Rémi Coulon

Infinite periodic groups of even exponents

Keywords: Burnside groups, periodic groups, geometric group theory, small cancellation theory, hyperbolic geometry. MSC. 20F50, 20F65, 20F67, 20F06

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Introduction

Historical background

A group G has exponent n ∈ N, if g n = 1 for every g ∈ G. In 1902, Burnside asked whether every finitely generated group of finite exponent was necessarily finite [START_REF] Burnside | On an unsettled question in the theory of discontinuous groups[END_REF]. Despite its simplicity, this question remained open for a long time and motivated many developments in group theory. The class of groups of exponent n forms a group variety whose free elements are the free Burnside groups of exponent n. More concretely, the free Burnside group of rank r and exponent n, that we denote by B r (n), admits the following presentation B r (n) = a 1 , . . . , a r | x n , ∀x .

A major breakthrough in the subject was achieved by Novikov and Adian in 1968 [START_REF] Novikov | Infinite periodic groups[END_REF]. They proved that B r (n) is infinite provided r 2 and n is a sufficiently large odd exponent. Later Ol'shanskȋ provided an alternative proof of the same result [START_REF] Ol'shanskii | The Novikov-Adyan theorem[END_REF]. Despite these progresses the case of even exponents held up longer. It was only in the early 90's that Ivanov [START_REF] Ivanov | The free Burnside groups of sufficiently large exponents[END_REF] and Lysenok [START_REF] Lysenok | Infinite Burnside groups of even period[END_REF] independently proved that free Burnside groups of sufficiently large even exponents are also infinite.

The aforementioned results rely (more or less explicitly) on an iterated version of small cancellation theory (using combinatorics on words and/or reasoning on van Kampen diagrams). Historically, the respective works of Novikov-Adian and Ol'shanskiȋ appeared before the theory of hyperbolic spaces/groups was formalized and developed by Gromov in [START_REF] Gromov | Hyperbolic groups[END_REF]. However working with groups acting on hyperbolic spaces provides a perfect framework offering new insights into small cancellation theory. See for instance [START_REF] Guirardel | Geometric small cancellation[END_REF][START_REF] Coulon | Théorie de la petite simplification: une approche géométrique[END_REF] for a survey on the topic. Using this geometric point of view, Delzant and Gromov revisited the Burnside problem making an explicit use of hyperbolic geometry [START_REF] Delzant | Courbure mésoscopique et théorie de la toute petite simplification[END_REF]. Nevertheless their work only applies to odd exponents.

Main results. In this article we provide a new approach to the free Burnside groups of even exponents based on the geometrical ideas of Delzant and Gromov [START_REF] Delzant | Courbure mésoscopique et théorie de la toute petite simplification[END_REF]. More precisely we prove the following statement (compare with Ivanov [START_REF] Ivanov | The free Burnside groups of sufficiently large exponents[END_REF] and Lysenok [START_REF] Lysenok | Infinite Burnside groups of even period[END_REF]).

Theorem 1.1. Let r 2. There exists a critical exponent n 0 ∈ N such that for every integer n n 0 , the free Burnside group B r (n) is infinite.

Not only is our approach substantially shorter than the one of Lysenok and Ivanov (200 and 300 pages respectively) it also gives a way to produce (partially) periodic quotients of many groups as soon as they carry a certain form of negative curvature, far beyond the single instance of hyperbolic groups. Let us mention a few examples. The next theorem is originally due to Ol'shanskiȋ and Ivanov [START_REF] Ol'shanskii | Periodic quotient groups of hyperbolic groups[END_REF][START_REF] Ivanov | Hyperbolic groups and their quotients of bounded exponents[END_REF] answering a question of Gromov [START_REF] Gromov | Hyperbolic groups[END_REF]. Given an arbitrary group G, we write G n for the (normal) subgroup of G generated by the n-th power of all its elements. Theorem 1.2 (Ol'shanskiȋ-Ivanov [START_REF] Ivanov | Hyperbolic groups and their quotients of bounded exponents[END_REF]). Let G be a non-elementary hyperbolic group. There exist p, n 0 ∈ N, such that for every integer n n 0 that is a multiple of p, the quotient G/G n is infinite. Moreover

n 1 G n = {1}.
More generally if G is a group acting acylindrically on a Gromov hyperbolic space X (see Section 5.2 for a precise definition) then for arbitrarily large exponents n ∈ N, we are able to produce a partially n-periodic quotient of G (Theorem 5.7), i.e. a quotient Q of G such that (i) every elliptic subgroup of G (for its action on X) embeds in Q, (ii) for every q ∈ Q, either q is the image of an elliptic element of G or q n = 1.

Applied to the mapping class group of a surface acting on the curve graph it yields the following statement Theorem 1.3. Let Σ be a compact surface of genus g with k boundary components such that 3g + k -3 > 1. There exist p, n 0 ∈ N such that for every integer n n 0 which is a multiple of p, there exists a quotient Q of the mapping class group MCG(Σ) with the following properties.

(i) If E is a subgroup of MCG(Σ) that does not contain a pseudo-Anosov element, then the projection MCG(Σ) Q induces an isomorphism from E onto its image.

(ii) Let f be a pseudo-Anosov element of MCG(Σ). Either f n = 1 in Q or f coincide in Q with a periodic or a reducible element.

(iii) There are infinitely many elements in Q which are not the image of a periodic or reducible element of MCG(Σ). Any non-trivial element in the kernel of MCG(Σ) Q is pseudo-Anosov.

Bass-Serre theory also provides examples of groups acylindrically on a tree for which our approach works (Theorem 5.15). For instance if G = A * B is a free product, the corresponding quotient Q corresponds to the n-periodic product of A and B, see for instance [START_REF] Adian | Periodic products of groups. Akademiya Nauk Soyuza Sovetskikh Sotsialisticheskikh Respublik[END_REF]. Note that the same strategy could also be used to study the outer automorphism group of free Burnside groups of even exponents, extending some other work of the author [START_REF] Coulon | Outer automorphisms of free Burnside groups[END_REF]. Nevertheless to limit the length of the article we decided not to detail that part.

A geometrical approach

Let us highlight a few important ideas involved in the proofs. For simplicity we restrict our attention to free Burnside groups as this case already covers all the difficulties. As shown by Ivanov and Lysenok, free Burnside groups of (sufficiently large) odd or even exponent have a considerably different algebraic structure. For instance if n is odd, every finite subgroup of B r (n) is cyclic. By contrast if n is even, B r (n) contains arbitrarily long direct products of dihedral groups. Nevertheless the global strategy to study those groups remains the same.

A sequence of approximation groups. Let n ∈ N be a large exponent. All known strategies for studying Burnside groups start in the same way: one produces by induction an approximation sequence of hyperbolic groups

F r = G 0 G 1 G 2 • • • G k G k+1 . . . (1)
whose direct limit is exactly B r (n). At each step G k+1 is obtained from G k by adding new relations of the form h n = 1, where h runs over the set of all "small" loxodromic elements of G k . The goal is to prevent this sequence to collapse to a finite group. This is achieved by small cancellation arguments. The novelty of our method is to use a geometric point of view on small cancellation à la Delzant-Gromov in the context of torsion groups of even exponent.

Geometric small cancellation. Let S be a finite set and R a collection of cyclically reduced words of F(S). Assume for simplicity that R is invariant under taking cyclic permutations and inverses and write for the length of its shortest element. Given λ ∈ (0, 1), the group presentation S|R satisfies the classical C (λ) small cancellation condition1 if every prefix u of two distinct relations in R has length at most |u| < λ . For small values of λ, one understands precisely the properties of the corresponding group Ḡ = F(S)/ R . For instance if λ 1/6, then Ḡ is hyperbolic. The C (λ) condition can advantageously be reformulated as follows. Let X be the Cayley graph of F(S) with respect to S. The presentation S|R satisfies the C (λ) condition if for every distinct r 1 , r 2 ∈ R, the overlap between the respective axes of r 1 and r 2 has length less then λ , where is also the smallest translation length of an element in R.

With this idea in mind, one can extend the small cancellation theory to the context of hyperbolic groups [START_REF] Gromov | Hyperbolic groups[END_REF][START_REF] Ol'shanskii | On residualing homomorphisms and G-subgroups of hyperbolic groups[END_REF][START_REF] Delzant | Sous-groupes distingués et quotients des groupes hyperboliques[END_REF] (or more generally of groups acting on a hyperbolic space). Let G be a non-elementary group acting properly cocompactly by isometries on a hyperbolic space X and R a subset of G, which is invariant under conjugation. Roughly speaking we will say that R satisfies a small cancellation condition if given any two distinct r 1 , r 2 ∈ R, the length ∆(r 1 , r 2 ) on which the respective axes of r 1 and r 2 fellow travel is very small compare to the translation lengths r 1 and r 2 (see Figure 1). In this sit-

∆ ( r1 , r2 ) x r 1 x r 2 x
Axis(r 1 ) Axis(r 2 )

Figure 1: Overlap between two relations seen in the hyperbolic space X. The translation length r i of r i is roughly the distance between x and r i x.

uation, the quotient Ḡ = G/ R is still a non-elementary hyperbolic group [START_REF] Gromov | Hyperbolic groups[END_REF][START_REF] Delzant | Sous-groupes distingués et quotients des groupes hyperboliques[END_REF][START_REF] Ol'shanskii | On residualing homomorphisms and G-subgroups of hyperbolic groups[END_REF]. Under this hypothesis, Gromov explains in [START_REF] Gromov | Mesoscopic curvature and hyperbolicity[END_REF] how to let this group act on a hyperbolic space X whose geometry is finer than the one of the Cayley graph of Ḡ, see also [START_REF] Delzant | Courbure mésoscopique et théorie de la toute petite simplification[END_REF][START_REF] Coulon | Asphericity and small cancellation theory for rotation families of groups[END_REF][START_REF] Coulon | On the geometry of Burnside quotients of torsion free hyperbolic groups[END_REF]. Assume for instance that Ḡ is a quotient of the form Ḡ = G/ h n where h is a loxodromic element and n a (large) exponent. Then M = X/ Ḡ can be seen as an orbifold (whose fundamental group is Ḡ and universal cover is X) and comes with an analog of Margulis' thin/thick decomposition for hyperbolic manifolds. The thin part corresponds to the neighborhood of a single singular point whose isotropy group is exactly the maximal finite subgroup F ⊂ Ḡ containing the image of h. The pre-image in X of the thin part is roughly speaking the collection of all Ḡ-translates of an F -invariant hyperbolic disc D ⊂ X. Moreover there exists a natural map q : F → D n , where D n , is the dihedral group of order 2n, such that the action of F on D is identified via q to the natural action of D n on the disc.

Adopting this point of view, we associate to each approximation group G k in (1) a hyperbolic space X k on which G k acts properly co-compactly. The goal will be to prove that, at each step, the new relations defining G k+1 will satisfy a small cancellation condition in the above sense (i.e. relative to the action of G k on the space X k ). It has the following main advantage: almost every needed property of the relations defining G k is captured by the hyperbolicity of X k . Consequently when studying the quotient map G k G k+1 one can completely forget the relations defining G k and rely only on the geometry of X k . Following Delzant-Gromov [START_REF] Delzant | Courbure mésoscopique et théorie de la toute petite simplification[END_REF], this allows us to formulate -unlike in [START_REF] Novikov | Infinite periodic groups[END_REF][START_REF] Ol'shanskii | The Novikov-Adyan theorem[END_REF][START_REF] Lysenok | Infinite Burnside groups of even period[END_REF][START_REF] Ivanov | The free Burnside groups of sufficiently large exponents[END_REF] -the induction hypothesis used to build the approximation sequence (1) in a rather compact form (see Proposition 5.1).

A Margulis' lemma. As mentioned above, the main challenge when building the approximation sequence [START_REF] Adian | Periodic products of groups. Akademiya Nauk Soyuza Sovetskikh Sotsialisticheskikh Respublik[END_REF] is to make sure that G k is not eventually finite. This will not happen if, at each step, the relations (of the form h n = 1) used to define G k+1 from G k satisfy a small cancellation condition. Therefore, given any two loxodromic elements g 1 , g 2 ∈ G k which do not generate an elementary subgroup, one needs to control uniformly, independently of k, the ratio

∆(g 1 , g 2 ) max { g 1 , g 2 } ( 2
)
where ∆(g 1 , g 2 ) measures the "overlap" between the respective axes of g 1 and g 2 in X k (see Figure 1). If X k was a simply connected manifold with pinched negative sectional curvature, such estimate would follow from Margulis' Lemma.

However hyperbolicity only provides an upper bound for the curvature of the space. To bypass this difficulty, one usually uses assumptions on the action of the group (e.g. the fact that the action is proper co-compact or acylindrical).

For instance, a first (naive) attempt to bound the ratio (2) could work as follows. Suppose that g 1 , g 2 ∈ G k are two loxodromic elements such that ∆(g 1 , g 2 ) > N max{ g 1 , g 2 }. It is a standard exercise of hyperbolic geometry to see that there exists a point x ∈ X k such that for every i ∈ 0, N -1 , the commutator [g 1 , g i 2 ] moves x by at most 100δ k (where δ k is the hyperbolicity constant of X k ), see Figure 2. In particular, if N exceeds the number of elements in the

g -i 2 g -1 1 g i 2 g 1 Axis(g 1 )
Axis(g 2 )

x set {u ∈ G k : |ux -x| 100δ k }, then g 1 commutes with a power of g 2 , thus g 1 , g 2 is non-elementary. So, roughly speaking, the ratio (2) is bounded above by the cardinality of the (almost) stabilizers of points in X k . This strategy has a major weakness though: if n is an even exponent, the cardinality of finite subgroups is not uniformly bounded along the sequence (G k ). During the process we will indeed encounter points in X k with arbitrarily large stabilizers. Hence this method cannot be used to keep the ratio (2) uniformly bounded. Any refinement of the above argument using acylindrical actions of G k on X k -see for instance [START_REF] Hull | Small cancellation in acylindrically hyperbolic groups[END_REF][START_REF] Dahmani | Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces[END_REF] -shall fail in the same way.

To bypass this difficulty one associates to the action of G k on X k several numerical invariants: (i) A(G k , X k , d) is characterized as follows: if S is any finite subset of G k generating a non-elementary subgroup, then the set of points in X k which are moved by a distance at most d by every element of S has diameter at most A(G k , X k , d) (see Definition 3.2).

(ii) ν(G k , X k ) is the smallest integer m with the following property: let g, h ∈ G k with h loxodromic . If g, hgh -1 , h 2 gh -2 , . . . , h m gh -m generate an elementary subgroup, then so do g and h (see Definition 3.4).

The quantity A(G k , X k , d) can be thought of as a local version of the ratio [START_REF] Bowditch | Notes on Gromov's hyperbolicity criterion for path-metric spaces[END_REF]. Indeed, its definition only involves "small" elements. Combined with the ν-invariant one recovers the following global analogue of Margulis' Lemma: if g 1 , g 2 ∈ G k do not generate an elementary subgroup, then

∆(g 1 , g 2 ) [ν(G k , X k ) + 2] max { g 1 , g 2 } + A(G k , X k , 400δ k ) + 1000δ k ,
see for instance [START_REF] Coulon | Partial periodic quotients of groups acting on a hyperbolic space[END_REF]Proposition 3.34] or Lemma 3.6. Consequently, in order to make sure that for every k ∈ N, the relations defining G k+1 from G k satisfy a suitable small cancellation condition, it suffices to control (among others) the values of A(G k , X k , 400δ k ) and ν(G k , X k ) all along the approximation sequence [START_REF] Adian | Periodic products of groups. Akademiya Nauk Soyuza Sovetskikh Sotsialisticheskikh Respublik[END_REF]. This was done in [START_REF] Coulon | Partial periodic quotients of groups acting on a hyperbolic space[END_REF] in the absence of even torsion. As soon as even torsion is involved, the situation becomes much more delicate. In particular, the ν-invariant does not behave very well when passing to a quotient (see for instance the discussion and examples at the beginning of Section 4.7.2). It results from the fact that the algebraic structure of finite subgroups of B r (n) is rather intricate, see for instance Lysenok [START_REF] Lysenok | Burnside structures of finite subgroups[END_REF].

Remark 1.4. Note that Hull also developed a small cancellation theory in the context of acylindrically hyperbolic groups [START_REF] Hull | Small cancellation in acylindrically hyperbolic groups[END_REF]. However this work does not provide the necessary tools to control the small cancellation parameters along the sequence [START_REF] Adian | Periodic products of groups. Akademiya Nauk Soyuza Sovetskikh Sotsialisticheskikh Respublik[END_REF] and thus to build infinite torsion groups with bounded exponents. As we explained above, it is not possible to control the acylidricity parameters for the action of G k on X k . The purpose of this article is precisely to develop the implements needed to bypass this difficulty.

Structure of elementary subgroups. As we mentioned earlier, if n is odd, then every maximal finite subgroup of B r (n) is isomorphic to the cyclic group Z n . Moreover finite subgroups "stabilize" along the approximation sequence [START_REF] Adian | Periodic products of groups. Akademiya Nauk Soyuza Sovetskikh Sotsialisticheskikh Respublik[END_REF]. More precisely we have the following property: if F 0 is a non-trivial finite subgroup of some G k , then for every k, every finite subgroup of G containing the image of F 0 actually comes from a finite subgroup F of G k which already contains F 0 . By contrast, if n is even, finite subgroups may "grow" when taking successive quotients. Let us illustrate this fact with the following toy example.

Example 1.5. Assume that n is a (large) even exponent. Start with the free group G 0 = F 2 generated by a and b and set

G 1 = G 0 / a n , b n , (ba) n .
In G 1 the elements s 0 = a n/2 , s 1 = b n/2 and s 2 = (ba) n/2 all generate a subgroup isomorphic to Z 2 . Consequently in the quotient

G 2 = G 1 / (s 0 s 1 ) n , (s 0 s 2 ) n
s 0 and s 1 generate a subgroup isomorphic to the dihedral group D n of order 2n. In particular, s 0 commutes with the involution u 1 = s 0 (s 0 s 1 ) n/2 . Similarly s 0 commutes with the involution u 2 = s 0 (s 0 s 2 ) n/2 . Form now the quotient

G 3 = G 2 / (u 1 u 2 ) n .
Note that s 0 , u 1 and u 2 generate a subgroup isomorphic to Z 2 × D n . In this example,

F 1 = s 0 = Z 2 is a finite subgroup of G 1 . Its image in G 2 (respectively G 3 ) embeds in F 2 = s 0 , s 1 = D n (respectively F 3 = s 0 , u 1 , u 2 = Z 2 × D n ). However F 2 (respectively F 3 ) is not the image of a finite subgroup of G 1 (respectively G 2
). We stopped our example after three steps. However one can proceed further and embeds F 0 in an arbitrarily large product of the form

Z 2 × • • • × Z 2 × D n .
The previous example suggests that G k -and thus B r (n) -contains nested copies of dihedral groups. As mentioned above, one of the advantages of the space X k (compare to the Cayley graph of G k ) is that one "sees" some of these dihedral groups acting as the isometry group of a disc. Unfortunately this is not sufficient to capture all the properties of finite subgroups of G k . To sort a bit this nested structure, we introduce the notion of dihedral germ. A dihedral germ of G k is an elliptic subgroup C (for its action on X k ) containing a subgroup C 0 which is normalized by a loxodromic element and such that [C : C 0 ] is a power of 2. As suggested by the terminology, the dihedral germs are exactly the finite subgroups of G k that may eventually grow in G k+1 , i.e. be embedded in a larger finite subgroup of G k+1 that does not come from G k . This typically arises as follows.

Example 1.6. Assume that A is a finite subgroup of G k and C a subgroup of A of index 2 which is normalized by a loxodromic element h ∈ G k . In particular, A is a dihedral germ. Suppose for simplicity that

h n is trivial in G k+1 . Let a ∈ A \ C. Seen in G k+1 , the group C has index 2 in both A = C, a and B = C, h n/2 .
In particular, A and B generate an elementary subgroup E of G k+1 which is (most of the time) isomorphic to E = A * C B and such that E/C = D ∞ . As an element of G k+1 the product t = ah n/2 has infinite order. However, since B r (n) is the direct limit of (G k ), we have

t n = 1 in G for some > k + 1.
The image in G of E is actually isomorphic to E/ t n which is a finite group that strictly contains the dihedral germ A. Nevertheless there is no finite subgroup F of G k containing A such that the canonical quotient map G k G induces an isomorphism from F onto E/ t n .

It turns out that the dihedral germs of G k are exactly its 2-subgroups (when studying general periodic quotients different from free Burnside groups, those dihedral germs are slightly more complicated). A careful analysis of dihedral germs allows us to prove that every finite subgroup of G k embeds in a direct product of the form

D n × D n2 × • • • × D n2
where n 2 is the largest power of 2 dividing n, see also [START_REF] Ivanov | The free Burnside groups of sufficiently large exponents[END_REF][START_REF] Lysenok | Infinite Burnside groups of even period[END_REF]. In particular, finite subgroups of G k share many identities with finite dihedral groups. Those identities can be used to control a variation on the ν-invariant which captures both geometric and algebraic features of the groups G k and behaves better when taking quotients (see Definition 3.12). Once we control the ν-invariant along the sequence (1), a uniform estimate of the quantity A(G k , X k , d) follows rather easily (Proposition 3.5). Those two invariants (together with the injectivity radius of G k acting on X k ) provide a sufficient control to show that each group G k+1 is actually a small cancellation quotient of G k . Therefore it is hyperbolic and non-elementary, which ensures that at the limit B r (n) is infinite.

Critical exponent. All these arguments actually only work provided n is divisible by a large power of 2, namely 128 for free Burnside groups. Nevertheless, given any two integers p, n ∈ N, the group B r (pn) naturally maps onto B r (n). Since Burnside groups of large odd exponents are known to be infinite, we can conclude that free Burnside groups of sufficiently large exponents are infinite. The works of Ivanov [START_REF] Ivanov | The free Burnside groups of sufficiently large exponents[END_REF] and Lysenok [START_REF] Lysenok | Infinite Burnside groups of even period[END_REF] have a similar restriction. They require n to be divisible by 2 9 and 16 respectively. Using [START_REF] Lysenok | Burnside structures of finite subgroups[END_REF] as a "black box" our proof can be adapted for large exponents n which are only divisible by 16. However for completeness and simplicity we preferred to detail our own understanding of finite subgroups of B r (n).

According to Theorem 1.1, there exists a critical exponent N 0 such that for every integer n N 0 , the group B r (n) is infinite. In our method, N 0 is directly related to the parameters of the Small cancellation Theorem, see Equations ( 21)- [START_REF] Harvey | Boundary structure of the modular group[END_REF]. Since our approach to small cancellation is qualitative we do not provide an explicit value of N 0 . Nevertheless, for a general group G we stress how this critical exponent depends on the action of G on a hyperbolic space X (see Theorem 5.4). An interested reader could go through all the arguments with a quantitative point of view to get an estimate of N 0 . However the resulting N 0 would most likely be very large.

Outline of the paper

The proof that we present here is essentially self-contained. Beside hyperbolic geometry, the arguments only rely on geometrical small cancellation theory which is now well understood. See for instance [START_REF] Coulon | Théorie de la petite simplification: une approche géométrique[END_REF][START_REF] Guirardel | Geometric small cancellation[END_REF] for a survey on the topic. For the benefit of the reader we did not attempt to write the shortest possible proof. In particular, we added in the course of the article numerous discussions, examples and figures to highlight the main difficulties and illustrate the important results.

In Section 2 we make a short review of hyperbolic geometry. We define in Section 3 all the geometric and algebraic invariants needed to control the small cancellation parameters when building the approximation sequence [START_REF] Adian | Periodic products of groups. Akademiya Nauk Soyuza Sovetskikh Sotsialisticheskikh Respublik[END_REF]. In Section 4 we first review the main properties of small cancellation theory. Given a group G acting on a hyperbolic space, the goal is to understand the properties of the quotient Ḡ obtained from G by adjoining relations of the form h n = 1, where n is a large even integer. In particular, we study the elementary subgroups of Ḡ (Section 4.6) as well as the geometric/algebraic invariants of Ḡ (Section 4.7). This section follows closely the work of Delzant-Gromov [START_REF] Delzant | Courbure mésoscopique et théorie de la toute petite simplification[END_REF]. Section 5 collects all the previous work. We first state and prove the induction hypothesis used to produce the approximation sequence [START_REF] Adian | Periodic products of groups. Akademiya Nauk Soyuza Sovetskikh Sotsialisticheskikh Respublik[END_REF], see Proposition 5.1. Then we apply our main result (Theorem 5.4) to various examples (Section 5.3) such as free Burnside groups, periodic quotients of hyperbolic groups, etc.

Acknowledgment. The author is grateful to the Centre Henri Lebesgue ANR-11-LABX-0020-01 for creating an attractive mathematical environment. He acknowledges support from the Agence Nationale de la Recherche under Grant Dagger ANR-16-CE40-0006-01. He is greatly indebted to Thomas Delzant for his valuable comments and suggestions that helped improving the exposition of the article.

Hyperbolic geometry

We recall here a few basic facts about hyperbolic spaces in the sense of Gromov [START_REF] Gromov | Hyperbolic groups[END_REF]. A reader familiar with the subject can directly jump to Section 3 where we define the important invariants associated to the action of a group on a hyperbolic space. We included precise references for the quantitative results. Some of the them only provides a proof in the context of geodesic metric spaces. However, by relaxing if necessary the constants, which we do here, the same arguments work in the more general setting of length spaces. For the rest, we refer the reader to Gromov's original article [START_REF] Gromov | Hyperbolic groups[END_REF] or the numerous literature on the subject, e.g. [START_REF] Coornaert | Géométrie et théorie des groupes[END_REF][START_REF] Ghys | Sur les groupes hyperboliques d'après Mikhael Gromov[END_REF][START_REF] Bowditch | Notes on Gromov's hyperbolicity criterion for path-metric spaces[END_REF][START_REF] Bridson | Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF].

General facts

Four point inequality. Let X be a metric length space. In this article all the paths are rectifiable and parametrized by arc length. Given two points x, y ∈ X, we write |x -y| X or simply |x -y| for the distance between them. The Gromov product of three points x, y, z ∈ X is defined as

x, y z = 1 2 (|x -z| + |y -z| -|x -y|) .
Let δ ∈ R * + . We assume that X is δ-hyperbolic, i.e. for every x, y, z, t ∈ X we have

x, y t min { x, z t , z, y t } -δ.

or equivalently

|x -y| + |z -t| max {|x -z| + |y -t| , |x -t| + |y -z|} + 2δ. (4)
In this context, the Gromov product has the following useful interpretation. For every x, y, z ∈ X, the quantity y, z x is roughly the distance between x and any geodesic [y, z] between y and z. More precisely, we have

where k is the number of points in {y, z} which belong to ∂X. Moreover, for every t ∈ X, for every x, y, z ∈ X ∪ ∂X, the four point inequality (3) leads to

x, z t min { x, y t , y, z t } -δ.

The isometry group of X naturally acts on ∂X preserving Gromov's products.

Busemann cocycles. To every point ξ ∈ ∂X, we would like to associate a Busemann cocycle. However the space X is neither locally compact, nor geodesic. To that end we proceeds as follows. Given any point ξ ∈ ∂X and a base point o ∈ X, we first define a map b :

X → R by b(x) = o, ξ x -x, ξ o and then let c : X × X → R (x, y) → b(x) -b(y).
The map c is obviously a cocycle, i.e.

c(x 1 , x 3 ) = c(x 1 , x 2 ) + c(x 2 , x 3 ), ∀x 1 , x 2 , x 3 ∈ X.
We call c a Busemann cocycle at ξ (based at o). Note that c does depend on the base point o. Nevertheless for every x, y ∈ X, c(x, y) -y, ξ x -x, ξ y 3δ.

In particular, any two cocycles at ξ differ by at most 6δ. Moreover c is almost

1-Lipschitz, i.e. |c(x, y)| |x -y| + 2δ, ∀x, y ∈ X. (8)
Lemma 2.1. Let ξ ∈ ∂X and c a Busemann cocycle at ξ. For every x, y, y ∈ X we have |y -y | |c(y, y )| + 2 max x, ξ y , x, ξ y + 8δ.

Proof. Let (z n ) be a sequence of points of X converging to ξ. It follows from the four point inequality that for every n ∈ N,

|y -y | ||y -z n | -|y -z n || + 2 max x, z n y , x, z n y + 2δ
y , z n y -y, z n y + 2 max x, z n y , x, z n y + 2δ see for instance [START_REF] Coulon | On the geometry of Burnside quotients of torsion free hyperbolic groups[END_REF]Lemma 2.2 (ii)]. The conclusion follows by taking the limit and applying [START_REF] Bridson | Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF].

We denote by ∂ h X the set of all Busemann cocycles obtained as above. The isometry group of X naturally acts on ∂ h X: if g is an isometry of X and c a Busemann cocycle at ξ ∈ ∂X, then the map gc : X × X → R defined by

(gc)(z, z ) = c(g -1 z, g -1 z ) is a Busemann cocycle at gξ. Quasi-geodesics. Let κ ∈ R * + and ∈ R + . A (κ, )-quasi-isometric em- bedding is a map f : X 1 → X 2 between two metric spaces such that for every x, x ∈ X 1 , κ -1 |x -x | - |f (x) -f (x )| κ |x -x | + . A (κ, )-quasi-geodesic is a (κ, )-quasi-isometric embedding γ : I → X from an interval I of R into X.
Recall that all the paths we consider are rectifiable by arc length. Hence, if γ : I → X is a (κ, )-quasi-geodesic, we have the following more accurate inequalities:

κ -1 |s -t| - |γ(s) -γ(t)| |s -t| , ∀s, t ∈ I.
A path is an L-local (κ, )-quasi-geodesic if its restriction to any interval of length L is a (κ, )-quasi-geodesic. If γ : R + → X is (κ, )-quasi-geodesic, then there exists a unique point ξ ∈ ∂X such that for every sequence of real numbers (t n ) diverging to infinity we have lim n→∞ γ(t n ) = ξ. We view ξ as the endpoint at infinity of γ and write ξ = γ(∞). In this article, we mostly work with local (1, )-quasi-geodesics. Therefore we use the version of the stability of local quasi-geodesics below. We refer to [START_REF] Bridson | Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]Theorem 1.13] for the proof. (i) For every s, t, t ∈ I, with t s t , we have γ(t), γ(t ) γ(s) /2 + 2δ.

(ii) For every x ∈ X, for every y, y ∈ X, lying on γ, we have d(x, γ) y, y x + /2 + 4δ.

(iii) The path γ is a (global) (κ, )-quasi-geodesic, where κ = L L-2( +2δ) .
In particular, the Hausdorff distance between two L-local (1, )-quasi-geodesics with the same endpoints (eventually in ∂X) is at most + 6δ.

Although the space X is not geodesic, its boundary satisfies a visibility property: for every x ∈ X, ξ ∈ ∂X, for every > 10δ and L 0, there exists an L-local (1, )-quasi-geodesic γ : R + → X (which is also a global quasi-geodesic) such that γ(0) = x and γ(∞) = ξ, compare with [11, Lemma 2.9]. Lemma 2.3. Let ∈ R + and L > 4 + 8δ. Let γ : R + → X be an L-local (1, )-quasi-geodesic and ξ = γ(∞) its endpoint at infinity. Let c be a Busemann cocycle at ξ. For every s, t ∈ R + with t s, we have

|c(γ(s), γ(t)) -|γ(s) -γ(t)|| + 8δ.
Proof. The lemma directly follows from (7) and Proposition 2.2 (i).

Quasi-convex subsets.

Let α ∈ R + . A subset Y of X is α-quasi-convex if for every x ∈ X, for every y, y ∈ Y , d(x, Y ) y, y x + α. Assume now that Y is connected by rectifiable paths. The length metric on Y induced by the restriction of | . | X to Y is denoted by | . | Y . We say that Y is strongly quasi-convex if it is 2δ-quasi-convex and for every y, y ∈ Y , |y -y | X |y -y | Y |y -y | X + 8δ. (9)
A (1, )-quasi-geodesic is ( /2 + 2δ)-quasi-convex, compare with the proof of Proposition 2.2 (ii). More generally, every

L-local (1, )-quasi-geodesic is ( /2 + 4δ)-quasi-convex, provided L > 4 + 8δ, see Proposition 2.2 (ii). If Y is an α-quasi-convex subset of
diam Y +A 1 ∩ . . . ∩ Y +A m diam Y +α1+3δ 1 ∩ . . . ∩ Y +αm+3δ m + 2A + 4δ.

Isometries

An isometry g of X is either elliptic (its orbits are bounded) parabolic (its orbits admit exactly one accumulation point in ∂X) or loxodromic (its orbits admit exactly two accumulation points in ∂X). In order to measure the action of g on X we used the translation length and the stable translation length respectively defined by

g X = inf x∈X |gx -x| and g ∞ X = lim n→∞ 1 n |g n x -x| .
If there is no ambiguity, we will omit the space X from the notations. These lengths are related by

g ∞ g g ∞ + 8δ, (10)
compare with [7, Chapitre 10, Proposition 6.4]. In addition, g is loxodromic if and only if g ∞ > 0. In such a case the accumulation points of g in ∂X are

g -= lim n→∞ g -n x and g + = lim n→∞ g n x.
They are the only points of X ∪ ∂X, fixed by g.

Lemma 2.6. Let g be a loxodromic isometry of X. Let , L ∈ R + , with L > 4 + 8δ. Let γ : R → X be a bi-infinite L-local (1, )-quasi-geodesic between g - and g + . Let Y be a non-empty g -invariant α-quasi-convex subset of X. Then γ lies in the (α + /2 + 4δ)-neighborhood of Y .

Proof. Let x be a point on γ. It follows from the stability of quasi-geodesics that g -, g + x /2 + 2δ (Proposition 2.2). We fix a point y ∈ Y . Since Y is α-quasi-convex, we have d(x, Y ) g -n y, g n y x + α, for every n ∈ N. We pass to the limit as n approaches infinity and use [START_REF] Bridson | Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF] to get

d(x, Y ) g -, g + x + α + 2δ α + /2 + 4δ.
The next lemma is a weak variation on the quasi-convexity of the distance function in a δ-hyperbolic space.

Lemma 2.7 (See [START_REF] Coulon | On the geometry of Burnside quotients of torsion free hyperbolic groups[END_REF]Lemma 2.26]). Let x, x and y be three points of X. If g is an isometry of X, then |gy -y| max {|gx -x|, |gx -x |} + 2 x, x y + 6δ.

Let S be a set of isometries of X. Its energy is defined by

λ(S) = inf x∈S max s∈S |sx -x| .
Given a number d ∈ R + , we denote by Fix(S, d) the set of points which are moved by S by a distance less than d, i.e.

Fix(S, d) = {x ∈ X : ∀g ∈ S, |gx -x| d} . (11)
If the set S = {g} is reduced to a single isometry, then λ(S) = g and we simply write Fix(g, d) for Fix(S, d).

Lemma 2.8. Let S be a set of isometries and d > max{λ(S), 5δ}. Then Fix(S, d) is non-empty and 8δ-quasi-convex. Moreover it satisfies the following properties.

(i) For every x ∈ X \ Fix(S, d), we have

sup g∈S |gx -x| 2d(x, Fix(S, d)) + d -10δ. (ii) Let x ∈ X and A ∈ R + . If |gx -x| d + 2A for every g ∈ S, then x is (A + 5δ)-close to Fix(S, d).
Remark. The "converse" of (ii) is obvious. Lemma 2.9. Let , L ∈ R + , with L > 4 + 8δ. Let g be a loxodromic isometry of X. Let γ : R → X be a bi-infinite L-local (1, )-quasi-geodesic joining g -to g + .

(i) For every d > max{ g , 5δ}, the path γ is contained in Fix(g, d + + 24δ).

(ii) Conversely, if g > 8δ, the for every d ∈ R + , the set Fix(g, d) is contained in the A-neighborhood of γ, where

A = 1 2 (d -g ) + 1 2 + 13δ.
Proof. The fist part of the statement is a consequence of Lemma 2.6 applied to Y = Fix(g, d). Let us focus on the second part. Without loss of generality we can assume that Fix(g, s) is non-empty. Let η > 0 such that g > 8η + 8δ. Let y ∈ X such that |gy -y| g + η. Consider ν : [0 , L] → X be a (1, η)-quasigeodesic from y to gy. We extend this path to g -invariant L-local (1, 2η)-quasigeodesic ν : R → X by letting ν(mL+t) = g m ν(t) for every m ∈ Z and t ∈ [0, L). Let x ∈ Fix(g, d) and p = ν(t) a projection of x on ν. Since ν is g -invariant, gp = ν(t + L) is also a projection of gx on ν. Moreover |gp -p| g > 8η + 8δ. Recall that ν restricted to [t , t + M ] is (η + 2δ)-quasi-convex. It follows from the projection a quasi-convex (Lemma 2.4) that

d |gx -x| 2 |x -p| + g -4η -10δ
Note that L g > 8η + 8δ. By stability of quasi-geodesics (Proposition 2.2) we get g -, g + p η + 2δ. Since γ is ( /2 + 4δ)-quasi-convex, we get

d(x, γ) g -, g + x + /2 + 6δ |x -p| + g -, g + p + /2 + 6δ 1 2 (d -g ) + /2 + 3η + 13δ.
This holds for every sufficiently small η, whence the result.

Lemma 2.10. Let ξ ∈ ∂X and c a Busemann cocycle at ξ. Let g be an isometry of X fixing ξ. There exists ε ∈ {±1}, such that for every x ∈ X, we have

|c(gx, x) + ε g ∞ | 6δ. Proof. Let x ∈ X. Let n ∈ N. Observe that c(g n x, x) = n-1 k=0 c(g k+1 x, g k x) = n-1 k=0 g -k c(gx, x).
Since g fixes ξ, for every k ∈ N, the map g -k c is a Busemann cocycle at ξ, and therefore differs from c by at most 6δ. Thus |c(g n x, x) -nc(gx, x)| 6nδ. As Busemann cocycles are almost 1-Lipschitz, we get

|c(gx, x)| 1 n |c(g n x, x)| + 6δ 1 n |g n x -x| + 6 + 2 n δ.
Taking the limit yields |c(gx, x)| g ∞ + 6δ. In particular, the result holds if g is either elliptic or parabolic.

Assume now that g is loxodromic. There exists ε ∈ {±1} such that ξ is the attractive point of g ε . We fix η > 0. Note that g n > 8η + 8δ, for every sufficiently large n ∈ N. We fix such an exponent n and write h = g εn . Let y ∈ X be a point such that |hy -y| h + η. We choose a (1, η)-quasi-geodesic γ : [0 , L] → X joining y to hy and extend γ to a bi-infinite path γ : R → X as follows: for every t ∈ [0, L), for every n ∈ Z, we let γ(nL + t) = hγ(t). It follows from our choice of y that γ is an L-local (1, 2η)-quasi-isometry from h - to h + = ξ. Applying Lemma 2.3, we get |c(y, hy) -|hy -y|| 2η + 8δ.

(

) 14
Observe that h -1 c and c are two cocycles at ξ (since h fixes ξ), hence they differ by at most 6δ. The same arguments can be used to prove the following lemma.

Lemma 2.12. Let g be a loxodromic isometry. Let L, ∈ R + , with L > 4 +8δ. Let γ : R + → X be an L-local (1, )-quasi-geodesic from g -to g + . Then for every t ∈ R, for every n ∈ Z, we have

|γ(t + n g ∞ ) -g n γ(t)| 2 + 20δ.

Group action

Y = {x ∈ X : d(x, Γ) < 20δ} . ( 15
)
It is a strongly quasi-convex subset of X, see [START_REF] Coulon | Partial periodic quotients of groups acting on a hyperbolic space[END_REF]Lemma 3.13]. By construction

∂Y = {g -, g + } = Λ(G).
Lemma 2.14. Let g ∈ G be a loxodromic element. Let F be an elliptic subgroup of G normalized by g. Let , L ∈ R + , with L > 4 + 8δ. Let γ be an L-local (1, )-quasi-geodesic from g -to g + . Then γ is contained in Fix(F, + 30δ).

Proof. Since g normalizes F , the set Fix(F, 6δ) is a g -invariant 8δ-quasi-convex subset (Lemma 2.8). By Lemma 2.6, the path γ is contained in the ( /2 + 12δ)neighborhood of Fix(F, 6δ).

Non-elementary action. The next lemma is an improved version of the classical ping-pong argument. It provides a simple criterion to ensure that a group is non-elementary.

Lemma 2.15 (See [START_REF] Coulon | Partial periodic quotients of groups acting on a hyperbolic space[END_REF]Lemma 3.24]). Let A 0. Let x ∈ X. Let G be a group of isometries of X generated by two elements u and v such that

(i) 2 u ±1 x, v ±1 x x < min{|ux -x|, |vx -x|} -A -8δ, (ii) 2 ux, u -1 x x < |ux -x| + A, (iii) 2 vx, v -1 x x < |vx -x| + A.
Then G is non-elementary.

Gentle action. In order to complete the description of loxodromic groups started above, we introduce a (harmless) additional assumption.

Definition 2.16. The action of G on X is gentle if every loxodromic subgroup H preserving the orientation splits as a semi-direct product H = F Z where F consists exactly of all elliptic elements of H.

If every loxodromic subgroup is virtually cyclic, then the action of G is automatically gentle. From now on we assume that the action of G on X is gentle. Let H be a loxodromic subgroup of G and H + the subgroup of H fixing pointwise Λ(H). Let F be the set of all elliptic elements of H + . It follows from our assumption that F is an elliptic normal subgroup of H and is maximal for these properties. The quotient H/F is either isomorphic to Z if H preserves the orientation (i.e. H = H + ) or the infinite dihedral group D ∞ otherwise. Observe that if H is generated by two elliptic subgroups, then H cannot preserve the orientation.

Lemma 2.17. Let H be a loxodromic subgroup of G. If p : H → D ∞ is a morphism whose kernel is elliptic, then this kernel is exactly the maximal normal elliptic subgroup F of H.

Proof. By assumption the kernel of p is an elliptic normal subgroup of H, hence it is contained in F . Let us prove the other inclusion. We fix a loxodromic element h ∈ H. According to our assumption, the pre-image under p of any finite subgroup of D ∞ is elliptic (as a finite extension of an elliptic subgroup). Hence p(h) belongs to Z * ⊂ D ∞ . Observe then that p(F ) does not contain any element of Z \ {0}. Indeed otherwise there would exist m ∈ Z \ {0} and u ∈ F such that p(h m ) = p(u). Since the kernel of p is contained in F , the element h m should belong to F which contradicts the fact that h is loxodromic. Hence p(F ) is a finite subgroup of D ∞ . As h normalizes F , its image p(h) normalizes p(F ) which forces p(F ) to be trivial.

Given a loxodromic element g ∈ G, we write E(g) for the subgroup of G preserving {g -, g + }. It is the maximal elementary subgroup of G containing g. The group E + (g) stands for the maximal subgroup of E(g) fixing pointwise {g -, g + }.

Definition 2.18. (Primitive element) Let g ∈ G be a loxodromic element. Let F be the maximal normal elliptic subgroup of E(g). We say that g is primitive if its image in E + (g)/F ≡ Z generates the group.

Invariants of a group action

Let X be a δ-hyperbolic length space and G a group acting gently by isometries on X (see Definition 2.16). Note that for the moment, we have not made any serious assumption on the group G or the space X. In order to study the action of G on X we define several numerical invariants. Those quantities will be useful later to estimate the small cancellation parameters needed to run the induction leading to the infiniteness of Burnside groups. We define two types of invariants. The first kind, namely the injectivity radius inj (G, X), the acylindricity constant A(G, X) as well as the ν-invariant ν(G, X) are purely geometric. Those invariants (or some variations of them) already appeared in [START_REF] Delzant | Courbure mésoscopique et théorie de la toute petite simplification[END_REF]Définition 2.4.1] and [START_REF] Coulon | Partial periodic quotients of groups acting on a hyperbolic space[END_REF]Definition 3.40]. Unfortunately they are not sharp enough to handle even torsion. More precisely the ν-invariant, does not behave well when passing to quotient. Therefore we also define (among others) a strong variation ν stg (G, X) of the ν-invariant, which has a mixed nature: it reflects both the geometric and algebraic features of G.

Geometric invariants

Definition 3.1 (Injectivity radius). The injectivity radius of G on X is the quantity inj (G, X) = inf { g ∞ X : g ∈ G loxodromic} Definition 3.2 (Acylindricity). Let d ∈ R + . The acylindricity parameter at scale d, denoted by A(G, X, d), is A(G, X, d) = sup S⊂G diam (Fix(S, d)) ,
where S runs over all subsets of G generating a non-elementary subgroup. The previous definition is designed to compensate for the fact that we do not control from below the curvature of X. Its definition is modeled on the Margulis lemma for manifolds with pinched negative curvature For our next invariant, we adopt the following terminology borrowed from Lysenok [START_REF] Lysenok | Infinite Burnside groups of even period[END_REF]. A chain of length m is a tuple C = (g 0 , . . . , g m ) of elements of G for which there exists h ∈ G such that for every k ∈ 0, m -1 , we have g k+1 = hg k h -1 . The element h is called a conjugating element of C. Note that such an element is not necessarily unique. Definition 3.4 (ν-invariant). The quantity ν(G, X) is the smallest integer ν with the following property: if C = (g 0 , . . . , g ν ) is a chain of length ν generating an elementary subgroup and h a loxodromic conjugating element of C, then g 0 , h is elementary.

The ν-invariant is useful to prove the following local-to-global phenomenon. Proposition 3.5. For every d ∈ R + , we have

A(G, X, d) [ν(G, X) + 3] d + A(G, X, 400δ) + 24δ.
In particular, if A(G, X, 400δ) and ν(G, X) are finite, then the action of G on X is weakly acylindrical.

Before proving Proposition 3.5 we focus on the following lemmas. Lemma 3.6. Fix d 1 = 320δ and d 2 = 400δ. Let g and h be two elements of G which generate a non-elementary subgroup.

(i) Assume that h is loxodromic. Let L ∈ R + , with L > 12δ and γ : R → X be an L-local (1, δ)-quasi-geodesic from h -to h + . Then diam (Fix(g, d 1 ) ∩ γ) ν(G, X) h + A(G, X, d 2 ) + 2δ.
(ii) Assume that both g and h are loxodromic. There exists L > 12δ, with the following property. If γ g , γ h : R → X are two L-local (1, δ)-quasi-geodesics from g -to g + , and h -to h + respectively, then

diam γ +8δ g ∩ γ +8δ h g + h +ν(G, X) max { g , h }+A(G, X, d 2 )+20δ.
Remark. A similar statement is proved in [START_REF] Coulon | Partial periodic quotients of groups acting on a hyperbolic space[END_REF] closely following the ideas of Delzant and Gromov [START_REF] Delzant | Courbure mésoscopique et théorie de la toute petite simplification[END_REF]. For completeness, we reproduce it here.

Proof. For simplicity we let ν = ν(G, X). We start with Point (i). Assume that contrary to our claim,

diam (Fix(g, d 1 ) ∩ γ) > ν h + A(G, X, d 2 ) + 2δ.
In particular, there exist x = γ(s) and

x = γ(s ) lying in Fix(g, d 1 ) such that |x -x | > ν h + A(G, X, d 2 ) + 2δ.
Without loss of generality we can assume that s < s , so that

s -s |x -x | > ν h + A(G, X, d 2 ) + 2δ. Since Fix(g, d 1 ) is 8δ-quasi-convex (Lemma 2.8), γ restricted to [s , s ] lies en- tirely in the 11δ-neighborhood of Fix(g, d 1 ). We now fix t = s+A(G, X, d 2 )+2δ. Let r ∈ [s , t] and k ∈ 0, ν . Note that r k = r + k h ∞ belongs to [s , s ]. Thus |gγ(r k ) -γ(r k )| d 1 + 22δ. Applying Lemma 2.12 we obtain h -k gh k γ(r) -γ(r) gh k γ(r) -h k γ(r) |gγ(r k ) -γ(r k )| + 44δ d 1 + 66δ.
In other words the restriction of γ to [s

, t] is contained in Fix(S, d 2 ), where S is the set S = {g, h -1 gh, . . . , h -ν gh ν }. Consequently the diameter of Fix(S, d 2 ) is larger that A(G, X, d 2 )
, and thus S generate an elementary subgroup. Recall that h is loxodromic. It follows from the definition of ν that g and h generate an elementary subgroup which contradicts our assumption. We now focus on Point (ii). Up to permuting g and h, we can assume that h g . We fix

= ν h + A(G, X, d 2 ) + 4δ and L = + g + h .
Let γ g , γ h : R → X as in the statement of Point (ii). Assume that contrary to our claim we have

diam γ +8δ g ∩ γ +8δ h > g + (ν + 1) h + A(G, X, d 2 ) + 20δ L + 16δ.
We fix two points x, y ∈ X lying in the 8δ-neighborhood of both γ g and γ h such that |x -y| > L + 16δ.

Up to changing the origin of γ g and γ h we can assume that γ g (0) and γ h (0) are projections of x on γ g and γ h respectively. We write γ g (s) and γ h (s ) for projections of y on γ g and γ h respectively. Note that s, s ∈ (L, ∞).

We claim that |γ g (r)

-γ h (r)| 57δ, for every r ∈ [0 , L]. Since γ g is an L-local (1, δ)-quasi-geodesic, we have γ g (0), γ g (s) γg(r) 3δ (Proposition 2.2), thus x, y γg(r)
19δ. Similarly x, y γ h (r) 19δ. Moreover, the quantities |γ g (r) -x| and |γ h (r) -x| differ by at most 17δ. It follows from the four point inequality -see for instance [10, Lemma 2.2 (2)] -that |γ g (r) -γ h (r)| 57δ, which completes the proof of our claim.

According to Lemma 2.12, g (respectively h) acts on γ g (respectively γ h ) almost like a translation of length g ∞ (respectively h ∞ ). Hence for every r ∈ R + , such that r ,

|ghγ h (r) -hgγ h (r)| 316δ d 1 , compare with Figure 2. Consequently the path γ h restricted to [0 , ] is contained in Fix(u, d 1 )
where u = g -1 h -1 gh. Thus, applying Lemma 2.5 we get

diam (Fix(u, d 1 ) ∩ γ h ) -δ > ν h + A(G, X, d 2 ) + 2δ.
It follows from the previous discussion that u and h generates an elementary subgroup. Hence g -1 hg and h generate an elementary subgroup. Since h is loxodromic, g fixes h -and h + , therefore g and h generate an elementary subgroup, which contradicts our assumption.

Lemma 3.7. Fix d 2 = 400δ. Let g and h be two elements of G which generate a non-elementary subgroup and set S = {g, h}. If h 28δ, then for every

d ∈ R + we have diam (Fix(S, d)) [ν(G, X) + 3] d + A(G, X, d 2 ) + 24δ.
Proof. For simplicity we write ν for ν(G, X). Without loss of generality we can assume that d > λ(S). In particular, d > max{ g , h }.

Assume first that g < 28δ. We fix L > 12δ and an L-local (1, δ)-quasigeodesic γ h from h -to h + . It follows from Lemma 2.9, that Fix(h, d) lies in the d/2-neighborhood of γ h . On the other hand Fix(g, d) lies in the d/2neighborhood of Fix(g, 28δ). Combining these observations with Lemma 2.5 we get

diam (Fix(S, d)) diam Fix(g, 10δ) +d/2 ∩ γ +d/2 h diam Fix(g, 28δ) +11δ ∩ γ +8δ h + d + 4δ diam (Fix(g, 66δ) ∩ γ h ) + d + 20δ
It follows then from Lemma 3.6 that

diam (Fix(S, d)) (ν + 1)d + A(G, X, d 2 ) + 22δ.
Assume now that g 28δ. In particular g is loxodromic. By Lemma 3.6 we can find γ g and γ h two L-local (1, δ)-quasi-geodesics with L > 12δ, joining g - to g + and h -to h + respectively such that

diam γ +8δ g ∩ γ +8δ h (ν + 2)d + A(G, X, d 2 ) + 20δ.
It follows from Lemma 2.9, that Fix(g, d) and Fix(h, d) lies in the d/2-neighborhood of γ g and γ h respectively. Using Lemma 2.5 as above we get

diam (Fix(S, d)) diam γ +d/2 g ∩ γ +d/2 h diam γ +8δ g ∩ γ +8δ h + d + 4δ (ν + 3)d + A(G, X, d 2 ) + 24δ.
Proof of Proposition 3.5. For simplicity we write ν for ν(G, X). We distinguish two cases. Assume first that g < 28δ, for every g ∈ S. It follows from Lemma 2.8 that Fix(g, d) is contained in the d/2-neighborhood of Fix(g, 28δ), for every g ∈ S. Combined with Lemma 2.5 we get

diam (Fix(S, d)) diam   g∈S Fix(g, 28δ) +11δ   + d + 4δ diam   g∈S Fix(g, 50δ)   + d + 4δ A(G, X, 50δ) + d + 4δ.
Assume now that there exists h ∈ S such that h 28δ. In particular h is loxodromic. It follows from our assumption that there exists g ∈ S such that g and h do not generate an elementary subgroup. Applying Lemma 3.7, we get

diam (Fix(S, d)) diam (Fix({g, h}, d)) (ν + 3)d + A(G, X, 400δ) + 24δ.
In both cases, the diameter of Fix(S, d) is bounded above by A(G, X, 400δ)+24δ, whence the result. Lemma 3.8. Let P be a parabolic subgroup of G. Let ξ ∈ ∂X be the unique accumulation point of P . If the action of G on X is weakly acylindrical, then Stab(ξ) is parabolic as well.

Proof. For simplicity we let E = Stab(ξ). Assume that contrary to our claim that E is not parabolic. In particular, E contains a loxodromic element h. Up to replacing h by a power of h we can assume that h > 8δ. Let g ∈ P . We set S = {g, h} and L = h + 100δ. Note that Fix(S, L) has infinite diameter (Lemma 2.11). It follows from the definition of weak acylindricitay that the subgroup of G generated by g and h is elementary. Consequently P is contained in the maximal elementary subgroup E(h) of G containing h. Nevertheless the subgroups of E(h) are either elliptic or loxodromic, which contradicts our assumption.

Mixed invariants

As explained in the previous section, the combination of the acylindricity parameter A(G, X, 400δ) and the ν-invariant provides a useful substitute to the Margulis lemma. Nevertheless the latter invariant does no behave well when passing to quotient (see Section 4.7). To bypass this difficulty we consider a stronger version of the ν-invariant whose mixed nature combines both geometric and algebraic features of G. More precisely the algebraic part captures the properties of a special class of elementary subgroups that we define now.

Dihedral germs and dihedral pairs. Recall that D ∞ stands for the infinite dihedral group. Given m ∈ N, we denote by

D m = s, r | s 2 , (sr) 2 , r m
the dihedral group of order 2m and by Z m the cyclic group of order m. Note that D 1 = Z 2 . By convention D 0 is the trivial group. We think of D m as the isometry group of the plane preserving a regular m-gon. This motivates the following terminology. The subgroup r is a normal subgroup called the rotation subgroup. Its elements are also called orientation preserving. The signature is the morphism ε : D m → Z 2 , where Z 2 is the quotient of D m by the rotation subgroup. An element of D m that does not preserve the orientation is called a reflection.

We adopt a similar terminology for D ∞ . In particular, its rotation subgroup (or translation subgroup) is the maximal subgroup isomorphic to Z. If m = 2, the rotation subgroup of D m is algebraically completely determined: it is the unique cyclic subgroup of order m. Otherwise it should be thought an implicit piece of information attached to D 2 . Definition 3.9. A subgroup C of G is called a dihedral germ if it contains an elliptic subgroup C 0 which is normalized by a loxodromic element and such that

[C : C 0 ] is a power of 2.
Note that dihedral germs are elliptic. Being a dihedral germ is invariant under conjugation. Without any further assumption on the structure of loxodromic subgroups, it is not true in general that being a dihedral germ is invariant by taking subgroup. Definition 3.10. A dihedral pair is a pair (E, C) of subgroups such that C is a dihedral germ which is also normal in E and E/C embeds in a (finite or infinite) dihedral group. A subgroup E of G has dihedral shape if there exists a subgroup C such that (E, C) is a dihedral pair.

Every subgroup with dihedral shape is elementary. Indeed such a group is virtually the extension of an elliptic subgroup by a cyclic group. Note that the morphism from E/C to a dihedral group is in general not unique. Lemma 3.11. Let E be a loxodromic subgroup of G and C a subgroup of E.

Then (E, C) is a dihedral pair if and only if C is the maximal elliptic normal subgroup of E.

Proof. Assume that (E, C) is a dihedral pair. In particular, E/C embeds in a dihedral group. Note that this dihedral group cannot be finite. Indeed otherwise E would be a finite extension of the elliptic subgroup C, hence an elliptic subgroup as well. Consequently E/C embeds in D ∞ . It follows from Lemma 2.17 that C = F . The converse statement is obvious.

Strong ν-invariant. Definition 3.12 (strong ν-invariant). The quantity ν stg (G, X) is the smallest integer ν with the following property: if C = (g 0 , . . . , g ν ) is a chain generating an elementary subgroup and h a conjugating element of C such that • either h is loxodromic,

• or g 0 , . . . , g ν-1 is contained in a dihedral germ, then g 0 , h is elementary with dihedral shape.

One observes easily that ν(G, X) ν stg (G, X). Let us mention an example where these two invariants are not equal.

Example 3.13. Observe first that if G = G 1 * G 2 is a free product acting on its Bass-Serre tree T , then ν(G, T ) 2. Consider indeed g, h ∈ G with h loxodromic such that the subgroup E = g, hgh -1 , h 2 gh -2 is elementary. Without loss of generality we can assume that g is non trivial. We first claim that the subgroup E 0 = g, hgh -1 cannot be elliptic. Assume on the contrary that E 0 fixes a point say x ∈ T . As G is a free product, x is the unique fixed point of g. Nevertheless hgh -1 also fixes x (as it belongs to E 0 ), hence g fixes h -1 x. This forces hx = x which contradicts the fact that h is loxodromic and completes the proof of the claim. Since T is a tree, G does not admit any finitely generated parabolic subgroup, hence E 0 is loxodromic. Observe now that the elementary subgroup E is generated by E 0 and hE 0 h -1 . Consequently h necessarily belongs to the maximal elementary loxodromic subgroup containing E 0 . Therefore g and h generate an elementary subgroup. This proves that ν(G, T ) 2 as announced.

In this setting, every elliptic subgroup which is normalized by a loxodromic element is trivial. Hence a subgroup of G is a dihedral germ if an only if it is a finite 2-group. Let us now consider a more precise example. We fix m ∈ N and let A = Z m+1 2 . For every i ∈ 0, m , we write g i for a generator of the i-th factor Z 2 in A. We denote by G 1 the following HNN extension of A

G 1 = A, h | hg i h -1 = g i+1 , ∀i ∈ 0, m -1 . Let G = G 1 * Z. It follows from the construction that g 0 , hg 0 h -1 , . . . , h m g 0 h -m = g 0 , . . . , g m = A
is a dihedral germ. On the other hand, the subgroup g 0 , h corresponds to G 1 which is not virtually cyclic, thus it cannot have dihedral shape. This shows that ν stg (G, T ) m. In particular, if m > 2, then ν(G, T ) < ν stg (G, T ). Note that in this example, the difference between ν(G, X) and ν stg (G, X) comes from the algebraic structure of elliptic subgroups. It emphases the fact that ν stg (G, X) is not a purely geometric invariant.

Model collections. As we will see later, controlling the strong ν-invariant is a key ingredient to handle even torsion and which was not needed to study free Burnside groups of odd exponents. It requires a fine understanding of the structure of dihedral pairs. We complete this section by a last notion designed to describe those subgroups.

A model collection is a family E of (abstract) torsion groups. Its exponent µ = µ(E) is the smallest positive integer such that g µ = 1, for every E ∈ E, for every g ∈ E.

Definition 3.14. Let p ∈ N and E be a model collection. We say that a dihedral pair (E, C) has type (E, p) if there exist k ∈ N and a morphism ϕ : E → E, where E ∈ E such that the map ϕ extends to an embedding from E into E/C ×D k p ×E.

Remark. A reader only interested in free Burnside groups can read the entire article by taking for E the collection that consists only of the trivial group. The exponent of this trivial model collection is 1.

We now fix an integer p ∈ N and a model collection E and write µ = µ(E) for its exponent. Saying that (E, C) has type (E, p) means that, up to a residual factor E ∈ E, the group E essentially embeds into a direct product of dihedral groups. In particular, we can exploit the algebraic identities of dihedral groups to recover information about E. The next two statements give simple but essential examples of this idea. Other applications will arise later in the article. Proposition 3.15. Let (E, C) be a dihedral pair with type (E, p). Let C = (g 0 , g 1 , . . . , g µ+2 ) be a chain of G and h a conjugating element of C. If g 0 and h belong to a subgroup E, then g µ+2 belongs to g 0 , g 1 , . . . , g µ+1 .

Proof. By assumption, there exist k ∈ N, a group E ∈ E, and a morphism ϕ : E → E, such that ϕ extends to an embedding

E → E/C × D k p × E. For every i ∈ 0, 3 we let u i = g i g i+1 • • • g i+µ-1 .
Note that it suffices to prove that u 3 u -1 2 u 0 u -1 1 = 1. To that end, we have to check that this identity holds in every factor of E/C × D k p × E. It was observed by Lysenok [START_REF] Lysenok | Infinite Burnside groups of even period[END_REF]Proposition 15.10] that if x and y are two elements of D ∞ then

y 3 xy -3 y 2 x -1 y -2 x yx -1 y -1 = y 2 , [y, x] = 1.
Hence the identity u 3 u -1 2 u 0 u -1 1 = 1 holds in E/C as well as in any factor D p . On the other hand, we observe that u 0 = (g 0 h) µ h -µ . By the very definition of the exponent µ, the element ϕ(u 0 ) ∈ E is trivial and thus so are its conjugates ϕ(u 1 ), ϕ(u 2 ) and ϕ(u 3 ). Hence the identity u 3 u -1 2 u 0 u -1 1 = 1 also holds in E, and the proof is complete.

Let n ∈ N. Let Π = D p1 × • • • × D p k × E be a direct product of dihedral groups with some E ∈ E where p i divides n for every i ∈ 1, k . The signature ε i : D pi → Z 2 induces a morphism Π → (Z 2 ) k , whose kernel is Π + = Z p1 ו • •× Z p k × E is the pure rotation subgroup. Given a subgroup A of Π, its reflection rank is the dimension of the image of A in Π/Π + (seen as a Z 2 -vector space).
The next lemma is a variation on Ivanov [START_REF] Ivanov | The free Burnside groups of sufficiently large exponents[END_REF]Lemma 16.2]. Lemma 3.16. Let A be a subgroup of Π and r its reflection rank. We assume that 2 r+3 µ divides n. For every h ∈ Π, normalizing A, there exists a ∈ A with the following properties.

(i) h n/4 a -1 centralizes A. (ii) [h n/4 a -1 , b] centralizes Π, for every b ∈ Π.
Proof. By assumption, there exist s 1 , . . . , s r ∈ A such that A is generated by s 1 , . . . , s r and A ∩ Π + . We let

a = (ε1,...,εr)∈{0,1} r [r, s ε1 1 . . . s εr r ] , where r = h 2 -(r+2) n .
Since h normalizes A, the element a belongs to A. It is sufficient to check that in each factor of Π the image of h n/4 a -1 satisfies the announced properties. Since the exponent of E divides 2 -(r+2) n, the elements h n/4 and a are trivial in E. Thus so is h n/4 a -1 . Hence (i) and (ii) hold in E.

We now focus on the dihedral factors. Let i ∈ 1, k . Assume first that the image of A is D pi is contained in the rotation group Z pi . Note that 2 divides 2 -(r+2)n . Hence (the image of) r lies in the rotation group of D pi . Thus a is trivial in D pi , while h n/4 belongs to Z pi . Consequently (the image of) h n/4 a -1 centralizes Z pi and thus (the image of) A. Moreover for every b ∈ Π, (the image of) [h n/4 a -1 , b] which coincides with (the image of) [h n/4 , b], centralizes D pi . Assume now that the image of A in D pi contains a reflection. By construction every element of A ∩ Π + is mapped to the rotation subgroup Z pi . Without loss of generality we can assume that s 1 is mapped to a reflection of D pi . Let Ω be the subset of all tuples (ε 1 , . . . , ε r ) ∈ {0, 1} r such that s ε1 1 . . . s εr r is mapped to a reflection in D pi . As previously the image of r in D pi is a rotation. Hence, seen in D pi , we have

[r, s ε1 1 . . . s εr r ] = r 2 if (ε 1 , . . . , ε r ) ∈ Ω 1 otherwise Consequently we get in D pi a = r 2|Ω| = h 2 -(r+1) |Ω|n .
Observe that the cardinality of Ω is 2 r-1 . Indeed the map sending (ε 1 , . . . , ε r ) to (ε 1 + 1, . . . , ε r ) induces a bijection from Ω onto {0, 1} r \ Ω. It follows that h n/4 and a coincide in D pi . Thus (i) and (ii) hold in D pi .

Small cancellation theory

Let us recall the main strategy to study the free Burnside group B r (n). Starting from the free group F r , we are going to build a sequence of non-elementary hyperbolic groups

F r = G 0 G 1 G 2 • • • G k G k+1 . . .
whose directly limit is exactly B r (n). The group G k+1 is obtained from G k by adjoining relations of the form h n = 1 where h runs over a subset of "small" loxodromic elements of G k . The main difficulty is to make sure that G k+1 remains a non-elementary hyperbolic group although the exponent n has been fixed in advance. In this article, we achieve this by using a geometric approach of small cancellation theory à la Delzant-Gromov [START_REF] Delzant | Courbure mésoscopique et théorie de la toute petite simplification[END_REF].

In this section we focus on a single step G k G k+1 . We present first an overview small cancellation theory and develop later the required additional material. For proving infiniteness of Burnside groups we only need to consider relations of the form h n = 1. Nevertheless we start our study in a slightly more general setting as the intermediate results are of independent interest.

General setting

Let X be a δ-hyperbolic length space and G a group acting gently by isometries on X. Let Q be a collection of pairs (H, Y ) where H is a subgroup of G and Y an H-invariant strongly quasi-convex subset of X. We assume that Q is invariant under the action of G defined by g •(H, Y ) = (gHg -1 , gY ), for every (H, Y ) ∈ Q and every g ∈ G. We denote by K the (normal) subgroup of G generated by all H where (H, Y ) runs over Q. The goal is to study the quotient Ḡ = G/K. To that end, we define two parameters ∆(Q, X) and inj (Q, X) which play the role of the lengths of the largest piece and the smallest relation respectively.

∆(Q, X) = sup diam Y +5δ 1 ∩ Y +5δ 2 : (H 1 , Y 1 ) = (H 2 , Y 2 ) ∈ Q , inj (Q, X) = inf { h : h ∈ H, (H, Y ) ∈ Q} .
Remark. As explained above, we will later focus on a particular set of relations. More precisely the collection Q will be of the form

Q = {(Y h , h n ) : h ∈ S}
where n is a large integer and S a subset of "small" loxodromic elements of G, which is invariant under conjugation. Assuming that ∆(Q, X) is finite will automatically imply that h n is normal in Stab(Y h ), for every h ∈ S.

We now fix once for all a number ρ ∈ R * + . Its value will be made precise later (see Theorem 4.7). It should be thought of as a very large distance.

ch |x -x | Z(Y ) = ch r ch r -sh r sh r cos θ(y, y ), (16)
where θ(y, y ) is the angle at the apex defined by

θ(y, y ) = min π, |y -y | Y sh ρ .
The distance between two points x = (y, r) and x = (y , r ) of Z(Y ) has the following geometric interpretation. Consider a geodesic triangle in the hyperbolic plane H 2 such that the lengths of two sides are respectively r and r and the angle between them is θ(y, y ). According to the law of cosines, |x -x | is exactly the length of the third side of the triangle (see Figure 3). In order to compare the metric of Y and Z(Y ), we use the map µ : R + → R + characterized as follows ch µ(t) = ch 2 ρsh 2 ρ cos min π, t sh ρ , ∀t ∈ R + , so that for every y, y ∈ Y we have

|ι(y) -ι(y )| Z(Y ) = µ(|y -y | Y ).
The next proposition summarizes the properties of µ.

Proposition 4.3. The map µ is continuous, concave, and non-decreasing. In addition, for all t ∈ [0 , π sh ρ], we have t π sh(µ(t)/2).

We complete this part with a useful tool to compare two cones.

Lemma 4.4. Let f : Y 1 → Y 2 a (1, ) quasi-isometric embedding between two metric spaces. The map Z(Y 1 ) → Z(Y 2 ) sending (y, r) to (f (y), r) is again a (1, )-quasi-isometric embedding.
Proof. The result is a direct consequence of the geometric interpretation of the metric on the cones. ). For every x, x ∈ X, we have

µ (|x -x | X ) |x -x | Ẋ |x -x | X . Lemma 4.6 ([10, Lemma 5.7]). Let (H, Y ) ∈ Q. Let x ∈ Z(Y ). Let d(x, Y ) be the distance between x and ι(Y ) computed with | . | Z(Y ) . For all x ∈ Ẋ, if |x -x | Ẋ < d(x, Y ) then x belongs to Z(Y ). Moreover |x -x | Ẋ = |x -x | Z(Y ) .
Let v be the apex of Z(Y ). It follows from the lemma that, as a set, the ball

B(v, ρ) (for the metric of Ẋ) is nothing but Z(Y ) \ ι(Y ). Moreover the metrics | . | Ẋ and | . | Z(Y ) coincide on B(v, ρ/3).
The quotient space X. The action of G on X naturally extends to an action by isometries on Ẋ as follows. Let (H, Y ) ∈ Q. For every g ∈ G, for every x = (y, r) in Z(Y ), we define gx to be the point of Z(gY ) given by gx = (gy, r). The space X is the quotient X = Ẋ/K. The metric on Ẋ induces a pseudometric on X. We write ζ : Ẋ → X for the canonical projection from Ẋ to X. The quotient Ḡ = G/K naturally acts by isometries on X. We denote by V the image of V in X. For every x ∈ Ẋ, we usually write x for its image in X.

Small cancellation theorem. The next statement is a combination of Proposition 6.4, Proposition 6.7, Corollary 3.12 and Proposition 3.15 in [START_REF] Coulon | On the geometry of Burnside quotients of torsion free hyperbolic groups[END_REF]. See also [16, Theorem 4.7. There exist δ 0 , δ 1 , ∆ 0 , ρ 0 ∈ R * + , which do not depend on X, G or Q, with the following property. Assume that ρ ρ 0 . If δ δ 0 , ∆(Q, X) ∆ 0 and inj (Q, X) 10π sh ρ, then the following holds

(i) The cone-off space Ẋ is δ-hyperbolic with δ δ 1 . (ii) The quotient space X is δ-hyperbolic with δ δ 1 . (iii) Let (H, Y ) ∈ Q. Let v be the image in X of the apex v of Z(Y ). The subgroup Stab(v) ⊂ Ḡ is isomorphic to the quotient Stab(Y )/H. Moreover the projection ζ : Ẋ → X induces an isometry from B(v, ρ/2)/H onto B(v, ρ/2).
(iv) For every r ∈ (0, ρ/20], for every x ∈ Ẋ, if d(x, V) 2r, then the projection ζ : Ẋ → X induces an isometry from B(x, r) onto B(x, r).

(v) For every x ∈ Ẋ for every g ∈ K \ {1}, we have |gx -x| Ẋ min{2r, ρ/5}, where r = d(v, V). In particular, K acts freely on Ẋ \ V. Moreover, the projection

ζ : Ẋ → X induces a covering map Ẋ \ V → X \ V.
Remark. Note that the constants δ 0 and ∆ 0 (respectively ρ 0 ) can be chosen arbitrarily small (respectively large). From now on, we will always assume that ρ 0 > 10 20 δ 1 whereas δ 0 , ∆ 0 < 10 -10 δ 1 . These estimates are absolutely not optimal. We chose them very generously to ensure that all the inequalities which we might need later will be satisfied. What really matters is their orders of magnitude recalled below.

max {δ 0 , ∆ 0 } δ 1 ρ π sh ρ.
An other important point to remember is the following. The constants δ 0 , ∆ 0 and π sh ρ are used to describe the geometry of X whereas δ 1 and ρ refers to the one of Ẋ or X. From now on and until the end of Section 4 we assume that X, G and Q are as in Theorem 4.7. In particular, Ẋ and X are respectively δand δhyperbolic. Up to increasing one constant or the other, we can actually assume that δ = δ. Nevertheless we still keep two distinct notations, to remember which space we are working in.

Notations. In this section we work with three metric spaces namely X, its coneoff Ẋ and the quotient X. Since the map X → Ẋ is an embedding we use the same letter x to designate a point of X and its image in Ẋ. We write x for its image in X. Unless stated otherwise, we keep the notation | . | (without mentioning the space) for the distances in X or X. The metric on Ẋ will be denoted by | . | Ẋ .

A few additional facts regarding the cone-off space

Radial projection. The radial projection p : Ẋ \ V → X is defined as follows.

Its restriction to X is the identity. Given any (H, Y ) ∈ Q, the restriction of p to Z(Y ) \ {v}, where v stands for the apex of Z(Y ), coincides with the radial projection defined in the previous paragraph. This map is G-equivariant.

Observe that |x -p(x)| Ẋ ρ, for every x ∈ Ẋ \ V.

Proposition 4.8. Let x, x ∈ X such that x, x v > 0, for every v ∈ V (here the Gromov product is computed in Ẋ). Then

|x -x | Ẋ |x -x | X π sh ρ 2ρ |x -x | Ẋ .
Proof. In this proof all the Gromov products are computed in Ẋ. The first inequality directly follows from the fact that the embedding X → Ẋ is 1-Lipschitz.

Let us focus on the second inequality. Let η > 0 and γ : [a , b] → Ẋ be a (1, η)quasi-geodesic from x to x . According to our assumption, up to decreasing η we can assume that for every (H, Y ) ∈ Q, the diameter of γ ∩ Z(Y ) is less than 2ρ. Consequently there exists a partition

t 0 = a t 1 • • • t m = b of [a , b] such that (i) γ(t i ) belongs to X for every i ∈ 0, m ; (ii) |γ(t i+1 ) -γ(t i )| Ẋ < 2ρ, for every i ∈ 0, m -1 .
Lemma 4.5 combined with the concavity of the map µ tells us that

2ρ π sh ρ |x -x | 2ρ π sh ρ m-1 i=0 |γ(t i+1 ) -γ(t i )| m-1 i=0 µ |γ(t i+1 ) -γ(t i )| m-1 i=0 |γ(t i+1 ) -γ(t i )| Ẋ |x -x | Ẋ + η.
This inequality holds for every sufficiently small η > 0, hence the result. Corollary 4.9. Let Z be a subset of Ẋ such that z, z v > 2 δ for every z, z ∈ Z and v ∈ V (here the Gromov product is computed in Ẋ). Then the radial projection p : Ẋ \ V → X restricted to Z is a quasi-isometric embedding.

Proof. In this proof all the Gromov products are computed in Ẋ. Let z, z ∈ Z. Let y, y ∈ X be the radial projections of z and z respectively. It follows from the triangle inequality that |z -z | Ẋ and |y -y | Ẋ differ by at most 2ρ. In view of Proposition 4.8 it is sufficient to prove that y, y v > 0 for every v ∈ V. The four point inequality (3) applied in Ẋ gives y, y v min y, z v , z, z v , z , y v -2 δ.

Assume that y, z v 2 δ. Then z necessarily belongs to the cone Z(Y ) for some (H, Y ) ∈ Q. Indeed otherwise z = y is a point of X, and thus y, z v 2ρ. It follows the from the definition of the radial projection and Lemma 4.6 that z lies on a geodesic between y and the apex of Z(Y ). As the distance between two apices is at least 2ρ, the point v is necessarily the apex of Z(Y ). Hence

z, z v = |v -z| Ẋ = y, z v
is bounded above by 2 δ, which contradicts our assumption. We prove in the same way that y , z v > 2 δ. On the other hand, according to our assumption we have z, z v > 2 δ. Thus y, y v > 0.

Parabolic subgroups. Lemma 4.10. Let P be a subgroup of G. If P is parabolic for its action on Ẋ, then so is its action on X.

Proof. Since the embedding X → Ẋ is 1-Lipschitz, P cannot be elliptic for its action on X. Hence it suffices to prove that P does not contain any loxodromic element (for its action on X). We denote by ξ the unique point of Λ(P ) ⊂ ∂ Ẋ. Let L > 100 δ and γ : R + → Ẋ be an L-local (1, 11 δ)-quasi-geodesic ray whose endpoint at infinity is ξ. Let g ∈ P . By Lemma 2.11, there is t 0 ∈ R + such that for every t t 0 , we have |gγ(t) -γ(t)| Ẋ < 2ρ. Since γ is infinite, there exists t t 0 such that γ(t) belongs to X. It follows then from Lemma 4.5 that

µ (|gγ(t) -γ(t)|) |gγ(t) -γ(t)| Ẋ < 2ρ.
Thus |gγ(t) -γ(t)| π sh ρ (Proposition 4.3). Consequently g π sh ρ, for every g ∈ P . In particular, P does not contain any loxodromic element for its action on X.

Apex stabilizer in the quotient space.

As we mentioned in the introduction the quotient space M = X/ Ḡ can be seen as an orbifold, whose fundamental group is Ḡ [START_REF] Delzant | Courbure mésoscopique et théorie de la toute petite simplification[END_REF]. Although this is not the point of view we adopted here, it is a great source of inspiration. According to Theorem 4.7 (iii), for every (H, Y ) ∈ Q, the quotient Stab(Y )/H embeds in Ḡ, which basically means that M is developable, so that its universal cover is X. This orbifold M also comes with an analog of Margulis' thin/thick decomposition for hyperbolic manifolds. The thin part corresponds to the neighborhood of the cone points (or more precisely their images in M ). In particular, if

x is point in a ball B(v, r) centered at a cone point v ∈ V and S a subset of Ḡ moving x by at most ρ -2r, then the triangular inequality tells us that every element in S fixes v, hence S generates an elliptic subgroup of Ḡ.

In this section we study the structure of X around the apices. In particular, we prove that the isotropy group of such a point locally acts as a dihedral group on a hyperbolic disc. To that end we make the following assumption (we refer to Section 2.3 for the definitions).

1 → F → Stab (Y ) q -→ L → 1, where L is either Z or D ∞ .
Since H is generated by a loxodromic element, its image in L is nZ for some n ∈ N \ {0}. We write L n for L/nZ, i.e.

L n = D n if L = D ∞ and L n = Z n , if L = Z.
Let v be the apex of Z(Y ) and v its image in X. Recall that, according to the small cancellation theorem (Theorem 4.7) the subgroup Stab(v) is isomorphic to Stab(Y )/H. After taking the quotient by H we get the following commutative diagram

1 F Stab(Y ) L 1 1 F Stab(v) L n 1 ∼ π
where the horizontal lines are short exact sequences. Note that Stab(v) → L n is a well-defined map. Indeed if (H , Y ) is another pair of Q such that v is this image of the apex v of Z(Y ), then there exists an element u ∈ K such that (H , Y ) = (uHu -1 , uY ). Thus the maps q : Stab(Y ) → L and q : Stab(Y ) → L differ at the source by the conjugation by u.

By analogy with singularities, the integer n is called the order of the cone point v. As we explained before L n can be either Z n or D n . In any case it embeds in D n . We call the map q v : Stab(v) → D n obtained in this way the geometric realization of Stab(v). Although it is not made explicit in the notation, we allow for the moment the order to be different from one apex to the other. Recall that the elements of D n are called rotations or reflections according to their action on the regular n-gon (see Section 3.2). This allows us to define similar notions for the elements of Stab(v). More precisely, we say that an element ḡ ∈ Stab(v) is a rotation (respectively a reflection, locally trivial ) at v if its image under q v is a rotation (respectively a reflection, trivial). A rotation at v is strict if it does not belong to F . A central half-turn at v is a strict rotation at v which is an involution and centralizes Stab(v) (note that the existence of such a half-turn forces n to be even). Given a reflection x ∈ L n , the pre-image under q v of x is called a reflection group at v.

Remark.

Being a reflection at v is a local property. Given two distinct apices v, v ∈ V, an element ḡ ∈ Ḡ can be simultaneously a reflection at v and locally trivial at v . For instance, consider the hyperbolic group

G = a, b, c | a 2 , b 2 , [b, c] = Z 2 * (Z 2 × Z)
where the left factor Z 2 is generated by a, whereas the right factor Z 2 × Z is generated by b and c. We consider the action of G on its Bass-Serre tree and blow up every vertex associated to (a conjugate of) Z 2 × Z to a line (on which Z 2 acts trivially). The resulting space X is a tree on which G acts properly co-compactly by isometries. Fix now a large integer n and define

Ḡ = G/ (ab) n , c n
One checks easily that Ḡ is a small cancellation quotient of G. Let v, v ∈ X be the apices of the cones attached to the relations (ab) n and c n respectively. One observes that the image b of b in Ḡ is a reflection at v but locally trivial at v . This subtlety is a source of difficulty when studying the strong ν-invariant ν stg ( Ḡ, X).

From now on, we make the following assumption. Assumption 4.12 (Central half-turn). For every apex v ∈ V, if the image of the geometric realization map q v : Stab(v) → D n has even torsion, then Stab(v) contains central half-turn at v.

Remark. Let us explain quickly how such an assumption can be satisfied. Later, when building the approximation sequence of B r (n), we will see that every loxodromic subgroup of G can be assumed to embed in a product of the form

D ∞ × D n × • • • × D n . In particular, if g ∈ Stab(Y ) is a primitive element of E, then g n/2
is almost central: it commutes with every element in E + and anticommutes with the ones of E\E + (i.e. ug n/2 u -1 = g -n/2 , for every u ∈ E\E + ). Consequenlty, if H is the subgroup generated by h = g n , then the image of g n/2 in Stab(Y )/H is a central half-turn.

Geometric realization. As suggested by the above terminology, the projection q v : Stab(v) → D n captures how Stab(v) acts geometrically on the ball B(v, ρ). To make this idea more precise, we are going to build a quasi-isometry between B(v, ρ) and a comparison hyperbolic cone D (endowed with the obvious action of D n ) which is almost q v -equivariant.

We first define a morphism L → Isom(R). Let ξ be one of the endpoints at infinity of Y . Let h 0 be a primitive element of Stab(Y ) whose attractive point is ξ.

• If L = Z, then we map the positive generator t of L to the translation by h 0 ∞ .

• If L is the dihedral group D ∞ = x, y | x 2 , y 2 , then we map x to the symmetry at 0 and y to the symmetry at h 0 ∞ /2. In particular, t = xy is mapped to the translation by h 0 ∞ .

Note that the resulting morphism L → Isom(R) does not depend on the choice of h 0 (any two primitive elements in Stab(Y ) have the same stable translation length). We write C for the quotient of R by the image of nZ in Isom(R). It is a circle whose perimeter is = n h 0 ∞ . We denote by D the cone of radius ρ over C and write o for its apex. The action of L on R induces an action by isometries of L n on D which fixes o. Observe that the space D is a hyperbolic cone (i.e. with constant sectional curvature equal to -1 everywhere except maybe at the apex) whose total angle at the apex o is

Ω = n h 0 ∞ sh ρ .
It follows from the small cancellation assumption that Ω > 10π. Said differently D can be decomposed into n copies of a sector of the hyperbolic disc of radius ρ whose angle is h 0 ∞ / sh ρ, so that D n is the group of isometries of D preserving this decomposition (see Figure 4). Let us now compare the hyperbolic cone D to the ball B(v, ρ). Let c ξ be a Busemann cocycle at ξ. Recall that H is cyclic. This allows us to build an H-invariant cocycle c : X × X → R which is at bounded distance from c ξ . Indeed as H is amenable there exists an H-invariant mean M : ∞ (H) → R. For every x, y ∈ X, we write f x,y : H → R for the map sending h to hc ξ (x, y) and define c(x, y) as the mean of f x,y . One checks that c is an H-invariant cocycle. Recall that H fixes ξ, hence hc ξ and c ξ differ by at most 6δ, for every h ∈ H. Consequently c and c ξ differ by at most 6δ as well. In particular, Lemma 2.10 yields |c(hx, x)| = h ∞ , for every x ∈ X and h ∈ H.

We now fix an arbitrary base point y 0 ∈ Y . If L is the infinite dihedral group, we choose y 0 in Fix(A, 15δ) where A ⊂ Stab(Y ) is the pre-image of x by q. Such a point always exists by Lemma 2.13. Recall that Y is contained in the 27δ-neighborhood of any L-local (1, δ)-quasi-geodesic joining the endpoints of Y , with L > 12δ. It follows from Lemma 2.3 that the map ϕ : Y → R sending y to c(y 0 , y) is an H-equivariant (1, 150δ)-quasi-isometric embedding. Moreover, this application is almost q-equivariant, in the sense that for every y ∈ Y , for every g ∈ Stab(Y ), we have

|ϕ(gy) -q(g)ϕ(y)| 200δ. Consequently ϕ induces a map φ : Y /H → C, such that for every ḡ ∈ Stab(Y )/H, for every ȳ ∈ Y /H, | φ(ḡ ȳ) -q v (ḡ) φ(ȳ)| 200δ.
By Lemma 4.4, φ induces a (1, 150δ)-quasi-isometric embedding Z(Y /H) → D, that we again still denote φ, so that for every ḡ ∈ Stab(Y )/H, for every

x ∈ Z(Y /H), we have | φ(ḡx) -q v (ḡ) φ(x)| D 200δ. (17)
Roughly speaking, this means that Stab(Y )/H acts on Z(Y /H) as L n does on D.

Note that Z(Y /H) -which is actually isometric to Z(Y )/H -is endowed here with the metric defined by [START_REF] Delzant | Courbure mésoscopique et théorie de la toute petite simplification[END_REF]. Although, as a set of points, Z(Y )/H can be identified with the closed ball of X of radius ρ centered at v (Theorem 4.7), the distance we considered so far is not the exactly the one coming from X.

Nevertheless the embedding Z(Y ) → Ẋ is 1-Lipschitz. It follows that the map φ : B(v, ρ) → D induced by φ : Z(Y /H) → D is such that for every x, x ∈ B(v, ρ) we have |x -x | X | φ(x) -φ(x )| D + 150δ. (18)
As we observed previously, the metrics of Z(Y ) and Ẋ coincide on B(v, ρ/3).

It follows that the metric on Z(Y /H) and X coincide on B(v, ρ/3). Hence the map φ :

B(v, ρ) → D is a (1, 150δ)-quasi-isometric embedding when restricted to B(v, ρ/3). Proposition 4.13. Let v ∈ V. (i) If ḡ ∈ Stab(v) is locally trivial at v, then B(v, ρ) is contained in Fix(ḡ, δ).
(ii) If Ā is a reflection group at v, then there exists a point x ∈ Fix( Ā, δ) with |v -x| = ρ such that for every z ∈ Fix( Ā, δ) ∩ B(v, ρ/3) we have

min { x, v z , ḡx, v z } δ,
where ḡ is a central half-turn at v.

(iii) If ḡ ∈ Stab(v) is a strict rotation at v, then there exists k ∈ Z, such that Fix(ḡ k , δ) is non-empty and contained in the δ-neighborhood of v. In particular, v is the unique vertex fixed by ḡ.

Remark 4.14. Roughly speaking Point (ii) is saying that any point of B(v, ρ/3) that is fixed by A lies on the geodesic [x, ḡx] -which goes through v by Point (iii). Nevertheless, in our setting, X does not need to be geodesic. Thus a rigorous statement is the one formulated above. Remark 4.15. It follows from Point (iii) that if Ē is an elliptic subgroup of Ḡ containing a strict rotation at v, then Fix( Ē, 10 δ) is contained in B(v, 14 δ). In particular, Ē is a subgroup of Stab(v).

Proof. We use the comparison map φ : B(v, ρ) → D defined during the previous discussion. Assume first that ḡ is locally trivial at v, i.e. q v (ḡ) = 1. In other words q v (ḡ) acts trivially on D. Combining ( 17) and ( 18) we get |ḡx -x| X 350δ, for every x ∈ B(v, ρ). Hence B(v, ρ) is contained in Fix(ḡ, δ), which completes the proof of (i).

Assume now that ḡ is a strict rotation at v. For simplicity, we let r = q v (ḡ). Since r is a non trivial rotation, one checks easily that there exists k ∈ Z such that r k acts on D as a rotation centered at o whose angle belongs to [Ω/4, 3Ω/4] We noticed before that thanks to the small cancellation assumption Ω > 10π.

In particular, for every x ∈ D, the angle at o between x and r k x is larger than π. Consequently

r k x -x D = 2 |x -o| D .
Recall that ϕ induces an almost q v -equivariant (1, 150δ)-quasi-isometric embedding from B(v, ρ/3) into D. Hence for every x ∈ B(v, ρ/3),

ḡk x -x X 2 |x -v| X -750δ.
In particular, Fix(ḡ k , δ) ∩ B(v, ρ/3) is contained in B(v, δ). Since Fix(ḡ k , δ) is 10 δ-quasi-convex (Lemma 2.8) the set Fix(ḡ k , δ) is entirely contained in B(v, δ), which completes the proof of (iii). We are left to prove Point (ii). Let Ā be a reflection group at v. Without loss of generality we can assume that q v (A) = x . It follows from Assumption 4.12 that n is even and Stab(v) contains a central-half turn ḡ. We write r for its image in L n . Recall that y 0 is a base point in Y ∩ Fix(A, 10δ) chosen to define the map φ. Let ȳ0 its image in X. It follows from the construction that the set of fixed point of x is exactly the geodesic of D between φ(ȳ 0 ) and r φ(ȳ 0 ). Note that this geodesic passes through o as the angle Ω at the apex of D is larger that 2π. Consequently for every d 0, for every x ∈ B(v, ρ/3), such that

|x φ(x) -φ(x)| d, we have either o, φ(ȳ 0 ) φ(x) d/2 or o, r φ(ȳ 0 ) φ(x) d/2.
We carry again this observation in X using the map φ : B(v, ρ) → D to get the conclusion of (ii).

Vocabulary. In view of the previous statement, we can say that an element ḡ ∈ Ḡ is a strict rotation if there is an apex v such that ḡ is a strict rotation at v. Indeed in such a case, ḡ cannot be locally trivial or a reflection at any other vertex. Note that being a strict rotation is invariant under conjugation.

Lifting properties

In Theorem 4.7 (iv) we mention a very important fact: small cancellation does not affect the small scale geometry of the space. More precisely the projection ζ : Ẋ → X is an isometry when restricted on small ball lying sufficiently far away from apices. This is a key ingredient to lift several figures from X to Ẋ. We complete this picture with other properties of the map ζ : Ẋ → X. Exceptionally, in this section all the distances are measured either in Ẋ or X. The first step is to explain how one can lift isometrically in Ẋ a quasi-convex subset Z ⊂ X as well as its (partial) stabilizer, provided it stays far away from the apex set V. Proof. In this proof all the distances are measure in Ẋ or X. We first claim that x, ȳ v 10 δ, for every v ∈ V. To that end we fix η > 0 and a (1, η)-quasigeodesic γ 1 : [a 1 , b 1 ] → Ẋ joining x to y. Note that γ stays far for V. Indeed, d(v, γ 1 )

x, y v -η/2, for every v ∈ V. It follows from Theorem 4.7 (iv) that if η is sufficiently small, then the image γ1 :

[a 1 , b 1 ] → X of γ 1 in X is an L-local (1, η)-quasi-geodesic joining x to ȳ for some L > 4η + 8 δ. Let v ∈ V.
Applying the stability of quasi-geodesics to γ1 (Proposition 2.2) we get

12 δ -η/2 inf g∈K d(gv, γ 1 ) d(v, γ1 ) x, ȳ v + η/2 + 2 δ.
This inequality holds for every sufficiently small η, which completes the proof of our first claim. Let η ∈ (0, δ) and y ∈ Ẋ be a pre-image of ȳ such that |x -y | |x -ȳ| + η. In particular, x, y v

x, ȳ v -η/2, for every v ∈ V. We are going to prove that y and y are very close, provided η is small enough. Let γ 2 : [a 2 , b 2 ] → Ẋ be a (1, η)-quasi-geodesic joining y to y . Let v ∈ V. Applying the four point inequality (3) in Ẋ, we observe that

y, y v min { x, y v , x, y v } -δ min { x, y v , x, ȳ v } -δ -η/2 > 8 δ.
Reasoning as previously we see that for a sufficiently small value of η, the image γ2 : [a 2 , b 2 ] → X of γ 2 in X is an L-local (1, η)-quasi-geodesic from ȳ to ȳ , where L > 4η +8 δ does not depend on η. By Proposition 2.2, it is also a (global) (κ, η)quasi-geodesic joining ȳ to itself, where κ can be chosen independentely of η. Therefore |y -y | |b 2 -a 2 | κη as we announced. Applying the triangle inequality we get |x -y| |x -y | + κη |x -ȳ| + (κ + 1)η. This holds for every sufficiently small η > 0, hence |x -y| |x -ȳ|. The converse inequality follows from the fact that ζ : Ẋ → X is 1-Lipschitz. Lemma 4.17. Let Z be a subset of Ẋ such that z, z v > 13 δ, for every z, z ∈ Z and every v ∈ V. The map ζ : Ẋ → X induces an isometry from Z onto its image Z. In addition, the following holds.

We now prove that z, z v > 12 δ, for every z, z ∈ Z and every v ∈ V. It follows from the very definition of Z that z, z 0 v z, z0 v and z , z 0 v z , z0 v . Combining the four point inequality (3) with our assumption on Z, we get

z, z v min { z, z 0 v , z , z 0 v } -δ min { z, z0 v , z , z0 v } -δ > 12 δ.
It now follows from Lemma 4.16 that the projection ζ : Ẋ → X induces an isometry from Z onto its image, i.e. Z. This proves the existence of the set Z. The uniqueness directly follows from the definition of Z. Since Z → Z is an isometry, z, z v z, z v , for every z, z ∈ Z, for every v ∈ V.

The previous statements explain how to lift in Ẋ a quasi-convex subset Z ⊂ X, as well as its (partial) stabilizer, as soon as it stays away from the apex set V. In particular, it applies to any (1, η)-quasi-geodesic path of X that avoids the cone points. We focus now on a more delicate operation which consists in lifting paths of X (and their almost stabilizers) going through one or several apices. The next statement follows [START_REF] Coulon | Partial periodic quotients of groups acting on a hyperbolic space[END_REF]Proposition 5.13].

Proposition 4.19. Let x and y be two points of X. Let γ : [a , b] → Ẋ be a path from x to y whose image γ : [a , b] → X is a (1, δ)-quasi-geodesic. Let S be a subset of G and S its image in Ḡ. We assume that |gx -x| Ẋ ρ/100 and |ḡ ȳ -ȳ| ρ/100, for every g ∈ S. In addition we suppose that for every apex v ∈ V satisfying x, ȳ v ρ/4, the set S lies in the local kernel at v. Then |gy -y| Ẋ = |ḡ ȳ -ȳ| for every g ∈ S.

Proof. Since γ is a (1, δ)-quasi-geodesic and the projection ζ : Ẋ → X is 1-Lipschitz, the path γ is a (1, δ)-quasi-geodesic. Let v 1 , . . . , v m be the apices of V which are ρ/5-close to γ. For every j ∈ 1, m , we denote by γ(c j ) a projection of v j on γ. By reordering the apices we can always assume that c 1 c 2 • • • c m . For simplicity of notation we put c 0 = a and c m+1 = b. Let j ∈ 1, m . Since γ is a (1, δ)-quasi-geodesic, we can find b j-1 ∈ (c j-1 , c j ] and a j ∈ [c j , c j+1 ) with the following properties.

(i) |v j -γ(b j-1 )| Ẋ = 9ρ/10 and |v j -γ(a j )| Ẋ = 9ρ/10,

(ii) γ ∩ B(v j , 4ρ/5) is contained in the image of γ restricted to (b j-1 , a j )
In addition, we let a 0 = a, b m = a m+1 = b (see Figure 5). We claim that for every j ∈ 0, m + 1 , for every g ∈ S, we have

|ḡγ(a j ) -γ(a j )| = |gγ(a j ) -γ(a j )| .
The proof is by induction on j. If j = 0 then γ(a j ) = x. The claim follows from the fact that the map ζ : Ẋ → X induces an isometry from B(x, ρ/20) onto B(x, ρ/20) -see Theorem 4.7 (iv). Assume now that our claim is true for some j ∈ 0, m . Since γ is a local quasi-geodesic, a j b j . We denote by γ j the restriction of γ to [a j , b j ] and by γj its image in X. In addition we write Z j (respectively Zj ) for the ρ/50-neighborhood of γ j (respectively γj ). It follows from the construction that γ j and thus γj stay away from any cone point. Consequently ζ : Ẋ → X is an isometry when restricted to the Z j (Lemma 4.17). Moreover, it induces an isometry from Z j onto Zj (Lemma 4.18). Let g ∈ S. By assumption its image ḡ in Ḡ moves x and ȳ by at most ρ/100. Recall that we required γ to be (1, δ)-quasi-geodesic. It follows from Lemma 2.7 that ḡ moves γ(a j ) and γ(b j ) by at most ρ/50. In particular, ḡγ(a j ) and ḡγ(b j ) belongs to Zj . Nevertheless, according to our induction assumption, gγ(a j ) is the (unique) lift of ḡγ(a j ) that belongs to Z j . Applying Lemma 4.17 In the previous statement, we assumed that any isometry ḡ ∈ S which hardly move the endpoints of γ, is actually in the local kernel of every vertex v lying close to γ. We now explore the situation where some element ḡ might be a reflection at v. Proposition 4.20. Let x and y be two points of X. Let S be a subset of G and S its image in Ḡ. We assume that |gx -x| ρ/100 and |ḡ ȳ -ȳ| ρ/100, for every g ∈ S. In addition we suppose that for every apex v ∈ V, the set S ∩ Stab(v) is contained in a reflection group at v. Then there exists an element u ∈ G with the following properties.

v 1 v j v j+1 v m x y (c 1 ) (c j ) (c j+1 ) (c m ) (b j 1 ) (a j ) (b j ) (a j+1 ) B(v j , ⇢) B(v j+1 , ⇢)
(i) ū commutes with every element in S;

(ii) |guy -uy| Ẋ = |ḡ ȳ -ȳ| for every g ∈ S;

(iii) either ū is trivial, or S lies in a reflection group at some apex v ∈ V.

Remark 4.21. Note that if S ∩Stab(v) is not contained in a reflection group at v, then there is a strict rotation at v which is the product of at most two elements of S. Indeed given any reflection x of D n , the geometric realization

q v : Stab(v) → D n cannot map S ∩ Stab(v) into x . Consequently the image of S ∩ Stab(v)
either contains a non-trivial rotation or two distinct reflections, whence the claim. This observation will be useful later to check that the assumptions of the proposition are fulfilled.

Proof. Assume first that for every apex v ∈ V satisfying x, ȳ v ρ/4, the set S lies in the local kernel at v. We fix ε > 0 and u ∈ K such that |x -uy| Ẋ |x -ȳ| + ε. In addition we take a (1, ε)-quasi-geodesic γ : [a , b] → Ẋ from x to uy. It follows from our choice of u that the image γ : [a , b] → X of γ is a (1, 2ε)quasi-geodesic from x to ȳ. Hence if ε is sufficiently small, Proposition 4.19 applies, which completes the proof.

Assume now that there exists a vertex v ∈ V satisfying x, ȳ v ρ/4 such that the set S does not lie in the local kernel at v. Any element ḡ ∈ S moves x and ȳ by at most ρ/100. Hence ḡ moves v by at most ρ (Lemma 2.7). It follows that S is contained in Stab(v). According to our assumption S is contained in a reflection group at v.

We now denote by U the set of all elements u ∈ G whose image ū in Ḡ commutes with S. This set is non-empty as it contains the identity. We chose u 0 ∈ U such that |ū 0 ȳ -x| |ūȳ -x| + δ, for every u ∈ U. We are going to prove that for every v ∈ V, if x, ū0 ȳ v ρ/4, then S lies in the local kernel at v. Consider indeed an apex v ∈ V such that x, ū0 ȳ v ρ/4. As ū0 commutes with S, the distance |ḡū 0 ȳ -ū0 ȳ| = |ḡ ȳ -ȳ| is bounded above by ρ/100 for every ḡ ∈ S. We prove as above that S is contained in a reflection group at v. Suppose now that contrary to our claim S is not in the local kernel at v. We fix s ∈ S a reflection at v. According to Proposition 4.13 (ii), there exists a point z0 ∈ X with |v -z0 | = ρ such that for every z ∈ Fix( S, δ) ∩ B(v, ρ/3) we have

min z0 , v z , hz 0 , v z δ,
where h is a central half-turn at v. Observe that ū1 = hū 0 belongs to U. Indeed, as h centralizes Stab(v), it commutes with S. We now claim that which completes the proof our claim and contradicts the minimality of u 0 . Consequently for every v ∈ V, if x, ū0 ȳ v ρ/4, then S lies the local kernel at v. The conclusion now follows from the discussion at the beginning of the proof.

|ū 1 ȳ -x| < |ū 0 ȳ -x| -2ρ/15. Let γ : [a , b] → X be a (1, δ)-quasi-

The action of Ḡ on

X

We now study the general properties of the action of Ḡ on X.

Proposition 4.22. The action of Ḡ on X is gentle.

Remark. Recall that the action of Ḡ on X is gentle if every loxodromic subgroup Ē preserving the orientation splits as Ē = F Z, where F is the set of all elliptic elements of Ē.

Proof. Let Ē be a loxodromic subgroup of Ḡ, preserving the orientation and Z its cylinder (see Section 2.3 for the definition). Assume first that there exists an apex v ∈ V such that d(v, Z) 20 δ. Let F be the set of all elliptic elements of Ē. Since Ē preserves the orientation, Z is contained in Fix( F , 100 δ) (Lemma 2.14). It follows that |ūv -v| < 2ρ, for every ū ∈ F . Consequently F = Ē ∩ Stab(v) and is therefore a (normal) subgroup of Ē. We now prove that Ē/ F is cyclic. To that end, we fix a loxodromic element ḡ0 ∈ Ē such that ḡ0 ḡ + δ, for every loxodromic element ḡ ∈ Ē. Since ḡ0 is a loxodromic element of Ē, it sends v to a distinct apex. It follows that ḡ0 ρ. Let ḡ ∈ Ē. The element ḡ0 acts on Z by translation of length approximately ḡ0 (Lemma 2.9). Hence there exists k ∈ Z such that ḡk 0 ḡ ḡ0 /2 + ρ/10 < ḡ0 -ρ/10. It follows from our choice of ḡ0 , that ḡk 0 ḡ is elliptic and thus belongs to F . Hence Ē/ F is a cyclic group generated by the image of ḡ0 .

Assume now that d(v, Z) > 20 δ for every apex v ∈ V. Since Z is 2 δ-quasiconvex, there exists a subset Z of Ẋ such that the projection ζ : Ẋ → X induces an isometry from Z onto Z (Lemma 4.18). It follows then from Lemma 4.17 that there exists a subgroup E of G such that π : G → Ḡ induces an isomorphism from E onto Ē. By construction z, z v > 13 δ, for every z, z ∈ Z, for every apex v ∈ V (the Gromov product is computed in Ẋ here). Consequently the radial projection p : Ẋ \ V → X induces an E-equivariant quasi-isometry from Z onto p(Z) (Corollary 4.9). This produces a π E -equivariant quasi-isometry from p(Z) to Z where π E stands for the map π restricted to E. Since Ē is loxodromic and preserves the orientation, the same holds for E. Moreover if F (respectively F ) stands for the set of elliptic elements of E (respectively Ē), then π sends F onto F . As the action of G is gentle, E splits as E = F Z. Hence Ē splits as well as E = F Z. Proof. In this proof all the distances are measured in Ẋ or X. Let v 1 be an apex of V and v1 its image in X. We denote by v 2 another apex such that

|v 1 -v 2 | |v 1 -v| + δ, for every v ∈ V \ {v 1 }. We are going to prove that the image v2 of v 2 in X is distinct from v1 . Let γ : [a 1 , a 2 ] → Ẋ be a (1, δ)-quasi- geodesic from v 1 to v 2 . Let b 1 = a 1 + ρ/4 and b 2 = a 2 -ρ/4. For simplicity we write x 1 = γ(b 1 ) and x 2 = γ(b 2 ). Note that |x 1 -x 2 |
3ρ/2. Moreover, x 1 , x 2 v > 12 δ, for every v ∈ V. Indeed otherwise v would be a cone point distinct from v 1 but much closer to v 1 than v 2 . According to Lemma 4.16, we have

|x 1 -x2 | = |x 1 -x 2 |, hence |x 1 -x2 | 3ρ/2.
Combined with the triangle inequality we obtain |v 1 -v2 | ρ. Proposition 4.24. Assume that V contains two distinct apices whose order is at least 3. Then the action of Ḡ on X is non-elementary.

Proof. Let v1 and v2 be those apices. We fix a point x ∈ X such that v1 , v2 x δ whereas |x -v1 | ρ and |x -v2 | ρ. Reasoning as in Proposition 4.13 (iii) we see that Stab(v i ) contains a rotation ḡi at vi such that Fix( Si , δ) is contained in B(v i , δ), where Si = {ḡ i , ḡ2

i }, see Figure 7. Applying Lemma 2.8, we get Combined with the four point inequality (3) it yields ḡ±1 1 x, ḡ±1 2 x x 3 δ. It follows from Lemma 2.15 that ḡ1 and ḡ2 generate a non-elementary subgroup. Hence Ḡ is non-elementary.

Structure of elementary subgroups

An important step to study further the action of Ḡ on X (and its invariants) is to understand the algebraic structure of its elementary subgroups. As the map X → X is 1-Lipschitz, the projection π : G Ḡ maps every elementary subgroup of G to an elementary subgroup of Ḡ. More precisely it sends every elliptic (respectively parabolic, loxodromic) to an elliptic (respectively elliptic or parabolic, elementary) subgroup of Ḡ. However the nature of these subgroups (i.e. whether they are elliptic, parabolic, or loxodromic) may change. Indeed given any element g ∈ G, there always exists h ∈ K such that gh is a loxodromic element of G. Thus gh is loxodromic whereas its image ḡ in Ḡ can be anything. New elementary subgroups may also appear in Ḡ. This motivates the following definition. Definition 4.25. Let Ē be an elliptic (respectively parabolic, loxodromic) subgroup of Ḡ. We say that Ē can be lifted if there exists an elliptic (respectively parabolic, loxodromic) subgroup E of G (for its action on X) such that the quotient map π : G Ḡ induces an isomorphism from E onto Ē. In this situation E is a lift of Ē.

Note that in this definition we ask E and Ē to have the same nature. The idea is that the subgroups of Ḡ that can be lifted are as "easy" as the elementary subgroups of G. Complicated algebraic structures necessarily come for the "new" elementary subgroups. In the next paragraphs we discuss whether an elementary subgroup of Ḡ can be lifted. If not we use the geometry of small cancellation theory to describe its properties.

Comparing lifts. We start by proving that the lift of an elliptic subgroup of Ḡ is essentially unique. More precisely, if F 1 and F 2 are two lifts of the same elliptic subgroup F ⊂ Ḡ, then F 1 and F 2 are conjugated. This result is a particular case of a more general statement (see Corollary 4.28) that allows to consider simultaneously elliptic and parabolic subgroups. Lemma 4.26. Let S be a subset of G such that Fix(S, ρ/10) is non-empty. Then the projection π : G Ḡ is one-to-one when restricted to S. In particular, if E is an elliptic or a parabolic subgroup, the projection π induces an isomorphism from E onto its image.

Proof. The first part of the statement is a consequence of Theorem 4.7 (v). Since E is elliptic or parabolic, Fix({1, g}, ρ/10) is non-empty for every g ∈ E -see for instance [START_REF] Coulon | On the geometry of Burnside quotients of torsion free hyperbolic groups[END_REF]. Hence the result.

Proposition 4.27. Let E be an elliptic or a parabolic subgroup of G (for its action on X). Let S 1 be a subset of E such that Fix(S 1 , ρ/100) is non-empty and S1 its image in Ḡ. Let h ∈ Ḡ. Let S 2 be a pre-image in G of h S1 h-1 such that Fix(S 2 , ρ/100) is non-empty. Then there exists h 0 ∈ G with the following properties (i) For every g ∈ S 1 , the element h 0 gh -1 0 is the (unique) pre-image of hḡ h-1 in S 2 .

(ii) If h is loxodromic, then either h 0 is loxodromic, or S1 is contained in a reflection group at some vertex v ∈ V.

Remark.

Observe that h 0 is not necessarily the pre-image of h.

Proof. Let h ∈ G be an arbitrary pre-image of h. We fix two points x 1 , x 2 ∈ X lying respectively in Fix(S 1 , ρ/100) and Fix(S 2 , ρ/100). Note that both x1 and h-1 x2 belongs to Fix( S1 , ρ/100). We claim that S1 ∩ Stab(v) is contained in a reflection group at v, for every v ∈ V. Assume on the contrary that it is not the case. There exists g ∈ E whose image ḡ is a strict rotation (Remark 4.21).

According to Proposition 4.13 (iii) there exists k ∈ N such that Fix(ḡ k , δ) is contained in B(v, δ). On the other hand, since g belongs to E, the element g k is elliptic or parabolic (as an isometry of X). Hence there exists x ∈ X such that |g k x -x| 10δ -see for instance [START_REF] Coulon | On the geometry of Burnside quotients of torsion free hyperbolic groups[END_REF] -and thus |ḡ k x -x| δ. This contradicts the previous point and completes the proof of our claim. It follows from Proposition 4.20 applied with x = x 1 and y = h -1 x 2 that there exists u ∈ G, such that ū centralizes S1 and

guh -1 x 2 -uh -1 x 2 Ẋ = ḡh -1 x2 -h-1 x2 , ∀g ∈ S 1 . ( 19
)
Moreover either ū is trivial or S1 lies in a reflection group. We let h 0 = hu -1 . Let g ∈ S 1 and g the (unique) pre-image of hḡ h-1 in S 2 . It follows from ( 19) that h 0 gh -1 0 and g are two pre-images of hḡ h-1 that move x 2 by at most ρ/100. Thus h 0 gh -1 0 = g , which proves (i). Assume now that h is loxodromic. If ū is trivial, then h 0 is a pre-image of h, hence a loxodromic element (recall that ζ : X → X is 1-Lipschitz). On the contrary if ū is not trivial, then S1 is contained in a reflection group.

Corollary 4.28. Let F 1 and F 2 be two subgroups of G. We assume that F 1 is elliptic and F 2 generated by a set S 2 such that Fix(S 2 , ρ/100) is non-empty. Let F1 and F2 be their respective images in Ḡ. If F1 = F2 , then there exists u ∈ G whose image in Ḡ centralizes F1 and such that

F 2 = uF 1 u -1 .
Remark. Note that the assumption on F 2 is automatically satisfied if F 2 is elliptic. In particular, if F 1 and F 2 are two elliptic subgroups of G whose images in Ḡ coincide, then they are conjugate.

Proof. Let S be the image of S 2 in F1 = F2 and S 1 the pre-image of S in F 1 . Note that Fix(S 1 , ρ/100) is non-empty (Lemma 2.13). According to Proposition 4.27 applied with S 1 and S 2 , there exits u ∈ G such that for every s ∈ S 1 , the element usu -1 is the pre-image of s in S 2 . In particular, ū commutes with S, hence F1 . Moreover since S 2 generates F 2 , the group u -1 F 2 u is contained in F 1 . Nevertheless the projection π : G → Ḡ is one-to-one when restricted to F 1 (Lemma 4.26). Thus F 2 = uF 1 u -1 . Corollary 4.29. Let F 1 and F 2 be two subgroups of G. We assume that F 1 is elliptic and F 2 generated by a set S 2 such that Fix(S 2 , ρ/100) is non-empty. Let F1 and F2 be their respective images in Ḡ. If F1 and F2 , are conjugated in Ḡ, then so are F 1 and F 2 in G.

Lifting elliptic subgroups. We now characterize the elliptic subgroups of Ḡ that can be lifted and explore the structure of the one that cannot be lifted. The statement generalizes [16, Lemme 5.10.2]. Proposition 4.30 (Lifting elliptic subgroups). An elliptic subgroup F of Ḡ cannot be lifted if and only if it contains a strict rotation. In this case, F fixes an apex v ∈ V. Moreover, Fix( F , δ) is contained in B(v, δ). In particular, v is the only apex fixed by F .

Proof. Recall that Fix( F , 10 δ) is a non-empty 8 δ-quasi-convex subset of X (Lemmas 2.13 and 2.8). Assume first that there exists a point x ∈ Fix( F , 10δ) such that d(x, V) ρ/3. By Proposition 4.13 (iii), F does not contain a strict rotation. We are going to prove that F can be lifted. We write Z for the Forbit of x. It is F -invariant and its diameter is at most 10 δ. It follows that z, z v ρ/4, for every z, z ∈ Z, for every v ∈ V. According to Lemmas 4.18 and 4.17, there exist a subgroup F of G and an F -invariant subset Z of Ẋ with the following properties: the map ζ : Ẋ → X induces an isometry from Z onto Z; the projection π : G → Ḡ induces an isomorphism from F onto F ; moreover z, z v > 13 δ for every z, z ∈ Z, for every v ∈ V (the Gromov product is computed in Ẋ here). In particular, Z is bounded. It follows from Corollary 4.9 that the radial projection p(Z) is a bounded F -invariant subset of X. Hence F is an elliptic subgroup of G (for its action on X), lifting F . Assume now that d(x, V) < ρ/3 for every x ∈ Fix( F , 10 δ). Since the set Fix( F , 10 δ) is 8 δ-quasi-convex, there exists v ∈ V such that Fix( F , 10 δ) is contained in B(v, ρ/3). In particular, F fixes v. In addition, F cannot be lifted. Indeed if F was a lift of F , then the image in X of Fix(F, 10δ) ⊂ X would be contained in Fix( F , 10 δ) \ B(v, ρ/3). We now claim that F contains a strict rotation ḡ at v. If it was not the case, then F would be contained in a reflection group at v. Thus, there would exist a point x ∈ Fix( F , 10 δ) such that |x -v| > ρ/2, see Proposition 4.13 (ii), which contradicts the previous observation. It follows then from Proposition 4.13 (iii) that Fix( F , δ) is contained in B(v, δ). Proof. Let F be the maximal elliptic normal subgroup of Stab(Y ) and F its image in Ḡ. Recall that we have the following commutative diagram

1 F Stab(Y ) L 1 1 F Stab(v) L n 1 q v ∼ π where (L, L n ) is either (Z, Z n ) or (D ∞ , D n ).
Since C can be lifted, its image under q v does not contain a non-trivial rotation (Proposition 4.30). The result follows from diagram chasing.

We continue with a study of dihedral germs. Let F be an elliptic subgroup of G and F its image in Ḡ. Observe that if F is a dihedral germ, then the same does not necessarily hold for F . Indeed it may happen that the only loxodromic elements that are normalizing (a finite index subgroup of) F became elliptic in Ḡ. Nevertheless the converse statement holds. This is the aim of the next lemmas. Proof. For simplicity we let E = Stab(Y ). Let E + be the subgroup of E fixing pointwise ∂Y and F the maximal elliptic subgroup of E + . Let C be an elliptic subgroup of E. The intersection C 0 = C ∩ E + is a subgroup of C with index at most 2. Let h be a (loxodromic) element in H. Let c ∈ C 0 . It follows from the small cancellation assumption that H is a normal subgroup of E. In particular, chc -1 = h k for some k ∈ Z. Recall that E + /F is isomorphic to Z. Pushing the previous identity in Z, we get that k = 1. In other words h commutes with C 0 . Hence C is a dihedral germ. The next lemma is formally not needed. However it illustrates the role played by dihedral germs. As we observed earlier, every elliptic subgroup F of G yields an elliptic subgroup F of Ḡ. However it could happen that F is strictly contained in an elliptic subgroup, which does not already come from a subgroup of G containing F . In this case F is necessarily a dihedral germ. As suggested by the name, dihedral germs are exactly the elliptic subgroups of G which can eventually "grow" when passing to the quotient Ḡ. Proof. This is just a reformulation of Lemma 4.33. Indeed according to Proposition 4.30, there exists v ∈ V such that Ā is contained in Stab(v). Since C can be lifted, it does not contain a strict rotation at v (Proposition 4.30) and thus lies in a reflection group at v.

We complete our discussion on elliptic subgroups with some preparatory work for the study of loxodromic subgroups. If such a group Ē does not preserve the orientation, it can be decomposed as Ē = Ā * C B, where C has index 2 in both Ā and B. As the cylinder of Ē is contained in Fix( C, 100 δ) (Lemma 2.14), C can also be lifted. We describe in this context what is the structure of Ā or B.

Lemma 4.36. Let Ā be an elliptic subgroup of Ḡ. Assume that Ā contains a subgroup C of index 2 that can be lifted. Let ā ∈ Ā \ C. Then there exists ū ∈ Ḡ such that (i) C, ū is an elliptic subgroup that can be lifted;

(ii) ā-1 ū centralizes C; and

(iii) ā2 = ū2 . Remark. Observe that if ū is trivial, then Ā is isomorphic to C × ā = C × Z 2 .
In general the map ā → ū extends to an (abstract) epimorphism from Ā onto C, ū .

Proof. If Ā can be lifted, then the statement obviously holds. Assume now that Ā cannot be lifted. There exists v ∈ V such that Ā is contained in Stab(v) (Proposition 4.30). Let q v : Stab(v) → D n be corresponding the geometric realization. Since C can be lifted, q v ( C) is either trivial or equal to x where x is a reflection of D n . Let t be a generator of the rotation group Z n ⊂ D n .

Recall that C has index 2 in Ā and Ā cannot be lifted in G. It follows that n is even and

q v ( Ā) = q v ( C), t n/2 .
In particular, q v (ā)t n/2 belongs to q v ( C). According to Assumption 4.12, Stab(v) contains a central half-turn at v that we denote by ḡ. Note that q v maps ḡ to t n/2 . We let ū = āḡ. We observe that ā-1 ū = ḡ centralizes C and ā2 = ū2 . By construction C, ū and C have the same image under q v . Consequently C, ū is contained in a reflection group at v, hence can be lifted, which completes the proof.

Another crucial ingredient to describe loxodromic subgroups of Ḡ, is to understand the normalizer of elliptic subgroups that can be lifted, see for instance Proposition 4.42. This is the purpose of the next proposition. Proposition 4.37 (Lifting normalizer). Let F be an elliptic subgroup of G and F its image in Ḡ. For every h ∈ Norm( F ) there exists h 0 ∈ Norm(F ) such that h-1h 0 centralizes F . If in addition h2 belongs to F , then one can choose h 0 such that h 2 0 ∈ F and h2 = h2 0 .

hence g k X 10δ. Thus there exists x ∈ X such that |g k x -x| 10δ, thus x belongs to Fix(ḡ k , δ) \ B(v, δ), which yields another contradiction. Proposition 4.40 (Lifting parabolic subgroups). Let P be parabolic subgroup of Ḡ. Assume that there exist d ∈ R + and a subset S generating P such that Fix( S, d) is non-empty. Then P can be lifted.

Proof. Let ξ be the unique point of ∂ X fixed by P and x be a point in the set Fix( S, d). Let L > 100 δ and γ : R + → X be an L-local (1, 11 δ)-quasi-geodesic ray starting at x whose endpoint at infinity is ξ. According to Lemma 2.11, there exists t 0 such that for every t t 0 , for every g ∈ S, we have |ḡγ(t) -γ(t)| 42 δ. It follows that d(γ(t), V) ρ/2, for every t t 0 . Indeed otherwise, there would exist v ∈ V such that S, and thus P , is contained in Stab(v), which contradicts our assumption.

Let Z be the ρ/10-neighborhood of γ restricted to [t 0 , ∞). It is a 2 δ-quasiconvex subset. According to our claim that z, z v > ρ/10 for every z, z ∈ Z and v ∈ V. It follows from Lemma 4.18 that there exist a subset Z of Ẋ such that the map ζ : Ẋ → X induces an isometry from Z onto Z. Moreover z, z v > ρ/10 for every z, z ∈ Z and v ∈ V (the Gromov product are computed in Ẋ here). In particular, there exists an L-local (1, 11 δ)-quasi-geodesic ray γ : [t 0 , ∞) → Ẋ contained in Z such that π • γ = γ and Z is its ρ/10 neighborhood. We write ξ ∈ ∂ Ẋ for the endpoint at infinity of γ.

We now claim that P is contained in the image of Stab(ξ) by the projection π : G → Ḡ. Let ḡ ∈ P . As we observed, |ḡγ(t) -γ(t)| 42 δ, for every t t 0 . It follows from Lemma 4.17 applied to Z that there exists a pre-image g ∈ G of ḡ such that for every t t 0 , we have |gγ(t) -γ(t)| Ẋ 42 δ. In particular, g fixes ξ, which completes the proof our first claim.

We denote by P the pre-image of P in Stab(ξ). We now claim that P is parabolic for its action on X. To that end, it suffices to show that P is parabolic for its action on Ẋ (Lemma 4.10). We are going to prove that P is not elliptic and does not contain a loxodromic element. As P is parabolic, it has unbounded orbits. The map ζ : Ẋ → X being 1-Lipschitz, the group P has unbounded orbits, and thus cannot be elliptic (for its action on Ẋ). Let g ∈ P and ḡ its image in P . According to Lemma 2.11 there exist t 1 t 0 and ε ∈ {±1} such that for every t t 1 we have |gγ(t) -γ(t + ε g ∞ Ẋ )| Ẋ 42 δ. In particular, if t is sufficiently large both γ(t) and gγ(t) belong to Z. Since ζ : Ẋ → X is an isometry when restricted to Z, we get

g Ẋ |gγ(t) -γ(t)| Ẋ |ḡγ(t) -γ(t)| 42 δ.
This inequality holds for every g ∈ P . Therefore P cannot contain a loxodromic element for its action on Ẋ, which completes the proof of our second claim. As P is a parabolic subgroup, the projection π : G → Ḡ is one-to-one when restricted to P (Lemma 4.26). Hence P is a lift of P .

Lifting loxodromic subgroups. It Ḡ does not contain any element of order 2, then one can prove that all its loxodromic subgroups can be lifted. This is typically what is happening when studying Burnside groups of odd exponent. As shown by the next example this is unfortunately no more the case in the presence of even torsion.

Example 4.41. Assume for instance that Ḡ is the group defined by

Ḡ = Z n * Z n = a, b | a n = b n = 1 .
If n is a sufficiently large even integer, it is a small cancellation quotient of the free group F 2 generated by a and b. The subgroup a n/2 , b n/2 ≡ D ∞ is loxodromic but cannot be lifted.

In general we show that every loxodromic subgroup of Ḡ is an (abstract) subdirect product of an elementary subgroup of G and either Z or D ∞ . Proposition 4.42 (Lifting loxodromic subgroups). Let Ē be a loxodromic subgroup of Ḡ and F the maximal normal elliptic subgroup of Ē. There exist a lift F of F , an elementary subgroup E of G containing F , and an epimorphism θ : Ē E with the following properties.

(i) (E , F ) is a dihedral pair.

(ii) The morphism π • θ is the identity when restricted to F .

(iii) The map θ induces an embedding from Ē into Ē/ F × E .

Proof. Since the action of Ḡ on X is gentle (Proposition 4.22), the group Ē fits in a short exact sequence

1 → F → Ē q -→ L → 1,
where L is either Z or D ∞ . The cylinder Z of Ē is contained in Fix( F , 100 δ) (Lemma 2.14). In particular, Fix( F , 100 δ) ∩ ζ(X) is non-empty. It follows from Proposition 4.30 that F admits a lift in G that we denote by F . The subgroup F is a dihedral germ, hence so is F (Lemma 4.34). We now claim that there exists an elementary subgroup E of G containing F as a normal subgroup such that the canonical section F → F extends to an epimorphism θ : Ē E . To that end we distinguish two cases.

Case 1. Assume that Ē preserves the orientation, i.e. L = Z. Then Ē splits as a semi-direct product Ē = F L. Let h be a primitive element of Ē (i.e. an element whose image under q generates L). According to Proposition 4.37 there exists h 0 ∈ Norm(F ) such that h-1h 0 centralizes F . Let E be the subgroup of G generated by F and h 0 . The canonical section F → F extends to an epimorphism θ : Ē → E sending h to h 0 .

Case 2. Assume that Ē does not preserve the orientation, so that L = D ∞ . Let x 1 , x 2 ∈ D ∞ be two reflections generating L. Let ā1 , ā2 ∈ Ē be pre-images of x 1 and x 2 respectively. By construction āi normalizes F and ā2 i belongs to F . According to Proposition 4.37 there exists b i ∈ Norm(F ) with the following properties: ā-1 i bi centralizes F ; b 2 i belongs to F ; and ā2 i = b2 i . Let E be the subgroup of G generated by F , a 1 , and a 2 . The canonical isomorphism F → F extends to an epimorphism θ : Ē → E sending āi to b i .

In both cases we have build the map announced in the claim. As θ extends the canonical section F → F , the composition π•θ is the identity when restricted to F . Moreover θ induces an epimorphism Ē/ F E /F . Any quotient of a dihedral group is still a dihedral group. Hence (E , F ) is a dihedral pair. In particular, E is elementary. One checks that the map Ē → Ē/ F × E induced by θ is an embedding.

In the remainder of this section we revisit the previous statement and explore further the structure certain loxodromic subgroups that cannot be lifted. As suggested by Example 4.41 such a group Ē often does not preserves the orientation. In particular, it splits as Ē = Ā * C B where C is the maximal elliptic normal subgroup of Ē and has index 2 in both Ā and B. If Ē could be lifted, then obviously so would Ā and B. Nevertheless the converse is false. This is the purpose of the first proposition. The second one discusses the case where Ā or B cannot be lifted. (ii) The subgroup A∩uBu -1 contains the pre-image of C in A, and A, uBu -1 is elementary.

Proof. Recall that the quotient map π : G Ḡ induces an isomorphism from A and B onto Ā and B respectively (Lemma 4.26). Let C A (respectively C B ) be the pre-image of C in A (respectively B). According to Corollary 4.28, there exits u

∈ G whose image ū in Ḡ centralizes C such that C A = uC B u -1 . It follows that C A ⊂ A ∩ uBu -1
. Moreover this elliptic subgroup has index 2 in both A and uBu -1 , hence E u is elementary.

Before moving to the case where Ā or B cannot be lifted, let us illustrate the previous statement with an example.

Example 4.44. Let G be the group defined by

G = a 1 , a 2 , b, c | a 2 1 , a 2 2 , b 2 , c 2 , [a 1 , c], [a 2 , c] = (D ∞ × Z 2 ) * Z2 D ∞ .
acting on its Cayley graph X. In this description the elements a 1 , a 2 and c (respectively b and c) generate the factor D ∞ × Z 2 (respectively D ∞ ). In particular, the amalgamated subgroup is Z 2 = c . Set s = a 1 a 2 and r = bc. If n is a sufficiently large integer the group Ḡ = G/ s n , r n is a small cancellation quotient of G. Assume in addition that n is even. It follows that ū = rn/2 commutes with c. Consequently the subgroup Ē generated by ā1 , ū-1 ā1 ū and c is loxodromic and isomorphic to D ∞ × Z 2 . Note that Ē cannot be lifted in G.

Indeed since Z 2 = c is malnormal in D ∞ = b, c every loxodromic subgroup of G is isomorphic to either Z or D ∞ .
Observe that Ē also splits as Ē = Ā * C B where Ā = ā1 , c , B = ū-1 ā1 ū, c , and C = c . As ū commutes with b, the group B is actually B = ū-1 Āū. We are in a configuration where both Ā and B are elliptic subgroups which can be lifted.

As described in Proposition 4.43, a partial conjugation by ū maps Ē to a new elementary subgroup Ēu = Ā which is not necessarily loxodromic. In this precise example, it turns out that Ēu can be lifted in G. This is not always the case though. Indeed we can run the same construction with Ē = ā1 , ū-1 ā2 ū, c . It is a loxodromic subgroup of Ḡ isomorphic to D ∞ × Z 2 . On the other hand, it splits as Ē = Ā C B where B = ū-1 ā2 ū, c . In this case Ē u = ā1 , ā2 , c is an elliptic subgroup of Ḡ, which is isomorphic to D n × Z 2 , and thus cannot be lifted. Nevertheless, in both cases, Ēu or Ē u is the image (by the natural quotient map) of an elementary subgroup of G, which was not the case for Ē or Ē . Proposition 4.45. Let Ē be a loxodromic subgroup of Ḡ that splits as Ē = Ā * C B, where C is the maximal elliptic normal subgroup of Ē and has index 2 in both Ā and B. Assume that there exists v ∈ V such that Ā contains a strict rotation at v. Let q v : Stab(v) → D n be the associated geometric realization map and r ∈ D n a generator of the rotation group. If n is divisible by 4, then one of the following holds (i) q v ( C) is trivial and

q v Ā = r n/2 ,
(ii) q v ( C) is a reflection group generated by say x ∈ D n and

q v Ā = x, r n/4 xr -n/4 .
Suppose now that there exist a subgroup Ē0 ⊂ Ā and an element h ∈ Ḡ such that h Ē0 h-1 is contained in Ā and Ā = Ē0 , C . Then either Ē0 contains a strict rotation, in which case h fixes v or the first case above fails and q v maps Ē0 onto r n/4 xr -n/4 .

Remark.

Our assumption on Ā exactly means that Ā cannot be lifted (Proposition 4.30). It follows from Remark 4.15 that Ā is contained in Stab(v), hence the image under q v of C or Ā is well defined. It is important to note that in the second part of the statement h is not necessarily an element of Stab(v). In particular, if h does not fix v, the last conclusion tells us that C and Ē0 are two reflection groups at v; geometrically we can think that one is the conjugate of the other by a quarter-turn at v.

Proof. The first part of the proof is essentially a variation on Lemma 4.36. Since the cylinder of Ē is contained in Fix( C, 100 δ) (Lemma 2.14) the subgroup C can be lifted and thus does not contain a strict rotation (Proposition 4.30). In particular, C is either locally trivial at v or a reflection group at v. Assume first that q v maps C to the trivial group. By assumption Ā contains a strict rotation at v whereas [ Ā : C] = 2, which forces

q v ( Ā) = r n/2 .
Assume now that q v maps C to a reflection group generated by say x ∈ D n . Reasoning as above we observe that

q v ( Ā) = x, r n/2 = x, r n/4 xr -n/4 .
This completes the first part of the statement.

Let us focus now on the second half of the proposition. Suppose first that Ē0 contains a strict rotation (which is necessarily at v). Then h Ē0 h-1 contains a strict rotation at hv, which as an element of Ā has to fix v. Strict rotations having a single fixed vertex (Proposition 4.13) it yields hv = v. Suppose now that Ē0 does not contain a strict rotation. In particular, q v maps both Ē0 and C to subgroups of D n which are trivial or reflection groups. Since Ē0 and C generates Ā, the only possible configuration for q v ( Ā) to contain a non-trivial rotation is the one where q v ( Ē0 ) and q v ( C) are two distinct reflection groups. Consequently the first case above fails, and q v ( Ē0 ) = r n/4 xr -n/4 , see Figure 8 Ē0 C v Figure 8: The action of Ē on B(v, ρ). One assumes here that Ē0 does not contain a strict rotation. The shaded areas represent Fix( C, δ) and Fix( Ē0 , δ) respectively.

Invariants of Ḡ acting on

X

This section is devoted to the study of the numerical invariants associated to the action of Ḡ on X, namely inj Ḡ, X , A( Ḡ, X, d) and ν stg ( Ḡ, X) (see Section 3 for the definitions). As we explained earlier, the first two are purely geometric, whereas the last one has a mixed nature and captures both geometric and algebraic features of Ḡ.

Geometric invariants

The injectivity radius. Proposition 4.46 (Compare with [START_REF] Delzant | Courbure mésoscopique et théorie de la toute petite simplification[END_REF]Lemma 5.11.1]). Let N be a subgroup of G containing K and N its image in Ḡ. We denote by the infimum over the stable translation length (in X) of loxodromic elements of N which do not belong to Stab(Y ) for some (H, Y ) ∈ Q. Then inj N , X min κ , δ , where κ = δ/π sh(10 δ).

Remark. By convention, if N does not contain any loxodromic element then inj N , X is infinite, in which case the statement is void.

Proof. Let ḡ be a loxodromic element of N . We need to show that ḡ ∞ min κ , δ . By [START_REF] Coulon | On the geometry of Burnside quotients of torsion free hyperbolic groups[END_REF] we have m ḡ ∞ ḡm -8 δ, for every m ∈ N. Therefore it suffices to find an integer m such that ḡm m min κ , δ + 8 δ.

We denote by m the largest integer satisfying m min κ , δ 2 δ. Assume that ḡm is smaller than m min κ , δ + 8 δ. In particular, ḡm < 10 δ. Thus Fix(ḡ m , 10 δ) is non empty. Moreover d(x, V) ρ -5 δ, for every x ∈ Fix(ḡ m , 10 δ). Indeed if it was not the case, ḡm would fix an apex v ∈ V which contradicts the fact that ḡ is loxodromic. Hence Z = Fix(ḡ m , 20 δ) contains a point in ζ(X). Note also that Z is a 8 δ-quasi-convex subset. Thus z, z v > ρ/2, for every z, z ∈ Z and v ∈ V. By Lemma 4.18, there exists a subset Z of Ẋ such that the map ζ : Ẋ → X induces an isometry from Z onto Z and the projection π : G Ḡ induces an isomorphism from Stab(Z) onto Stab( Z). Observe that ḡ preserves Z. We denote by g the preimage of ḡ in Stab(Z). Since the kernel K is contained in N , the element g belongs to N . By construction g is a loxodromic element which does not belong to any Stab(Y ) where (H, Y ) ∈ Q. Hence g ∞ . As we noticed before Z contains a point in z ∈ ζ(X). Let z ∈ X be its pre-image in Z. It follows from Lemma 4.5, that

µ (|g m z -z| X ) |g m z -z| Ẋ |ḡ m z -z| X 20 δ < 2ρ. By Proposition 4.3, m m g ∞ |g m z -z| X π sh(10 δ) = κ -1 δ,
which contradicts the maximality of m.

If Ḡ has no even torsion, one can prove using only geometrical arguments the following dichotomy -see for instance the proof of [START_REF] Coulon | Partial periodic quotients of groups acting on a hyperbolic space[END_REF]Proposition 5.28]. Given any chain C = (ḡ 0 , . . . , ḡm ) generating an elementary subgroup of Ḡ and h ∈ Ḡ a loxodromic conjugating element of C, (i) either ḡ0 , h is elementary, (ii) or the chain C can be lifted to a chain C = (g 0 , . . . , g m ) in G which generates an elementary subgroup of G and where one of the conjugating elements h of C is a pre-image of h.

In the latter case, h is necessarily loxodromic. If in addition m ν(G, X), we can conclude that g 0 , h and thus ḡ0 , h is elementary. It follows then that ν( Ḡ, X) ν(G, X).

Unfortunately this strategy fails in the presence of even torsion. In Proposition 4.37 we observed the following phenomenon. Let F be an elliptic subgroup of G and F its image in Ḡ. If h is an element of Ḡ normalizing F , then there exists an element h 0 ∈ G normalizing F whose action by conjugation on F coincide with the one of h on F . However h 0 is not necessarily a pre-image of h. In particular, if h is loxodromic, there is no reason that h 0 should be loxodromic as well. The same issue arises when lifting chain. If a chain in Ḡ admits a loxodromic conjugating element, there is no reason that its lift in G (provided it exists) has a loxodromic conjugating element. This motivates the definition of the strong ν-invariant (see Definition 3.12). However this is not the only obstruction. As illustrated by the next example, the above dichotomy may fail anyway. Note that ḡ0 , ḡ1 , ḡ2 is a subgroup of Ē which is also isomorphic to Z 2 × D ∞ (see Figure 9). As t commutes with ḡ1 , we observe that ḡ1 = hḡ 0 h-1 and ḡ2 = hḡ 1 h-1 , where h = tr n/4 .

Hence (ḡ 0 , ḡ1 , ḡ2 ) is a chain which generates an elementary subgroup of Ḡ. The reader can check that ḡ0 , h is not an elementary subgroup of Ḡ. It cannot either be lifted to a chain which generates an elementary subgroup of G.

In the previous example, the difficulty comes from the fact that the subgroup ḡ0 , ḡ1 , ḡ2 generated by the chain is a loxodromic subgroup of Ḡ that cannot be lifted. Note that ḡ0 , ḡ1 is an elliptic subgroup that cannot be lifted either. As described in Proposition 4.45, ḡ1 is obtained from ḡ0 by conjugation by a quarter turn.

This discussion suggests that the conjugation by a quarter turn rotation plays an important role. This is the place where algebra enters the stage. Take v ∈ V. Let F be the kernel of the geometric realization map q v D n . Assume as in our example that n is divisible by 4. Let r be strict rotation at v. From a geometric point of view, conjugating F by r is not a trackable operation. Indeed this will send an element ḡ which is locally trivial at v to rḡr -1 which is still locally trivial at v. Hence one cannot distinguish ḡ and rḡr -1 from their action on B(v, ρ). This does not mean that ḡ and r commutes though. Nevertheless if we had a better understanding of the algebraic structure of Stab(v), more precisely if we knew that Stab(v) is essentially a subproduct of dihedral groups, it could be possible to find a suitable quarter turn at v which truly commutes with a prescribed subset of F (see Lemma 3.16).

With additional algebraic hypotheses we are actually able to prove that if m is sufficiently large, any chain C = (ḡ 0 , . . . , ḡm ) will satisfy a variation on the above dichotomy, which salvages the original strategy.

Additional assumption. We now begin a systematic study of the mixed invariants. As we pointed out above, we require some additional hypothesis on the algebraic structure of elementary subgroups.

We fix once for all an even integer n and write n 2 for the largest power of 2 dividing n. We also choose a model collection E, i.e. a family of (abstract) torsion groups and assume that its exponent µ = µ(E) divides n. We suppose in addition that for every E ∈ E, the exponent of E/Z(E) divides n/2, where Z(E) stands for the center of E. From now on, we substitute Assumption 4.12 on vertex stabilizers for the following stronger hypotheses. Recall that a dihedral pair (E, C) has type (E, n 2 ) if there exist k ∈ N, and a morphism ϕ : E → E, where E ∈ E such that ϕ extends to an embedding from E into E/C × D k n2 × E (see Definition 3.14). Let us now mention a few consequences of these new assumptions. Lemma 4.51. Every elliptic subgroup of a loxodromic subgroup of G is a dihedral germ.

Proof. Let E be a loxodromic subgroup of G and F its maximal elliptic normal subgroup. Let g ∈ E be a loxodromic element. According to Assumption 4.49 there exists a group E of exponent n such that E embeds in E/F × E . Hence g n centralizes F . We conclude as in Lemma 4.32. Remark. In particular, Assumption 4.12 holds.

Proof. Fix (H, Y ) ∈ Q such that π : G → Ḡ maps E = Stab(Y ) onto Stab(v).
Let F be the maximal elliptic normal subgroup of E. According to Assumption 4.49, there exist an integer k ∈ N, and a morphism ϕ : E → E, where E ∈ E, such that ϕ extends to an embedding from E into E/F × D k n2 × E. We write α : E → D k n2 ×E for the map obtained by composing the embedding

E → E/F × D k n2 × E with the natural projection onto D k n2 × E.
Note that α is one-to-one when restricted to F . Recall that E/F is either Z or D ∞ . We denote by t a generator of the maximal infinite cyclic subgroup of E/F . By Assumption 4.50, there exists a pre-image g ∈ E of t such that H = g n . In particular, n is the order of v. By assumption, the exponent of D k n2 ×E divides n. Hence α(H) is trivial. It follows that α induces a map ᾱ : Stab(v) → D k n2 × E whose restriction to F (the image of F in Ḡ) is an embedding. Hence the morphism Stab(v) → D n × D k n2 × E given by q v and ᾱ is an embedding.

We are left to prove existence of a central half turn. To that end we claim that for every u ∈ E, g n/2 ug -n/2 ∈ uH.

Let u in E. Recall that the exponents of E and E/Z(E) respectively divide n and n/2, hence g n/2 ug -n/2 = u = ug -n , where g = ϕ(g) and u = ϕ(u).

The same identities are also satisfied by the images of g and u is any factor D n2 .

Assume now that u belongs to E + (the maximal subgroup of E preserving the orientation). Then g n/2 commutes with u (one checks indeed that in each factor of E/F × D k n2 × E, we have g n/2 ug -n/2 = u). Assume now that u ∈ E \ E + . In other words the image of u in E/F is a reflection. We similarly check that g n/2 ug -n/2 = ug n , which completes the proves of our claim. As we already observed α(g n ) = 1, hence the image of g n/2 in Stab(v) is a non trivial rotation of order 2 at v. It follows from [START_REF] Gromov | Hyperbolic groups[END_REF] that it is also central in Stab(v), thus it is a central half-turn at v.

Model collection for Ḡ. We now prove that the elementary subgroups of Ḡ satisfy a condition similar to Assumption 4.49. Proposition 4.53. Dihedral pairs of Ḡ (for its action on X) have type (E, n 2 ). Remark 4.54. Note that the model collection E is the same as the one of Assumption 4.49. In particular, we deduce as in Lemma 4.51 that every elliptic subgroup contained in a loxodromic subgroup of Ḡ is a dihedral germ.

Proof. Let ( Ē, C) be a dihedral pair. We distinguish several cases. Assume first that Ē is either an elliptic subgroup that can be lifted in G or a parabolic subgroup. Note that in the latter case Ē can also be lifted in G. Indeed Ē is the extension of an elliptic subgroup, namely C, by a finitely generated subgroup. Thus there exist d ∈ R + and a finite subset S generated Ē such that Fix( S, d) is non-empty. It follows then from Proposition 4.40 that Ē can be lifted. Let E be a lift of Ē and C the pre-image of C in E. According to Lemma 4.34, C is a dihedral germ. It follows that (E, C) is a dihedral pair. Hence the result follows from Assumption 4.49 applied to (E, C).

Assume now that Ē is a loxodromic subgroup. In particular, C is its maximal elliptic normal subgroup (Lemma 3.11). According to Proposition 4.42 there exist a dihedral pair (E , C) in G where C is an elliptic subgroup lifting C and an epimorphism θ : Ē E with the following properties.

(i) The morphism π • θ is the identity when restricted to C.

(ii) The map θ induces an embedding from Ē into Ē/ C × E .

Our assumption applied to (E , C) says that there exist k ∈ N, and a morphism ϕ : E → E where E ∈ E, which extends to an embedding

E → E /C ×D k n2 ×E. We claim that ϕ • θ : Ē → E extends to a embedding Ē → Ē/ C × D k n2 × E.
For simplicity we write ψ :

E → D k n2 × E for the composition of E → E /C × D k n2 × E with the canonical projection onto D k n2 × E. It suffices to shows that ψ • θ : Ē → D k n2 × E induces an embedding from Ē into Ē/ C × D k n2 × E. Consider an element ḡ ∈ Ē which is trivial in Ē/ C × D k n2 × E.
In particular, ḡ belongs to C, hence θ(ḡ) ∈ C. Moreover ψ • θ(ḡ) is trivial. Since ψ extends to an embedding from E → E /C × D k n2 × E, the element θ(ḡ) is trivial. On the other hand, θ induces an embedding from Ē into Ē/ C × E , thus ḡ = 1, which completes the proof our claim.

We finally assume that Ē is an elliptic subgroup that cannot be lifted in G. In particular, there exists an apex v ∈ V such that Ē is contained in Stab(v) (Proposition 4.30). According to Lemma 4.52 there exist k ∈ N, and a morphism φ : Stab(v) → E, where E ∈ E, which combined with the geometric realization q v : Stab(v) → D n provides an embedding from Stab(v) into D n ×D k n2 ×E. For simplicity we write ψ :

Stab(v) → D k n2 × E for the composition of Stab(v) → D n × D k n2 × E with the natural projection onto D k n2 × E.
Composing the geometric realization q v : Stab(v) → D n with the canonical projection D n → D n2 leads to a morphism that we denote q v : Stab(v) → D n2 . We are going to prove that q v and ψ extend to an embedding from

Ē into Ē/ C × D n2 × D k n2 × E. Let F be the kernel of q v : Stab(v) → D n . We first claim that [ C : C ∩ F ] is a power of 2.
Since C is a dihedral germ, it contains a subgroup C0 which is normalized by a loxodromic element and such that [ C : C0 ] = 2 m for some m ∈ N. Observe that C0 is contained in a reflection group at v. Indeed otherwise, any element normalizing C0 would belong to Stab(v) thus they would be no loxodromic element centralizing C0 . In particular, [ C0 : C0 ∩ F ] is at most 2. On the other hand

[ C : C ∩ F ][ C ∩ F : C0 ∩ F ] = [ C : C0 ∩ F ] = [ C : C0 ][ C0 : C0 ∩ F ] Consequently [ C : C ∩ F ] divides [ C : C0 ∩ F ].
In particular, it is a power of 2 which completes the proof of our claim.

Consider now ḡ ∈ Ē whose image in Ē/ C × D n2 × D k n2 × E is trivial. First observe that ḡ belongs to C. It follows from the previous claim that the order of q v (ḡ) is a power of 2. Nevertheless the kernel of the projection D n → D n2 , which contains q v (ḡ), consists only of element with odd order. Therefore q v (ḡ) is trivial, i.e. ḡ belongs to F . Observe that the map ψ :

Stab(v) → D k n2 × E is an embedding when restricted to F . Since ψ(ḡ) = 1, the element ḡ is trivial. This shows that Ē embeds in Ē/ C × D n2 × D k n2 × E.
The strong ν-invariant. We now start our study of the strong ν-invariant.

The ultimate goal is to prove the following statement.

Proposition 4.55. Assume that 2 ν+2 µ divides n where ν = ν stg (G, X). Then ν stg ( Ḡ, X) max {ν stg (G, X), µ + 4}

For simplicity we adopt the following terminology.

Proof. There exists a (unique) apex v ∈ V such that Ē is a subgroup of Stab(v) (Proposition 4.30). Let q v : Stab(v) → D n be the canonical geometric realization map. Assume first that Ē0 = ḡ0 , . . . , ḡm-1 contains a strict rotation at v. Consequently h Ē0 h-1 contains a strict rotation at hv. Since strict rotations fix a unique cone point we get hv = v. Thus ḡ0 , h is contained in Stab(v).

Assume now that Ē0 does not contain a strict rotation at v. Since Ē cannot be lifted, q v maps ḡ0 and ḡm to two distinct reflections, and ḡ1 , . . . , ḡm-1 to the identity. We denote by C the intersection of Ē with the kernel of q v and let Ā = ḡ0 , C and B = ḡm , C . Note that C is a subgroup of index 2 in both Ā and B. Let (H, Y ) ∈ Q such that the projection π : G → Ḡ maps Stab(Y ) onto Stab(v). We choose a lift A (respectively B) of Ā (respectively B) contained in Stab(Y ) so that A ∩ B is a lift of C that we denote by C (see Corollary 4.31). Let a 0 , . . . , a m-1 (respectively b 1 , . . . , b m ) the pre-images of ḡ0 , . . . , ḡm-1 (respectively ḡ1 , . . . , ḡm ) in A (respectively B). As we already observed ḡ1 , . . . , ḡm-1 belong to C, thus a k = b k , for all k ∈ 1, m -1 . Applying Proposition 4.27 with S 1 = A and S 2 = B, we get that there exists h 0 ∈ G such that for every k ∈ 0, m -1 we have b k+1 = h 0 a k h -1 0 . If follows that C = (a 0 , a 1 , . . . , a ν-1 , b ν ) is a chain and h 0 a conjugating element of C. In addition this chain generates a subgroup of Stab(Y ), which is therefore elementary. Recall that a 0 , . . . , a ν-1 is an elliptic subgroup of Stab(Y ). Hence it is a dihedral germ (Lemma 4.32). Consequently C is a strong chain of G lifting C. (ii) There exists a strong chain C = (ḡ 0 , . . . , ḡ m ) which can be lifted and such that (ḡ 1 , . . . , ḡ m-1 ) = (ḡ 1 , . . . , ḡm-1 ).

Proof. Assume firs that Ē0 = ḡ0 , . . . , ḡm-1 contains a loxodromic element say t. Since Ē0 and h Ē0 h-1 generate an elementary subgroup, namely Ē, both ḡ0 and h are contained in the maximal elementary subgroup containing t. In particular, ḡ0 , h is loxodromic.

Assume now that Ē0 is elliptic (a subgroup of a loxodromic subgroup cannot be parabolic). Let C be the maximal normal elliptic subgroup of Ē. Since Ē is generated by two elliptic subgroups (namely Ē0 and h Ē0 h-1 ) it does not preserve the orientation. Hence the quotient Ē/ C is isomorphic to D ∞ . We write

1 → C → Ē q -→ D ∞ → 1
for the corresponding short exact sequence. One observes that q maps ḡ0 and ḡm to two distinct reflections, while C is the normal subgroup of Ē generated by ḡ2 0 , ḡ1 , . . . , ḡm-1 , ḡ2 m . We let Ā = ḡ0 , C and B = ḡm , C so that Ē is isomorphic to Ā * C B. We now distinguish two cases.

Case 1. Assume first that both Ā and B can be lifted in G. We denote by A and B a lift of Ā and B respectively. According to Proposition 4.43 there exists u ∈ G whose image ū in Ḡ centralizes C such that E u = A, uBu -1 is elementary. We let ḡ k = ḡk , for every k ∈ 0, m -1 and ḡ m = ūḡ m ū-1 . Since ū centralizes C we observe that C = (ḡ 0 , . . . , ḡ m ) is a chain and h0 = ūh a conjugating element of C . Note also that C and C only eventually differ on the last element. We now focus on this new chain. Let a 0 , . . . , a m-1 be the lift of ḡ 0 , . . . , ḡ m-1 in A and b 1 , . . . , b m the lifts of ḡ 1 , . . . , ḡ m in uBu -1 . We now proceed exactly as in the proof of Lemma 4.59. We first observe that C = (a 0 , a 1 , . . . , a m-1 , b m ) is a chain for some conjugating element h 0 ∈ G. Moreover it generates a subgroup of E u which is therefore elementary. Recall that Ē0 = ḡ0 , . . . , ḡm-1 is an elliptic subgroup of the loxodromic subgroup Ē, therefore it is a dihedral germ (Lemma 4.51). Hence C is a strong chain. Moreover, as a lift of ḡ0 , . . . , ḡm-1 , the subgroup a 0 , . . . , a m-1 is a dihedral germ as well (Lemma 4.34). Hence C is a strong chain of G lifting C . Case 2. Assume that either Ā or B cannot be lifted in G. Up to replacing (ḡ 0 , . . . , ḡm ) by (ḡ m , . . . , ḡ0 ), we can assume that Ā cannot be lifted in G. In particular, there exists a (unique) apex v ∈ V such that Ā contains a strict rotation at v (Proposition 4.30). We write q v : Stab(v) → D n for the geometric realization map associated to v and r ∈ D n for a generator of the rotation group. Let Ē1 = ḡ1 , . . . , ḡm-2 . Observe that Ā is generated by h-1 Ē1 h and C. Moreover Ē1 is contained in Ā. If follows from Proposition 4.45 that either Ē1 contains a strict rotation, in which case ḡ0 , h is a subgroup of Stab(v), or there exists a reflection x ∈ D n such that q v ( C) = (i) There exists v ∈ V such that ḡ0 , h is contained in Stab(v).

(ii) The subgroup ḡ0 , h is loxodromic, (iii) There exists a strong chain C = (ḡ 0 , . . . , ḡ m ) of Ḡ which can be lifted such that (ḡ 1 , . . . , ḡ m-1 ) = (ḡ 1 , . . . , ḡm-1 ).

We study each case separately. Assume first that ḡ0 , h is contained in Stab(v) for some v ∈ V. Note that h cannot be loxodromic (it fixes v). By the very definition of strong chain ḡ0 , . . . , ḡm-1 is a dihedral germ. According to Lemma 4.52 there exist k ∈ N and E ∈ E such that Stab(v) embeds in D n × D k n2 × E. Since m µ + 2, it follows from Proposition 3.15 that ḡ0 and ḡm respectively belong to ḡ1 , . . . , ḡm and ḡ0 , . . . , ḡm-1 . In other words h normalizes ḡ0 , . . . , ḡm-1 . Hence ḡ0 , h is a cyclic extension of the dihedral germ ḡ0 , . . . , ḡm-1 , therefore it has dihedral shape.

Assume now that ḡ0 , h is loxodromic. Then it automatically has dihedral shape (Lemma 3.11).

We are left with the last case. Let C = (g 0 , . . . , g m ) be a strong chain of G lifting the chain C given by Point (iii). Let h 0 be a conjugating element of C . Recall that m ν. Thus g 0 , h 0 is an elementary subgroup with dihedral shape. Recall that by Assumption 4.49 every dihedral pair of G has type (E, n 2 ). Since m -2 µ + 2, it follows from Proposition 3.15 that g 1 and g m-1 respectively belong to g 2 , . . . , g m-1 and g 1 , . . . , g m-2 . Recall that C and C coincide everywhere but except maybe on the first and the last element. Pushing the previous observation in Ḡ we get that ḡ1 and ḡm-1 respectively belong to ḡ2 , . . . , ḡm-1 and ḡ1 , . . . , ḡm-2 . Hence h normalizes ḡ1 , . . . , ḡm-2 and a fortiori ḡ0 , . . . , ḡm-1 . In particular, ḡ0 , h is elementary. If h is loxodromic, then ḡ0 , h has automatically dihedral shape (Lemma 3.11) Otherwise ḡ0 , h is a cyclic extension of the dihedral germ ḡ0 , . . . , ḡm-1 , and thus it has dihedral shape.

Periodic groups

Induction step

The next proposition will play the role of the induction step in the final induction (see Theorem 5.4). It is the generalization of [16, Lemma 6.2.1] in the presence of even torsion. Proposition 5.1. There exist positive constants δ 1 , C 0 and C 1 such that for every positive integer ν 0 , there exists a critical exponent N 0 ∈ N with the following properties. Let E be a model collection of abstract groups whose exponent µ = µ(E) is finite. Let N 1 N 0 and n N 1 be a multiple of 2 ν0+2 µ.

Observe that δ 1 C 0 ρ C 1 . For every integer N ∈ N we define a rescaling parameter ε N as follows

ε N = C 1 √ N Let ν 0 ∈ N.
The sequence (ε N ) converges to 0 as N tends to infinity. Therefore there exists a critical exponent N 0 ∈ N, such that for every integer N N 0 we have

ε N δ 1 δ 0 , (21)
ε N (ν 0 + 5)C 0 min{∆ 0 , C 0 } (22) ε N κ 1/2. ( 23
)
Let E be a collection of (abstract) groups and µ its exponent. We now fix N 1 N 0 . For simplicity we write ε instead of ε N1 . Let n N 1 be a multiple of 2 ν0+2 µ. In particular, µ divides n/2. Consequently for every E ∈ E, the exponents of E and E/Z(E) respectively divide n and n/2, which means that the model collection E satisfies the assumptions stated in Section 4.7.2.

Let G be a group acting on a δ-hyperbolic space X such that (G, X) satisfies the induction hypotheses relative to (n, E). We denote by P the set of all primitive loxodromic elements h ∈ G such that h 10δ 1 . Let K be the (normal) subgroup of G generated by {h n : h ∈ P } and Ḡ the quotient of G by K. If P is empty, then Ḡ = G. Thus X = εX obviously satisfies the conclusion of the proposition. Otherwise, we are going to prove that Ḡ is a small cancellation quotient of G. To that end we consider the action of G on the rescaled space εX. According to [START_REF] Gromov | Mesoscopic curvature and hyperbolicity[END_REF] this space is εδ-hyperbolic where εδ δ 0 . We define the family Q by

Q = h n , Y h : h ∈ P .
Lemma 5.2. The family Q satisfies the small cancellation hypotheses, i.e. ∆(Q, εX) ∆ 0 and inj (Q, εX) 10π sh ρ.

Proof. We start with the upper bound of ∆(Q, εX). Let h 1 and h 2 be two elements of P such that ( h n 1 , Y h1 ) and ( h n 2 , Y h2 ) are distinct. We first claim that h 1 and h 2 generate a non-elementary subgroup. Assume on the contrary that it is not the case. Let E be the maximal elementary subgroup containing h 1 and h 2 . This subgroup is necessarily loxodromic. We denote by F its maximal elliptic normal subgroup, so that (E, F ) is a dihedral pair. According to our assumption there exists k ∈ N and E ∈ E such that E embeds in E/F ×D k n2 ×E, where n 2 is the largest power of 2 dividing n. Recall that h 1 and h 2 are primitive. Hence up to replacing h 2 by its inverse, we may assume that h 1 and h 2 have the same image in E/F . Since the exponents of D n2 and E divides n, the images of h n 1 and h n 2 are trivial in D n n2 × E. Consequently h n 1 = h n 2 and thus Y h1 = Y h2 . This contradicts the fact that ( h n 1 , Y h1 ) and ( h n 2 , Y h2 ) are distinct and completes the proof of our claim.

Recall that h i moves the points of Y hi by at most h i εX +65εδ (Lemma 2.9), while h i εX 10εδ. Consequently

Y +5εδ h1 ∩ Y +5εδ h2 ⊂ Fix({h 1 , h 2 }, 85εδ).
Since h 1 and h 2 generate a non-elementary subgroup we obtain

diam Y +5εδ h1 ∩ Y +5εδ h2 A(G, εX, 400εδ) εA(G, X, 400δ) ε(ν 0 + 5)C 0 .
Using [START_REF] Guirardel | Geometric small cancellation[END_REF] we get ∆(Q, εX) ∆ 0 . Let us now focus on inj (Q, εX). It follows from our assumption on inj (Q, X) that

inj (G, εX) ε inj (G, X) 10C 0 π sh ρ √ N 1 1 C 0 √ N 1 10π sh ρ N 1 10π sh ρ n .
Let (H, Y ) ∈ Q. By construction, any element g ∈ H is the n-th power of a loxodromic element of G. Consequently

g εX n inj (G, εX) 10π sh ρ.
It follows that inj (Q, εX) 10π sh ρ.

On account of the previous lemma, we can now apply the small cancellation theorem (Theorem 4.7) to the action of G on the rescaled space εX and the family Q. We denote by Ẋ the space obtained by attaching on εX for every (H, Y ) ∈ Q, a cone of radius ρ over the set Y . The space X is the quotient of Ẋ by K. According to Theorem 4.7, X is a δ-hyperbolic length space with δ δ 1 and Ḡ acts by isometries on it. As usual we write V for the set of apices in Ẋ and V for its image in X. We now prove that the action of Ḡ on X satisfies the induction hypotheses relative to (n, E). This action is gentle (Proposition 4.22) and non-elementary (Proposition 4.24), which provides (i). In addition dihedral pairs of Ḡ have type (E, n 2 ) (Proposition 4.53). Thus (ii) holds. Point (iii) is a consequence of the following lemma.

Lemma 5.3. The parameters A( Ḡ, X), inj Ḡ, X and ν stg ( Ḡ, X) satisfy

(i) A( Ḡ, X, 400 δ) (ν 0 + 5)C 0 ; (ii) inj Ḡ, X 1/C 0 √ N 1 ; (iii) max{ν stg ( Ḡ, X), µ + 4} ν 0 .
Proof. We start with the upper bound of A( Ḡ, X). Proposition 4.47 yields A( Ḡ, X, 400 δ) A(G, εX, d) + 1200 δ where d = π sh(800 δ). Applying Proposition 3.5 in εX, we obtain that A( Ḡ, X, 400 δ) A(G, εX, 400εδ) + [ν(G, X) + 4] π sh(800δ 1 ).

Since ν(G, X) is bounded above by ν stg (G, X), hence by ν 0 , we get that A( Ḡ, X) ε(ν 0 + 5)C 0 + (ν 0 + 4)C 0 .

Using [START_REF] Guirardel | Geometric small cancellation[END_REF] we obtain A( Ḡ, X) (ν 0 + 5)C 0 . We now focus on the injectivity radius of Ḡ. Let g be a loxodromic isometry of G. Since dihedral pairs have type (E, n 2 ) we can write g = g k 0 u where k is a positive integer, g 0 a primitive element and u an elliptic element centralized by some large power of by g 0 . In particular, g ∞ εX g 0 ∞ εX . Assume now that g does not stabilize any cylinder Y h , where h ∈ P . It follows that g 0 does not belong to P . Thus by [START_REF] Coulon | On the geometry of Burnside quotients of torsion free hyperbolic groups[END_REF]

g 0 ∞ εX g 0 εX -8εδ 2εδ 1 Proposition 4.46 applied with N = G yields inj Ḡ, X min {2εκδ 1 , δ 1 } Combined with (23) we obtain inj Ḡ, X 2εκδ 1 1 C 0 √ N 1 .
The upper bound for ν stg ( Ḡ, X) directly follows from Proposition 4.55.

We now study the properties of the projection π : G → Ḡ. Recall that the map ζ : εX → X is 1-Lipschitz. Hence X → X is ε-Lipschitz (and π-equivariant by construction). If F is an elliptic (respectively parabolic) subgroup of G, then it follows from Lemma 4.26 that π : G → Ḡ induces an isomorphism from F onto its image F . Moreover, F is elliptic (respectively elliptic or parabolic). Let F be an elliptic subgroup of Ḡ. If F is not the isomorphic image of an elliptic subgroup of G, then there exists an apex v ∈ V such that F is contained in Stab(v) (Proposition 4.30). On the other hand, there exist k ∈ N and E ∈ E such that Stab(v) embeds in D n ×D k n2 ×E (Lemma 4.52). Since the exponent of E divides n, the exponent of F is finite and divides n as well. By Proposition 4.40 any finitely generated parabolic subgroup of Ḡ is the isomorphic image of a parabolic subgroup of G. Let F 1 and F 2 be two subgroups of G. Assume that F 1 is elliptic and F 2 is generated by a finite set S 2 such that Fix(S 2 , 100δ 1 ) is non-empty. It follows from our choice of ρ, that 100δ 1 ρ/100. Thus, if the respective images F1 and F2 are conjugated in Ḡ, then so are F 1 and F 2 (Corollary 4.29). We have checked all the announced properties of the projection π : G → Ḡ, and the proof of the proposition is completed.

Construction of periodic groups

The number of variables in the next statement can be confusing at first sight. Basically we are stating the fact that the critical exponent N 1 does not depend on the group G, but only on certain parameters related to its action on a hyperbolic space X. More precisely N 1 is a function of • the hyperbolicity constant δ of X;

• the invariants inj (G, X), ν stg (G, X), and A(G, X, d) for some appropriate value of d ;

• the structure of subgroups of G with dihedral shape.

A fine understanding of these dependencies can be crucial sometimes, see for instance [START_REF] Coulon | Small cancellation theory over Burnside groups[END_REF].

Theorem 5.4. Let δ, r ∈ R * + , ν, µ ∈ N and set ν 1 = max{ν + 2, µ + 6}. There exist N 1 ∈ N such that for every integer n N 1 which is a multiple of 2 ν1 µ, the following holds.

Let E be a model collection of groups whose exponent divides µ. Let G be a group acting on a δ-hyperbolic length space X such that

• the action of G on X is gentle and non-elementary;

• for every dihedral pair (E, C) the group E embeds in E/C × E for some E ∈ E.

• A(G, X, 400δ) r, ν stg (G, X) ν, and inj (G, X) 1/r

Then there exists a quotient Q of G with the following properties.

(i) For every elliptic (respectively parabolic) subgroup F of G, the projection G Q induces an embedding from F into Q.

(ii) For every q ∈ Q, either q n = 1 or q is the image of an elliptic or parabolic element of G.

(iii) The projection G G/G n induces an epimorphism Q G/G n . In particular, if G has no parabolic element, and every elliptic element of G has finite order dividing n, then Q = G/G n .

(iv) For every x ∈ X, the map G → Q is one-to-one when restricted to {g ∈ G : |gx -x| < r} .

(v) There are infinitely many elements of Q which are not the image of an elliptic or a parabolic element of G.

(vi) The kernel K of G Q is purely loxodromic (i.e. all its non-trivial elements are loxodromic). As a normal subgroup K is not finitely generated.

Proof. The main ideas of the proof are the following. Using Proposition 5.1 we construct by induction a sequence of groups G = G 0 → G 1 → G 2 → . . . where G k+1 is obtained from G k by adding new relations of the form h n where h is a primitive element of G. Then we choose for the quotient Q = G/K the direct limit of these groups.

Critical exponent. Let us define first all the parameters leading to the critical exponent. For simplicity we let ν 0 = max{ν, µ + 4} and ν 1 = ν 0 + 2. The parameters δ 1 , C 0 , C 1 , and N 0 are the one given by Proposition 5.1. We choose ε > 0 and an integer N 1 N 0 such that εδ δ 1 , εr min {(ν 0 + 5)C 0 , 50δ 1 } , ε r

1 C 0 √ N 1 , and
C 1 √ N 1 < 1.
We now a fix an integer n N 1 which is divisible by 2 ν1 µ.

The initialization. Let E be a model collection of groups and G be a group acting on a δ-hyperbolic length space X as in the theorem. Let X 0 be the space X whose metric has been rescaled by ε. It follows from our choice of ε and N 1 that X 0 is εδ-hyperbolic where εδ δ 1 , A(G, X 0 , 400εδ) (ν 0 + 5)C 0 , and inj (G, X 0 ) 1/C 0 √ N 1 . In addition max{ν stg (G, X 0 ), µ + 4} ν 0 . In other words, if G 0 = G, then (G 0 , X 0 ) satisfies the induction hypotheses relative to (n, E).

The induction step. Let k ∈ N. We assume that we already constructed the group G k and the space X k such that (G k , X k ) satisfies the induction hypotheses relative to (n, E). We denote by P k the set of primitive loxodromic elements h ∈ G k such that h X k 10δ 1 . Let K k be the normal subgroup of G k generated by {h n , h ∈ P k }. We write G k+1 for the quotient of G k by K k . According to Proposition 5.1, there exists a metric space X k+1 such that (G k+1 , X k+1 ) satisfies the induction hypotheses relative to (n, E). Moreover X k+1 comes with a C 1 / √ N 1 -Lipschitz map X k → X k+1 which is π k -equivariant, where π k : G k G k+1 is the canonical projection, and fulfills the following properties.

(P1) If F is an elliptic (respectively parabolic) subgroup of G k , then π k induces an isomorphism from F onto its image which is also elliptic (respectively elliptic or parabolic).

(P2) Any elliptic subgroup of G k+1 is either isomorphic to an elliptic subgroup of G k or a finite group whose exponent divides n.

(P3) Any finitely generated parabolic subgroup of G k+1 is the isomorphic image of a parabolic subgroup of G k .

(P4) Let F 1 and F 2 be two subgroups of G k . Assume that F 1 is elliptic and F 2 is generated by a finite set S 2 such that Fix(S 2 , 100δ 1 ) is non-empty.

If the images of F 1 and F 2 are conjugated in G k+1 , then F 1 and F 2 are conjugated in G k .

Direct limit. The direct limit of the sequence (G k ) is a quotient Q = G/K of G. We claim that this group satisfies the announced properties. Let E be a subgroup of G which is either elliptic or parabolic. A proof by induction on k ∈ N using (P1) shows that for every k ∈ N, the map G G k induces an isomorphism from E onto its image which is either elliptic or parabolic for the action of G k on X k . It follows that G Q induces an isomorphism from E onto its image, which proves (i).

A proof by induction on k ∈ N using (P2) and (P3) shows that if g ∈ G k is elliptic or parabolic (for its action on X k ) then either g n = 1 or g is the image of an elliptic or a parabolic element of G (for its action on X). Let q ∈ Q and g ∈ G be a pre-image of q. For simplicity we still write g for the image of g in G k . Since the map X k → X k+1 is C 1 / √ N 1 -Lipschitz, we get for every k ∈ N,

g ∞ X k C 1 √ N 1 k g ∞ X .
As

C 1 / √ N 1 < 1, there exists k ∈ N such that g ∞ X k < 1 C 0 √ N 1 inj (G k , X k ) .
Consequently g is elliptic or parabolic as an element of G k . It follows from the previous observation that one of the following holds.

• The element g coincide in G k with an elliptic or a parabolic element of G, hence q is the image of an elliptic or a parabolic element of G.

• We have g n = 1 (in G k ), hence q n = 1.

This completes the proof of (ii).

All the relation we added to built the sequence of groups (G k ) have the form h n = 1. Hence the projection G → Q induces an epimorphism Q → G/G n , which gives (iii).

Let g be an elliptic or a parabolic element of K. It follows from (i) that the map G G/K induces an isomorphism from g onto its image. Hence g is trivial. Consequently K is purely loxodromic. For every k ∈ N, the action of G k on X k is non-elementary. It follows that the sequence (G k ) does not ultimately stabilize. Indeed, otherwise (ii) would fail. Thus K is infinitely generated as a normal subgroup, which completes the proof of (vi).

Let x ∈ X. Let g 1 , g 2 ∈ G such that |g i x -x| X < r. It follows from our choice of ε that Fix(g, 100δ 1 ) ⊂ X 0 is non empty, where g = g -1 1 g 2 . Assume now that g 1 and g 2 have the same image in Q, i.e. g is trivial in Q. In particular, there exists i ∈ N such that the image of g in G i is trivial. Recall that the map X k → X k+1 is 1-Lipschitz for every k ∈ N. In particular, Fix(g, 100δ 1 ) ⊂ X k is non-empty for every k ∈ N. A proof by induction using (P4) show that g = 1.

Hence the quotient map G Q is one-to-one when restricted to the set {g ∈ G : |gx -x| < r} , whence (iv). We are left to prove (v). Let S be the collection of all elements of Q which are not the image of an elliptic or a parabolic element of G. Assume contrary to our claim that S is finite. Let S 0 be a finite pre-image of S in G 0 . Using the same argument as above we observe that there exists i ∈ N, such that the image of S 0 in G i only consists of elliptic and parabolic elements. As we already observed, the sequence is (G k ) is not ultimately constant. Consequently there exists j i such that P j is non-empty (recall that P j is a set of primitive elements of G j such that h Xj 10δ 1 ). We fix g ∈ P j . We claim that g does not coincide in Q with an elliptic or a parabolic element of G j . Assume on the contrary that it is the case. There exist an elliptic or a parabolic element u ∈ G j as well as an index k > j such that g and u coincide in G k . Note that the set Fix(g, 100δ 1 ) ⊂ X is non empty, for every j. A proof by induction using (P4) shows that g and u are conjugated as elements of G j . It contradicts the fact that g is loxodromic and u is not, hence the claim is proved. Our claim has two consequences for the image q of g in Q. First q is not the image of an elliptic or parabolic element of G, hence q ∈ S. Since every element of S 0 is elliptic in G j , q does not belong to S, a contradiction.

Examples

One source of examples comes from groups acting acylindrically on a δ-hyperbolic length space X. Let us recall first the definition of acylindricity. For our purpose we need to keep in mind the parameters that appear in the definition. Since X is a hyperbolic space, one can decide whether an action is acylindrical by looking at a single value of d. Assume now that the action of G on X is (100δ, L, N )-acylindrical. There exist parameters C inj (δ, L, N ), C ν (δ, L, N ) and C A (δ, L, N ), which only depend on δ, L and N , that control the various invariants defined in Section 3. More precisely exactly the virtually cyclic subgroups of G and the ones which are conjugated to a subgroup of some P j . As in the case of groups with an acylindrical action, one can prove that inj (G, X) is positive whereas ν(G, X) and A(G, X, 400δ) are finite. Proceeding as in Theorem 5.7 we get the following result. Theorem 5.12. Let G be a group and {P 1 , . . . , P m } be a collection of subgroups of G such that G is hyperbolic relatively to {P 1 , . . . , P m }. Assume that there are only finitely many isomorphism classes of finite subgroups with dihedral shape. There exist p, N 1 ∈ N such that every integer n N 1 multiple of p, there exists a quotient Q of G with the following properties.

(i) if E is a finite subgroup of G or conjugated to some P j , then the projection G Q induces an isomorphism from E onto its image;

(ii) for every element g ∈ Q, either g n = 1 or g is the image a non-loxodromic element of G;

(iii) there are infinitely many elements in Q which do not belong to the image of an elementary non-loxodromic subgroup of G.

Remark 5.13. Another possible strategy is to consider the action of G on its coned-off Cayley Γ -see Bowditch [START_REF] Bowditch | Relatively hyperbolic groups[END_REF] -which has the following properties : the elliptic subgroup for the action of G on Γ are precisely the elliptic and parabolic subgroups for the action of G on X; the action of G on Γ is acylindrical -see Osin [START_REF] Osin | Acylindrically hyperbolic groups[END_REF].

Mapping class groups. Let Σ be a compact surface of genus g with k boundary components. In the rest of this paragraph we assume that its complexity 3g + k -3 is larger than 1. The mapping class group MCG(Σ) of Σ is the group of orientation preserving self homeomorphisms of Σ defined up to homotopy. A mapping class f ∈ MCG(Σ) is (i) periodic, if it has finite order;

(ii) reducible, if it permutes a collection of essential non-peripheral curves (up to isotopy);

(iii) pseudo-Anosov, if there exists an homeomorphism in the class of f that preserves a pair of transverse foliations and rescale these foliations in an appropriate way.

It follows from Thurston's work that any element of MCG(Σ) falls into one these three categories [START_REF] Thurston | On the geometry and dynamics of diffeomorphisms of surfaces[END_REF]Theorem 4]. The complex of curves X is a simplicial complex associated to Σ. It has been first introduced by Harvey [START_REF] Harvey | Boundary structure of the modular group[END_REF]. A dsimplex of X is a collection of d + 1 homotopy classes of curves of Σ that can be disjointly realized. Masur and Minsky proved that this new space is hyperbolic [START_REF] Masur | Geometry of the complex of curves. I. Hyperbolicity[END_REF]. By construction, X is endowed with an action by isometries of MCG(Σ). Moreover Bowditch showed that this action is acylindrical [START_REF] Bowditch | Tight geodesics in the curve complex[END_REF]Theorem 1.3]. This action provides an other characterization of the elements of MCG(Σ). An element of MCG(Σ) is periodic or reducible (respectively pseudo-Anosov) if and only it is elliptic (respectively loxodromic) for the action on the complex of curves [START_REF] Masur | Geometry of the complex of curves. I. Hyperbolicity[END_REF]. Recall that MCG(Σ) contains only finitely many conjugacy classes of finite subgroups [START_REF] Farb | A primer on mapping class groups[END_REF]Theorem 7.14]. Hence the next statement is a direct application of Theorem 5.7.

Theorem 5.14. Let Σ be a compact surface of genus g with k boundary components such that 3g + k -3 > 1. There exist p, N 1 ∈ N such that for every integer n N 1 which is a multiple of p, there exists a quotient Q of MCG(Σ) with the following properties.

(i) If E is a subgroup of MCG(Σ) that does not contain a pseudo-Anosov element, then the projection MCG(Σ) Q induces an isomorphism from E onto its image.

(ii) Let f be a pseudo-Anosov element of MCG(Σ). Either f n = 1 in Q or f coincide in Q with a periodic or a reducible element.

(iii) There are infinitely many elements in Q which are not the image of a periodic or reducible element of MCG(Σ). Any non-trivial element in the kernel of MCG(Σ) Q is pseudo-Anosov.

Amalgamated product. Let G be a group. A subgroup H of G is malnormal if for every g ∈ G, we have gHg -1 ∩ H = {1} unless g belongs to H. (ii) For every g ∈ Q, if g is not a conjugate of an element of A or B then g n = 1.

(iii) There are infinitely many elements in Q which are not conjugate of elements of A or B.

Proof. We denote by X the Bass-Serre tree associated to the amalgamated product G = A * C B [START_REF] Serre | Arbres, amalgames, SL2[END_REF]. As C is malnormal in A or B, the action of G on X is acylindrical. Moreover every loxodromic subgroup is either Z or D ∞ . Hence every dihedral germ is necessarily a 2-group. Consequently, a finite group with dihedral shape is isomorphic to the extension of a 2-group by a finite cyclic or dihedral group. Being finite, such a group is contained in a conjugate of A or B. Thus is follows from our assumption that G admits only finitely many isomorphism classes of finite subgroups with dihedral shape. The conclusion follows from Theorem 5.7.

Figure 2 :

 2 Figure 2: The point x is hardly moved by the commutator [g 1 , g i 2 ].

Proposition 2 . 2 (

 22 Stability of quasi-geodesics). Let , L ∈ R + and γ : I → X be an L-local (1, )-quasi-geodesic. If L > 4 + 8δ, then the following holds.

Definition 3 . 3 .

 33 The action of G on the space X is weakly acylindrical if the map d → A(G, X, d) is bounded above by an affine function of d.

  Cones. Let (H, Y ) ∈ Q. The cone of radius ρ over Y , denoted by Z ρ (Y ) or simply Z(Y ), is the quotient of Y × [0, ρ] by the equivalence relation that identifies all the points of the form (y, 0). The equivalence class of (y, 0), denoted by v, is called the apex or cone point of Z(Y ). By abuse of notation, we still write (y, r) for the equivalence class of (y, r). The map ι : Y → Z(Y ) that sends y to (y, ρ) provides a natural embedding from Y to Z(Y ). The radial projection p : Z(Y ) \ {v} → Y is the map sending (y, r) to y. We denote by | . | Y the length metric on Y induced by the restriction of | . | to Y . This cone Z(Y ) can be endowed with a metric as described below. Proposition 4.1. [5, Chapter I.5, Proposition 5.9] The cone Z(Y ) is endowed with a metric characterized in the following way. Let x = (y, r) and x = (y , r ) be two points of Z(Y ) then

Figure 3 :

 3 Figure 3: Geometric interpretation of the distance in the cone.

  The cone-off space Ẋ. The cone-off of radius ρ over X relative to Q denoted by Ẋρ (Q) (or simply Ẋ) is obtained by attaching for every (H, Y ) ∈ Q, the cone Z(Y ) on X along Y according to ι. The subset of X consisting of all apices of the cones is denoted by V. We endow Ẋ with the largest pseudo-metric | . | Ẋ for which all the maps X → Ẋ and Z(Y ) → Ẋ -where (H, Y ) runs over Q -are 1-Lipschitz. It turns out that this pseudo-distance is a length metric on Ẋ [10, Proposition 5.10]. The next lemmas detail the relationship between the metrics of X and Ẋ. Lemma 4.5 ([10,Lemma 5.8]

  Théorèmes 5.2.5 et 5.5.2].

Assumption 4 . 11 (

 411 Cyclic relations). For every (H, Y ) ∈ Q, the group H is loxodromic and Y is its cylinder. Let (H, Y ) ∈ Q. According to our small cancellation assumption, any nontrivial element in H has a very large translation length. Thus H is necessarily a cyclic group generated by a loxodromic element. Local classification of isometries. Let (H, Y ) ∈ Q. Note that Stab(Y ) is the maximal loxodromic subgroup containing H. We write Stab + (Y ) for the subgroup of Stab(Y ) fixing pointwise ∂Y . Its index in Stab(Y ) is at most 2. Since the action of G on X is gentle, the set F of all elliptic elements of Stab + (Y ) is a normal subgroup of Stab(Y ). Moreover Stab + (Y )/F is isomorphic to Z while Stab(Y )/F embeds in D ∞ . In other words we have a short exact sequence

Figure 4 :

 4 Figure 4: The comparison D cone for n = 8.

Lemma 4 . 16 .

 416 Let x, y ∈ Ẋ such that x, y v > 12 δ, for every v ∈ V. Then |x -y| Ẋ = |x -ȳ|.

Figure 5 :

 5 Figure 5: The cones intersecting γ.

1 .

 1 (i) we observe that gγ(b j ) is the unique pre-image in Z j of ḡγ(b j ). In particular, |gγ(b j ) -γ(b j )| Ẋ = |ḡγ(b j ) -γ(b j )| ρ/50. If j = m, then a m+1 = b m , thus the claim holds for j + 1. Otherwise, |v j+1γ(b j )| = 9ρ/10, thus g necessarily belongs to Stab(v j+1 ). Moreover x, ȳ vj+1 d(v j+1 , γ) + 4 δ d(v j+1 , γ) + 4 δ ρ/4, see for instance by [10, Lemma 2.4]. Since g moves the point γ(b j ) ∈ B(v j+1 , ρ) by at most ρ/50, it is the (unique) elliptic preimage of ḡ (Theorem 4.7 (v)). Therefore it moves all the points of B(v j+1 , ρ) by a distance at most δ, see Lemma 2.14. In particular, |gγ(a j+1 ) -γ(a j+1 )| Ẋ δ. However, the map ζ : Ẋ → X induces an isometry from the ball B(γ(a j+1 ), ρ/20) onto its image, hence |gγ(a j+1 ) -γ(a j+1 )| Ẋ = |ḡγ(a j+1 ) -γ(a j+1 )|. This proves our claim for j + The statement of the lemma follows from our claim for j = m + 1.

Figure 6 : 7 δ

 67 Figure 6: The path γ going through the ball B(v, ρ). The grey area corresponds to Fix( S, δ) ∩ B(v, ρ). ρ/10. Up to permuting z0 and hz 0 , it forces z0 , v ȳ-ρ/100 + 8 δ, and z0 , v hȳ+ = hz 0 , v ȳ+ ρ/100 + 8 δ. Hence |ȳ --hȳ + | ρ/50 + 18 δ [10, Lemma 2.2 (ii)]. On the other hand, since ȳ-and ȳ+ lie on a (1, δ)-quasi-geodesic between x and ū0 ȳ we have |x -ū0 ȳ| |x -ȳ-| + |ȳ --ȳ+ | + |ȳ + -ū0 ȳ| -δ |x -ȳ-| + |ȳ + -ū0 ȳ| + ρ/6 -7 δ Combined with the triangle inequality, it yields |x -ū1 ȳ| |x -ȳ-| + ȳ--hy + + |ȳ + -ū0 ȳ| |x -ū0 ȳ| -2ρ/15.

Lemma 4 . 23 .

 423 The set V contains at least two apices.

2 xFigure 7 :

 27 Figure 7: The element ḡ1 and ḡ2 acting on x. The shaded discs represent B(v 1 , ρ) and B(v 2 , ρ) respectively.

Corollary 4 . 31 .

 431 Let (H, Y ) ∈ Q. Let v be the image in X of the apex v of Z(Y ).Let C be a subgroup of Stab(v). If C can be lifted, then it admits a lift which is contained in Stab(Y ).

Lemma 4 . 32 .

 432 Let (H, Y ) ∈ Q. Every elliptic subgroup of Stab(Y ) is a dihedral germ.

Lemma 4 . 33 .

 433 Let C be an elliptic subgroup of G (for its action onX). Let v ∈ V. If the image of C in Ḡ is contained in a reflection group at v, then C is a dihedral germ. Proof. Let C be the image of C in Ḡ. Let (H, Y ) ∈ Q such that the apex v of the cone Z(Y ) is a pre-image of v.There exists an elliptic subgroup C of Stab(Y ) such that the projection π : G → Ḡ maps C onto C (Corollary 4.31). In other words C and C are two lifts of C, hence they are conjugated (Corollary 4.28). Being a dihedral germ is invariant under conjugacy. Thus the conclusion follows from Lemma 4.32.

Lemma 4 . 34 (

 434 Lifting dihedral germs). Let C be an elliptic subgroup of G (for its action on X) and C its image in Ḡ. If C is a dihedral germ, then so is C.Proof.By assumption there exists a subgroup C0 of C which is normalized by a loxodromic element, say h, and such that [ C : C0 ] = 2 k for some k ∈ N. We write C 0 for the pre-image of C0 in C. Note that [C : C 0 ] = 2 k . It follows from Proposition 4.27 applied with S 1 = S 2 = C 0 that there exists h 0 ∈ G normalizing C 0 . Moreover either h 0 is loxodromic or C0 is contained in a reflection group at some apex v ∈ V. If h 0 is loxodromic, then C is automatically a dihedral germ. Assume now that C0 is contained in a reflection group at v. If follows from Lemma 4.33 that C 0 is a dihedral germ. Hence it contains a subgroup C 1 which is normalized by a loxodromic element of G and such that [C 0 : C 1 ] = 2 m for some m ∈ N. Thus [C : C 1 ] = 2 k+m and C is a dihedral germ.

Lemma 4 . 35 .

 435 Let C be an elliptic subgroup of G (for its action on X) and C its image in Ḡ. Assume that there exists an elliptic subgroup Ā containing C which cannot be lifted. Then C is a dihedral germ.

Proposition 4 .

 4 43. Let A and B be two elliptic subgroups of G. Denote by Ā and B their respective images in Ḡ. Assume that their intersection C = Ā ∩ B has index 2 in both Ā and B so that Ē = Ā * C B is elementary. There exists u ∈ G with the following properties. (i) The image ū of u in Ḡ centralizes C.

Example 4 .Figure 9 :

 49 Figure 9: The action of Ē and rn/4 on the space X. The shaded discs respectively represent B(v, ρ) and B( tv, ρ) where v is the cone point associated to the relation rn = 1.

Assumption 4 .

 4 49 (Structure of elementary subgroups). Every dihedral pair of G has type (E, n 2 ).

Assumption 4 .

 4 50 (Relations). For every (H, Y ) ∈ Q, there exists a primitive element g ∈ G such that H = g n .

Lemma 4 .

 4 52. Let v ∈ V. The cone point v has order n. The group Stab(v) contains a central half-turn at v. There exist k ∈ N and a morphism φ : Stab(v) → E, where E ∈ E such that the geometric realization q v : Stab(v) → D n together with φ extend to an embedding from Stab(v) into D n × D k n2 × E.

Lemma 4 .

 4 60. Let m 3. Let C = (ḡ 0 , . . . , ḡm ) be a strong chain generating a loxodromic subgroup Ē of Ḡ. Let h be a conjugating element of C. If 2 m+2 µ divides n then one of the following holds (i) The subgroup ḡ0 , h is either loxodromic or contained in Stab(v) for some v ∈ V.

Figure 10 :

 10 Figure 10: Lifting chains. The shaded areas represent Fix( Ā, 10 δ), Fix( B, 10 δ), and Fix( C, 10 δ) respectively. The isometry ḡm-1 is not on the picture. It is not clear a priori whether it is a reflection or locally trivial at v. there exist k ∈ N, and an abstract group E ∈ E, such that Stab(v) embeds in D n × D k n2 × E (Lemma 4.52). Let F0 be the normal subgroup of Stab(v)

Definition 5 . 5 (

 55 Acylindrical action). Let N, L, d ∈ R * + . The group G acts (d, L, N )-acylindrically on X if the following holds: for every x, y ∈ X with |x-y| L, the number of elements u ∈ G satisfying |ux-x| d and |uy-y| d is bounded above by N . The group G acts acylindrically on X if for every d > 0 there exist N, L > 0 such that G acts (d, L, N )-acylindrically on X.

Proposition 5 . 6 (

 56  Proposition 5.31]). The action of G on X is acylindrical if and only if there exist N, L > 0 such that the action is (100δ, L, N )-acylindrical.Remark. Dahmani, Guirardel and Osin work in a class of geodesic spaces. Nevertheless, following the proof of [14, Proposition 5.31] one observes that the statement also holds for length spaces. Moreover one gets the following quantitative statement. Assume that the action of G on X is (100δ, L, N )-acylindrical, then for every d > 0 the action is (d, L(d), N (d))-acylindrical whereL(d) = L + 4d + 100δ,

Theorem 5 . 15 .

 515 Let A and B be two groups. Let C be a subgroup of A and B malnormal in A or B. Assume that there exists M ∈ N such that every subgroup of A (respectively B) that is isomorphic to the extension of a 2-group by finite cyclic or dihedral group contains at most M elements. There exist p, N 1 such that for every integer n N 1 which is a multiple of p, there exists a quotient Q of A * C B with the following properties.(i) The natural projection A * C B Q induces an embedding of A and B into Q.

  We adopt the convention that the diameter of the empty set is zero, whereas the distance from a point to the empty set is infinite. , . . . , Y m be a collection of subsets of X such that Y j is α j -quasi-convex, for every j ∈ 1, m . For all A 0, we have

	Remark. Lemma 2.4 (Compare with [7, Proposition 2.1] or [10, Lemma 2.12]). Let
	α ∈ R + and Y an α-quasi-convex subset of X. Let x, x ∈ X.
	(i) If p is an η-projection of x on Y , then for every y ∈ Y , we have x, y p
	α + η.
	(ii) If p and p are respectively η-and η -projection of x and x on Y then
	|p -p | max{|x -x | -|x -p| -|x -p | + 2ε, ε},
	where ε = 2α + δ + η + η .
	Lemma 2.5 (Compare with [16, Lemme 2.2.2] or [10, Lemma 2.13]). Let
	Y 1
	i.e. the set of points x ∈ X such
	that d(x, Y ) < A, is strongly quasi-convex [11, Lemma 2.13].
	Let x be a point of X. A point y ∈ Y is an η-projection of x on Y is
	|x -y| d(x, Y ) + η. A 0-projection is simply called a projection.

X, then for every A α, its A-neighborhood, that we denote by Y +A , is 2δ-quasi-convex [10, Proposition 2.13]. Similarly, for every A > α + 2δ, the open A-neighborhood of Y ,

  Fix(S, d) is quasi-convex. Let y and y be two points of Fix(S, d). Let x be a point of X. Lemma 2.7 yields for every g ∈ S, |gx -x| max {|gy -y| , |gy -y |} + 2 y, y x + 6δ d + 2 y, y x + 6δ.

	now fix ε ∈ (0, η) such that |gy -y| + ε < d and choose a (1, ε)-quasi-geodesic
	γ : I → X joining y to gy, so that γ is entirely contained in Y (this is where
	the strict inequality in the definition of Y plays a role). In particular, y and gy
	are respective η-projections of x and gx on γ, which is (ε/2 + 2δ)-quasi-convex.
	Consequently Lemma 2.4 yields
	d -2η |gy -y| max {|gx -x| -2 |x -y| + 6η + 10δ, 3η + 5δ} .	(12)
	Recall that d > 5δ. Taking η > 0 arbitrarily small leads to
	|gx -x| 2d(x, Fix(g, d)) + d -10δ.	(13)
	We now prove Point (i) for a general set S. Let x ∈ X \ Fix(S, d). Note that
	Fix(S, d) is exactly the intersection of all Fix(g, d) where g runs over S. Hence
	there exists g ∈ S such that x does not belong to Fix(g, d). Applying (13) we
	get
	|gx -x| 2d(x, Fix(g, d)) + d -10δ 2d(x, Fix(S, d)) + d -10δ.
	and Point (i) follows. Point (ii) is a direct consequence of Point (i). We are left
	to prove that It follows then from Point (ii) that d(x, Fix(S, d))	y, y x + 8δ.
	Indeed the A-neighborhood of Fix(S, d)
	is contained in Fix(S, d + 2A), for every A ∈ R + . Although the objects are
	defined in a slightly different way, the proof works verbatim as in [10, Proposi-
	tion 2.28]. Nevertheless for completeness we reproduce it here.
	Proof. We first prove Point (i) when S is reduced to a single element, say g. We
	denote by Y the "open version" of Fix(g, d), i.e.
	Y = {x ∈ X : ∀g ∈ S, |gx -x| < d} .
	Since d > λ(S), the set Y is non-empty. It is also g -invariant and contained in
	Fix(g, d). Let x ∈ X \ Fix(g, d). Let η > 0 and y be an η-projection of x on Y .
	Since x does not belongs to Fix(g, d), one observes that |gy -y| d -2η. We

  Lemma 2.11. Let x ∈ X and ξ ∈ ∂X. Let , L ∈ R + with L > 4 + 8δ. R + such that for every t t d , we have |γ(t)-x| > d+ /2+3δ. Let g be an isometry fixing ξ such that |gx -x| d. It follows from the triangle inequality that gx, x γ(t) > /2 + 3δ whenever t t d . Let c be a Busemann cocycle at ξ. According to Lemma 2.10 there exists ε ∈ {±1} such that for every x ∈ X, we have |c(gx, x) + ε g ∞ | 6δ. Let t t d . For simplicity we write y 1 = γ(t) and y 2 = γ(t + ε g ∞ ). Since γ is an L-local (1, )-quasi-geodesic, c(y 1 , y 2 ) differs from ε g ∞ by at most + 8δ (Lemma 2.3). Hence

	Let γ : R + → X be an L-local (1, )-quasi-geodesic ray from x to ξ. For every
	d ∈ R + , there exists t d ∈ R + with the following property: if g is an isometry
	fixing ξ and satisfying |gx -x|	d, then there exists ε ∈ {±1} such that for
	every t t d we have
			|γ(t + ε g	∞ ) -gγ(t)| 2 + 20δ.
	Proof. The path γ is a global quasi-geodesic (Proposition 2.2) thus there exists
	t d ∈ |c(gy 1 , y 2 )| |c(gy 1 , y 1 ) + c(y 1 , y 2 )|	+ 14δ.
	By Proposition 2.2, x, ξ y2		/2 + 2δ and gx, ξ gy1	/2 + 2δ. The four point
	inequality (6) yields
			min gx, ξ y2 , gx, x y2	x, ξ y2 + δ	/2 + 3δ.
	It follows from our choice of t d that the minimum cannot by achieved by
	gx, x y2 , hence gx, ξ y2		/2 + 3δ. Applying Lemma 2.1 we get
	|gy 1 -y 2 | |c(gy 1 , y 2 )| + 2 max x, ξ gy1 , x, ξ y2 + 8δ 2 + 20δ.
			The cocycle property yields |c(g n x, x) + εc(y, hy)| 12δ. Thus
	(14) becomes |c(g n x, x) + ε|g n y -y||	2η + 20δ. Recall that c(g n x, x) and
	nc(gx, x) differs by at most 6nδ. Hence
	c(gx, x) +	ε n	|g n y -y|	1 n	|c(g n x, x) + ε |g n y -y|| + 6δ 6δ +	1 n	(2η + 20δ).
	The result follows by taking the limit as n approaches infinity.

  Moreover every element of G preserves {g -, g + }. We denote by G + the subgroup of G fixing pointwise {g -, g + }. It has index at most 2 in G. If G = G + we say that G is preserves the orientation.Let Γ be the union of all L-local (1, δ)-quasi-geodesics from g -to g + with L > 12δ. The cylinder Y of G is the set

	Elliptic action. Even though X is not necessarily locally compact, a group
	G is elliptic if and only if its orbits are bounded [11, Proposition 3.5]. Elliptic
	groups actually have very small orbits.
	Lemma 2.13 (Compare with [16, Proposition 2.3.4] or [10, Corollary 2.38]).
	Let G be an elliptic group of isometries of X. The set Fix(G, 5δ) is non-empty.
	Moreover if Y is a non-empty G-invariant α-quasi-convex subset of X, then
	Fix(G, 10δ) intersects the α-neighborhood of Y .
	Loxodromic action. Let G be a loxodromic group. In particular, it contains
	a loxodromic isometry, say g [11, Proposition 3.6]. Note that g -and g + are
	the two points of Λ(G).
	Classification of actions. Let G be a group acting by isometries on X. Its
	limit set Λ(G) is the set of accumulation points in ∂X of some (hence any) orbit
	of G. The action of G on X is elliptic (respectively parabolic, loxodromic, non-
	elementary) if Λ(G) is empty (respectively contains exactly 1 point, exactly 2
	points, at least 3 points). If there is no ambiguity regarding the action, we simply
	say that G is elliptic (respectively parabolic, loxodromic, non-elementary).

  Since any element of S moves the endpoint of γ by at most ρ/100, the path γ restricted to [s -, s + ] is contained in Fix(S, ρ/50) (Lemma 2.7) hence in the (ρ/100 + 7 δ)-neighborhood of Fix( S, δ) (Lemma 2.8). Hence for every s ∈ (s -, s

	x
	ḡx
	hū 0	ȳ	z0	v	h ρ	hz 0	ḡū 0 ū0 ȳ ȳ

geodesic from x to ū0 ȳ. Let γ(t) be the projection of v onto γ. By assumption x, ū0 ȳ v ρ/4, hence |γ(t) -v| ρ/4 + 4 δ, see for instance [10, Lemma 2.4]. Let s -= sup {s ∈ [a , t] : |γ(s) -v| ρ/3} , s + = inf {s ∈ [t , b] : |γ(s) -v| ρ/3} , so that ȳ-= γ(s -) and ȳ+ = γ(s + ) are at distance exactly ρ/3 from v and |ȳ --ȳ+ | ρ/6 -8 δ. + ) we have min z0 , v γ(s) , hz 0 , v γ(s)

ρ/100 + 8 δ.

See Figure

6

. By continuity it also applies to s -and s + . Recall that |ȳ --ȳ+ |

The more classical small cancellation condition C (λ) requires that the length of any piece u contained in a relation r satisfies |u| < λ|r|. We use here the stronger, uniform condition C (λ) where the length of pieces is small compared to any relation in R.

(i) Let ḡ ∈ G and z 1 , z 2 ∈ Z such that ḡz 1 = z2 . Then there exists a unique pre-image g ∈ G of ḡ such that gz 1 = z 2 . Moreover for every z, z ∈ Z, if ḡz = z , then gz = z .

(ii) The projection π : G → Ḡ induces an isomorphism from Stab(Z) onto Stab( Z).

Remark. The statement applies in particular if Z is α-quasi-convex and satisfies d(v, Z) > α + 13 δ, for every v ∈ V. This slightly weaker version will be more flexible for later use, though.

Proof. In this proof all the distances are measure in Ẋ or X. By Lemma 4.16, the projection ζ : Ẋ → X induces an isometry from Z onto Z. Let ḡ ∈ G and z 1 , z 2 ∈ Z such that ḡz 1 = z2 . By the very definition of X, there exists a preimage g ∈ G of ḡ, such that gz 1 = z 2 . Uniqueness follows from the fact that K acts freely on Ẋ \ V -see Theorem 4.7 (v). We now prove that g satisfies the announced property. Let z, z ∈ Z such that ḡz = z . Let v ∈ V. Applying the four point inequality (3) in Ẋ we have

Note that z 1 , z 2 , z and z belongs to Z. Hence, if follows from our assumption that gz, z v > 12 δ, for every v ∈ V. By Lemma 4. [START_REF] Delzant | Courbure mésoscopique et théorie de la toute petite simplification[END_REF], we get that |gz -z | = |ḡz -z | = 0. This completes the proof of (i). Point (ii) follows directly from (i).

The next two statements are a variation on [16, Lemme 5.10.1].

Lemma 4.18. Let Z be a subset of X such that z, z v > 13 δ, for every z, z ∈ Z and every v ∈ V. Let z0 be a point of Z and z 0 ∈ Ẋ a pre-image of z0 Then there exists a unique subset Z of Ẋ containing z 0 such that the projection ζ : Ẋ → X, induces an isometry from Z onto Z. In particular, z, z v z, z v for every z, z ∈ Z and every v ∈ V.

Remark. Note that Lemma 4.17 applies to the lifted set Z. Hence, we can lift any isometry ḡ ∈ Ḡ which (partially) preserves Z to an isometry g ∈ G with the same properties. Lemma 4.18 holds in particular if Z is α-quasi-convex and satisfies d(v, Z) > α + 13 δ, for every v ∈ V, in which case one can prove that Z is quasi-convex as well. Nevertheless we will not use this fact here.

Proof. In this proof all the distances are measure in Ẋ or X. We define Z as the set of points z ∈ Ẋ being the pre-image of a point z ∈ Z and such that |z -z 0 | = |z -z0 |. We claim that every z ∈ Z has a pre-image in Z. Let z ∈ Z. We fix a pre-image z ∈ Ẋ of z such that |z 0 -z| |z 0 -z| + 2 δ. In particular for every v ∈ V, we have z, z 0 v z, z0 v -δ, hence z, z 0 v > 12 δ. It follows from Lemma 4.16 that |z 0 -z| = |z 0 -z|. In other words z belongs to Z, which completes the proof of our claim. Hence the projection ζ : Ẋ → X maps Z onto Z.

Remark 4.38. The result can be reformulated in the following way. Let N (F ) and C(F ) be the respective normalizer and centralizer of F in G. We define N ( F ) and C( F ) in the same way. The projection π : G Ḡ does not necessarily map N (F ) onto N ( F ). Nevertheless it induces an epimorphism from N (F )/C(F ) onto N ( F )/C( F ). Actually Lemma 4.26 implies that this map is an isomorphism, but we will not need this fact here.

Proof. Applying Proposition 4.27 with S 1 = S 2 = F , we see that there exists h 0 ∈ G such that for every g ∈ G, the element h 0 gh -1 0 is the pre-image of hḡ h-1 in F . In particular, h 0 normalizes F and h-1h 0 centralizes F . Let us now focus on the second part of the statement. We assume that h ∈ Ḡ normalizes F and h2 ∈ F . In particular, F = F , h is an elliptic subgroup of Ḡ. Observe that there exists h1 ∈ Ḡ such that h-1h

1 centralizes F , h2 = h2 1 , and F , h1 is an elliptic subgroup that can be lifted. Indeed, if F , h can be lifted, it suffices to take h1 = h, otherwise the conclusion follows from Lemma 4.36. Let F 1 be a lift of F 1 = F , h1 and h 1 the pre-image of h1 in F 1 . There exists u ∈ G such that ū centralizes F and uF u -1 is the pre-image of F in F 1 (Corollary 4.28). We choose h 0 = u -1 h 1 u. By construction h 1 normalizes uF u -1 and h 2 1 belongs to uF u -1 . Consequently h 0 normalizes F and h 2 0 belongs to F . Since ū centralizes F , it commutes with h2

Lifting parabolic subgroups. The first result is not needed for the rest of the study. However, we believe that it may help the reader by clarifying the structure of parabolic subgroups in G. As we mentioned earlier, since ζ : X → X is 1-Lipschitz, the image by the projection π : G Ḡ of a parabolic subgroup of G is either elliptic or parabolic. The next statement tells us that the former case does not happen. Lemma 4.39. Let E be an elementary subgroup of G and Ē its image in Ḡ. Assume that there exist d ∈ R + and a subset S generating E such that Fix(S, d) is non-empty. If E is parabolic (for its action on X) then Ē is parabolic (for its action on X).

Remark. Note that the assumption automatically holds if E is finitely generated.

Proof. Since E is parabolic, it has a unique fixed point in ∂X. Thus according to Lemma 2.11 we can assume that Fix(S, ρ/100) is non-empty. We just need to prove that Ē cannot be elliptic. Assume on the contrary that it is. We distinguish two cases. Suppose first that Ē can be lifted and let E be a lift of Ē (recall that by definition E is elliptic). Applying Corollary 4.28 with S 2 = S, we get that E and E are conjugate, which contradicts the fact that E and E have different nature. Suppose now that Ē cannot be lifted. In particular, there exists g ∈ E whose image ḡ in Ē is a strict rotation (Proposition 4.30). By Proposition 4.13 (iii), there exists k ∈ Z such that Fix(ḡ k , δ) is contained in B(v, δ) for some apex v ∈ V. On the other hand, g k cannot be loxodromic, Acylindricity. Proposition 4.47 (Compare with [16, Lemma 5.9.5]). For every d 0 ∈ [0 , ρ/10], we have A( Ḡ, X, d 0 ) A(G, X, d 1 ) + 3d 0 , where d 1 = π sh(2d 0 ).

Proof. Let S be a subset of Ḡ generating a non-elementary subgroup. Let d 0 ∈ [0 , ρ/10]. The goal is to bound from above the diameter of Z0 = Fix( S, d 0 ). Without loss of generality we can assume that this set is non-empty. Observe that d(x, V) ρd 0 /2, for every x ∈ Z. Indeed if it was not the case, every element of S would fix a common apex v ∈ V, contradicting the fact that S generates a non-elementary subgroup. Let Z be the d 0 -neighborhood of Z0 . Since Z0 is 8 δ-quasi-convex (Lemma 2.8), we have z, z v > ρ/4, for every z, z ∈ Z and v ∈ V. According to Lemma 4.18 there exists a subset Z of Ẋ such that the projection ζ : Ẋ → X induces an isometry from Z onto Z. Moreover z, z v > ρ/4, for every z, z ∈ Z and v ∈ V (the Gromov product is computed here in Ẋ). We denote by Z 0 the pre-image of Z0 in Z. Let ḡ ∈ S. By construction ḡz belongs to Z for every z ∈ Z0 . Consequently there exists a (unique) g ∈ G such that for every z ∈ Z 0 we have |gz -z| Ẋ = |ḡz -z| (Lemma 4.17). We denote by S the set of all g ∈ G obtained in this way. Note that S does not generate an elementary subgroup, otherwise so would S. Let z ∈ Z0 and z ∈ Ẋ its pre-image in Z 0 . Let y = p(z) be the radial projection of z. Since Z lies in the 3d 0 /2-neighborhood of ζ(X) we have |z -y| Ẋ 3d 0 /2. Combining the triangle inequality with Lemma 4.5, we get for every g ∈ S,

By Proposition 4.3, we get that y lies in Fix(S, d 1 ) ⊂ X. Assume that z is another point in Z0 . As previously we denote by z ∈ Ẋ its pre-image in Z 0 and by y = p(z ) the radial projection of z . In particular, y also belongs to Fix(S, d 1 ) ⊂ X and |z -y | Ẋ 3d 0 /2. It follows from the triangle inequality that

This inequality holds for every z, z ∈ Fix( S, d 0 ). Hence diam Fix( S, d 0 ) diam (Fix(S, d 1 )) + 3d 0 .

Mixed invariants

In view of Propositions 4.47 and 3.5, the ν-invariant of G can be used to control the acylindricity invariant A( Ḡ, X, d) of the quotient Ḡ. If we want to iterate the procedure we need to control as well the ν-invariant of Ḡ. Let us start with an informal discussion to emphasizes the difficulties that may arise along the way. For simplicity let us assume that G (hence Ḡ) does not contain any parabolic subgroup. Indeed those subgroups will not be a source of trouble.

Definition 4.56. A chain C = (g 0 , . . . , g m ) of G is a strong chain if it satisfies the following holds (i) g 0 , . . . , g m generate an elementary subgroup of G (for its action on X).

(ii) either C admits a loxodromic conjugating element or g 0 , . . . , g m-1 is a dihedral germ.

We define strong chains of Ḡ in the exact same way. A strong chain C = (ḡ 0 , . . . , ḡm ) can be lifted if there exists a strong chain C = (g 0 , . . . , g m ) of G such that the quotient map π : G Ḡ sends g k to ḡk for every k ∈ 0, m . In this situation we also say that C lifts C. As suggested at the beginning of this section we first prove the following dichotomy. (i) There exists v ∈ V such that ḡ0 , h is contained in Stab(v).

(ii) The subgroup ḡ0 , h is loxodromic.

(iii) There exists a strong chain C = (ḡ 0 , . . . , ḡ m ) of Ḡ which can be lifted and such that (ḡ 1 , . . . , ḡ m-1 ) = (ḡ 1 , . . . , ḡm-1 ).

We split the proof into several lemmas depending on the nature of the group generated by C. Proof. We first claim that Ē can always be lifted even if Ē is parabolic. Indeed since Ē is finitely generated Proposition 4.40 applies. Let E be a lift of Ē.

For every k ∈ 0, m , we denote by g k the pre-image of ḡk in E. Note that, contrary to (ḡ 0 , . . . , ḡm ), the tuple (g 0 , . . . , g m ) could not be a chain. Nevertheless, according to Proposition 4.27 applied with S 1 = {g 0 , . . . , g m-1 } and S 2 = {g 1 , . . . , g m }, there exists h 0 ∈ G with the following properties (i) for every k ∈ 0, m -1 , we have

Thus C = (g 0 , . . . , g ν ) is actually a chain and h 0 a conjugating element of C.

Obviously C generates an elementary subgroup of G. Note that either h 0 is loxodromic or g 0 , . . . , g ν-1 is a dihedral germ (Lemmas 4.33 and 4.34) hence the proof is complete.

Lemma 4.59. Let C = (ḡ 0 , . . . , ḡm ) be a strong chain of Ḡ which generates an elliptic subgroup Ē of Ḡ which cannot be lifted. Let h be a conjugating element of C. Then either Ē0 = ḡ0 , . . . , ḡm-1 contains a strict rotation and ḡ0 , h is contained in Stab(v) for some v ∈ V, or C can be lifted.

generated by ḡ2 0 , ḡ1 , . . . , ḡm-1 . Seen as a subgroup of D n ×D k n2 ×E, the reflection rank of F0 is at most m -1 (see Section 3.2). Recall that 2 m+2 µ divides n. According to Lemma 3.16 there exists a pre-image ā ∈ Stab(v) of r n/4 such that

We now let ḡ 0 = āḡ 0 ā-1 and ḡ k = ḡk for every k ∈ 1, m . By construction ā commutes with ḡ1 , . . . , ḡm-1 . Hence C = (ḡ 0 , . . . , ḡ m ) is a chain with h0 = hā -1 as conjugating element. Note also that C and C only differ on the first element. Identity (ii) also tells us that given c ∈ C and ε ∈ {±1}, we have [[ā, ḡε 0 ], c] = 1, which can be reformulated as

As ḡ0 normalizes C, so does ḡ 0 . Since ā commutes with ḡ2 0 , we have

In particular, Ē = Ā , B is elementary. Hence C generates an elementary subgroup. Recall that ā is a preimage of r n/4 . Hence

On the other hand, q v maps C to x . Consequently q v ( Ā ) = x and Ā can be lifted in G. As observed above C generates a subgroup Ē , which is either elliptic, parabolic or loxodromic. We let Ē 0 = ḡ 0 , . . . , ḡ m-1 . Note that Ē 0 = ā Ē0 ā-1 . Since Ē0 is an elliptic subgroup of the loxodromic group Ē it is a dihedral germ (Remark 4.54), hence so is Ē 0 . Consequently C is a strong chain. We now distinguish again two cases.

Case 2.1 Assume that Ē is not loxodromic. Note that Ē 0 is contained in Ā . Since Ā can be lifted Ē 0 does not contain a strict rotation. If Ē is elliptic or parabolic, then Lemmas 4.58 and 4.59 tell us that there exists a strong chain C lifting C . Case 2.2 Assume that Ē is loxodromic. Observe that Ē 0 does not contain a loxodromic element (it lies in the elliptic subgroup Ā ). We rerun the previous discussion replacing C and Ē = Ā * C B by C and Ē = Ā * C B. In particular, if B can be lifted, we are back to Case 1. This means that there exists a strong chain C which can be lifted and which coincides with C except maybe for the first or the last term. Assume now that B cannot be lifted. Note that Ē 1 = ḡ 1 , . . . , ḡ m-1 coincides with Ē1 , and therefore does not contain a strict rotation. We permute in Case 2 the role of Ā and B and produce a new strong chain C = (ḡ 0 , . . . , ḡ m ) such that (ḡ 1 , . . . , ḡ m-1 ) = (ḡ 1 , . . . , ḡm-1 ) which generates an elementary subgroup of the form Ē = Ā * C B where both Ā and B are elliptic subgroups which can be lifted. Following Case 1. we observe that there exists a strong chain C which can be lifted and which coincides with C except maybe for the first or the last term.

Let G be a group acting by isometries on a δ-hyperbolic length space X for some δ δ 1 and satisfying the following assumptions.

(i) The action of G on X is gentle and non-elementary.

(ii) Dihedral pairs of G have type (E, n 2 ), where n 2 is the largest power of 2 dividing n.

We denote by P the set of all primitive loxodromic elements h ∈ G such that h 10δ 1 . Let K be the (normal) subgroup of G generated by {h n : h ∈ P } and Ḡ the quotient of G by K. We write π : G Ḡ for the corresponding quotient map.

Then there exists a δ-hyperbolic length space X, with δ δ 1 , on which Ḡ acts by isometries satisfying (i)-(iii). In addition there exists a π-equivariant map X → X with the following properties.

• The map

• If F is an elliptic (respectively parabolic) subgroup of G, then π induces an isomorphism from F onto its image F which is also elliptic (respectively elliptic or parabolic).

• Any elliptic subgroup of Ḡ, is either the isomorphic image of an elliptic subgroup of G, or has finite exponent dividing n.

• Any finitely generated parabolic subgroup of Ḡ is the isomorphic image of a parabolic subgroup of G.

• Let F 1 and F 2 be two subgroups of G. Assume that F 1 is elliptic and F 2 is generated by a set S 2 such that Fix(S 2 , 100δ 1 ) is non-empty. If the images of F 1 and F 2 are conjugated in Ḡ, then F 1 and F 2 are conjugated in G.

Vocabulary. Assume that ν 0 , n and the model collection E have been already fixed. If G is a group acting on a metric space X satisfying the assumptions of the proposition, including Points (i)-(iii), we say that (G, X) satisfies the induction hypotheses relative to (n, E). The proposition says among others that if (G, X) satisfies the induction hypotheses relative to (n, E), then so does ( Ḡ, X)

Proof. We start by defining the various constants appearing in the statement. Let δ 0 , δ 1 , ∆ 0 , and ρ 0 be the parameters given by the small cancellation theorem (Theorem 4.7). We define κ = δ 1 /π sh(10δ 1 ) (so that we can apply Proposition 4.46). We fix C 0 and C 1 as follows.

C 0 = π sh(800δ 1 ) and [START_REF] Coulon | Partial periodic quotients of groups acting on a hyperbolic space[END_REF]Lemma 6.14].

Recall that given a group E, its holomorph is the semi-direct product Hol(E) = Aut(E) E. If E 0 stands for a collection of groups, we let

+ and N ∈ N. Let E 0 be a finite collection of finite groups. We write µ for the exponent of Hol(E 0 ). There exist ν 1 , N 1 ∈ N such that for every integer n N 1 which is a multiple of 2 ν1 µ the following holds.

Let G be a group acting by isometries on a δ-hyperbolic length space X. We assume that this action is (100δ, L, N )-acylindrical and non-elementary. In addition we suppose that every finite subgroup of G with dihedral shape is isomorphic to a group of E 0 . Then there exists a quotient Q of G with the following properties.

(i) For every elliptic subgroup F of G, the projection G Q induces an embedding from F into Q.

(ii) For every q ∈ Q, either q n = 1 or q is the image of an elliptic element of G.

(iii) The projection G G/G n induces an epimorphism Q G/G n .

(iv) For every x ∈ X, the map G → Q is one-to-one when restricted to {g ∈ G : |gx -x| < r} .

(v) There are infinitely many elements of Q which are not the image of an elliptic element of G.

(vi) The kernel K of G Q is purely loxodromic. As a normal subgroup K is not finitely generated.

Proof. We are going to apply Theorem 5.4. To that end we let we denote by M the cardinality of the biggest group in E 0 . Up to replacing r by a largest value, we can assume that

Recall that µ is the exponent of Hol(E 0 ). We now set ν = max{C ν (δ, L, N ), M + 1} and ν 1 = max{ν + 2, µ + 6}.

and denote by N 1 the critical exponent given by Theorem 5.4. Let n N 1 be an integer divisible by 2 ν1 µ.

Let G be a group acting on a δ-hyperbolic length space X as in the theorem. It follows from our previous discussion that A(G, X, 400δ) r, inj (G, X) 1/r and ν(G, X) ν. Let us prove that ν stg (G, X) ν. Let g, h ∈ G and m ν. For every k ∈ N, we write g k = h k gh -k . Suppose that E = g 0 , . . . , g m is elementary. Assume first that h is loxodromic. According to our choice of ν, we have m ν(G, X), thus the elements g and h generate an elementary subgroup of G. Note that this group is necessarily loxodromic, hence has dihedral shape. Assume now that E 0 = g 0 , . . . , g m-1 is a dihedral germ. Since the action of G is acylindrical, every loxodromic subgroup of G is virtually cyclic. Hence E 0 is finite. As a dihedral germ it also has dihedral shape. Consequently E 0 is isomorphic to a group in E 0 hence contains at most M element. Since m M + 1, there exist i, j ∈ 0, m -1 with i < j such that g i = g j . In particular, g j-i = g 0 . It follows that h normalizes g 0 , . . . , g m-1 . Hence the subgroup generated by g and h is elementary. More precisely, it is a cyclic extension of the dihedral germ g 0 , . . . , g m-1 , thus it has dihedral shape. This proves that ν ν stg(G,X) and completes the proof of our claim.

We now build an appropriate model collection E. Let (E, C) be a dihedral pair where E is infinite. Since C is finite, it fits into the following short exact sequence

where L is either Z or D ∞ . We choose an element g ∈ E whose image in L generates the maximal infinite cyclic subgroup of L. Recall that µ is the exponent of Hol(E 0 ). We claim that g µ is a normal subgroup of E. Let u ∈ E. There exist ε ∈ {±1} and c ∈ C such that ugu -1 = g ε c. The value of ε depends whether the image of u in L is a reflection or not. Since g normalizes C its action by conjugation on C induces an automorphism of C that we denote by ϕ. One checks that for every p ∈ N,

However in the holomorph Hol(C), whose exponent divides µ, we have

Hence ug µ u -1 = g εµ . Consequently g µ is a normal subgroup of E as we announced. Note also that the exponent of E/ g µ divides µ. Moreover E embeds in E/C × E/ g µ . We denote by E 1 the collection of quotients E/ g µ obtained as above, where (E, C) runs over all dihedral pairs with E infinite. In addition we let E = E 0 ∪E 1 . It follows from the construction that the exponent of E divides µ. Moreover, for every dihedral pair (E, C) there exists E ∈ E such that E embeds in E/C × E. All the assumptions of Theorem 5.4 are satisfied and the result follows.

Free Burnside groups and periodic groups. Theorem 5.8. Let r 2. There exists N 1 ∈ N such that for every integer n N 1 that is a multiple of 128, the free Burnside group B r (n) is not finitely presented and therefore infinite. Moreover if n 2 stands for the largest power of 2 dividing n, then every finite subgroup of B r (n) embeds in D n × D k n2 for some k ∈ N.

Proof. Let X be the Cayley graph of the free group F r of rank r. It is δhyperbolic with δ ∈ R * + such that 400δ < 1. It follows that A(F r , X, 400δ) = 0. Moreover inj (F r , X) 1. Every dihedral germ is trivial. Hence ν stg (G, X) = 1. We choose for E the class of group reduced to the trivial one. Its exponent µ = µ(E) is 1. Every subgroup with dihedral shape is trivial or infinite cyclic. Hence the assumption of Theorem 5.4 are fulfilled. Note that ν 1 = max{ν stg (G, X) + 2, µ + 6} = 7. Hence there exists N 1 ∈ N such that for every n N 1 that is a multiple of 128, the group B r (n) is not finitely presented, hence infinite. We now prove the second assertion. Let

be the sequence of of groups produced in the proof of Theorem 5.4, whose direct limit is B r (n). Let F be a finite subgroup of B r (n). By construction there exist ∈ N and an elliptic subgroup F of G such that G B r (n) maps F onto F . We assume that is the smallest for this property. Recall that each map G k G k+1 is one-to-one when restricted to an elliptic subgroup. Hence F is actually isomorphic to F . According to our choice of , the subgroup F cannot be lifted. Therefore there exists v in the vertex set V of X such that F is contained in Stab(v). It follows from Lemma 4.52 that F embeds in D n × D k n2 for some k ∈ N.

Remark. Although we have not written the details here, a careful reader can follow the induction in the proof of Theorem 5.4 to show, just as in Ivanov [START_REF] Ivanov | The free Burnside groups of sufficiently large exponents[END_REF] or Lysenok [START_REF] Lysenok | Infinite Burnside groups of even period[END_REF], that the word and the conjugacy problems are solvable in B r (n). With the previous notations, the solution of the word problem is based on the following observation. Given an element g ∈ F r , there exists k ∈ N, which only depends on the word length of g, such that g is trivial in B r (n) if and only if so is it in G k . Hence it suffices to run the solution of the word problem in the hyperbolic group G k . The conjugacy problem can be solved as follows. Given g 1 , g 2 ∈ F r , there exists k ∈ N, which only depends on the word length of g 1 and g 2 , such that both g 1 and g 2 are elliptic in G k . A proof by induction based on Corollary 4.29 shows that g 1 and g 2 are conjugated in B r (n) if and only if so are they in G k . Hence it suffices to run the solution of the conjugacy problem in G k . Corollary 5.9 (Compare with Ivanov [START_REF] Ivanov | The free Burnside groups of sufficiently large exponents[END_REF] and Lysenok [START_REF] Lysenok | Infinite Burnside groups of even period[END_REF]). Let r 2. There exists N 1 ∈ N, such that for every integer n N 1 , the group B r (n) is infinite.

Proof. Recall that free Burnside groups of sufficiently large odd exponents are infinite, see for instance [START_REF] Delzant | Courbure mésoscopique et théorie de la toute petite simplification[END_REF][START_REF] Coulon | On the geometry of Burnside quotients of torsion free hyperbolic groups[END_REF] for a geometric proof. Hence it suffices to observe that given two integers p, n ∈ N, the group B r (pn) maps onto B r (n).

Theorem 5.10 (Compare with Ol'shankiȋ-Ivanov [START_REF] Ivanov | Hyperbolic groups and their quotients of bounded exponents[END_REF]). Let G be a non-elementary hyperbolic group. There exist p, N 1 ∈ N such that for every integer n N 1 that is a multiple of p, the quotient G/G n is infinite. Moreover

Proof. Let E 0 be the collection of all isomorphism classes of finite subgroups of G. Since G is hyperbolic, E 0 is finite. We write µ for its exponent. Let X be the Cayley graph of G relative to some finite generating set. The action of G on this hyperbolic space is acylindrical hence the assumptions of Theorem 5.7 holds. Let ν 1 , N 1 ∈ N be the parameters given by Theorem 5.7 and set p = 2 ν1 µ. Observe that the order of every elliptic element of G divides µ, hence p. Assume now that n N 1 is a multiple of p. According to Theorem 5.7 there exists an infinite quotient Q of G such that the projection G G/G n induces a map Q G/G n . Moreover every element q ∈ Q satisfies the following dichotomy. Either q n = 1, or q is the image of an elliptic element of G. In the latter case we have q p = 1, and thus q n = 1. In other words Q is a quotient of G/G n , hence Q is isomorphic to G/G n which is therefore infinite.

According to Theorem 5.7 the quotient map G → G/G n can be made one-toone on arbitrarily large balls by enlarging the value of n. Hence the intersection of all the subgroups G n , when n runs over N \ {0} is trivial.

Remark. As for free Burnside groups, one can give a precise description of the finite subgroups of G/G n , provided n is sufficiently large and divisible by p. One can also prove that the word and the conjugacy problem are solvable in these periodic quotients.

Relatively hyperbolic groups. Since Gromov's original paper [START_REF] Gromov | Hyperbolic groups[END_REF], several different definitions of relatively hyperbolic groups have emerged, see for instance [START_REF] Bowditch | Relatively hyperbolic groups[END_REF][START_REF] Farb | Relatively hyperbolic groups[END_REF]. These definitions have been shown to be almost equivalent [START_REF] Bowditch | Relatively hyperbolic groups[END_REF][START_REF] Szczepański | Relatively hyperbolic groups[END_REF][START_REF] Hruska | Relative hyperbolicity and relative quasiconvexity for countable groups[END_REF]. For our purpose we will use the following one. Definition 5.11 ([24,Definition 3.3]). Let G be a group and {P 1 , . . . , P m } be a collection of subgroups of G. We say that G is hyperbolic relative to {P 1 , . . . , P m } if there exist a proper geodesic δ-hyperbolic space X and a collection Y of pairwise disjoint open horoballs satisfying the following properties. The action of G on the space X given by Definition 5.11 is not acylindrical. Indeed the subgroups P j can be parabolic. This cannot happen with an acylindrical action [START_REF] Bowditch | Tight geodesics in the curve complex[END_REF]Lemma 2.2]. More generally, the elementary subgroups of G are