Curve graphs for Artin-Tits groups of type $B$, $\widetilde A$ and $\widetilde C$ are hyperbolic

For $n\geqslant 3$, we prove that the graphs of irreducible parabolic subgroups of the Artin-Tits groups of type $A_n$ and $B_n$ are both isomorphic to the curve graph of an $(n+1)$-times punctured disk. For $n\geqslant 2$, we show that the graphs of irreducible parabolic subgroups of the Artin-Tits groups of type $\widetilde A_n$ and $\widetilde C_n$ are isomorphic to some subgraphs of the curve graph of the $(n+2)$-times punctured disk which are not quasi-isometrically embedded. We prove nonetheless that all these graphs are hyperbolic.


Introduction and background
This presentation can be encoded by a Coxeter graph Γ: the vertices are in bijection with the set S and two distinct vertices a, b are connected by an edge labeled m ab if m ab > 2 and labeled by ∞ if a, b satisfy no relation. The Artin-Tits group defined by the Coxeter graph Γ will be denoted A Γ . Because all the relations in the given presentation of A Γ are balanced, there is a homomorphism Γ : A Γ −→ Z assigning to each element g ∈ A Γ the exponent sum of any word on S representing it. The group A Γ is said to be irreducible if Γ is connected and dihedral if Γ has two vertices. The quotient by the normal subgroup generated by the squares of the elements in S is a Coxeter group denoted by W Γ . The Artin-Tits group A Γ is said to be of spherical type if W Γ is finite. A proper subset ∅ = X S generates a proper standard parabolic subgroup of A Γ which is naturally isomorphic to the Artin-Tits group A Ξ defined by the subgraph Ξ of Γ induced by the vertices in X [21]. A subgroup P of A Γ is parabolic if it is conjugate to a standard parabolic subgroup. The flagship example of an Artin-Tits group (of spherical type) is the braid group on (n + 1) strands -i.e. the Artin-Tits group defined by the graph A n shown in Figure 1(a). This group is isomorphic to the Mapping Class Group of an (n + 1)-times punctured closed disk D n+1 , that is Figure 1: Coxeter graphs: (a) of type A n ; (b) of type B n ; (c) of type A n ; (d) of type C n . As usual, we omit the label m ab whenever m ab = 3. The given labeling of the standard generators will be used throughout the paper.
of Artin-Tits groups closely related to Artin's braid groups, whose defining Coxeter graphs are depicted in Figure 1 (b)-(c)-(d). Here is a brief summary of the results in the paper. Firstly, we recall that the Artin-Tits group of (spherical) type B n can be realized as a finite index subgroup of Artin's braid group on (n + 1) strands A An [13]. We shall prove that this inclusion induces a graph isomorphism between the respective graphs of irreducible parabolic subgroups: Theorem 1.3. For n 3, the graphs C parab (B n ) and C parab (A n ) are isomorphic; hence both are hyperbolic.
Secondly, we focus on the Artin-Tits group of type A n . This group can be embedded in the Artin-Tits group of type B n+1 and A Bn+1 can be decomposed as a semi-direct product A An Z [13]. From this we obtain the following  (ii) The graph C parab ( A n ) is isomorphic to a subgraph of C parab (B n+1 ).
(iii) The graph C parab ( A n ) has infinite diameter.
It turns out that the inclusion of C parab ( A n ) in C parab (B n+1 ) is not a quasi-isometric embedding -see Proposition 4.8; however using Theorems 1.3 and 1.2, we are able to see C parab ( A n ) as a subgraph of the curve graph of D n+2 . This allows us to prove, using results from [22]: Theorem 1.5. Let n 2. The graph C parab ( A n ) is hyperbolic.
Finally, we turn attention to the Artin-Tits group of type C n . We recall that it embeds as a finite index subgroup of Artin's braid group on (n + 2) strands A An+1 in a very similar way as A Bn embeds in A An [1]. We prove the following Theorem 1.6. Let n 2.
(i) The graph C parab ( C n ) is connected.
(ii) The graph C parab ( C n ) is isomorphic to a subgraph of C parab (A n+1 ).
(iii) The graph C parab ( C n ) has infinite diameter.
By contrast with the relation between C parab (B n ) and C parab (A n ), it turns out (Remark 5.10) that the inclusion of C parab ( C n ) in C parab (A n+1 ) is not a quasi-isometric embedding. We nevertheless will show, again seeing C parab ( C n ) as a subgraph of the curve graph of D n+2 and using results from [22]: Theorem 1.7. Let n 2. The graph C parab ( C n ) is hyperbolic.
It is important to note that our proofs of the hyperbolicity of C parab (B n ), C parab ( A n ) and C parab ( C n ) strongly depend on the hyperbolicity of curve graphs; it would be highly desirable to obtain independent algebraic proofs. The paper is arranged as follows. In Section 2 we review some results on parabolic subgroups of Artin-Tits groups, we explain the isomorphism given by Theorem 1.2 and we introduce a result from [22] which allows to establish the hyperbolicity of some subgraphs of the curve graph. Theorem 1.3 is proved in Section 3. Section 4 is devoted to the proofs of Theorems 1.4 and 1.5 while Theorems 1.6 and 1.7 are established in Section 5. Finally, Section 6 contains some closing remarks and open questions; in particular it is shown in Corollary 6.9 that the union of the normalizers of standard parabolic subgroups (the union of standard parabolic subgroups and the center, respectively) are hyperbolic structures on A Bn and that these structures are not equivalent -see [7].

Artin-Tits groups and Coxeter groups
Let A Γ be any Artin-Tits group with standard generators S. Let W Γ = A Γ / s 2 | s ∈ S be the associated Coxeter group. The canonical projection π : A Γ W Γ admits a set section τ defined as follows -see for instance [4, Theorem 3.3.1(ii)]. For s ∈ S, denote bys its image in W Γ and S = {s | s ∈ S}. Let w ∈ W Γ and lets 1 . . .s r be a reduced expression for w, meaning a shortest word representative for w onS; then τ (w) = s 1 . . . s r . The kernel of the projection π is called the pure Artin-Tits group (or coloured Artin-Tits group) and is denoted P A Γ . Given a subset X of S, the standard parabolic subgroup of A Γ generated by X is denoted by A X . In the rest of this section, we assume that A Γ is of spherical type. In this case, W Γ contains a unique longest element w 0 (see for instance [11,Lemma 4.6.1]). Denote its lift τ (w 0 ) in A Γ by ∆ Γ . Whenever Γ is connected, it is known that A Γ has cyclic center generated by ∆ Γ or ∆ 2 Γ [5, Theorem 7.2]. Any proper irreducible parabolic subgroup P of A Γ is itself an irreducible Artin-Tits group of spherical type. The center of P is a cyclic group generated by an element z P (actually we have the generators z P and z −1 P and we choose z P so that its exponent sum Γ (z P ) is positive). We will always refer to this particular element as the central element of P . The following two results will be used throughout. The first one says in particular (with g = 1) that the element z P determines completely the subgroup P (and conversely). Proposition 2.3. [19, Theorem 5.2] Let A Γ be an Artin-Tits group of spherical type. Let P, Q be two irreducible parabolic subgroups of A Γ and let g ∈ A Γ . Then Q = P g if and only if z Q = z g P .
The second result reduces the definition of adjacency in the graph of irreducible parabolic subgroups to a very simple commutation condition between the respective central elements.

Correspondence between curves and parabolic subgroups
Recall that the braid group on (n + 1) strands -or Artin-Tits group A An -can be identified with the Mapping Class Group of a closed disk with (n + 1) punctures D n+1 . Assume that D n+1 is the closed disk in the complex plane of radius n+2 2 centered at n+2 2 and the punctures are at the integer numbers 1 i n + 1. For 1 i n, the standard generator σ i of A An corresponds to a clockwise half-Dehn twist along the horizontal segment [i, i + 1]. The group A An naturally acts -on the right-on the set of isotopy classes of essential simple closed curves in D n+1 . In the sequel we will simply write "essential curve" or even "curve" instead of "isotopy class of essential simple closed curve"; accordingly we will say that two curves are disjoint if the corresponding isotopy classes admit disjoint representatives. The result of the action of a braid y on a curve C will be denoted by C y . Finally, note that a curve C in D n+1 divides the disk in two connected components naturally seen as the interior and the exterior of C. Let I be a proper subinterval of [n] = {1, . . . , n}, that is This defines a proper irreducible standard parabolic subgroup of A An , generated by {σ i | i ∈ I}; denote this subgroup by A I . As explained in [10,Section 2], there is a one-to-one correspondence which induces the graph isomorphism of Theorem 1.2. To a curve C in D n+1 , we associate the subgroup f(C) of A An consisting of all isotopy classes of automorphisms of D n+1 whose support is enclosed by C; this is a proper irreducible parabolic subgroup. In particular, given a proper subinterval I of [n], with m = min(I) and k = #I, the standard parabolic subgroup A I is the image f(C I ) of the circle C I surrounding the k + 1 punctures m, . . . , m + k -such a curve is called standard or round. The inverse correspondence is given by the -well-defined-formula A y I → C y I , for any proper subinterval I of [n] and any y ∈ A An . Let us see that the adjacency condition given in Proposition 2.4 turns f into a graph isomorphism. Let C be a curve in D n+1 , let P = f(C) and let z P be the central element of P . If C surrounds at least three punctures, then z P is the Dehn twist around the curve C. Otherwise, z P is the half-Dehn twist along an arc connecting the two punctures in the interior of C and which does not intersect C. Now, given two parabolic subgroups P 1 = f(C 1 ) and P 2 = f(C 2 ), z P1 and z P2 commute if and only if C 1 and C 2 are disjoint.

Hyperbolicity for some graphs of curves
In this section, we present a specialization of a theorem of Kate Vokes, which we will use as a criterion for proving the hyperbolicity of some subgraphs of the curve graph of the punctured disk. Consider the n-times punctured disk D n . A subsurface of D n is (the isotopy class of) a connected subsurface X of D n so that every boundary component of X is either ∂D n or an essential curve in D n . A simple closed curve in X is essential (in X) if it cannot be isotoped in X to a point, a puncture or a boundary component of X. By an annulus in D n we mean the subsurface consisting of a tubular neighborhood of some essential curve in D n . Given a curve C in D n and a subsurface X of D n we say that C and X are disjoint if they admit disjoint representatives; X is said to be a witness for C if C and X are not disjoint. In particular, X is not a witness for any of its boundary components. Two subsurfaces are disjoint if they admit disjoint representatives.
Theorem 2.5. [22, Corollary 1.5] LetĈ be a family of curves in D n ; let K be the subgraph of CG(D n ) induced byĈ, equipped with the combinatorial metric d K (each edge has length one). Let X be the set of witnesses for K, that is the set of all subsurfaces of D n which are a witness for every element ofĈ. Suppose: (ii) The action of P A An−1 on D n induces an isometric action of P A An−1 on K.
(iii) X contains no annulus.
(iv) No two elements of X are disjoint.
Then K is hyperbolic.
Remark 2.6. Let us check that the hypothesis of Theorem 2.5 match the hypothesis of [22,Corollary 1.5], namely that K is a twist-free multicurve graph having no pair of disjoint witnesses. The second half is exactly our clause (iv). The definition of a twist-free multicurve graph ([22, Definition 2.1]) consists of clauses (1)- (5). To see clauses (2) and (4), observe that each vertex of K being a curve in D n is in particular a multicurve and each pair of adjacent vertices in K are disjoint curves. Clauses (1) and (5) correspond to (i) and (iii) of Theorem 2.5, respectively. Clause (3) is adapted into clause (ii) of Theorem 2.5. The results in [22] work for compact surfaces (possibly with boundary). However, according to [16,Section 2.3], punctures can be treated as boundary components, so the results of [22] apply to punctured surfaces as well. In the case of the punctured disc, we have to replace the whole braid group by the pure braid group since mapping classes in [22] are required to fix the boundary pointwise. In the sequel we will find it more convenient to maintain the difference between punctures of the disk and "real" boundaries, thinking of punctures as "distinguished boundary components".
3 The graph C parab (B n ) A proper irreducible standard parabolic subgroup of A Bn is determined by a proper subinterval of [n]: for any proper subinterval I of [n], we denote by B I the proper irreducible standard parabolic subgroup of A Bn generated by {τ i | i ∈ I}. Notice that in A Bn , by Lemma 2.2, the standard generators fall into two conjugacy classes, namely each τ i , i 2 is conjugate to each other and not conjugate to τ 1 . In view of Lemma 2.1, each proper irreducible parabolic subgroup P of A Bn is exactly one of the following types: • Type A if P is conjugate to B I for I ⊂ {2, . . . , n}, There is a monomorphism The image of η is the subgroup P 1 of (n + 1)-strands 1-pure braids, that is the subgroup of all (n + 1)-strands braids in which the first strand ends in the first position. In other words, a braid y on (n + 1) strands belongs to P 1 if and only if π y (1) = 1, where π y = π(y) is the permutation in S n+1 = W An associated to y. A presentation for P 1 was given by Wei-Liang Chow [8] in 1948; for a proof that η defines an isomorphism between A Bn and P 1 , the reader may consult [13].
The proof of the next lemma follows from an easy computation left to the reader.
Let I, J be proper subintervals of [n] and let g ∈ A Bn . The following are equivalent: Proof. Assume (i). According to Proposition 2.3, we have z g Conversely, assume (ii). By Proposition 2.3, we get z η(g) if and only if J = {1}, as η(g) is 1-pure. In this case it follows that (σ 2 1 ) η(g) = σ 2 1 ; as η is injective, we get τ g 1 = τ 1 , which is to say B g I = B J . If on the contrary I = {1}, we can immediately pull back to A Bn the relation z η(g) A I = z A J using Lemma 3.1(ii) and we get z g B I = z B J which, by Proposition 2.3, implies (i). Now, observe that P 1 has index (n + 1) in A An . For 1 i n, define a i = σ i . . . σ 1 and a 0 is the trivial braid. Notice that π ai (i + 1) = 1, for all 0 i n. Given y ∈ A An , there is a unique i ∈ {0, . . . , n} so that ya i ∈ P 1 . In Figure 2(i)-(iii) are depicted a 2 , a 8 , a 5 ∈ A A9 , respectively.
Proof. The contents of Lemma 3.3 are depicted in Figure 2(i)-(iii). Only the third case might need a short proof: it suffices to observe that the crossings σ i0 , . . . , σ m I fix the curve C I as they are inner to it, so it only remains the action of σ m I −1 · · · σ 1 = a m I −1 .
For each I, we define C I = C am I −1 I . Observe that C I is not a standard curve whenever m I > 1 (see the bottom part of Figure 2(iii)); however it can be transformed into a standard curve by the action of a 1-pure braid. To be precise, the action of the 1-pure braid transforms the curve C I into the round curve C [1,k I ] which surrounds the k I + 1 first punctures -see an example in Figure 2(iv).
Using the correspondence f between curves and proper irreducible parabolic subgroups (Section 2.2), we have shown that is a standard parabolic subgroup in the first two cases and that in the third case, there is ξ I ∈ P 1 such that (A ai 0 Proof. Let ζ be any braid such that P ζ is a standard parabolic subgroup, say A J , and write m = min(J) and k = #J. [1,k] is, and we can take I = [1, k], α = ζa i0 ξ J , which is 1-pure as ζa i0 and ξ J are 1-pure.
, where I is a proper subinterval of [n] and g ∈ A Bn , defines a bijective map H between the respective sets of vertices of C parab (A n ) and C parab (B n ).
Proof. By Proposition 3.2, the map H is well-defined and injective; by Proposition 3.4, it is surjective.  According to [13], there is a monomorphism where the action of ρ is given by conjugation, as in (ii). k + 1 < l n, then (θ( A I )) ρ n−l+2 = τ 2 , . . . , τ k+n−l+3 , whence θ( A I ) is again conjugate to a proper irreducible standard parabolic subgroup. Finally, if P is not standard, there is some g ∈ A An such that P g = P 0 is standard; by the above discussion, θ(P 0 ) is a proper irreducible parabolic subgroup of A Bn+1 and from θ(P ) θ(g) = θ(P 0 ), we deduce that θ(P ) is itself a proper irreducible parabolic subgroup of A Bn+1 . Proposition 4.2 allows us to define a map Θ from the set of vertices of C parab ( A n ) to the set of vertices of C parab (B n+1 ), which sends a proper irreducible parabolic subgroup of A An to its image under the monomorphism θ. We first describe the image of Θ. Recall from Section 3 that each proper irreducible parabolic subgroup P of A Bn+1 is either of type A (P is conjugate to B I with 1 / ∈ I) or of type B (P is conjugate to B I with 1 ∈ I). If P is not cyclic, P is of type A if and only if P is an Artin-Tits group of type A k , k 2 and P is of type B if and only if P is an Artin-Tits group of type B k , k 2. (i) There exists a proper irreducible parabolic subgroup P of A An such that Q = θ(P ), Proof. Suppose that Q = θ(P ), for some proper irreducible parabolic subgroup P of A An . If P is cyclic, P is conjugate in A An to σ i for some 0 i n; by definition of θ, we then see that Q is conjugate to τ 2 (recall that in A Bn+1 all τ i , i 2, are conjugate) so Q is of type A. If P is not cyclic, P is isomorphic to a braid group A A k for 2 k n. Note that Q = θ(P ) and P are isomorphic; therefore Q must be of type A. Conversely, suppose that Q is of type A. Assume first that Q is standard; that is Q = B I , for is the image of a proper irreducible parabolic subgroup of A An . If on the contrary ζ is not in the image of θ, there is some r ∈ Z and x ∈ A An such that ζρ r = θ(x) -see Proposition 4.1(iii). Using where in the last term all indices are taken modulo n + 1. But this is equivalent to saying that Q = θ( A x −1 {i+r | i∈I } ), which achieves the proof.
Let K be the subgraph of C parab (B n+1 ) induced by the vertices of the form θ(P ), where P is a proper irreducible parabolic subgroup of A An ; equivalently, according to Lemma 4.3, K is the subgraph of C parab (B n+1 ) induced by those proper irreducible parabolic subgroups of type A.
Proposition 4.4. The map Θ defines an isomorphism between C parab ( A n ) and K.
Proof. The injectivity of Θ at the level of vertices follows from the injectivity of θ; by Lemma 4.3, Θ is surjective onto the vertices of K. Applying the monomorphism θ, it is easily seen that whenever P and Q are adjacent proper irreducible parabolic subgroups of A An (Definition 1.1), then θ(P ) and θ(Q) are adjacent as well. It follows that the map Θ induces a graph homomorphism. It remains to show that the inverse map is also a graph homomorphism, that is for every proper irreducible parabolic subgroups P, Q of A An , the condition that θ(P ) and θ(Q) are adjacent in C parab (B n+1 ) implies that P and Q are adjacent in C parab ( A n ). But this again follows from the injectivity of the homomorphism θ.
The last ingredient for Theorem 1.4 is to prove that the subgraph K is 1-dense in C parab (B n+1 ), namely: Lemma 4.5. For every proper irreducible parabolic subgroup Q of A Bn+1 , there exists a proper irreducible parabolic subgroup P of A An so that d Bn+1 (Q, θ(P )) 1.
Proof. By Lemma 4.3, we may assume that Q is of type B, otherwise Q is already the image of a proper irreducible parabolic subgroup of A An . Suppose first that Q is standard, that is Q = τ 1 , . . . , τ k , for 1 k n − 1. We shall see that d Bn+1 (Q, θ( σ e )) = 1, for e = 1 or e = 2. Indeed, we have d Bn+1 ( τ 1 , θ( σ 2 ) = 1 and for k 2, d Bn+1 ( τ 1 , . . . , τ k , θ( σ 1 )) = 1. Suppose now that Q is not standard; let ζ ∈ A Bn+1 such that Q ζ is standard. As we have just seen, by taking e = 1 or 2, we have d Bn+1 (Q ζ , θ( σ e )) = 1. Let r ∈ Z and x ∈ A An be such that ζρ r = θ(x) (Proposition 4.1(iii)). Taking P = σ e+r x −1 , we obtain (using Proposition 4.1(ii) for the second equality), We are now in position to prove Theorem 1. It would be conceivable that the subgraph K is quasi-isometric to C parab (B n+1 ), which would imply at once, in view of Theorem 1.3, the hyperbolicity of C parab ( A n ). However, this is not the case, as we will now see. In the sequel, we shall identify C parab (B n+1 ) with CG(D n+2 ), the curve graph of an (n + 2)-times punctured disk, thanks to Theorems 1.3 and 1.2: The following describes the curves which correspond to vertices of K under this identification. (ii) C does not surround the first puncture.
Proof. Suppose that C = f −1 H(P ), where P is a proper irreducible parabolic subgroup of type A of A Bn+1 . There is g ∈ A Bn+1 such that P = B g I , for I ⊂ {2, . . . , n + 1}. We have Notice that C I does not surround the first puncture and that η(g) is 1-pure. It follows that C does not surround the first puncture either. Conversely, suppose that C does not surround the first puncture. Write Q = f(C). By Proposition 3.4, there is a 1-pure braid α and a proper subinterval I of [n + 1] such that Q α = A I is standard. We then also have C α = C I . As α is 1-pure, C I does not surround the first puncture, so that I does not contain 1. Let x ∈ A Bn+1 be such that η( Let K be the subgraph of CG(D n+2 ) induced by the curves which do not surround the first puncture; by Proposition 4.4 and Lemma 4.6, we have graph isomorphisms Our next task is to show that K matches hypothesis (i)-(iii) of Theorem 2.5. (iii) Given an essential curve C in D n+2 , we will see that there always exists some curve c in K which is disjoint from the annulus determined by C. Assume first that C does not surround the first puncture, so that C is a curve in K ; then C itself can be isotoped so that it does not intersect the annulus it determines. Suppose on the contrary that C surrounds the first puncture; if the exterior of C contains at least 2 punctures, we can take c to be any curve in the exterior of C. Otherwise the interior of C contains n + 1 3 punctures and we can choose c to be any curve surrounded by C and not enclosing the first puncture.
Denote by d CG the distance in the curve graph of D n+2 , by d K the distance in the graph K and by d K the distance in the graph K . As the next proposition is not needed in the sequel, its proof is only sketched and we refer the reader to [22] for a precise statement of the results used throughout.
Proposition 4.8. The subgraph K is not quasi-isometrically embedded in C parab (B n+1 ). More precisely, given any M > 0, there exists a pair of parabolic subgroups P, Q of type A of A Bn+1 with the following properties: • P, Q are adjacent to B [1] so that d Bn+1 (P, Q) = 2; We are now ready for the proof of Theorem 1.5. As C parab ( A n ) is isomorphic to K it suffices to prove that K is hyperbolic. We need to check the remaining hypothesis of Theorem 2.5. The next lemma describes all possible witnesses for K ; its proof will achieve the demonstration of Theorem 1.5. Throughout, p 1 denotes the first puncture of D n+2 . Lemma 4.9. Let X be a subsurface of D n+2 . Then X is a witness for K if and only if one of the following holds.
where D is the interior of an essential curve surrounding p 1 and exactly one other puncture.
(ii) X is the interior of an essential curve surrounding p 1 and n other punctures.
(iii) X = X \ D, where X is the interior of an essential curve surrounding p 1 and n other punctures and D is the interior of an essential curve surrounding p 1 and exactly one other puncture.
We will say that X is a witness of type (i), (ii) or (iii). Two witnesses for K are never disjoint.
Proof. The three types of subsurfaces in Lemma 4.9 are depicted in Figure 3. First, we check that all subsurfaces (i)-(iii) are witnesses for K : we see that the only curves which can fail to be witnessed by X must surround the first puncture. Conversely, let X be a witness for K . We shall distinguish two cases. First case. Suppose that ∂D n+2 is a boundary component of X. Assume that X = D n+2 . Therefore, there is at least some essential curve C of D n+2 which is a boundary component of X.
Assume that C is another essential curve of D n+2 which is a boundary component of X. Notice that C and C cannot be nested as X has to be connected. Then at least one of C or C does not surround p 1 and this provides a particular curve of K for which X is not a witness, a contradiction. Therefore X has exactly one essential curve C of D n+2 as a boundary component and C must surround p 1 . Moreover, C must surround exactly 2 punctures, otherwise there would exist a curve c in the interior of C, not surrounding p 1 , and X would fail to be a witness for this curve c. Letting D be the interior of C, we have shown that whenever X has ∂D n+2 as a boundary component, X = D n+2 \ D has to be of type (i).
Second case. Suppose that the boundary ∂D n+2 is not a boundary component of X. As X is connected, X has exactly one outermost boundary component which is an essential curve C of D n+2 . Again, C must surround p 1 , otherwise C is a curve from K which is disjoint from X and X fails to be a witness for K . Moreover, C must surround n + 1 punctures, otherwise the outer component of D n+2 \ C would contain at least two punctures and there would exist a curve c from K entirely contained in D n+2 \ X, contradicting that X is a witness for K . If X has no other boundary component, we have shown that X is of type (ii). Finally, suppose that X has another boundary component. This must be an essential curve C of D n+2 which is nested in C. Let C be another putative boundary component of X nested in C.
Then C and C cannot be nested, as X is connected. Therefore only one of C , C can surround p 1 : one of C , C is a curve from K for which X is not a witness, contradiction. Therefore there is exactly one boundary component C of X nested in C and C must surround p 1 . Moreover, C must surround exactly 2 punctures, for the same reasons as in the first case. Taking X to be the interior of C and D to be the interior of C , we have shown that X = X \ D is of type (iii). Finally, the last claim follows by a direct case-by-case inspection.
5 The graph C parab ( C n ) We start with a description of the proper irreducible standard parabolic subgroups of A Cn . Such a subgroup is determined by a proper subinterval of [n + 1]: if I is a proper subinterval of [n + 1], we denote by C I the proper irreducible standard parabolic subgroup of A Cn generated by Notice that, by Lemma 2.2, the standard generators of A Cn fall in three conjugacy classes, namely τ 1 ( τ n+1 , respectively) is the unique standard generator in its conjugacy class while each τ i , 2 i n, is conjugate to each other.
The following facts can be found in [1,Section 4]. There is a monomorphism The image of λ is the subgroup P of (n + 2)-strands braids in which the first strand ends in the first position and the (n + 2)nd strand ends in the (n + 2)nd position. In other words, a braid y on (n + 2) strands belongs to P if and only if π y (1) = 1 and π y (n + 2) = n + 2, where π y = π(y) is the permutation in S n+2 = W An+1 associated to y.
Although A Cn is not of spherical type, we observe that each proper irreducible parabolic subgroup of A Cn is an irreducible Artin-Tits group of spherical type. This allows to associate to each proper irreducible parabolic subgroup P of A Cn its central element z P , as in Section 2.1. Given I a proper subinterval of [n + 1], the formula for z C I is very similar to the formulae given in Section 3 and we do not write it explicitly. The following is the analogue of Lemma 3.1: As A Cn is not of spherical type, we do not know a priori the analogues of Propositions 2.3 and 2.4. However, these analogues hold, as it will follow from the next Propositions 5.2 and 5.3.
Proposition 5.2. Let I, J be proper subintervals of [n + 1] and let g ∈ A Cn . The following are equivalent.
Proof. Suppose (i). Then Z( C I ) g = Z( C J ); the exponent sum being invariant under conjugation, we get z g Let us show that (ii) implies (iii). Suppose first that I = {1}. Then z g C I = τ g 1 . Recall that = Cn is the exponent sum of words on { τ 1 , . . . , τ n+1 }. Note that (z g C I ) = 1. Now, observe that for any subinterval K of [n + 1], (z C K ) = 1 if and only if #K = 1. From z g C I = z C J we deduce (z C J ) = 1 and #J = 1. Therefore z C J = τ i for some i ∈ [n + 1]. We then have τ g 1 = τ i . By Lemma 2.2, i = 1 and we deduce τ g 1 = τ 1 . It follows that (σ 2 1 ) λ(g) = σ 2 1 ; from [12, Theorem 2.2] we deduce that σ Finally, assume (iii) and let us show (i). We have, as λ(g) ∈ P and using Lemma 5.1(i), and the injectivity of λ ensures that C g I = C J , as required. Proposition 5.3. Let I, J be proper subintervals of [n + 1] and let g ∈ A Cn . The following are equivalent.
(i) C g I and C J are adjacent in C parab ( C n ). Proof. Suppose (i). If C g I ⊂ C J , then z g C I must commute with z C J , which is central in C J ; similarly if C J ⊂ C g I . Otherwise, C g I ∩ C J = {1} and uv = vu for each u ∈ C g I and v ∈ C J ; in particular z g C I and z C J commute. This is (ii). Suppose (ii). By Lemma 5.1(ii) (and [12, Theorem 2.2] for the case when I or J = {1} or {n + 1}), we obtain that z λ(g) A I and z A J commute, which is to say, according to Proposition 2.4, that A λ(g) I and A J are adjacent in C parab (A n+1 ), whence (iii) Suppose (iii) and let us show (i). Suppose first that A λ(g) I ⊂ A J . Then we have, as λ(g) ∈ P and using Lemma 5.1(i), It follows that λ( C g I ) ⊂ λ( C J ) and injectivity of λ shows that C g I ⊂ C J . The proof when A J ⊂ A λ(g) I is similar. Assume finally that A λ(g) I ∩ A J = {1} and that both subgroups commute. Then using Lemma 5.1(i) and the fact that λ(g) ∈ P, By injectivity of λ, it follows that C g , λ( C J ) is a subgroup of A J and A λ(g) I and A J commute mutually, we obtain that λ( C g I ) and λ( C J ) commute mutually and again by injectivity of λ, C g I and C J commute. Therefore we have shown that C g I and C J are adjacent.
, where I is a proper subinterval of [n + 1] and g ∈ A Cn , defines an injective map Λ from the set of proper irreducible parabolic subgroups of A Cn to the set of proper irreducible parabolic subgroups of A An+1 . Moreover, the map Λ induces a graph isomorphism from C parab ( C n ) onto its image.
Proof. By Proposition 5.2, the given formula provides a well-defined map Λ which is injective. By Proposition 5.3, Λ is a graph isomorphism onto its image.
Contrary to what happened with the embedding of C parab (B n ) in C parab (A n ), the map Λ is not surjective, as shows the following example. . In other words, we would have C λ(g) −1 = C I . As C surrounds both the first and the last punctures and λ(g) ∈ P, we deduce that the essential standard curve C I must surround both the first and the last punctures; this is a contradiction.
This example suggests a description of the image of Λ : C parab ( C n ) −→ C parab (A n+1 ). This is stated in the next proposition. Proof. Suppose first that P is in the image of Λ; that is P = A λ(g) I for some proper subinterval I of [n+1] and some g ∈ A Cn . We have f −1 (P ) = f −1 (A I ) λ(g) = C λ(g) I . The curve C I cannot surround both the first and the last punctures as it is standard; assume for instance that it does not surround the first puncture (the other case is similar). Then as λ(g) ∈ P, we see that C λ(g) I = f −1 (P ) does not surround the first puncture either. Conversely, suppose that C = f −1 (P ) does not surround both the first and the last punctures. Suppose for instance that C does not surround the first puncture.We must show that P can be transformed into a standard parabolic subgroup (equivalently, that C can be transformed into a standard curve) by the action of some braid β in P. By Proposition 3.4, we know that there exists a 1-pure braid α and a proper subinterval I of [n + 1] such that P α = A I ; in other words C α = C I is a standard curve surrounding punctures m, . . . , m + k, for some m 2, k 1. Let j 0 = π α (n + 2) ∈ [2, . . . , n + 2], in such a way that α(σ j0 . . . σ n+1 ) ∈ P. The following analysis is analogue to Lemma 3.3. Suppose first that j 0 < m, that is the puncture numbered j 0 is to the left of C I . Then C α(σj 0 ...σn+1) = C we obtain that ξ I ∈ P and that C α(σj 0 ...σn+1)ξ I = C [n−k+2,n+1] is standard and we can choose β = α(σ j0 . . . σ n+1 )ξ I ∈ P. The proof when C does not surround the last puncture is similar.
Lemma 5.7. The image of Λ is 1-dense in C parab (A n+1 ); that is, for every proper irreducible parabolic subgroup Q of A An+1 , there exists a proper irreducible parabolic subgroup P of A Cn such that d An+1 (Q, λ(P )) 1.
Proof. Using the identification from Section 2.2, we can show the lemma in the curve graph of D n+2 . By Proposition 5.6, it suffices to show that given a curve C surrounding both the first and the last punctures, it is possible to find another curve c disjoint from C and such that c does not surround both the first and the last punctures. If the exterior of C contains at least 2 punctures, we take any curve c in the exterior of C. Otherwise, the interior of C contains n + 1 3 punctures and we can choose in the interior of C any curve which does not surround both the first and the last punctures.
From now on, we denote by G the subgraph of CG(D n+2 ) induced by those vertices which are curves not enclosing both the first and the last punctures. We are now able to prove Theorem 1.6.
Proof of Theorem 1.6. (i) The same argument as given in [17,Lemma 5.2] shows the connectivity of C parab ( C n ). (ii) This follows from Proposition 5.4. More specifically, by Proposition 5.6, the graph C parab ( C n ) is isomorphic to the subgraph f(G) of C parab (B n+1 ). (iii) This follows from Lemma 5.7 in the same lines as Theorem 1.4(iii) follows from Lemma 4.5.
To show Theorem 1.7 it will suffice to show that G is hyperbolic. In the next two lemmas, we show that G satisfies the hypothesis of Theorem 2.5, from which we conclude that G is hyperbolic. This achieves the proof of Theorem 1.7.
Lemma 5.8. The graph G is connected. The natural action of the pure braid group P A An+1 on D n+2 induces an action by isometries on G. No annulus in D n+2 is a witness for G.
Proof. In view of Theorem 1.6(i), the proof is identical to the proof of Lemma 4.7.
For the following lemma, we denote by p 1 the first puncture and by p n+2 the last puncture.
Lemma 5.9. Let X be a subsurface of D n+2 . Then X is a witness for G if and only if one of the following holds.
where D is the interior of an essential curve surrounding p 1 and p n+2 and no other puncture.
(ii) X is the interior of an essential curve surrounding p 1 , p n+2 and exactly n−1 other punctures.
(iii) X = X \ D, where X is the interior of an essential curve surrounding p 1 , p n+2 and exactly n − 1 other punctures and D is the interior of an essential curve surrounding p 1 and p n+2 and no other puncture.
We will say that X is a witness of type (i), (ii) or (iii). Two witnesses for G are never disjoint.
Proof. Mutatis mutandis, the proof is the same as the proof of Lemma 4.9.
Remark 5.10. The same argument as in the proof of Proposition 4.8 shows that the embedding of C parab ( C n ) in C parab (A n+1 ) is not quasi-isometric.

Additional results on parabolic subgroups of A An and A Cn
The work presented so far allows us to generalize to the Artin-Tits groups of type A n and C n some results which, to the best of our knowledge, were only known in the framework of Artin-Tits groups of spherical type. If P is a proper irreducible parabolic subgroup of A An or A Cn , P is an irreducible Artin-Tits group of spherical type so that we can define as in Section 2.1 the central element z P of P . The following two results are the analogues for A An of Propositions 2.3 and 2.4.
Proposition 6.1. Let P, Q be two proper irreducible parabolic subgroups of A An ; let g ∈ A An . Then P g = Q if and only if z g P = z Q .
Proof. The direct implication is obvious, considering the centers. Let us prove the converse. Given any proper irreducible parabolic subgroup P of A An , Proposition 4.2 says that θ(P ) is a proper irreducible parabolic of A Bn+1 . Note that θ induces an isomorphism between P and θ(P ); moreover, in the formulae defining θ we see that Bn+1 (θ(x)) = An (x) for every x ∈ A An . Therefore θ(z P ) = z θ(P ) , for every proper irreducible parabolic subgroup P of A An . Now assume z g P = z Q . We have θ(z g P ) = θ(z P ) θ(g) = z θ(g) θ(P ) = z θ(P ) θ(g) = z θ(P g ) , while θ(z Q ) = z θ(Q) . By Proposition 2.3, it follows that θ(P g ) = θ(Q). Finally, as θ is injective, P g = Q as desired.
Proposition 6.2. Let P, Q be two distinct proper irreducible parabolic subgroups of A An . Then P and Q are adjacent if and only if z P and z Q commute.
Proof. The argument given in the (i)⇒ (ii) part of the proof of Proposition 5.3 works for the direct implication. Conversely, suppose that z P and z Q commute. Then θ(z P ) = z θ(P ) and θ(z Q ) = z θ(Q) commute, whence by Proposition 2.4, θ(P ) and θ(Q) are adjacent in C parab (B n+1 ). By Proposition 4.4, this implies that P and Q are adjacent in A An .
Suppose first that ζ = θ(ς) for some ς ∈ A An . Then we have θ(P ς ) = θ(P ) ζ = θ( A I ) and θ(Q ς ) = θ(Q) ζ = θ( A J ). We deduce from the injectivity of θ, P ς = A I and Q ς = A J , showing our claim. If on the contrary ζ / ∈ Imθ, in view of Proposition 4.1(iii), we can write ζ = ζ 0 ρ r for some r ∈ Z and ζ 0 = θ(ς) for some ς ∈ A An . In view of Proposition 4.1(ii), we then have where I = {i − r | i ∈ I }, J = {j − r | j ∈ J } and the indices are taken modulo n + 1. We obtain that P ς = A I and Q ς = A J are standard, as needed.
Proposition 6.4. Let P, Q be adjacent proper irreducible parabolic subgroups of A Cn . Then there exists ς ∈ A Cn so that P ς and Q ς are standard.
Proof. By Proposition 5.3, Λ(P ) and Λ(Q) are adjacent in A An+1 . By Proposition 5.6, C 1 = f −1 (Λ(P )) and C 2 = f −1 (Λ(Q)) are disjoint essential curves in D n+2 which do not surround both the first and the last punctures. We shall prove that there exists ζ 0 in P = λ(A Cn ) and I, J proper subintervals of [n + 1] such that C ζ0 1 = C I and C 2 ζ0 = C J are standard curves; in other words Λ(P ) ζ0 = A I and Λ(Q) ζ0 = A J are standard parabolic subgroups of A An+1 . In this way, we will obtain ζ 0 = λ(ς), for ς ∈ A Cn , Λ(P ς ) = Λ(P ) ζ0 = Λ( C I ) and Λ(Q ς ) = Λ(Q) ζ0 = Λ( C J ), whence by injectivity of Λ (Proposition 5.4), P ς = C I and Q ς = C J are both standard as wanted. By [14,Proposition 4.4], there exist ζ ∈ A An+1 and I, J proper subintervals of [n + 1] such that C ζ 1 = C I and C 2 ζ = C J are standard curves. Let i 0 = π ζ (1) and j 0 = π ζ (n + 2). Notice that none of the curves C I , C J can surround the two punctures numbered i 0 and j 0 . If i 0 = 1 and j 0 = n + 2, we have nothing to do and we can take ζ 0 = ζ. If i 0 = 1 and j 0 < n + 2 (or j 0 = n + 2 and i 0 > 1, respectively) then define ζ = ζ(σ j0 . . . σ n+1 ) (or ζ = ζ(σ i0−1 . . . σ 1 ), respectively). In any case, ζ ∈ P. We follow the proof of Proposition 5.6 (Proposition 3.4, respectively). If C ζ e is not standard (e = 1, 2), which can happen only if the first puncture (the last puncture, respectively) is not enclosed by C ζ e , we can find a braid ξ ∈ P, with the first strand (the last strand, respectively) straight, so that C ζ ξ e is standard and we can choose ζ 0 = ζ ξ. We can therefore assume that i 0 > 1 and j 0 < n + 2. We then define Note that ζ ∈ P. The reader may check that, as none of C I and C J surrounds both punctures i 0 and j 0 , even if C ζ 1 or C ζ 2 is not standard we can always find ξ ∈ P so that C ζ ξ e is standard, for e = 1, 2, and ζ 0 = ζ ξ does the job as required.

Other graphs
In this section we show somme connections between the previous reIn [7] were described three different (infinite) generating sets of an Artin-Tits group A (of spherical type) and the corresponding Cayley graph of A was conjectured to be hyperbolic. A generating set X of a group G such that the associated Cayley graph Cay(G, X) is hyperbolic is called an hyperbolic structure for G. Suppose that A is of spherical type; these three sets are the following: • X N P (A) is the union of the normalizers of the proper irreducible standard parabolic subgroups of A.
• X P (A) is the union of the union of the proper irreducible standard parabolic subgroup of A and the cyclic subgroup generated by the square of the element ∆ (recall that ∆ is the lift of the longest element of the corresponding Coxeter group).
• X abs (A) is the set of absorbable elements as described in [6].
The definition of X abs (A) rests on the Garside structure of A and can be defined only for A of spherical type; this is the only one of the three sets which is known to be a hyperbolic structure for all A of spherical type [6,Theorem 1]. The definition of X P (A) and X N P (A) can be easily extended to any Artin-Tits group (simply dropping the powers of ∆ in the definition of X P ). When A is of type A n , both Cayley graphs Cay(A An , X N P (A An )) and Cay(A An , X P (A An )) are hyperbolic. Indeed, Cay(A An , X N P (A An )) is quasi-isometric to the curve graph of the (n + 1)times punctured disk D n+1 [7, Proposition 3.2] while Cay(A An , X P (A An )) is quasi-isometric to A ∂ (D n+1 ), the graph of arcs in D n+1 both of whose endpoints lie in the boundary ∂D n+1 [7, Proposition 3.4]. More generally, for each A of spherical type (except dihedral), Cay(A, X N P (A)) is quasi-isometric to the graph of irreducible parabolic subgroups of A [7,Proposition 4.4]. From the work done so far, we can deduce that this still holds true for A of type A n or C n : Proposition 6.5. Assume that Γ is either A n or C n (n 2). Then C parab (Γ) is quasi-isometric to Cay(A Γ , X N P (A Γ )).
Proof. This will be a consequence of [7, Lemma 2.5]. By Theorems 1.4(i) and 1.6(i), C parab (Γ) is connected. The finite set of standard parabolic subgroups of A Γ is a set of representatives of the orbits of vertices under the action of A Γ . By Propositions 6.3 and 6.4, the finite set of edges bounded by two standard parabolic subgroups is a set of representatives of the orbits of edges under the action of A Γ . We conclude applying [7, Lemma 2.5].
Our results thus give a partial answer to [7, Conjecture 4.7]: Proposition 6.6. Suppose that Γ is either B n , n 3, A n or C n , n 2. Then X N P (A Γ ) is a hyperbolic structure for A Γ .
We now focus on the graph Cay(A Bn , X P (A Bn )). Recall that the monomorphism η : A Bn −→ A An expresses A Bn as the subgroup P 1 of 1-pure braids on (n+1) strands and that P 1 has index (n+1) in A An . For 1 i n, recall that a i = σ i . . . σ 1 and a 0 = Id; for each y ∈ A An , there is a unique i such that ya i ∈ P 1 . We first prove a lemma (see Lemma 3.3): Lemma 6.7. Let I be a proper subinterval of [n]; let m = min(I) and k = #I. Let 0 i 0 , j 0 n. Let I = {i + 1 | i ∈ I} (whenever max(I) < n). Let g ∈ A I . Suppose that z = a −1 i0 ga j0 is 1-pure.
Here, ξ I ∈ P 1 is the braid defined in Section 3 which satisfies A am−1ξ I I = A [1,k] and sh denotes the shift homomorphism σ i → σ i+1 .