊂ U 1 ∩ U 2 .
To prove that this topology is second countable, we dene countable many open sets (U n ) n≥0 such that for every open set U in O EG and every x in U , there exist an integer m such that x ∈ U m ⊂ U .

Since EG is the realisation of a complex of spaces over a simplicial complex with countably many simplices, and with bres that have a CW-structure with countably many cells, it inherits a CW-complex structure with countably many cells. Thus its topology is second countable, and we can choose a countable basis of neighbourhoods (U n ), n ≥ 0 of EG. Since X is a simplicial complex with countably many cells, it is a separable space, hence so is the set Λ of points lying on a geodesic from v 0 to a point of ∂X (note that a given geodesic segment may not necessarily be extendable to a geodesic ray). Let Λ be a dense countable subset of Λ. Now the family of open sets V r,ε (η) for η ∈ ∂X, γ η (r) ∈ Λ and ε ∈ Q is a countable family, yielding a countable family of open neighbourhoods of EG, denoted (V n ) n≥0 . Note that (V n ) n≥0 contains a basis of neighbourhoods for every point of EG that belongs to ∂X.

A neighbourhood of a point ξ of ∂ Stab G is dened by choosing a constant ε ∈ (0, 1), a nite subcomplex of X (the domain of ξ), and for every vertex v of that subcomplex an open set of EG v . Since domains of points of ∂ Stab G are nite by Proposition 4.2, there are only countably many such subcomplexes. Furthermore, for every vertex v of X, EG v has a countable basis of neighbourhoods. It is now clear that we can dene a countable family (W n ) n≥0 of open neighbourhoods, containing a basis of neighbourhoods of every element of ∂ Stab G.

The family consisting of all the U n , V n , W n is now a countable basis of neighbourhoods of EG.

Finally, the subset EG, which is open by construction of the topology, is dense in EG since every open set in that basis of neighbourhoods meets EG by construction. Lemma 6.18. The topology of EG does not depend on the choice of a basepoint. Moreover, the action of G on EG continuously extends to ∂G.

Proof. Choose x 0 and x 1 two points of X (note that we do not assume these points to be vertices). Throughout this proof, we will indicate the dependence on the basepoint by indicating it in superscript, as explained in Denition 6.5. It is a well known fact that the topology of X does not depend on the basepoint, so it is enough to consider neighbourhoods of points in ∂ Stab G.

Recall that the number of simplices in a domain D(ξ), ξ ∈ ∂ Stab G is uniformly bounded by the constant d max dened in 4.3. Let ξ ∈ ∂ Stab G, U 0 a ξ-family for the topology centred at x 0 and ε > 0. Now let U 1 be a ξ-family for the topology centred at x 1 , which is 2d maxrened in U 0 . Let x be a point of Cone

x 1 U 1 ,ε (ξ).
Then the geodesic reattachment lemma 5.8 implies that [x 0 , x] meets D(ξ). We can thus apply Lemma 3.7 to subsegments of [x 0 , x] and [x 1 , x], and to simplices σ x 0 ξ,ε (x) and σ

x 1 ξ,ε (x). Since U 1 is 2d max -rened in U 0 , it follows that x ∈ Cone x 0 U 0 ,ε (ξ), hence Cone x 1 U 1 ,ε (ξ) ⊂ Cone x 0 U 0 ,ε (ξ). Moreover, since U 1 is contained in U 0 , we get V x 1 U 1 ,ε (ξ) ⊂ V x 0 U 0 ,ε (ξ).
We extend the G-action on EG to ∂G as follows. First note that the action naturally extends to ∂X. Indeed, G acts on the CAT(0) space X by isometries, and those isometries naturally extend to homeomorphisms of the visual boundary ∂X. Furthermore, we dened in Section 2 a G-action on ∂ Stab G. Thus we have an action of G on EG, which we now prove to be continuous.

Let g ∈ G.

Since EG is open in EG and the action of G on EG is continuous, it is enough to check the continuity at points of ∂G. For a point z ∈ ∂G, the element g sends a basis of neighbourhood of z for the topology centred at v 0 to a basis of neighbourhoods of g.z for the topology centred at g.v 0 . Since the topology does not depend on the basepoint by the above discussion, the action of g is continuous at points of ∂G.

V U (η)∩EG = p -1 (U ∩X) which is open in EG. For ξ ∈ ∂ Stab G, ε ∈ (0, 1)
and U a ξ-family, it was proven in Lemma 6.14 that V U ,ε (ξ) ∩ EG is open in EG.

We now prove that for any open set U in the basis of neighbourhoods

O EG , U ∩ ∂X is open in ∂X. For a point η ∈ ∂X and U a neighbourhood of η in X, we have V U (η) ∩ ∂X = U ∩ ∂X, which is open in ∂X. Now consider ξ ∈ ∂ Stab G , ε ∈ (0, 1) and U a ξ-family. If V U ,ε (ξ) ∩ ∂X is empty there is nothing to prove, otherwise let η ∈ V U ,ε (ξ) ∩ ∂X. By Lemma 6.13, let U be a neighbourhood of η in X such that V U (η) ⊂ V U ,ε (ξ). Thus, η ∈ U ∩ ∂X ⊂ V U ,ε (ξ) ∩ ∂X, and V U ,ε (ξ) ∩ ∂X is open in ∂X.
Before proving the analogous result for EG v , with v a vertex of X, we need the following lemma.

Lemma 6.20. Let ξ ∈ ∂ Stab G, U a ξ-family and ε ∈ (0, 1). We proved already that the topology of EG induces the natural topology on EG. Now using the ltration lemmas 6.12 and 6.15, it is enough to show, for every ξ ∈ ∂G v , every ε ∈ (0, 1) and every ξ-family U, that V U ,ε (ξ) ∩ EG v contains a neighbourhood of ξ in EG v . By Lemma 6.20, let U be a ξ-family contained in U and such that every point of

U v ∩ ∂G v ⊂ V U ,ε (ξ). Proof. Let ξ ∈ v∈D(ξ) U v ∩ ∂G v and x ∈ D(ξ ). If x is a vertex of D(ξ) ∩ D(ξ ),
U v ∩ ∂G v belongs to V U ,ε (ξ). Then we have ξ ∈ U v ⊂ V U ,ε (ξ) ∩ EG v , and so V U ,ε (ξ) ∩ EG v is open in EG v .
Thus the topology of EG induces the natural topology on EG v .

Finally, note that the map EG v → EG is injective by Proposition 4.4. As EG v is a compact space, that map is an embedding.

In the exact same way, we can prove the following: Lemma 6.21. Let σ be a closed cell of X. Then the quotient map σ × EG σ → EG is continuous.

Proof. Choose any neighbourhood of x in EG. This is by denition a neighbourhood of x in EG, to which ξ does not belong.

Lemma 7.7. Let ξ, ξ be two dierent points of ∂ Stab G. Then there exists a neighbourhood of ξ in EG that does not contain ξ .

Proof. If D(ξ) ∩ D(ξ ) = ∅, let v be a vertex in that intersection and let U v be a neighbourhood of ξ in EG v that does not contain ξ . Now we can take a ξ-family U small enough so that U v ⊂ U v and thus ξ / ∈ V U , 1 2 (ξ) by Proposition 6.19.

If D(ξ) ∩ D(ξ ) = ∅, let x ∈ D(ξ ).
There are two cases to consider:

• If [v 0 , x] does not meet then V U ξ , 1 2
(ξ) does not contain ξ by Corollary 5.9.

• Otherwise, [v 0 , x] meets D(ξ) and leaves it. Let σ be the rst simplex touched by [v 0 , x] after leaving D(ξ), v a vertex of σ ∩ D(ξ) and U v a neighbourhood of ξ in EG v that does not contain EG σ . Now let U be ξ-family such that U v ⊂ U v and U a ξ-family that is d ξ -nested in U . It then follows from the crossing lemma 5.4 that ξ / ∈ V U , 1 2 (ξ).

Regularity

In this paragraph, we prove the following:

Proposition 7.8. The space EG is regular, that is, for every open set U in EG and every point x ∈ U , there exists another open set U containing x and contained in U , and such that every point of EG \ U admits a neighbourhood that does not meet U .

Since we previously dened a basis of neighbourhoods for EG, it is enough to prove such a proposition for open sets U in that basis. As usual, the proof of Proposition 7.8 splits in many cases, depending on the nature of the open sets U and points of U involved. Lemma 7.9. Let x ∈ EG and U an open neighbourhood of x in EG. Then there exists a subneighbourhood U of EG containing x and such that every point in EG \ U admits a neighbourhood that does not meet U .

Proof. The space EG being a CW-complex, its topology is regular, so we can choose a neighbourhood U of x in EG whose closure (in EG) is contained in U . Let us call V that closure, and let x = p( x). Since EG is locally nite, we can further assume that p(V ) meets only nitely many simplices and that it is contained in st(σ x ). We now show that V is closed in EG, which implies the proposition.

A point of EG \ V clearly admits a neighbourhood in EG that does not meet V , since open subsets of EG are open in EG. For a point η ∈ ∂X, choosing any neighbourhood of η in X that does not meet p(V ) yields a neighbourhood of η in EG not meeting V . Thus the only case left is that of a point ξ ∈ ∂ Stab G. There are two cases to consider: If x ∈ D(ξ), then since p(V ) meets only nitely many simplices, it is easy to nd a ξ-family U such that W U , 1 2 (ξ) misses V , which implies that the whole V U , 1 2 (ξ) misses V .

If x /

∈ D(ξ), then Lemma 3.5 ensures the existence of a nite subcomplex K ⊂ X containing Geod(v 0 , p(V )). We dene a ξ-family U and a constant ε as follows. Let v be a vertex of D(ξ). For every σ ⊂ (st(v) ∩ K) \ D(ξ), let U v,σ be a neighbourhood of ξ in EG v which is disjoint from EG σ . We now set

U v = σ⊂(st(v)∩K)\D(ξ) U v,σ . Let U be a ξ-family which is contained in {U v , v ∈ V (ξ)}, and choose ε = min 1 3 dist(p(V ), D(ξ)), 1 ,
which is positive since p(V ) ⊂ st(σ x ).

We now show by contradiction that V U ,ε (ξ) ∩ V = ∅. Suppose there exists a point y in that intersection and let y = p( y). By Corollary 5.9, [v 0 , y] goes through D(ξ). But since y ∈ V , we have σ ξ,ε (y) ⊂ K, which contradicts the construction of U.

Thus every point of EG\V admits a neighbourhood missing V , so V is closed in EG.

Lemma 7.10. Let η ∈ ∂X and U be an open neighbourhood of η in X. Then there exists an open neighbourhood U of η in X such that every point not in V U (η) admits a neighbourhood that does not meet V U (η).

Proof. By Lemma 6.3, we rst choose a neighbourhood W of η in X contained in U and such that d(W ∩X, X \U ) > A+1, where A is the acylindricity constant. Since X is metrisable, hence regular, we can further assume that W ⊂ U . Finally, we can choose R > 0 and δ > 0 such that U = V R,δ (η) is contained in W and B(γ η (R), δ) is contained in the open star of the minimal simplex containing γ η (R) (recall that γ η is a parametrisation of the geodesic ray [v 0 , η)). We now show that every point not in V U (η) admits a neighbourhood that does not meet V U (η).

Let z ∈ EG \ V U (η). Then p(z) is not in U , hence not in U . Since U is closed in X, there exists an open set U of X containing p(z) and such that U ⊂ X \ U . Then p -1 (U ) is open in EG and p -1 (U ) does not meet V U (η). Let η ∈ ∂X \ V U (η). Then η / ∈ U ∩ ∂X hence η / ∈ U . Since U is closed in X, we choose an open set U in O X (η) disjoint from U . It is now clear that V U (η ) does not meet V U (η). Let ξ ∈ (∂ Stab G) \ V U (η).
To nd a neighbourhood of ξ that does not meet V U (η), is enough to nd a ξ-family U such that U ∩ Cone U , 1 2 (ξ) = ∅. We dene such a ξ-family as follows:

Let x = γ η (R). By Lemma 3.5, let K be the nite subcomplex of X spanned by open simplices meeting Geod(D(ξ), x). Let v be a vertex of D(ξ). For every simplex σ contained in (st(v) ∩ K) \ D(ξ), let U v,σ be an open neighbourhood of ξ in EG v disjoint from EG σ .

We then set

V v = σ⊂(st(v)∩K)\D(ξ) U v,σ .
Now take U to be a ξ-family contained in {V v , v ∈ V (ξ)}, and let U be a ξ-family that is 2-rened in U.

We now show by contradiction that U ∩ Cone U , 1 2

(ξ) = ∅ . Let y be an point of this intersection. Then [v 0 , y] meets D(ξ) (by Corollary 5.9) and B(x, δ)

∩ S(v 0 , R) (by construction of U ). Since d(U , X \ U ) ≥ A + 1 and D(ξ) meets X \ U , it follows that N(D(ξ)) ∩ U = ∅. Hence the geodesic segment [v 0 , y] enters D(ξ) before meeting B(x, δ) ∩ S(v 0 , R). Let y be the point of [v 0 , y] inside B(x, δ) ∩ S(v 0 , R). By construction of R and δ, it follows that σ y is in the open star of σ x . Now since x ∈ Cone U , 1 2
(ξ), the renement lemma 5.10 implies that σ y ⊂ Cone U , 1 2 (ξ), which contradicts the denition of U.

Lemma 7.11. Let ξ ∈ ∂ Stab G, ε ∈ (0, 1) and U a ξ-family. Then there exists a ξ-family U such that every point not in V U ,ε (ξ) admits a neighbourhood that misses V U ,ε (ξ).

Proof. Recall that domains of points of ∂ Stab G contain at most d max simplices (see Denition 4.3). Choose a ξ-family U which is d max -rened and nested in U. We now show that every point not in V U ,ε (ξ) admits a neighbourhood that misses V U ,ε (ξ).

Let x ∈ EG \ V U ,ε (ξ), and x = p( x).

• If x ∈ D ε (ξ), let v be a vertex of D(ξ) ∩ σ x . We have φ v,σx ( x) / ∈ U v , hence φ v,σx ( x)
/ ∈ U v . Let W x be a neighbourhood of φ v,σx ( x) in EG v that does not meet U v , and V be an open neighbourhood of x in X contained in st(σ x ). Let W be the neighbourhood of x consisting of those elements y ∈ EG whose projection p( y) is in V and such that φ v,σx ( y) belongs to W x . Since U is rened in U, it then follows that W is a neighbourhood of x which does not meet V U ,ε (ξ).

• If x / ∈ D ε (ξ), let V be an open neighbourhood of x in X \D ε (ξ) contained in st(σ x ). As U is rened in U and x / ∈ V U ,ε (ξ), Lemma 5.10 implies that p -1 (V ) is a neighbourhood of x that does not meet V U ,ε (ξ). Let η ∈ ∂X \ V U ,ε (ξ). We construct a neighbourhood V of η in X that does not meet Cone U ,ε (ξ). First, since D(ξ) is bounded, let R > 0 such that D(ξ) is contained in the R-ball centred at v 0 , and let x = γ η (R + 1). • If [v 0 , η) does not meet D(ξ), let δ = 1 2 dist γ η [0, R + 1] , D(ξ) > 0, and let V be a neighbourhood of η in X that is contained in V R+1,δ (η). For every y in V , [v 0 , y] does not meet D(ξ), hence V ∩ Cone U ,ε (ξ) = ∅. • If [v 0 , η) goes through D(ξ), then since x does not belong to Cone U ,ε (ξ), let v be a vertex of D(ξ) in σ ξ,ε (x) such that EG σ ξ,ε (x) U v in EG v . Lemma 5.11 yields a constant δ > 0 such that for every y ∈ B(x, δ), [v 0 , y] goes through D ε (ξ) and σ ξ,ε (y) ⊂ st σ ξ,ε (x) . Let V := V R+1,δ (η) and y ∈ V . Then [v 0 , y] goes through B(x, δ), hence σ ξ,ε (y) ⊂ st σ ξ,ε (x) . As U is nested in U and EG σ ξ,ε (x) U v in EG v , it follows that EG σ ξ,ε (y) U v , hence y / ∈ Cone U ,ε (ξ) and V ∩ Cone U ,ε (ξ) = ∅. Let ξ ∈ (∂ Stab G) \ V U ,ε (ξ).
To nd a neighbourhood of ξ that misses V U ,ε (ξ), it is enough, since cones are open subsets of X by Corollary 5.12, to nd a ξ -family U such that Cone U ,ε (ξ ) ∩ Cone U ,ε (ξ) = ∅ and such that for every vertex v of D(ξ) ∩ D(ξ ),

we have U v ∩ U v = ∅.
We dene such a ξ -family as follows. By Lemma 3.5, let K be a nite subcomplex containing Geod(v 0 , D(ξ)). Let v be a vertex of D(ξ ). For every σ ⊂ (st(v) ∩ K) \ D(ξ ), let U v,σ be a neighbourhood of ξ in EG v which is disjoint from EG σ , and set

U v = σ⊂(st(v)∩K)\D(ξ ) U v,σ .
If v is also in D(ξ), note that since the closure of U v is contained in U v , we can assume that U v ∩ U v = ∅. Furthermore, we can assume by the convergence property 4.8 that the only EG σ inside EG v meeting both U v and U v contain ξ and ξ . Now let U be a ξ -family which is

d max -rened in U v , v ∈ D(ξ ) .
Let us prove by contradiction that Cone U ,ε (ξ ) ∩ Cone U ,ε (ξ) = ∅. Let x be in such an intersection. Then, by Corollary 5.9, the geodesic [v 0 , x] goes through both D(ξ) and D(ξ ). Note that, by construction of the various neighbourhoods U v , the geodesic segment [v 0 , x] cannot leave D(ξ ) before leaving D(ξ); nor can it leave both D(ξ) and D(ξ ) at the same time. If D(ξ) ∩ D(ξ ) = ∅, it follows from the fact that U is d max -rened in U that D(ξ ) ⊂ Cone U ,ε (ξ) by Lemma 5.10, hence ξ ∈ V U ,ε (ξ), which is absurd. Otherwise, let x be the last point of D(ξ ) met by [v 0 , x] and let γ be a geodesic path in D(ξ ) from x to a point of D(ξ), such that γ meets D(ξ) in exactly one point. Let σ be the last simplex touched by γ before touching D(ξ). The fact that U is d max -rened in U implies that EG σ ⊂ U v for some (hence every) vertex v of σ ∩ D(ξ) by Lemma 5.10, hence ξ ∈ U v ⊂ V U ,ε (ξ), a contradiction.

Finally, for every vertex v of D(ξ) ∩ D(ξ ), we have U v ∩ U v = ∅ by construction of U v , hence the result.

Theorem 7.12. The space EG is separable and metrisable.

Proof. It is secound countable by Theorem 6.17, regular by Proposition 7.8 and satises the T 0 condition by Proposition 7.1. Thus it is Hausdor and the result follows from Urysohn's metrisation theorem.

Sequential Compactness.

In this subsection, we prove the following: Theorem 7.13. The metrisable space EG is compact.

First of all, note that since EG is dense in EG by Theorem 6.17, it is enough to prove that any sequence in EG admits a subsequence converging in EG. Let ( x n ) n≥0 ∈ (EG) N . For every n ≥ 0, let x n = p( x n ). Furthermore, to every x n we associate the nite sequence σ

(n) 0 = v 0 , σ (n)
1 , . . . , of simplices met by [v 0 , x n ]. Finally, let l n ≥ 1 be the number of simplices of such a sequence. Lemma 7.14. Suppose that for all k ≥ 0, σ (n) k , n ≥ 0 is nite.

• If (l n ) admits a bounded subsequence, then ( x n ) admits a subsequence that converges to a point of EG ∪ ∂ Stab G.

• Otherwise, ( x n ) admits a subsequence that converges to a point of ∂X.

Proof. Up to a subsequence, we can assume that there exist open simplices σ 0 , σ 1 , . . . such that for all k ≥ 0, (σ

(n)
k ) n≥0 is eventually constant at σ k . There are two cases to consider: (i) Up to a subsequence, there exists a constant m ≥ 0 such that each geodesic [v 0 , x n ] meets at most m simplices. This implies that the x n live in a nite subcomplex. Up to a subsequence, we can now assume that there exists a (closed) simplex σ of X such that x n is in the interior of σ for all n ≥ 0. This in turn implies that x n is in σ × EG σ (or more precisely in the image of σ × EG σ in EG) for all n ≥ 0. This space is compact since the canonical map σ × EG σ → EG is continuous by Lemma 6.21, hence we can take a convergent subsequence.

(ii) Up to a subsequence, we can assume that l n → ∞. For r > 0, let π r : X → B(v 0 , r) be the retraction on B(v 0 , r) along geodesics starting at v 0 . By assumption, we have that for every r > 0, the sequence of projections (π r (x n )) n≥0 lies in a nite subcomplex of X. A diagonal argument then shows that, up to a subsequence, we can assume that all the sequences of projections (π m (x n )) n≥0 converge in X for every m ≥ 0. As the topology of X is the topology of the projective limit B(v 0 , 1)

π 1 ← -B(v 0 , 2) π 2 ← -. . . ,
it then follows that (x n ) converges in X. As l n → ∞, (x n ) converges to a point η of ∂X. The denition of the topology of EG now implies that ( x n ) converges to η in EG.

Lemma 7.15. Suppose that there exists k ≥ 0 such that σ (n) k , n ≥ 0 is innite. Then (x n ) admits a subsequence that converges to a point of ∂ Stab G.

Proof. Without loss of generality, we can assume that such a k is minimal. Up to a subsequence, we can assume that there exist open simplices σ 1 , . . . , σ k-1 such that for all n ≥ 0, σ Since EG has a countable basis of neighbourhoods, it is enough to prove that for every ε ∈ (0, 1) and every ξ-family U there exists a subsequence of ( x n ) lying in V U ,ε (ξ). By construction of ξ, we have σ k-1 ⊂ D(ξ), and there exists a vertex v k of D(ξ) such that σ (n) k ⊂ st(v k ) for all n ≥ 0. Two cases may occur:

(n) 0 = σ 0 , . . . , σ (n) k-1 = σ k-1 ,
• Up to a subsequence, all the [v 0 , x n ] leave D ε (ξ) inside σ • Up to a subsequence, all the [v 0 , x n ] remain inside D ε (ξ) when inside σ (n) k . Up to a subsequence, we can further assume that all the σ (n) k+1 , n ≥ 0 are above a unique simplex of Y . Thus there exists a vertex v k+1 of D(ξ) ∩ st(v k ) such that σ (n) k+1 ⊂ st(v k+1 ) for all n ≥ 0.

In particular we have σ

(n) k ⊂ st(v k )∩ st(v k+1 ) and thus ξ ∈ ∂G v k+1 . Since U is a ξ-family, the fact that EG σ (n) k uniformly converges to ξ in EG v k implies that that EG σ (n) k
uniformly converges to ξ in EG v k+1 . Note that since the sequence (σ

(n) k ) n≥0
takes innitely many values, the niteness lemma 3.5 implies that (σ By iterating this algorithm, two cases may occur:

• There is a value k ≥ k such that, up to a subsequence, all the [v 0 , x n ] leave D ε (ξ) while being inside σ (n) k and the same argument as before shows that we can take a subsequence satisfying x n ∈ V U ,ε (ξ).

• Up to a subsequence, at every stage k ≥ k all the [v 0 , x n ] remain within D (ξ). In the latter case, the containment lemma 3. As a direct consequence, we get the following convergence criterion.

Corollary 7.16. Let (K n ) be a sequence of subsets of EG.

• K n uniformly converges to a point η ∈ ∂X if and only if the sequence of coarse projections p(K n ) uniformly converges to η in X.

• Suppose that there exists ξ ∈ ∂ Stab G such that, for n large enough, every geodesic from v 0 to a point of p(K n ) goes through D(ξ). For every such n and every z ∈ K n , choose x ∈ p(z) and let σ n,x be the rst simplex touched by the geodesic [v 0 , x] after leaving D(ξ). If there exists a vertex v ∈ D(ξ) contained in each σ n,x and such that for every neighbourhood U of ξ in EG v , there exists an integer N ≥ 0 such that for every (n, x) ∈ ∪ n≥N {n} × K n , we have EG σn,x ⊂ U , then (K n ) uniformly converges to ξ.

8

The properties of ∂G.

In this section we prove the following:

Theorem 8.1. (EG, ∂G) is an EZ-structure in the sense of FarrellLafont.

Lemma 8.5. Let η ∈ ∂X and U be a neighbourhood of η in X. Then there exists a subneighbourhood U ⊂ U of η in X such that the inclusion

V U (η) \ ∂G → V U (η) \ ∂G is null-homotopic.
Proof. Lemma 6.3 yields a neighbourhood U of η in X such that d(U ∩ X, X \ U ) > 1.

In particular, Span(U \ ∂X) ⊂ U , and p -1 (Span(U \ ∂X)) can be seen as the realisation of a complex of spaces over Span(U \ ∂X) the bres of which are contractible. Thus Proposition 1.10 implies that the projection p -1 (Span(U \ ∂X)) → Span(U \ ∂X) is a homotopy equivalence. Now Lemma 6.2 yields another neighbourhood U ⊂ U of η in X such that U \ ∂X is contractible. We thus have the following commutative diagram:

V U (η) \ ∂G p -1 (Span(U \ ∂X)) ? _ o o V U (η) \ ∂G ? _ o o Span(U \ ∂X) U \ ∂X. ? _ 0 o o Now since U \∂X is contractible, the inclusion V U (η)\∂G → V U (η)\∂G is null-homotopic.
Lemma 8.6. Let ξ ∈ ∂ Stab G, ε ∈ (0, 1) and U a ξ-family. Then there exists a ξ-family U such that V U ,ε (ξ) is a subneighbourhood of V U ,ε (ξ) and such that the inclusion

V U ,ε (ξ) \ ∂G → V U ,ε (ξ) \ ∂G is null-homotopic.
Lemma 8.7. There exists a ξ-family U , a subcomplex X of X with Cone U ,ε (ξ) ⊂ X ⊂ Cone U ,ε (ξ), and a subset C of EG with V U ,ε (ξ) \ ∂G ⊂ C ⊂ V U ,ε (ξ) \ ∂G, such that p(C ) ⊂ X and the projection map C → X is a homotopy equivalence.

Proof. Let U be a ξ-family that is 2-rened in U and d ξ -nested in U. It follows from the renement lemma 5.10 that Span Cone U ,ε (ξ) ⊂ Cone U ,ε (ξ). By Lemma 6.6, we have

V U ,ε (ξ) ⊂ V U ,ε (ξ). Let X = Span(Cone U ,ε (ξ)) ∪ D ε (ξ) ∩ Cone U ,ε (ξ) .
Note that it is possible to give D ε (ξ) ∩ Cone U ,ε (ξ) a simplicial structure from that of X such that a vertex of D ε (ξ) ∩ Cone U ,ε (ξ) for that structure either is a vertex of D(ξ) or belongs to an edge in X between a vertex of D(ξ) and a vertex of X \ D(ξ). Furthermore, we can give Span(Cone U ,ε (ξ)) a simplicial structure that is ner that that of X, whose vertices are the vertices of Span(Cone U ,ε (ξ)) and vertices of D ε (ξ) ∩ Cone U ,ε (ξ) (for its given simplicial structure), that is compatible with that of D ε (ξ), and which turns X into a simplicial complex such that an open simplex is completely contained either in D ε (ξ) or in X \ D ε (ξ) (see Figure 5). Thus X is endowed with a simplicial structure.

D ε (ξ)

Span(Cone U ,ε (ξ)) X Figure 5
We now dene a contratible open subset C σ of EG σ for every open simplex σ of X . This will allow us to dene the following subset of EG:

C = σ∈S(X ) σ × C σ .
Note that although C is not naturally the realisation of a complex of spaces in the sense of the rst section, it is nonetheless possible to endow it with one, so as to use Proposition 1.10.

We rst dene these spaces C σ for vertices of X . Let v be such a vertex.

• If v is a vertex of D(ξ), the compactication EG v is locally contractible so we can choose a contractible open set U v of EG v contained in U v and containing ξ, and set

C v = U v ∩ EG v . As ∂G v is a Z-boundary, C v is a contractible open subset. • If v does not belong to D ε (ξ), set C v = EG v . • If v is a vertex of D ε (ξ) \ D(ξ) (for the chosen simplicial structure of D ε (ξ) ⊂ X ),
then either v belongs to Span(Cone U ,ε (ξ)), in which case we set C v = EG v , or it does not, in which case v belongs to a unique edge e (for the simplicial structure of X) between a vertex v of D(ξ) and a vertex of X \ D(ξ). In that case, EG e is contained in U v since U is nested in U and we set C v = EG e .

We dene the space C = σ∈S(X ) σ × C σ . As explained above, the projection C → X is a homotopy equivalence. Furthermore, we can choose a ξ-family U small enough so that the subset V U ,ε (ξ) \ ∂G is contained in C . Proof of Lemma 8.6. We apply the previous lemma twice to get the following commutative diagram: 3) . ? _ 0 o o Since X (3) retracts by strong deformation (along geodesics starting at v 0 ) inside Cone U ,ε (ξ) on the contractible subcomplex D(ξ) (relatively to D(ξ)), the inclusion X (3)

V U ,ε (ξ) \ ∂G C ? _ o o V U ,ε (ξ) \ ∂G ? _ o o C (3) ? _ o o X Cone U ,ε (ξ) ? _ o o X (
→ Cone U ,ε (ξ) is nullhomotopic, hence C (3) → V U ,ε (ξ) \ ∂G is null-homotopic.
As there exists a ξ-family U (4) such that V U (4) ,ε (ξ) \ ∂G → C (3) , this concludes the proof.

Proof of Proposition 8.2: Thus, Theorem 7.13 and Lemma 8.4 together with Lemma 8.5 and Lemma 8.6 yield the desired result.

Compact sets fade at innity

Here we prove the following: Proposition 8.8. Compacts subsets of EG fade at innity in EG, that is, for every x ∈ ∂G, every neighbourhood U of x in EG and every compact K ⊂ EG, there exists a subneighbourhood V ⊂ U of x such that any G-translate of K meeting V is contained in U .

As usual, we split the proof in two parts, depending on the nature of the points considered. Proposition 8.9. Let η ∈ ∂X. For every neighbourhood U of η in X and every compact subset K ⊂ EG, there exists a neighbourhood U of η contained in U and such that any

G-translate of K meeting V U (η) is contained in V U (η). Proof. By Lemma 6.3, let U be a neighbourhood of η in X which is contained in U and such that d(U , X \ U ) > diam(p(K )). Let g ∈ G such that gK meets V U (η). Since G acts on X by isometries, we have diam (p(g.K)) = diam (g.p(K)) = diam (p(K)) , which implies that gK ⊂ V U (η).
The proof for points of ∂ Stab G is slightly more technical. We start by dening a class of compact sets of EG which are easy to handle. Denition 8.10. Let F be a nite subcomplex of X, together with a collection (K σ ) σ∈S(F ) of non empty compact subsets of EG σ for every simplex σ of F . Suppose that for every simplex σ of F and every face σ of σ, we have φ σ ,σ (K σ ) ⊂ K σ . Then the set

σ∈S(F ) σ × K σ .
is called a standard compact subset of EG over F . Every compact subset of EG obtained in such a way is called a standard compact of EG.

Note that the projection in X of any compact subset of EG meets nitely many simplices of X, so every compact subset of EG may be seen as a subset of a standard compact subset of EG. Denition 8.11. Let ξ ∈ ∂ Stab G and U a ξ-family. We dene W U (ξ) as the set of points x of EG whose projection x ∈ X belongs to the domain of ξ and is such that for some (hence any) vertex v of σ x ∩ D(ξ) we have

φ v,σx ( x) ∈ U v .
Before proving that compact sets fade near points of ∂ Stab G, we prove the following lemma. Lemma 8.12. Let ξ ∈ ∂ Stab G, ε ∈ (0, 1) and U a ξ-family. Let K be a compact subspace of EG. Then there exists a ξ-family U contained in U such that for every point g ∈ G, the following holds:

If gK meets W U (ξ), then gK ∩ p -1 (D(ξ)) is contained in W U (ξ).
Proof. Let L be a standard compact subset of EG over the (nite) full subcomplex of X dened by Span p(K). By choosing the L σ big enough, we can assume that L contains K.

Let N ≥ 0 be such that any two vertices of L can be joined by a sequence of at most N adjacent vertices.

Since D(ξ) and p(L) meet nitely many vertices of X, there are only nitely many elements of G such that g.p(L) meets D(ξ) up to left multiplication by an element of G v , v ∈ V (ξ). Let (g λ .p(L)) λ∈Λ be such a nite family of cosets. For every vertex v of V (ξ), {g λ L ∩ EG v , λ ∈ Λ} is a nite (possibly empty) collection of compact subsets of EG v . Since ∂G v is a Bestvina boundary for G v , compact subsets fade at innity in EG v , so there exists a subneighbourhood U v of U v such that any G v -translate of one of these g λ L meeting U v is contained in U v . Repeating this procedure N + 1 times, we get a sequence of ξ-families

denoted {U v , v ∈ V (ξ)} ⊃ U [N ] ⊃ U [N -1] ⊃ . . . ⊃ U [0] .
Let g ∈ G such that gK meets W U (ξ), and let w be a vertex of D(ξ) such that gK, hence gL, meets U

[0] w . In order to prove the lemma, it is enough to show by induction on k = 0, . . . , N the following:

(H k ) : For every chain of adjacent vertices w 0 = w, w 1 , . . . , w k of D(ξ) such that gL meets EG w 0 , . . . , EG w k , we have gL ∩ EG w k ⊂ U

[k+1] w . Since gL meets D(ξ), let λ ∈ Λ such that gL = g λ L pointwise. The result is true for k = 0 by denition of U [0] and U [1] . Suppose we have proven it up to rank k, and let w 0 = w, w 1 , . . . , w k+1 a chain of vertices of D(ξ) such that gL meets EG w 0 , . . . , EG w k . By induction hypothesis, we already have gL

∩ EG w k ⊂ U [k+1] w k . Since p(L) is a full subcom- plex of X, it
G-translate of K meeting V U ,ε (ξ) is contained in V U ,ε (ξ).
Proof. Let k be the number of simplices met by p(K), and let U be a ξ-family that is k-rened in U. Applying the previous proposition to V U ,ε (ξ) yields a ξ-family U . Finally, let U be a ξ-family that is k-rened in U .

Suppose that gK meets V U ,ε (ξ), and let x 0 ∈ gK ∩ V U ,ε (ξ). Let x ∈ gK, and let us prove that x ∈ V U ,ε (ξ). Since p(K) is connected, let γ be a path from x 0 = p( x 0 ) to x = p( x) in p(gK). This yields a path of open simplices σ 1 , . . . , σ n , with n ≤ k. If gK does not meet D(ξ), the renement lemma 5.10 implies that σ n ⊂ Cone U ,ε (ξ), and x ∈ V U ,ε (ξ).

Otherwise, let n 0 (resp. n 1 ) be such that σ n 0 (resp. σ n 1 ) is the rst (resp. the last) simplex contained in D(ξ). If x 0 is not in D(ξ), we can apply the renement lemma 5.10 to the path σ 1 , . . . , σ n 0 -1 , which implies σ n 0 -1 ⊂ N U (D(ξ)). In particular, we see that gK meets W U (ξ), which is also true if x 0 is in D(ξ). Now by denition of U , we have that gK ∩ p -1 (D(ξ)) ⊂ W U (ξ). If γ goes out of D(ξ) after σ n 1 , then σ n 1 +1 ⊂ N U (D(ξ)), and we can apply the renement lemma 5.10 to the path of simplices σ n 1 +1 , . . . , σ n . In any case, we get in the end x ∈ V U ,ε (ξ), which concludes the proof. Proof. The only thing to prove is that (EG, ∂G) is an EZ-structure in the sense of Carlsson Pedersen. We already know that it is an EZ-structure in the sense of FarrellLafont by Theorem 0.1 in the case of EZ-structures in the sense of FarrellLafont. Let H be a nite subgroup of G. In order to do this, rst notice that EG H is nothing but the complex of spaces over X H with bres the subcomplexes EG H σ of EG σ . Thus, it is possible to apply the exact same reasoning with X H in place of X and the EG H σ in place of the EG σ . As X H is a convex, hence contractible sucomplex of X, this is enough to recover the fact that EG

9

A high-dimensional combination theorem for hyperbolic groups.

In this section, we apply our construction of boundaries to get a generalisation of a combination theorem of Bestvina-Feighn to complexes of groups of arbitrary dimension.

Lemma 9.14. Let (g n ) be an injective sequence of elements of G, and suppose there exist vertices v 0 and v 1 of X such that g n v 0 = v 1 for innitely many n. Then there exist ξ + , ξ -∈ ∂G and a subsequence (g ϕ(n) ) such that for every compact subset K of ∂G \ {ξ -}, the sequence of translates g ϕ(n) K uniformly converges to ξ + .

Proof. It is enough to prove the result when g n v 0 = v 0 for innitely many n. Since G v 0 is hyperbolic, we can assume that there exists a subsequence of (g n ), that we still denote (g n ), and points ξ + , ξ -∈ ∂G v 0 such that for every compact subset K of EG v 0 \ {ξ -}, the sequence of translates g n K uniformly converges to ξ + . Throughout this proof, we choose v 0 as the basepoint.

Let σ be a simplex of X containing v 0 . If σ is not contained in D(ξ -), then the convergence property 4.8 implies that, up to a subsequence, we can assume that the sequence of g n ∂G σ uniformly converges to ξ + in ∂G v 0 .

If σ is contained in D(ξ -), then the subset ∂G σ ⊂ ∂G v 0 consists of at least two points among which there is ξ -. Since for any other point α of ∂G σ we have that g n α tends to ξ + , the convergence property 4.8 implies that one of the following situations happens:

• g n G σ only takes nitely many values of cosets, in which case we can nd a subsequence (g n ) such that g n ∂G σ is constant and contains ξ + . This means that we can write g n = g n .g where g is in the stabiliser of v 0 and g n in a sequence in the stabiliser of σ.

Up to replacing g n by g n , we can assume that g n xes σ.

• g n G σ takes innitely many values of cosets, in which case we can nd a subsequence (g n ) such that g n ∂G σ uniformly converges to ξ + .

As domains are nite subcomplexes of X by Proposition 4.2, we can iterate this procedure a nite number of times so as to obtain a subsequence (g n ) and a subcomplex F ⊂ D(ξ -) ∩ D(ξ + ) such that • F is xed pointwise under all the g n ,

• for every simplex σ in (st(F ) \ F ) and every vertex v of σ ∩ F , we have that g n ∂G σ uniformly converges to ξ + in ∂G v .

For every vertex v of D(ξ -), choose U v to be a neighbourhood of ξ -in ∂G v 0 . Choose a ξ --family U which is nested in {U v , v ∈ D(ξ -)}, and choose ε ∈ (0, 1). We can further assume that for every simplex σ of F and every vertex v of σ, the subset EG σ \ U v is innite. Let K = ∂G \ V U ,ε (ξ -).

We now prove that, up to a subsequence, the sequence of translates g n K uniformly converges to ξ + . Because of the denition of neighbourhoods of points of ∂ Stab G, we need to treat dierent cases.

Let σ be a simplex of F containing v 0 , so that G σ ⊂ G v 0 , and v a vertex of σ distinct from v 0 . Since G v is hyperbolic, there exists a subsequence of (g n ), that we still denote Proof. Choose an arbitrary vertex v and a point x of EG v . As ∂G is compact by Theorem 7.13 and (g n v) is bounded, we can choose a subsequence, still denoted (g n ), and points ξ + , ξ -∈ ∂ Stab G such that g n x → ξ + and g -1 n x → ξ -. We choose a vertex v 0 of D(ξ + ) to be the basepoint, and let x 0 ∈ EG v 0 . By Theorem 8.1, we still have g n x 0 → ξ + and g -1 n x 0 → ξ -.

Claim 1:

• For every η ∈ ∂X, the geodesic ray [g n v 0 , g n η) does not meet D(ξ + ) for n large enough.

• For every ξ ∈ ∂ Stab G, the subset Geod(g n v 0 , g n D(ξ)) does not meet D(ξ + ) for n large enough.

Let z ∈ ∂G. If z ∈ ∂X, we denote by D(z) the singleton {z}. By contradiction, suppose that there exists an innite number of n for which there exist y n ∈ D(ξ + ) and x n ∈ Geod(v 0 , D(z)) such that g n x n = y n . As (y n ) is bounded by Proposition 4.2, the assumption on (g n ) implies that (x n ) is bounded too. Since x n lies on Geod(v 0 , D(z)) for every n, the containment lemma 3.3 and the niteness lemma 3.5 imply that, up to a subsequence, we can assume that x n always lies in the same simplex σ of X. Furthermore, since D(ξ + ) is nite by Proposition 4.2, we can assume, up to a subsequence, that y n lies in a simplex σ of X for every n. As the action of G on X is without inversion, this implies that g n σ = σ for every n, which was excluded by assumption.

Claim 2: For every ξ in ∂G, the sequence g n ξ converges to ξ + .

Let U be a ξ + -family, U a ξ + -family that is 3d max -nested in U and ε > 0. Recall that, by assumption on (g n ), the vertex g n v 0 does not belong to D(ξ + ) for n big enough. Furthermore, since g n x 0 → ξ + , we have that EG σ ξ + ,ε (gnv 0 ) ⊂ U v for n large enough and for some (hence every) vertex v of D(ξ + ) ∩ σ ξ + ,ε (g n v 0 ). We split the proof of the claim in two cases.

Let η ∈ ∂X. For n large enough, the path [g n v 0 , g n η) does not meet D(ξ + ) by Claim 1. By Proposition 4.2 ,we can choose y ∈ D(ξ + ) which minimises the distance to Geod(g n v 0 , g n η). Let τ (resp. τ ) be a simplex of N(D(ξ + )) \ D(ξ + ) whose interior is crossed by [y, g n v 0 ] (resp. [y, g n η)) at a point u (resp. u ). By convexity of the function z → d(z, [g n v 0 , g n η)), it follows from the denition of y that the geodesic segment [u, u ] does not meet D(ξ + ), thus yielding a path of simplices of length at most d max between τ and τ in N(D(ξ + )) \ D(ξ + ). Lemma 3.7 implies that there exists a path of simplices of length at most d max between τ and the exit simplex σ ξ + ,ε (g n v 0 ) (resp. between τ and the exit simplex σ ξ + ,ε (g n η)) in N(D(ξ + )) \ D(ξ + ). Thus for n large enough, there is a path of simplices of length at most 3d max from σ ξ + ,ε (g n v 0 ) to σ ξ + ,ε (g n η) in 57 N(D(ξ + )) \ D(ξ + ). As EG σ ξ + ,ε (gnv 0 ) ⊂ U v for n large enough and for some (hence every) vertex v of D(ξ + ) ∩ σ ξ + ,ε (g n v 0 ), it follows from the fact that U is 3d max -nested in U that EG σ ξ + ,ε (gnη) ⊂ U v for n large enough and for some (hence every) vertex v of D(ξ + ) ∩ σ ξ + ,ε (g n η). It thus follows that (g n η) converges to ξ + .

Let ξ ∈ ∂ Stab G. For n large enough, Geod(g n v 0 , g n D(ξ)) does not meet D(ξ + ) by Claim 1. Let x ∈ D(ξ) and, by Proposition 4.2, let y be a point of D(ξ + ) which minimises the distance to Geod(g n v 0 , g n x). Using the same reasoning as above, we get, for n large enough, a path of simplices of length at most 3d max from σ ξ + ,ε (g n v 0 ) to σ ξ + ,ε (g n x) in N(D(ξ + )) \ D(ξ + ). As EG σ ξ + ,ε (gnv 0 ) ⊂ U v for n large enough and for some (hence every) vertex v of D(ξ + ) ∩ σ ξ + ,ε (g n v 0 ), it follows from the fact that U is 3d max -nested in U that, for n large enough and for every x ∈ D(ξ), EG σ ξ + ,ε (gnx) ⊂ U v for some (hence every) vertex v of D(ξ + ) ∩ σ ξ + ,ε (g n x). It thus follows that (g n ξ) converges to ξ + .

In the same way, we prove that for every ξ ∈ ∂G, the sequence g -1 n ξ converges to ξ -.

To conclude the proof of the lemma, it remains to show that this convergence can be made uniform away from ξ -: Claim 3: For every ξ = ξ -in ∂G, there is a subsequence (g n ) and a neighbourhood U of ξ in ∂G such that the sequence of g n U uniformly converges to ξ + .

Once again, we split the proof in two cases.

Let ξ ∈ ∂ Stab G. We already have that g n ξ → ξ + by Claim 2. In order to nd a ξ-family U and a constant ε such g n V U ,ε (ξ) uniformly converges to ξ + , it is enough, using the same reasoning as in Claim 2, to nd a ξ-family U and a constant ε such that for every x in D(ξ) ∪ Cone U ,ε (ξ), the geodesic from g n v 0 to g n x does not meet D(ξ + ). By Claim 1, we already have that for n large enough, no geodesic from g n v 0 to a point of g n D(ξ) meets D(ξ + ). As ξ = ξ -, we choose a ξ-family U, a ξ --family U and constants ε, ε ∈ (0, 1) such that the neighbourhoods V U ,ε (ξ) and V U ,ε (ξ -) are disjoint. Up to a subsequence, we have by the rst claim that g n D(ξ) does not meet D(ξ + ). It now follows from the denition of U and the fact that g -1 n ξ + → ξ -that Cone U ,ε (ξ) does not meet the sets g -1 n D(ξ + ), hence the sets g n Cone U ,ε (ξ) do not meet D(ξ + ). Now this implies that for every x in Cone U ,ε (ξ), the geodesic from g n v 0 to g n x does not meet D(ξ + ): indeed, this geodesic must meet g n D(ξ) since the geodesic from v 0 to a point of Cone U ,ε (ξ) must meet D(ξ), and we already proved that a geodesic segment from g n v 0 to a point of g n D(ξ) does not meet D(ξ + ). Now the same proof as in Claim 2 shows that g n V U ,ε (ξ) uniformly converges to ξ + .

Let η ∈ ∂X. We already know that g n η → ξ + by Claim 2. In order to nd a neighbourhood U of η in X such that such g n V U (η) uniformly converges to ξ + , it is enough, using the same reasoning as in Claim 2, to nd a neighbourhood U of η in X such that for every x in U , the geodesic from g n v 0 to g n x does not meet D(ξ + ). First, notice that the distance from the geodesic rays [g n v 0 , g n η) to D(ξ + ) is uniformly

  We denote by p Y : |C(X )| → |C Y (X )| the canonical projection, and simply p for p X : |C(X )| → X. In the same way, if Y ⊂ Y are subcomplexes of X, we denote by p Y,Y : |C Y (X )| → |C Y (X )| the canonical projection.

  Contractibility: Since the complex of spaces associated to Cl D(Y) is locally nite and has contractible bres, Cl D(Y) is contractible by Proposition 1.10. Cocompact action: For every simplex σ of Y , we choose a compact fundamental domain K σ for the action of G σ on D σ = EG σ . Now the image in Cl D(Y) of σ∈S(Y ) σ × K σ clearly denes a compact subset of Cl D(Y) meeting every G-orbit.

  Now, let F be a nite subcomplex of Cl D(Y) and S(F ) the (nite) set of pairs (τ, τ ) of cells of F that are in the same G-orbit. The set {g ∈ G | gF ∩ F = ∅} is contained in (τ,τ )∈S(F ) {g ∈ G | gτ = τ } , and {g ∈ G | gτ = τ } has the same cardinality as the set {g ∈ G | gτ = τ }, which is nite by the previous argument.

Figure 1

 1 Figure 1

Denition 4 . 1 .

 41 Let ξ ∈ ∂ Stab G. We dene D(ξ), called the domain of ξ, as the subcomplex of X spanned by simplices σ such that ξ ∈ ∂G σ . We denote by V (ξ) the set of vertices of D(ξ).The aim of this paragraph is to prove the following: Proposition 4.2. Domains are nite convex subcomplexes of X whose diameters are uniformly bounded above. The containment lemma 3.3 and Proposition 4.2 imply the following: Corollary 4.3. For every ξ ∈ ∂ Stab G, there exists an integer d ξ such that D(ξ) has at most d ξ simplices, and such that a geodesic segment in the open simplicial neighbourhood of D(ξ) meets at most d ξ open simplices. Furthermore, there exists an upper bound d max on the set of integers d ξ , ξ ∈ ∂ Stab G.

Lemma 4 .

 4 10 (nesting lemma). Let ξ ∈ ∂ Stab G, v a vertex of D(ξ) and U a neighbourhood of ξ in EG v . Then there exists a subneighbourhood of ξ in EG v , V ⊂ U , which is nested in U .

Denition 5 . 3 .

 53 For ξ ∈ ∂ Stab G and U a ξ-family (Denition 4.11), we call N U (D(ξ)) the subcomplex spanned by simplices σ ⊂ N(D(ξ)) such that for some (hence for every) vertex v of D(ξ) ∩ σ, we have, in EG v : EG σ ∩ U v = ∅. 5.1 The crossing lemma. Lemma 5.4 (crossing lemma). Let ξ ∈ ∂ Stab G, U, U two ξ-families, and σ 1 , . . . , σ n (n ≥ 1)

5. 2

 2 The geodesic reattachment lemma. Recall that Denition 4.3 yields for every ξ ∈ ∂ Stab G a constant d ξ ≤ d max such that D(ξ) contains at most d ξ simplices and such that a geodesic contained in the open simplicial neighbourhood of D(ξ) meets at most d max open simplices. Denition 5.5 (rened families). Let n ≥ 1. By Lemma 3.7, we can choose a constant m such that the following holds:

Figure 3 5. 4

 34 Figure 3

Lemma 6 . 9 .

 69 Let x, y ∈ EG and U ∈ O EG ( x) an open neighbourhood of x in EG. If y ∈ U , then there exists an open neighbourhood of y in EG, U ∈ O EG ( y) such that U ⊂ U . Proof. By denition of the topology, we can take U = U . Lemma 6.10. Let η, η ∈ ∂X and U ∈ O X (η) an open neighbourhood of η in X. If η ∈ V U (η), then there exists an open neighbourhood

  the denition of a ξ-family implies that ξ ∈ U x . Otherwise, since D(ξ ) is convex by Proposition 4.2, let γ be a geodesic path in D(ξ ) from x to D(ξ) and meeting D(ξ) at a single point. This yields a path of open simplices from a simplex σ ⊂ N(D(ξ)) \ D(ξ) to σ x of length at most d max in D(ξ ) \ D(ξ). Since ξ ∈ v∈D(ξ) U v ∩ ∂G v also belongs to ∂G σ , we have σ ⊂ N U (D(ξ)). Now since U is d max -rened in U, we get σ x ⊂ Cone U ,ε (ξ) by Lemma 5.10. Proof of Proposition 6.19. Let v be a vertex of X. We now prove that for every open set U in the basis of neighbourhood O EG , U ∩ EG v is open in EG v .

  EG σ (n) k uniformly converges to ξ in EG σ k-1 and thus in EG v k , we can assume, up to a subsequence, that EG σ (n) k ⊂ U v k inside EG σ k . This implies that x n ∈ V U ,ε (ξ), which is what we wanted.

  also takes innitely many values. Up to a subsequence, we can thus assume by the convergence property 4.8 that EG σ (n) k+1 uniformly converges in EG v k+1 . As EG σ (n) k uniformly con-verges to ξ in EG v k+1 , the same holds for EG σ (n) k+1 , and we are back to the previous situation.

Proof of Proposition 8 . 8 :

 88 This follows from Proposition 8.9 and Proposition 8.13. Proof of Theorem 8.1: This follows from Theorem 7.13, Lemma 6.18, Proposition 8.2, and Proposition 8.8. However, the previous reasoning does not show the contractibility of EG H . We now reformulate our main theorem in the setting of EZ-structures in the sense of Carlsson Pedersen.Denition 8.21. An EZ-complex of spaces in the sense of CarlssonPedersen (compatible with the complex of groups G(Y)) is a complex of spaces over a fundamental domain for the action satisfying the axioms of a compatible EZ-complex of spaces, with strong EZ-structures in the sense of CarlssonPedersen instead of EZ-structures in the sense of FarrellLafont.Theorem 8.22. The Combination Theorem for boundaries of groups remains true if ones replaces EZ-complexes of spaces with EZ-complexes of spaces in the sense of Carlsson Pedersen.

  H , the subcomplex (EG σ ) H of EG σ is nonempty and contractible. Thus EG σ that is xed under H ⊂ G σ . But this is absurd as (EG σ ) H = ∅ by assumption.

	Cl H D(Y) is the realisation of a complex of spaces over the contractible complex X H and with
	contractible bres, hence it is nonempty and contractible by Proposition 1.10.
	If H is an innite subgroup of G, we prove by contradiction that Cl H D(Y) is empty. If
	this was not the case, there would exist a simplex σ xed pointwise under H and a point x

of

  the set of all points on a geodesic between two points of [v 0 , . . . , v n ]. By the previous lemma, there is an innite subgroup H of G stabilising pointwise v 0 , . . . , v n . As X is CAT(0), H also stabilises pointwise every point of Y . As [v 0 , . . . , v n ] is contractible inside Y , the niteness lemma 3.5 implies that we can choose a nite 2-complex F such that the loop [v 0 , . . . , v n ] is contractible inside F , and such that F is pointwise xed by H. We call such a subcomplex a hull of the loop [v 0 , . . . , v n ]. Hence the result will follow from the following fact, which we now prove by induction.

  we can assume without loss of generality that its support [v 0 , . . . , v n ] contains all the vertices of σ x and σ x . By Lemma 4.7, this implies that the subgroup H = ∩ 0≤i≤p G v i is innite and that ξ ∈ ΛH ⊂ ∂G v 0 . Now since H xes pointwise all the vertices of σ x and σ x , and since X is CAT(0), H also xes pointwise the geodesic segment [x, x ]. But by Lemma 4.6, the xed-point set of H is contained in D(ξ), hence so is [x, x ]. Thus D(ξ) is convex.

  In this paragraph, we dene a topology on EG, by dening a basis of open neighbourhoods at every point. Since points of EG are of three dierent kinds (EG, ∂X and ∂ Stab G), we treat these cases separately. Denition 6.1. Let x ∈ EG. We dene a basis of neighbourhoods of x in EG, denoted O EG ( x), as the set of open sets of EG containing x.

  6.3 Induced topologies. Proposition 6.19. The topology of EG induces the natural topologies on EG, ∂X and EG v for every vertex v of X. rst prove that for any open set U in the basis of neighbourhoods O EG previously dened, U ∩EG is open in EG. For x ∈ EG, the result is obvious for points in O EG (x) since open sets in O EG (x) are open sets of EG by denition. For η ∈ ∂X and U a neighbourhood of η in X, we have

	We

  Recall that d max was dened in 4.3 as an integer such that domains of points of ∂ Stab G meet at most d max simplices. Let U be a ξ-family which is d max -rened in U. Then we have v∈D(ξ)

  and σ EG σ k-1 . By the convergence property 4.8, we can assume, up to a subsequence, that in EG σ k-1 the sequence of subspaces EG σ (n) k uniformly converges to a point ξ ∈ ∂G σ k-1 . Let us prove that ( x n ) n≥0 converges to ξ in EG.

	(n) k	n≥0	is injective. By cocompactness of the action,
	we can furthermore assume (up to a subsequence) that the σ	(n) k are above a unique simplex
	of Y . This corresponds to embeddings EG σ (n)

k →

  3 implies that there exists an integer m ≥ 0 such that each geodesic segment [v 0 , x n ] meets at most m simplices. Up to a subsequence, we can further assume that all the [v 0 , x n ] meet exactly m simplices. Thus we can iterate our algorithm up to rank m, which yields the existence of a vertex v m of D ε (ξ) such that σ ⊂ st(v m ) for all n ≥ 0 and such that EG σ (n) in EG vm . Up to a subsequence, we can furthermore assume that EG σ (n) m ⊂ U m in EG v k+1 for all n ≥ 0. This in turn implies x n ∈ W U ,ε (ξ), hence x n ∈ V U ,ε (ξ) and we are done.Proof of Theorem 7.13. This follows immediately from Theorem 7.12, Lemma 7.14 and Lemma 7.15.

	(n)	uniformly converges to
	m

m ξ

  follows from the fact that gL meets EG w k and EG w k+1 that gL also meetsEG [w k ,w k+1 ] . In particular, since gL ∩ EG w k ⊂ U [k+1]w k , it follows from the properties of ξ-families that gL ∩ EG w k+1 meets U w k+1 . This in turn implies that gL ∩ EG w k+1 ⊂ U Proposition 8.13. Let ξ ∈ ∂ Stab G, ε ∈ (0, 1) and U a ξ-family. Let K be a connected compact subset of EG. Then there exists a ξ-family U contained in U and such that every

	[k+1]	[k+2] v k+1 ,
	which concludes the induction.

  To prove that EG

	H	is contractible, we want to apply the lemma 8.3 of
	BestvinaMess to the pair (EG

H , EG H \ EG H ).

  The only properties that were required are the fact that X is a CAT(0) space, the convergence properties of the embeddings between the various classifying spaces, and the fact that ∂G σ is a Z-set in EG σ . But since X H is convex in a CAT(0) space, it is itself CAT(0). Moreover, the convergence properties of the embeddings are clearly still satised for simplices that are xed under H. Finally, by assumption, (∂G σ ) H is a Z-set in EG σ H . Thus, the same reasoning as in Lemma 8.5 and Lemma 8.6 shows that the lemma 8.3 of BestvinaMess applies, thus implying that (EG H , EG H \ EG H ) is a Z-compactication,

		H is
	contractible. Now, notice that, because of Lemma 8.20, EG H	is obtained from EG H by the same
	procedure as before, compatifying every EG H

σ (for σ a simplex xed under H) by EG σ H and adding the visual boundary of the CAT(0) subcomplex X H , ∂(X H ) = (∂X) H . We now briey indicate why this is enough to prove the Z-set property for (EG H , EG H \ EG H ). and we are done.

Proof. We prove the lemma by contradiction. Let x and σ be as in the statement of the lemma. Let z ∈ D(ξ) be such that x ∈ [v 0 , z] and z ∈ D(ξ) be such that the geodesic segment [x, z ] meets σ. Let σ be the last simplex touched by [v 0 , z] before meeting D(ξ), and v a vertex of σ .

We now dene the subsets C σ for simplices σ ⊂ X . Let σ be such a simplex, and let σ be the unique open simplex of X such that σ ⊂ σ as subsets of X. We set C σ = EG σ .

Proof. We prove the result by induction on n, by using the denition of nested families.

The result for n = 1 follows from the denition of a nested family. Suppose the result true for 1, . . . , n, and let σ 1 , . . . , σ n+1 be a path of simplices in N D(ξ) \ D(ξ) and U [0] ⊂ . . . ⊂ U [n+1] = U. By induction, the result is true for the path σ 1 , . . . , σ n and the ltration U [0] ⊂ . . . ⊂ U [n] , so the only inclusion to be proved is the aforementionned one for σ n+1 .

If σ n ⊂ σ n+1 , every vertex v of σ n is also a vertex of σ n+1 , so the result is already true for vertices of D(ξ) contained in σ n . Now by the denition of ξ-families (see Denition 4.11), this implies the result for every vertex of D(ξ) ∩ σ n+1 .

7

Metrisability of EG.

In this section, we prove that EG is a compact metrisable space. Recall that by the classical metrisation theorem, it is enough to prove that EG is a second countable Hausdor regular space (see below for denitions) which is sequentially compact.

Weak separation

In this paragraph, we prove the following: Proposition 7.1. The space EG satises the T 0 condition, that is, for every pair of distinct points, there is an open set of EG containing one but not the other.

Note that this property does not imply that the space is Hausdor. However, we will prove in the next subsection that EG is also regular, and it is a common result of pointset topology that a space that is T 0 and regular is also Hausdor. As usual, the proof of Proposition 7.1 splits in many cases. Lemma 7.2. Let x, y be two distinct points of EG ⊂ EG. Then x and y admit disjoint neighbourhoods.

Proof. Open sets in EG are open in EG by denition. The result thus follows from the fact that EG is a Hausdor space. Lemma 7.3. Let η, η be two distinct points of ∂X ⊂ EG. Then η and η admit disjoint neighbourhoods.

Proof. The space X is metrisable, hence Hausdor. Choosing disjoint neighbourhoods U of η in X (resp. U of η in X ) yield disjoint neighbourhoods V U (η), V U (η ). Lemma 7.4. Let x ∈ EG and η ∈ ∂X. Then x and η admit disjoint neighbourhoods.

Proof. Let x = p( x) ∈ X. Since X is a Hausdor space, let U be a neighbourhood of x in X and U be a neighbourhood of η in X that are disjoint. Then p -1 (U ) is a neighbourhood of x in EG and V U (η) is a neighbourhood of η in EG that is disjoint from p -1 (U ).

Lemma 7.5. Let ξ ∈ ∂ Stab G and η ∈ ∂X. Then there exists a neighbourhood of η in EG that does not contain ξ.

Proof. Since D(ξ) is bounded, let R > 0 such that the D(ξ) is contained in the R-ball centred at v 0 . Now take a neighbourhood U of η in X that does not meet that R-ball. The subset V U (η) is a neighbourhood of η in EG to which ξ does not belong. Lemma 7.6. Let x ∈ EG and ξ ∈ ∂ Stab G. Then there exists a neighbourhood of x in EG that does not contain ξ.

The Z-set property

Here we prove the following: Proposition 8.2. ∂G is a Z-set in EG.

Proving this property is generally technical. However, Bestvina and Mess proved in [START_REF] Bestvina | The boundary of negatively curved groups[END_REF] a useful lemma ensuring that a given set is a Z-set in a bigger set, which we now recall. Lemma 8.3 (BestvinaMess [4]). Let ( X, Z) be a pair of nite-dimensional metrisable compact spaces with Z nowhere dense in X, and such that X = X \ Z is contractible and locally contractible, with the following condition holding:

(*) For every z ∈ Z and every neighbourhood U of z in X, there exists a neighbourhood V contained in U and such that

Then X is an Euclidian retract and Z is a Z-set in X.

We now use this lemma to prove that the boundary ∂G is a Z-boundary in EG.

Lemma 8.4. EG and ∂G are nite-dimensional.

Proof. We have

Each vertex stabiliser boundary is a Z-boundary in the sense of Bestvina, hence nitedimensional, and they are closed subspaces of ∂G by Proposition 6.19. As the action of G on X is cocompact, their dimension is uniformly bounded above, so the countable union theorem implies that

space of nite geometric dimension, so its boundary has nite dimension by a result of Caprace [START_REF] Caprace | At innity of nite-dimensional CAT(0) spaces[END_REF]. Thus, the classical union theorem implies that ∂G is nite-dimensional. Now EG = EG ∪ ∂G. EG is a CW-complex that can be decomposed as the countable union of its closed cells, all of which have a dimension bounded above by dim(X ) • sup σ (dim EG σ ).

It follows from the countable union theorem in covering dimension theory that EG is nite dimensional, and the same holds for EG by the classical union theorem.

We now turn to the proof of the Z-set property, using the lemma of BestvinaMess recalled above. As usual, the proof splits in two cases, depending on the nature of the point of ∂G that we consider.

Proof of the main theorem.

We are now ready to conclude the proof of the Combination Theorem for boundaries of groups.

Lemma 8.14. Let X, Y and G as in the statement of the main theorem. Then for every simplex σ of Y , the embedding EG σ → EG realises an equivariant homeomorphism from ∂G σ to ΛG σ ⊂ ∂G. Moreover, for every pair H 1 , H 2 of subgroups in the family

Proof. The equivariant embedding EG σ → EG induces an equivariant embedding ∂G σ → ΛG σ ⊂ ∂G. But since EG σ is a closed subspace of EG by Proposition 6.19, and which is stable under the action of G σ , the reverse inclusion ΛG σ ⊂ ∂G σ follows. Now let σ 1 , . . . , σ n be simplices of X. The inclusion

Λ(

1≤i≤n

is clear, and the reverse inclusion follows directly from Lemma 4.7.

Lemma 8.15. Let X and G be as in the statement of the main theorem. Then for every simplex σ of X, the embedding EG σ → EG satises the convergence property 4.8.

Proof. Let (g n G σ ) be a sequence of distinct G-cosets. This yields an injective sequence of simplices (g n σ) of X. Let x be any point of EG σ . By compactness of EG, we can assume up to a subsequence that g n x converges to a point l ∈ EG. But it follows immediately from Lemma 7.14 and Lemma 7.15 that l ∈ ∂G and that g n EG σ uniformly converges to l.

Lemma 8.16. Let X and G be as in the statement of the main theorem. Then for every simplex σ of X, the group G σ is of nite height in G.

Thus the simplices g 1 σ, . . . , g n σ of X are distinct and such that the boundary of their stabilisers have a nonempty intersection in ∂ Stab G. The importance of such ner structures comes from the following implication.

Theorem 8.18 (CarlssonPedersen [START_REF] Carlsson | Controlled algebra and the Novikov conjectures for Kand L-theory[END_REF]). If G admits an EZ-structure in the sense of CarlssonPedersen, then G satises the integral Novikov conjecture.

In our context, we will need an additional assumption on these EZ-structures. This will be done by constructing an EZ-structure for G and proving that G is a uniform convergence group on its boundary. Note that this proof has the advantage of yielding a construction of the Gromov boundary of G.

In the following, G(Y) will be a complex of groups over a simplicial complex Y satisfying the conditions of the Combination Theorem for hyperbolic groups. We will denote by G the fundamental group of G(Y) and by X a universal covering. 9.1 A few facts about hyperbolic groups and quasiconvex subgroups.

We start by recalling here a few elementary facts about hyperbolic groups. There is an extensive litterature about such groups, and we refer the reader to [START_REF] Coornaert | Géométrie et théorie des groupes[END_REF], [START_REF] Gromov | Hyperbolic groups[END_REF] for more details.

Lemma 9.1.

• (Gromov [23,p.164]) Let H be a hyperbolic group, and

Corollary 9.2. Let Γ be a nite connected graph contained in the 1-skeleton of X, and

Proof. This follows from an easy induction on the number of vertices of Γ, together with Lemma 9.1.

Recall that in the case of a hyperbolic group H, there is a very explicit example of a classifying space for proper actions, namely the Rips complex. Moreover, there is a natural notion of boundary, namely the Gromov boundary of H (see [START_REF] Coornaert | Géométrie et théorie des groupes[END_REF]).

Construction of an EZ-complex of space compatible with G(Y).

We now dene an EZ-complex of spaces over Y as follows:

• We dene inductively sets of generators for the local groups of the complex of groups G(Y) induced over Y in the following way: Start with simplices σ of Y of maximal dimension, and choose for each of them a nite symmetric set of generators for G σ . Suppose we have dened a set of generators for local groups over simplices of dimension at most k. If σ is a simplex of dimension k -1, choose a nite set of generators which contains all the generators of local groups of simplices strictly containing σ. This allows us to dene for every simplex σ of Y a set of generator such that ψ σ,σ (S σ ) ⊂ S σ whenever σ ⊂ σ .

• Let n ≥ 1 be an integer. Dene D σ as the Rips complex P n (G σ ) associated to the set of generators S σ . Moreover, if σ ⊂ σ , let φ σ,σ be the equivariant embedding

• Since there are only nitely many hyperbolic groups involved, choose n ≥ 0 such that all the previously dened Rips complexes are contractible.

It follows from the above discussion that Proposition 9.4. The complex of spaces D(Y) is compatible with the complex of groups G(Y).

Lemma 9.5. The EZ-complex of spaces D(Y) satises the limit set property 2.9.

Proof. For every pair of simplices

by a result of Bowditch [START_REF] Bowditch | Convergence groups and conguration spaces[END_REF].

For every simplex σ of Y , the family

is contained in the family of quasiconvex subgroups of G σ . Indeed, let g 0 , . . . , g n be elements of G. Then, as X is CAT(0),

, where Γ is a graph containing all the vertices of the simplices g 0 σ, . . . , g n σ and contained in the convex hull of the g 0 σ, . . . , g n σ. For such subgroups, the equality ΛH 1 ∩ ΛH 2 = Λ(H 1 ∩ H 2 ) holds by Lemma 9.1.

Lemma 9.6. The EZ-complex of spaces D(Y) satises the convergence property 4.8.

Proof. This is Proposition 1.8 of Dahmani [START_REF] Dahmani | Combination of convergence groups[END_REF]. Note that this corollary does not use the hyperbolicity of X. 9.3 Background on convergence groups and hyperbolicity. Denition 9.9 (convergence group). A group Γ acting on a compact metrisable space M with more than two points is called a convergence group if, for every sequence (γ n ) of elements of Γ, there exists two points ξ + and ξ -in M and a subsequence (γ ϕ(n) ), such that for any compact subspace K ⊂ M \ {ξ -}, the sequence (γ ϕ(n) K) of translates uniformly converges to ξ + .

A hyperbolic group Γ is always a convergence group on Γ ∪ ∂Γ (see for instance [START_REF] Freden | Negatively curved groups have the convergence property[END_REF]).

A direct consequence is the following: Proposition 9.10. Let Γ be a hyperbolic group and EΓ an EZ-structure obtained as in Theorem 9.3. Then Γ is a convergence group on EΓ. Denition 9.11 (conical limit point). Let Γ be a convergence group on a compact metrisable space M . A point ζ in M is called a conical limit point if there exists a sequence (γ n ) of elements of Γ and two points ξ -= ξ + in M , such that γ n ζ → ξ -and γ n ζ → ξ + for every ζ = ζ in M . The group Γ is called a uniform convergence group on M if M consists only of conical limit points. Theorem 9.12 (Bowditch [5]). Let Γ be a uniform convergence group on a compact metrisable space M with more than two points. Then Γ is hyperbolic and M is Γ-equivariantly homeomorphic to the Gromov boundary of Γ.

A combination theorem.

We now prove that G is a hyperbolic group, by proving that it is a uniform convergence group on its boundary ∂G.

So far, the topology on EG and ∂G was dened by choosing a specic, although arbitrary, basepoint. In forthcoming proofs, we will choose neighbourhoods centred at points which are relevant to the geometry of the problem. Denition 9.13. Let δ ≥ 0 be such that the space X is δ-hyperbolic. We denote by ., .

the Gromov product on X and an extension to X. For z ∈ X, k ≥ 0 and x 0 ∈ X a basepoint, let

For η ∈ ∂X and k ≥ 0, the family of subsets (W k (η)) forms a basis of (not necessarily open) neighbourhoods of η in X. (g n ), and points ξ + , ξ -∈ ∂G v such that for every compact subset K of EG v \ ξ -, the sequence of translates g n K uniformly converges to ξ + . By denition of ξ + and ξ -, we already have that the sequence g n (EG v 0 \ U v 0 ) uniformly converges to ξ + in ∂G v 0 . We thus have that g n (EG σ \ U v 0 ) uniformly converges to ξ + in ∂G v . Since EG σ \ U v 0 is innite by construction, this implies that ξ + = ξ + . If we had ξ -= ξ -, then g n EG σ would uniformly converge to ξ + , contradicting the fact that g n EG σ = EG σ since g n xes σ. Therefore ξ -= ξ -. This implies that g n (∂G v \ U v ) uniformly converges to ξ + in ∂G v . Since F is nite, an easy induction shows that there exists a subsequence, still denoted (g n ), such that g n (∂G v \ U v ) uniformly converges to ξ + in ∂G v for every vertex v of F . Let x ∈ K, and x ∈ p( x) \ F . Let σ be the rst simplex touched by [v 0 , x] after leaving F . It follows from the denition of F that the sequence of simplices (g n σ) is such that for some (hence any) vertex v of σ ∩ F , the sequence of (∂G gnσ ) uniformly converges to ξ + in ∂G v . It follows from the convergence criterion 7.16 that the sequence (g n x) converges to ξ + . Since x / ∈ V U ,ε (ξ -), we have ∂G σ ⊂ U v for some (hence any

this being true for every x ∈ K and x ∈ p( x) \ F . We already have that for every vertex v of F , the sequence of g n .(∂G v \ U v ) uniformly converges to ξ + by the above discussion. As F is a nite subcomplex of X, the convergence criterion 7.16 now shows that the sequence (g n .K) uniformly converges to ξ + . Lemma 9.15. Let (g n ) be an injective sequence of elements of G. Suppose that for some (hence any) vertex v the sequence (g n v) is bounded, but there do not exist vertices v 0 and v 1 of X such that g n v 0 = v 1 for innitely many n. Then there exist ξ + , ξ -∈ ∂G and a subsequence (g ϕ(n) ) such that for every compact subset K of ∂G \ {ξ -}, the sequence of translates g ϕ(n) K uniformly converges to ξ + . bounded below: indeed, if this was not the case, the same reasoning as in Claim 1 would imply the existence of simplices σ, σ of X such that g n σ ∩σ = ∅. This in turn would imply that, up to a subsequence, there exist subsimplices τ ⊂ σ and τ ⊂ σ such that g n τ = τ , which was excluded. Thus, let ε > 0 be such a uniform bound. Let also

Now consider the neighbourhood V M,ε (η) of η in X. Let x ∈ X be in that neighbourhood, and let γ be a parametrisation of the geodesic from v 0 to x. Suppose by contradiction that the geodesic from g n v 0 to g n x does meet D(ξ + ). Then, by denition of M , the geodesic segment g n γ [0, M ] meets D(ξ + ). But as this geodesic segment is in the open ε-neighbourhood of [g n v 0 , g n η), we get our contradiction from the denition of ε. Thus for every x in V M,ε (η), the geodesic from g n v 0 to g n x does not meet D(ξ + ), and we are done. Lemma 9.16. Let (g n ) be an injective sequence of elements of G, and suppose that for some (hence every) vertex v 0 of X, d(v 0 , g n v 0 ) → ∞. Since (EG, ∂G) is an EZ-structure for G by Theorem 8.1, we can assume up to a subsequence that there exist ξ + , ξ -∈ ∂G such that for every compact subset K ⊂ EG, we have g n K → ξ + and g -1 n K → ξ -. Then there exists a subsequence (g ϕ(n) ) such that for every compact subset K of ∂G \ {ξ -}, the sequence of translates g ϕ(n) K uniformly converges to ξ + .

Proof. If ξ -∈ ∂X, let U be a neighbourhood of ξ -in ∂X and K = ∂G \ V U (ξ -). Since X has nitely many isometry types of simplices, it follows from Lemma 6.3 that we can choose a subneighbourhood U of U containing ξ -and such that any path from U ∩ X to X \ U meets at least d max simplices.

If ξ -∈ ∂ Stab G, let U be a ξ --family, and ε ∈ (0, 1), and let K = ∂G \ V U ,ε (ξ -). We also choose another ξ --family U which is 2d max -rened in U. We want to prove that (g n K) uniformly converges to ξ + . Recall that the sets W k (g n v 0 ) were dened in 9.13. Claim 1: For every k, the following holds:

We split the proof in two cases.

Suppose that ξ -∈ ∂X. First notice that since g -1 n v 0 → ξ -, there exists a constant C such that for every n ≥ 0 and every x / ∈ U , we have g -1 n v 0 , x v 0 ≤ C. Since we also have d(g -1 n v 0 , v 0 ) → ∞, the claim follows. Suppose now that ξ -∈ ∂ Stab G. We start by proving by contradiction that there exists a constant C such that for every n ≥ 0 and every x /

59

The containment lemma 3.3 yields a constant m such that a geodesic path of length at most δ meets at most m simplices, where δ is the hyperbolicity constant of X. Let U be a ξ --family that is m-nested in U . Since we are reasoning by contradiction, then, up to a subsequence, there exist points

for n large enough. Thus, for n large enough, there exist points

and a path of simplices of length at most m between a n and b n which is contained in X \ D(ξ -) (see Figure 7).

The renement lemma 5.10 now implies that a n and y n both are in Cone U ,ε (ξ -) for n large enough, a contradiction. Now the same reasoning as before shows that for every k ≥ 0, there exists N such that for every n ≥ N and every x /

Claim 2: For every k, we have g n p(K) ⊂ W k (g n v 0 ) for n large enough.

Suppose that ξ -∈ ∂X. By denition of U , we have that for every z ∈ K, p(z)∩U = ∅. Thus p(K) ⊂ X \ U and the result follows from Claim 1.

Suppose now that ξ -∈ ∂ Stab G, and let z ∈ K. Suppose by contradiction that p(z)

follows from the renement lemma 5.10 and Lemma 6.20 that z ∈ V U ,ε (ξ -), a contradiction. Thus p(K) ⊂ X \ Cone U ,ε (ξ -) and the result follows from Claim 1. Claim 3: g n K uniformly converges to ξ + .

Once again, we split the proof in two cases.

Suppose that ξ + ∈ ∂X. Then, as g n v 0 → ξ + , it follows from Claim 2 that for every k, g n p(K) ⊂ W k (ξ + ) for n large enough. It then follows from the convergence criterion 7.16 that g n K uniformly converges to ξ + .

Suppose now that ξ + ∈ ∂ Stab G. Let U + be a ξ + -family and ε ∈ (0, 1). Since X is δ-hyperbolic, let m be a constant such that a geodesic path of length at most δ meets at most m simplices, and let U + be a ξ + -family that is m-nested in U + . As g n x 0 → ξ + for any x 0 ∈ EG v 0 , we have g n v 0 ∈ Cone U + ,ε (ξ + ) for n large enough. For every T ≥ 0, we can choose n large enough so that the geodesic segments [v 0 , g n v 0 ] and [v 0 , g n x], x ∈ p(K), remain δclose up to time T (if we choose k large enough in Claim 2). In particular, we can choose k and N large enough so that, for every n ≥ N and every x ∈ p(K), there exists a path of simplices of length at most m in X \ D(ξ + ) between a point of [v 0 , g n v 0 ] ∩ Cone U + ,ε (ξ + ) and a point of [v 0 , g n x]. The renement lemma 5.10 now implies that g n p(K) ⊂ Cone U + ,ε (ξ + ) for n ≥ N , hence g n K ⊂ V U + ,ε (ξ + ) for n ≥ N . Thus, g n K uniformly converges to ξ + .

Corollary 9.17. The group G is a convergence group on ∂G.

Proof. This follows from Lemma 9.14, Lemma 9.15 and Lemma 9.16.

To prove that G is hyperbolic, it remains to show that every point of ∂G is conical. Lemma 9.18. Every point of ∂G is a conical limit point for ∂G.

Proof. Consider rst a point in ∂G v for some vertex v of X. It is a conical limit point for G v on ∂G v , since G v is hyperbolic. Therefore it is a conical point for G v on ∂G, hence for G since G is a convergence group on ∂G by Corollary 9.17. Now consider a point η ∈ ∂X. Since the action of G on X is cocompact, we can nd a sequence (g n ) of elements of G and a simplex σ such that (g n σ) uniformly converges to η in X and such that for every n, the geodesic ray [v 0 , η) meets the interior of g n σ. Let v be a vertex of σ and x ∈ EG v .

Claim : Up to multiplying each g n on the right by an element of G v and taking a subsequence, we can further assume that g -1 n x converges to a point ξ -∈ ∂G \ ∂G v .

Consider the rst simplex touched by the geodesic [v, g -1 n v] after leaving v. Since the action of G on X is cocompact, we can assume up to a subsequence that this sequence of simplices is in the same G-orbit. Now up to multiplying each g n by an element of G v , we can further assume that this sequence of simplices is constant at a unique simplex σ 1 . Up to a subsequence, we can further assume that all the geodesic segments [v, g -1

n v] leave σ 1 along the same open simplex τ 1 . Now consider the simplex σ

and let y n be a projection of x n on the quasiconvex subset Q 1 . By denition of Q 1 , there exists an element h n ∈ G σ 1 ⊂ G v such that h n y n = y. This implies that for every n, the subset h n EG σ (n) 2 contains a point that projects to y. In particular, no subsequence of h n EG σ (n) 2 converges to a point of ∂G σ 1 . Suppose by contradiction that there exists a subsequence of h n EG σ (n) 2 which converges to a point z ∈ ∂G τ 1 . Since G τ 1 is a convergence group on EG τ 1 by Proposition 9.10, it follows that for every x ∈ EG τ 1 except maybe one point, h n x converges to z. But as Q 1 is stable under all the h n , such a z belongs to ∂G σ 1 ⊂ ∂G τ 1 , contradicting the fact that no subsequence of h n EG σ (n) 2 converges to a point of ∂G σ 1 . Thus, no subsequence of h n EG σ (n) 2 converges to a point of ∂G τ 1 and the convergence property 4.8 now implies that, up to a subsequence, we can assume that

is constant. Up to a subsequence, we can further assume that σ (n) 2 is constant at σ 2 and every geodesic segment [v, g -1

n v] leaves σ 2 along the same open simplex τ 2 . In view of the above, we replace the sequence (g n ) by (g n h -1 n ). Now one of the following happens:

(i) Suppose that G σ 1 ∩ G σ 2 is nite. By applying the same reasoning as in the proof of the compactness lemmas 7.14 and 7.15, either there exists a subsequence of (g n ) such that g -1 n x converges to a point of ∂X and we are done, or the path of simplices σ 1 , σ 2 extends to a path of simplices σ 1 , . . . , σ m which are crossed by every geodesic segment [v, g -1

n v] and g -1 n x converges to a point ξ -∈ ∂G σm . As D(ξ -) is convex by Proposition 4.2 and G σ 1 ∩ G σ 2 is nite, it follows from Lemma 4.7 that ξ -/ ∈ ∂G v and we are done.

(ii

3 be the simplex touched by [v, g -1 n v] after leaving τ 2 , and let Q 2 be a G σ 1 ∩ G σ 2 -orbit in EG τ 2 . Note that Q 2 is quasiconvex in EG τ 2 by Lemma 9.2. We are thus back to the previous situation with EG τ 2 instead of EG τ 1 , EG σ (n)

We claim that this procedure eventually stops. Indeed, the containment lemma 3.3 yields a constant m such that every geodesic meeting at least m simplices has length at least A, where A is the acylindricity constant. Thus, after at most m applications of this algorithm, we get to situation (i), which concludes the proof of the claim.

By the above discussion, we already have that g -1 n x → ξ -for every x ∈ EG v . Thus, by Lemma 9.16, it is enough, in order to prove Lemma 9.18, to show that g -1 n η does not converge to ξ -, which we now prove by contradiction.

Suppose g -1 n η was converging to ξ -. For every n, let x n be a point of [g -1 n v, g -1 n η) that is contained in the interior of σ. Since the geodesic ray [g -1 n v, g -1 n η) meets σ for every n, the Gromov product g -1 n v, g -1 n η v is bounded. Thus, ξ -cannot belong to ∂X, and ξ -∈ ∂ Stab G.

62

Now since both g -1 n η and g -1 n x converge to ξ -∈ ∂ Stab G, both geodesics [v, g -1 n η) and [v, g -1

n v] must go through D(ξ -) for n large enough. But Lemma 3.7 and Lemma 5.8 imply that for n large enough and any x ∈ σ, both geodesic rays [x, g -1 n η) and [x, g -1 n v] also meet D(ξ -). In particular, [x n , g -1 n η) and [x n , g -1 n v] meet D(ξ -) for n large enough. As D(ξ -) is convex by Proposition 4.2, this implies that x n belongs to D(ξ -), hence so does v, which is absurd by construction of (g n ).

Corollary 9.19. G is a hyperbolic group and ∂G is G-equivariantly homeomorphic to its Gromov boundary.

Proof. The group G is a convergence group on ∂G by Corollary 9.17, and every point of ∂G is conical by Lemma9.18, thus the result follows from Theorem 9.12.

To conlude the proof of the Combination Theorem for hyperbolic groups, it remains to show that stabilisers embed as quasiconvex subsets. Proposition 9.20. Stabilisers of simplices of X are quasiconvex subgroups of G.

Proof. It is enough to prove the result for the stabiliser of a vertex v of X. Notice that, by Proposition 6.19, the boundary of G v embeds G v -equivariantly in ∂G, the latter being G-equivariantly homeomorphic to the Gromov boundary of G by Corollary 9.19. Hence, the result follows from a result of Bowditch [START_REF] Bowditch | Convergence groups and conguration spaces[END_REF] recalled in the introduction.