Extra-large type Artin groups are hierarchically hyperbolic

We show that Artin groups of extra-large type, and more generally Artin groups of large and hyperbolic type, are hierarchically hyperbolic. This implies in particular that these groups have finite asymptotic dimension and uniform exponential growth. We prove these results by using a combinatorial approach to hierarchical hyperbolicity, via the action of these groups on a new complex that is quasi-isometric both to the coned-off Deligne complex introduced by Martin-Przytycki and to a generalisation due to Morris-Wright of the graph of irreducible parabolic subgroups of finite type introduced by Cumplido-Gebhardt-Gonz\'alez-Meneses-Wiest.

Introduction Hyperbolic features of Artin groups. The geometry of Artin groups has seen an explosion of results in the last decade. While Artin groups remain in general much less understood than their Coxeter relatives, a driving theme behind current research has been to show that these groups are as well-behaved as Coxeter groups, and this has indeed been verified for several classes of Artin groups. On the geometric side, a popular theme has been to understand the "hyperbolic features" of groups, and Artin groups are conjectured to display such hyperbolic features. This rather vague notion comes in many flavours, a first one being the notion of acylindrically hyperbolic group. Loosely speaking, such groups can be described as having "hyperbolic directions" (see [Osi16] for the precise definition and its many consequences). It is conjectured that for an irreducible Artin group A Γ , its central quotient A Γ {ZpA Γ q is acylindrically hyperbolic. This question has been answered positively for most standard classes of Artin groups, such as right-angled Artin groups [CS11], Artin groups of finite type [CW:], Artin groups of Euclidean type [Cal20], Artin groups whose underlying presentation graph is not a join [RMW19], and two-dimensional Artin groups [Vas20].
While acylindrical hyperbolicity guarantees the existence of some hyperbolic directions, it does not provide much control on the overall geometry of the group. For instance, the free product A˚B, where A, B are the worst infinite groups you can think of, is acylindrically hyperbolic.
A notion of non-positive curvature that provides a much stronger control over the coarse geometry of the group is the notion of hierarchically hyperbolic group (or HHG). This notion was introduced by Behrstock-Hagen-Sisto [BHS17b,BHS19] and inspired by work of Masur-Minsky [MM99,MM00] as a framework that unifies and generalises the geometry of mapping class groups of hyperbolic surfaces and that of (all known) cocompactly cubulated groups [HS20]. The idea of hierarchical hyperbolicity is to describe the coarse geometry of a group/space using a "coordinate system" where the coordinates take values in various hyperbolic spaces.
Since this notion gives much better control than acylindrical hyperbolicity, it implies many results expected of non-positively curved groups that are not true for more general notions of non-positive curvature; for example, it gives quadratic isoperimetric inequality [Bow13,BHS19], solubility of the word and conjugacy problem [BHS19,HHP20], the Tits alternative [DHS17,DHS20], finite asymptotic dimension [BHS17a], semi-hyperbolicity [HHP20,DMS20], etc. In particular, since both braid groups and right-angled Artin groups belong to this family [BHS17b,BHS19], the following is a natural question that was for instance raised by Calvez-Wiest [CW19]: Question. Which Artin groups are hierarchically hyperbolic?
Building new examples of hierarchically hyperbolic groups is nontrivial; there are combination theorems [BHS19,BR20], and theorems about persistence of hierarchical hyperbolicity under various quotients [BHS17a,BHMS20], but the examples coming from "nature" are dominated by compact special groups, mapping class groups, and fundamental groups of certain 3-manifolds. The preceding question has only been previously answered positively for right-angled Artin groups and braid groups.
In this article, we add a large class of Artin groups to the "naturally occurring" hierarchically hyperbolic groups: Theorem A. Artin groups of large and hyperbolic type are hierarchically hyperbolic.
Note that Artin groups of large and hyperbolic type contain in particular all Artin groups of extra-large type (see Section 2.1 for precise definitions of these classes). We refer to Theorem 6.15 for a description of the HH structure.
The following results are new for Artin groups of large and hyperbolic type, and follow from the cited results about hierarchically hyperbolic groups: Corollary B. Let A Γ be an Artin group of large and hyperbolic type, with Γ connected and not a single vertex. Then: (1) A Γ has finite asymptotic dimension [BHS17a, Thm. A],  [Cal20], and Martin-Przytycki in the two-dimensional case [MP21].
A candidate for such a curve complex for Artin groups of finite type has been proposed recently by Cumplido-Gebhardt-Gonzales-Meneses-Wiest [CGGMW19], and generalised to all Artin groups by Morris-Wright [MW21] who studied this complex extensively for Artin groups of type FC. This is the notion of graph of irreducible proper parabolic subgroups of finite type of an Artin group.
It is conjectured that this complex is an infinite-diameter hyperbolic space (except in degenerate cases) for Artin groups of finite type [CGGMW19, Conjecture 2.4] and type FC [MW21,Conjecture 5.6]. We prove the analogous result for Artin groups of large and hyperbolic type as a corollary of Proposition 3.16 and Theorem A: Corollary C. Let A Γ an Artin group of large and hyperbolic type on at least three generators. Then A Γ admits an HHG structure whose maximal hyperbolic space is the graph of irreducible proper parabolic subgroups of finite type.
Note that the action of A Γ on this graph is acylindrical. This can be seen by combining the above corollary with [BHS17b, Theorem K], or by combining Proposition 3.16 with [MP21,Theorem A]. This corollary gives hope that this complex not only is hyperbolic for larger classes of Artin groups but also is the maximal hyperbolic space in an HHG structure.
Combinatorial approach to HHS, and strategy of proof. We will not strictly need the full definition of hierarchical hyperbolicity but, roughly, a hierarchically hyperbolic space (HHS) is a space X that comes with a family of hyperbolic spaces and projections from X to the various hyperbolic spaces satisfying various conditions [BHS19, Definition 1.1]. Additionally, a group is hierarchically hyperbolic if it acts geometrically on an HHS preserving the HHS structure, in a way expressed most simply in [PS20,p.4].
We will not need the full definition because we will verify a more combinatorial criterion from [BHMS20] for a group G to be hierarchically hyperbolic. We state this as Theorem 1.4, but for this discussion it suffices to know that criterion involves a hyperbolic simplicial complex X on which G acts, and maximal simplices of X are in bijection with the vertices of a graph quasi-isometric to G. The links of simplices in X give the hyperbolic spaces in the HHS structure, so the fine geometry of X is relevant for this purpose.
We now explain how to come up with a candidate X for a large-type Artin group A Γ of hyperbolic type. The strategy makes sense more generally and potentially could be applied to other classes of groups as well. First, we need a little more discussion on hierarchical hyperbolicity. An HHS contains a family of so-called standard product regions, which are HHS themselves and (coarsely) split as a product of HHS. Moreover, the standard product regions are "arranged hyperbolically" meaning that coning them off yields a hyperbolic space. (This is the "hierarchical structure" that gives the name.) In the context of the combinatorial criterion, said coned-off space is quasi-isometric to the hyperbolic simplicial complex X.
In view of this, a natural candidate for an action of a given group on a simplicial complex to which Theorem 1.4 can be applied is constructed as follows.
Step 1: Consider the family of subgroups that are maximal virtual products (these should give the standard product regions if the group is hierarchically hyperbolic). For our Artin groups, the tools to identify these subgroups come from the acylindricity of the action of A Γ on its coned-off Deligne complex and CAT(´1) geometry [MP21], see Section 2.4.
Step 2: Isolate the subgroups arising as minimal infinite intersections between such virtual product subgroups. Coarse intersections of standard product regions in an HHS are again coarse products of HHS, which are "smaller" than the original product subgroups. This should identify the simplest sub-HHS, if the group is to be hierarchically hyperbolic. For our Artin groups, here we have the cyclic subgroups conjugate to either the subgroup generated by a standard generators or centres of dihedral parabolic subgroups. See Section 3.1 for more discussion.
Step 3: To construct a hyperbolic space that the group acts on, it is natural to consider the graph encoding the intersections of candidates for the standard product regions. This graph is quasi-isometric to the commutation graph of the minimal subgroups above, see Section 3.1. This graph has a vertex for each such minimal subgroup and we put an edge when two such subgroups commute. In the case of right-angled Artin groups, we recover the extension graph, which is indeed hyperbolic [KK14]. In our case, there is a natural map between the commutation graph and the coned-off Deligne complex. This map is a quasi-isometry with very nice local properties, and this is crucial for our arguments in Section 5.
Step 4: The commutation graph as above is expected to have the right coarse geometry, but not the right local geometry, because stabilisers of maximal simplices are usually infinite. We remedy this by a blow-up construction, where we replace each vertex of the commutation graph by a quasi-isometric copy of the corresponding subgroup, preserving the A Γ -action. In our situation, the vertices of the commutation graph need to be blown up to quasi-lines. This creates a somewhat delicate situation where, for a certain cyclic subgroup H, we need to construct an action of N pHq on a quasiline with certain properties. Remarkably, this difficulty is circumvented using quasimorphisms (which are related to actions on quasilines by [ABO19]), in a similar way as in [HRSS21].
Once we have our hyperbolic complex X, we can build our quasi-isometry model W of A Γ . As mentioned above, the vertex set of W is the set of maximal simplices of X. The adjacency relation is defined in such a way as to guarantee that orbit maps A Γ Ñ W are quasi-isometries (Section 5.1). The remaining work is to verify the technical conditions of Theorem 1.4, and here we once again rely on the CAT(´1) geometry of the coned-off Deligne complex.
In the end, we get an HHS structure on A Γ where the maximal hyperbolic space -the HHS analogue of the curve graph in the mapping class group setting -is quasi-isometric to X and hence to the commutation graph.
As a final remark, our results do not cover the more general case of 2-dimensional Artin groups of hyperbolic type (that is, the case where edges with label 2 are allowed). In this case, the combinatorics of the commutation graph and associated spaces are more complex, with several statements in Section 3 and beyond needing more nuance since more cases have to be considered. However, we still believe that with more sophisticated arguments one can deal with the more general case using the same combinatorial HHS approach.
Outline of the paper. Section 1 contains the definitions and results from [BHMS20] that we will need. Section 2 contains background on Artin groups, and in particular the subclass of Artin groups considered in this paper, along with the coned-off Deligne complex. Section 3 is about the commutation graph, and also contains the discussion relating the commutation graph and the graph of irreducible parabolics. Section 4 contains the construction of the simplicial complex X -the blow-up of the commutation graph -along with some purely combinatorial facts about X, and the relationship between maximal simplices in X and coarse points in A Γ . In Section 5, we define the graph of maximal simplices of X, prove that it is quasi-isometric to A Γ , and study its combinatorial structure. Finally, in Section 6, we verify the remaining hypotheses of Theorem 1.4. The final subsection of Section 6 assembles the pieces into a proof of Theorem A and Corollary B.

A combinatorial criterion for hierarchical hyperbolicity
In this section, we recall the main combinatorial criterion introduced in [BHMS20] to show that a group is hierarchically hyperbolic by means of a map to a hyperbolic simplicial complex. We will not require these notions until Section 4, but we introduce them now to motivate the constructions in earlier sections. We refer the reader to [BHMS20, Section 1.2,1.5] for a more informal discussion of the various hypotheses in the definition of a combinatorial HHS.
Definition 1.1 (X-graph, augmented complex). Let X be a flag simplicial complex. An X-graph is a graph W whose vertex set is the set of maximal simplices of X. We say that two maximal simplices ∆, ∆ 1 of X are W -adjacent if the corresponding vertices of W are adjacent in W .
We denote by X`W the complex obtained from the 1-skeleton of X by adding edges between vertices that belong to W -adjacent maximal simplices. For a subcomplex X 0 of X, we denote by X`W 0 the full subcomplex of X`W induced by X 0 . We refer to these various objects X`W , X`W 0 , etc. as being augmented complexes. Definition 1.2 (Link, star, saturation of a simplex). Let X be a flag simplicial complex, and let ∆ be a simplex of X.
The star of ∆, denoted St X p∆q, is the union of all the simplices of X containing ∆. The link of ∆, denoted Lk X p∆q, is the full subcomplex of St X p∆q induced by St X p∆q´∆.
The saturation of ∆ is Sat X p∆q " ď tΣ:Lk X p∆q"Lk X pΣqu Σ p0q .
The following is Definition 1.8 in [BHMS20]: A combinatorial HHS is a pair pX, W q, where X is a simplicial complex and W is an X-graph, such that all of the following hold for some δ ă 8, n P N: (I) If ∆ 0 , . . . , ∆ m are simplices of X and Lk X p∆ i q Ĺ Lk X p∆ i`1 q for 0 ď i ď m´1, then m ď n. This condition will be called finite complexity. (II) Let ∆ be a non-maximal simplex of X. Let Cp∆q " Lk X p∆q`W and let Y ∆ " pX p0qŚ atp∆qq`W . Then Cp∆q is δ-hyperbolic and the inclusion Cp∆q ãÑ Y ∆ is a pδ, δq-quasiisometric embedding. We will call this condition hyperbolic links. (III) Let ∆ be a non-maximal simplex of X and let v, w P Lk X p∆q be distinct non-adjacent vertices. Suppose that v, w are contained in W -adjacent maximal simplices of X.
Then there exist maximal simplices Σ v , Σ w of Lk X p∆q, respectively containing v, w, such that Σ v ‹ ∆ and Σ w ‹ ∆ are W -adjacent. We will call this condition fullness of links.
(IV) Let Σ, ∆ be non-maximal simplices of X such that there exists a non-maximal simplex Γ with Lk X pΓq Ď Lk X pΣq X Lk X p∆q and diampCpΓqq ą δ. Then there exists a nonmaximal simplex Π Ă Lk X pΣq such that Lk X pΣ ‹ Πq Ď Lk X p∆q and all Γ as above satisfy Lk X pΓq Ă Lk X pΣ ‹ Πq. We call this condition the intersection condition.
The following criterion is immediate from [BHMS20, Theorem 1.18, Remark 1.19] and the fact that hierarchical hyperbolicity is a quasi-isometry invariant property [BHS19, Proposition 1.10]: Theorem 1.4. Let pX, W q be a combinatorial HHS. Then any quasigeodesic space quasiisometric to W is a hierarchically hyperbolic space. Moreover, suppose that the group G acts by simplicial automorphisms on X, and that the resulting G-action on the set of maximal simplices of X extends to a proper cobounded action of G on W . Suppose moreover that X contains finitely many G-orbits of subcomplexes of the form Lk X p∆q, for ∆ a simplex. Then G is a hierarchically hyperbolic group.
(In the statement, the notion of properness used is sometimes called metric properness, and what we mean is that given a ball in X there are only finitely many elements of G that do not map the ball to a disjoint ball.) Note that in addition to the properties of combinatorial HHS, Theorem 1.4 requires another condition, namely that there are finitely many orbits of links of simplices. For readers familiar with HHS terminology, we mention that this is so that the action of A Γ on the index set of the eventual HHG structure is cofinite, as the elements of the index set correspond to the links of the non-maximal simplices.
We will apply the above theorem to an Artin group of large and hyperbolic type A Γ by explicitly constructing pX, W q with W quasi-isometric to A Γ , and then verifying each of the properties from Definition 1.3.

Background on Artin groups and Deligne complexes
2.1. Artin groups. A presentation graph is a finite simplicial graph Γ such that every edge between vertices a, b P V pΓq is labelled by an integer m ab ě 2. The Artin group associated to Γ is the group A Γ given by the following presentation: Given an Artin group A Γ , the associated Coxeter group W Γ is obtained by further requiring that each generator a P V pΓq satisfies the relation a 2 " 1. An Artin group is said to be of hyperbolic type if the associated Coxeter group is hyperbolic, and of finite type if the associated Coxeter group is finite.
For a (possibly empty) full subgraph Γ 1 Ă Γ, the subgroup of A Γ generated by the vertices of Γ 1 is called a standard parabolic subgroup. Such a subgroup is isomorphic to the Artin group A Γ 1 by a result of Van der Lek [vdL83], and moreover we have A Γ 1 X A Γ 2 " A Γ 1 XΓ 2 for full subgraphs Γ 1 , Γ 2 of Γ. Conjugates of standard parabolic subgroups are called parabolic subgroups.
2.2. Structure of dihedral Artin groups. Since dihedral Artin groups appear as stabilisers of vertices of dihedral type in the modified Deligne complex and its cone-off, we mention some structural results that will be needed in this article.
A dihedral Artin group on two standard generators a, b will be denoted A ab for simplicity, even though the group depends on the coefficient m ab . Dihedral Artin groups come into two types: If m ab " 2, the group is a copy of Z 2 . The rest of this subsection focuses on the structure of dihedral Artin groups with m ab ě 3. We start by recalling the following definition: Definition 2.1. For a dihedral Artin group with m ab ě 3, the Garside element ∆ ab P A ab is defined as follows: BS72]). The centre of a dihedral Artin group A ab with m ab ě 3 is infinite cyclic and generated by the element z ab :" ∆ ab if m ab is even, and z ab :" ∆ 2 ab if m ab is odd. Lemma 2.3. Let A ab be a dihedral Artin group with m ab ě 3. The central quotient A ab {xz ab y is virtually a finitely generated non-trivial free group. In particular, A ab contains a finiteindex subgroup that splits as a direct product of the form xz ab yˆK, where K is a finitely generated free subgroup of A ab .
This virtual splitting is well-known to experts, see for instance [Cri05]. We give here a geometric proof of this result that uses objects that will be needed in Section 5. It follows from the existence and uniqueness of Garside normal forms (see for instance [JF06]) that elements of the quotient A ab {x∆ ab y are in bijection with left-weighted elements of the free monoid M ‚ on M , where x∆ ab y acts on A ab by right multiplication.
Definition 2.5. We denote by T ab the full subgraph of the Cayley graph CayleypM ‚ , M q spanned by left-weighted elements. The action of A ab on A ab {x∆ ab y by left multiplication induces an action of A ab on T ab (seen as an unlabelled graph).
Since M ‚ is a free monoid, the graph T ab is a quasi-tree, as already explained in [Bes99]. More precisely, the flag completion of T ab has a structure of tree of simplices of dimension m ab´1 glued along vertices (see Figure 1), where the simplices are either in the A ab -orbit of the simplex spanned by e, a, ab, . . . , aba¨¨loomoon  Figure 1. A portion of the quasi-tree T ab for m ab " 3. In red, the full subgraph spanned by the xay-orbit of e. In blue, the full subgraph spanned by the xby-orbit of e.
Proof of Lemma 2.3. The group A ab acts by left multiplication on T ab . Since the element z ab is central and is a power of ∆ ab , it follows that xz ab y acts trivially on that graph, hence the quotient A ab {xz ab y acts by left multiplication on T ab . The action is cocompact and proper, thus A ab {xz ab y is virtually free, and hence A ab is virtually a direct product of the form xz ab yˆK, where K is a finitely-generated free group. Note that a and b define elements of infinite order in A ab {xz ab y since their orbits in T ab span embedded lines (see Figure 1).
We recall that the syllabic length of an element g P A ab is the smallest non-negative integer n such that g can be written as a product of the form g " x k 1 1¨¨¨x kn n with k i P Z and x i P ta, bu for all 1 ď i ď n.
Lemma 2.6 ([Vas20, Proposition 4.6]). Let g be an element of A ab that can be written only with positive letters and that has syllabic length greater than 1. If m ab ě 3, then the syllabic length of g n goes to infinity as n goes to infinity.
Corollary 2.7. In a dihedral Artin group A ab with m ab ě 3, no non-trivial power of z ab is equal to a power of a standard generator.
Proof. By Lemma 2.6, the syllabic length of z n ab explodes as n grows, while the syllabic length of a n and b n is always equal to one.
Lemma 2.8. In a dihedral Artin group A ab with m ab ě 3, two distinct conjugates of standard generators never generate a subgroup isomorphic to Z 2 .
Proof. Let us consider a dihedral Artin group A ab with m ab ě 3. Up to conjugation, we can assume that the element a and a conjugate x :" gcg´1, with c P ta, bu, commute. Since the centraliser of a in A ab is xa, z ab y by [Cri05,Lemma 7], it follows that there exist integers , k such that x " a k z ab . We claim that necessarily " 0. Indeed, if that were not the case, then the syllabic length of the powers of z ab would go to infinity by Lemma 2.6, and since a k and z ab commute, so would the syllabic length of the powers of x (since the powers of a all have syllabic length 1). But since x is conjugate to a power of a generator, its powers have a uniformly bounded syllabic length, a contradiction. We thus have x " a k . By using the homomorphism A ab Ñ Z sending both generators to 1, we get that k " 1, hence x " a.
By taking the contrapositive, distinct conjugates of standard generators of A ab do not commute.
2.3. The modified Deligne complex. Parabolic subgroups of finite type of an Artin group are used to define a simplicial complex as follows: Definition 2.9 (Modified Deligne complex [CD95]). The cosets gA Γ 1 of standard parabolic subgroups of finite type of A Γ form a partially ordered set, for the partial order given by The modified Deligne complex (or Charney-Davis complex) D Γ of an Artin group A Γ is the geometric realisation of this poset. That is, vertices of D Γ correspond to cosets gA Γ 1 of standard parabolic subgroups of finite type, and for every chain of the form we add an n-simplex spanned by the vertices gA Γ 0 , gA Γ 1 , . . . , gA Γn . The group A Γ acts on its modified Deligne complex by left multiplication on left cosets.
Convention 2.10. From now on, we fix a large-type Artin group A Γ of hyperbolic type.
Since A Γ is assumed to be of large-type and of hyperbolic type here, its only parabolic subgroups of finite type are its parabolic subgroups on at most two generators. In particular, the Deligne complex of A Γ is a 2-dimensional simplicial complex.
It was shown in [CD95] that for an Artin group of large and hyperbolic type, there exists an A Γ -invariant piecewise hyperbolic metric that turns D Γ into a CAT(´1) space. From now on, we will assume that D Γ is endowed with such a metric.
Notation 2.11. For simplicity, we will often omit the 'modified' from the name and call D Γ the Deligne complex.
The vertex of D Γ corresponding to the standard dihedral parabolic subgroup A ab will be denoted v ab . Vertices of D Γ corresponding to cosets of dihedral parabolic subgroups are said to be of dihedral type.
Remark 2.12. We describe the stabilisers of vertices of the Deligne complex. Since a vertex of D Γ is a left coset of the form gA Γ 1 (with Γ 1 Ă Γ), its stabiliser is the conjugate gA Γ 1 g´1. In particular, we get the following description, for each type of vertices of D Γ : ‚ A vertex that corresponds to a left coset of the trivial subgroup has trivial stabiliser. ‚ A vertex that corresponds to a left coset of the form gxay, with a P V pΓq, has a stabiliser that is infinite cyclic. ‚ A vertex of dihedral type has a stabiliser that is isomorphic to a dihedral Artin group.
2.4. Standard trees and the coned-off Deligne complex. The structure of fixed-point sets of parabolic subgroups of A Γ play a crucial role. We start by a useful result: Lemma 2.13. The fixed-point set in D Γ of a parabolic subgroup on two generators of A Γ is a single vertex. In particular, such parabolic subgroups are self-normalising.
Proof. If a parabolic subgroup gA ab g´1 were to fix two distinct points of D Γ , then it would fix the unique CAT(0) geodesic of D Γ between them. Since stabilisers of edges and triangles of D are either trivial or Z, gA ab g´1 would embed in an infinite cyclic group, which is impossible as A ab is either Z 2 or contains a copy of Z 2 by Lemma 2.3.
Since an element of the normaliser of gA ab g´1 stabilises FixpgA ab g´1q, hence fixes the vertex gv ab , it follows that gA ab g´1 is self-normalising.
The fixed-point sets of infinite cyclic parabolic subgroup are much more interesting: Definition 2.14 (Standard trees [MP21, Definition 4.1]). For an element g P A Γ that is a conjugate of a standard generator, the fixed-point set Fixpgq is a convex subtree of D Γ that is contained in the 1-skeleton of D Γ .
Such subtrees are called standard trees of D Γ . Note in particular that all edges of a standard tree have the same infinite cyclic stabiliser.
We list here a few immediate results: Lemma 2.15. Two edges of D Γ have stabilisers that either intersect trivially or are equal. Moreover, if two edges of D Γ have the same non-trivial stabiliser, then they belong to the same standard tree.
Proof. Since a non-trivial element of A Γ stabilising two points fixes pointwise the unique CAT(0) geodesic between them, this result is a direct consequence of [MP21, Lemma 4.3].
Corollary 2.16. For every standard generator a and non-zero integer k P Z´t0u, the trees Fixpaq and Fixpa k q coincide.
Proof. The inclusion Fixpaq Ď Fixpa k q is clear, so let us show the other inclusion. Consider any point x in Fixpa k q, and any edge e of the tree Fixpaq. If x lies in e, we are done, otherwise we can consider a minimal length geodesic from x to e, which is a k -invariant. Since triangles of D Γ have trivial stabilisers, this geodesic is contained in the 1-skeleton and it intersects an edge e 1 in a non-trivial subpath containing x (if x is not a vertex we cannot say that the geodesic contains the edge). We have that e 1 is also a k -invariant, and since k ‰ 0 we have that the stabilisers of e and e 1 intersect non-trivially, and therefore by Lemma 2.15 e 1 , whence x, belongs to Fixpaq.
Corollary 2.17. Two points of D Γ have stabilisers that intersect non-trivially if and only if they are contained in a common standard tree.
Proof. If two points x, y of D Γ have a nontrivial common stabiliser H, then H fixes pointwise the unique CAT(0) geodesic γ between them. The geodesic γ cannot pass through the interior of a triangle because H is non-trivial, so γ is contained in the 1-skeleton and is contained in a minimal path of edges of the form e 1 , . . . , e n . So H fixes edges e 1 , . . . , e n , and by Lemma 2.15, it follows that e 1 , . . . , e n , hence x and y, are contained in the same standard tree. (If n " 1, note that an edge with non-trivial stabiliser belongs to a standard tree.) Corollary 2.18. Two distinct standard trees intersect in at most one vertex.
Proof. This follows directly from Lemma 2.15 and the convexity of standard trees. Notation 2.20. For a standard generator a P Γ, the standard tree Fixpaq (i.e. the standard tree containing the vertex xay of D Γ ) will be denoted T a , and the apex of the cone p T a over that tree will be denoted v a , see Figure 2. With this notation, the vertex gv a corresponds to the apex of the cone over the standard tree Fixpgag´1q containing gxay. The apex of a cone over a standard tree will be called a vertex of tree type. Remark 2.21. By work of Paris [Par97, Corollary 4.2], two standard generators are conjugated if and only if there is a path in the presentation graph Γ consisting of edges with odd labels connecting the corresponding vertices. Since distinct generators may be conjugated, it may happen that we have an equality of the form gv a " hv b for distinct standard generators a, b. The following result shows that this is the only case where such an equality happens.
Lemma 2.22. Let T be a standard tree of D Γ , let a, b be two standard generators, and let g, h P A Γ . If the two vertices gxay and hxby of D Γ are contained in T , then a and b are connected in Γ by a path with odd labels. (In particular, a and b are conjugated.) Proof. Since T is a connected tree, we consider a geodesic path e 1 , . . . , e k of T from gxay to hxby. Each edge e i joins a coset of a cyclic standard parabolic subgroup and a coset of parabolic subgroup of dihedral type. For each i, let a i be the unique standard generator such that e i contains a vertex that is a coset of xa i y. It is enough to show that for every 1 ď i ă k, a i and a i`1 are either equal or connected by an edge of Γ with odd label. Consider the vertex v of D Γ where e i and e i`1 meet. If v corresponds to a coset of cyclic standard parabolic subgroup, then a i " a i`1 . If v is a vertex of dihedral type, then it follows from [MP21,Lemma 4.3] that either a i " a i`1 , or a i and a i`1 are adjacent in Γ (since e i and e i`1 meet along a vertex that is a coset of A a i ,a i`1 , which must be an Artin group of finite type by construction of D Γ ) and the label of that edge is odd. This concludes the proof.
Remark 2.23. It was shown in [MP21, Proposition 4.8] that there exists an A Γ -invariant piecewise hyperbolic metric that turns p D Γ into a CAT(´1) space. From now on, we will assume that p D Γ is endowed such a metric from [MP21]. It should be noted that this metric depends on a constant ε ą 0 that can be chosen arbitrary small, see [MP21,Definition 4.7]. This constant is such that for an edge e of D Γ contained in a standard tree, the triangle of D Γ over e has angles at least π{2´ε at the two vertices of e. In this article, the choice of ε will be mostly irrelevant. We will only need to consider this constant in Lemma 2.25 below, where ε needs to be smaller than a certain constant depending only on the group A Γ .
We mention here a slight generalisation of the CAT(´1)-ness of p D Γ , which will be used in Section 5.
Lemma 2.24. Let Z be the full subcomplex of p D Γ whose vertex set is obtained from p D Γ by removing some vertices of tree type. Then Z is also CAT(´1) for the induced metric.
Proof. The complex Z can be thought of as being obtained from the simply connected complex D Γ by coning-off only certain (contractible) standard trees of D Γ , so Z is also simply connected. Moreover, for a point x P Z, the link Lk Z pxq is a subgraph of Lk , links of points are graphs with a systole of at least 2π, so the same is true for the subgraph Lk Z pxq. Thus, Z is locally CAT(´1). Since Z is simply-connected and locally CAT(´1), it is CAT(´1).
The standard trees of D Γ are convex in D Γ but become bounded in p D Γ . We mention the following intermediate result, which will be used in Section 5.1: Lemma 2.25. We can choose the constant ε ą 0 from Remark 2.23 small enough so that the following holds: Let T be a standard tree of D Γ , and let Z be the full subcomplex of p D Γ obtained by removing the vertex of tree type associated to T . Then T is convex in Z for the induced metric.
Proof. It follows from Lemma 2.25 that Z is CAT(´1) for the induced metric. Since T is a subtree of the CAT(´1) complex Z, it is enough to show that two edges of T that share a vertex v make an angle of at least π at v. (This follows from the local characterisation of geodesics in a CAT(0) space) This amounts to showing that for distinct vertices w, w 1 of T X Lk Z pvq, their distance in Lk Z pvq is at least π. By construction, the simplicial graph Lk Z pvq is obtained from Lk D Γ pvq by coning-off the sets T 1 X Lk D Γ pvq for each standard tree T 1 other than T , i.e. by constructing a simplicial cone over every such sets T 1 X Lk D Γ pvq. Moreover, each of these new edges has length at least π{2´ε by construction, see Remark 2.23. Since the action of A Γ on D Γ is cocompact, there is a uniform lower bound on the length of edges in the link of an arbitrary vertex of D Γ , and we can choose the constant ε ă π{10 smaller than half this uniform lower bound, which we now assume. We now claim that the distance in Lk Z pvq between w and w 1 is at least π. Indeed, assume that this were not the case. Since T is a convex subtree of D Γ by construction, two distinct points w, w 1 of Lk D Γ pvq that belong to T are at distance at least π in Lk D Γ pvq.
Therefore, if there was a geodesic of length less than π between w and w 1 in Lk Z pvq, it would have to go through at least two of the additional edges of length ě π{2´ε coming from the cone-off procedure. Moreover, since w and w 1 are not in the same standard tree (Corollary 2.18), said geodesic should also contain another edge, which has length at least 2ε. Therefore, the geodesic would have length at least 2ε`2pπ{2´εq ě π, a contradiction. Thus, T is convex in Z.
Convention 2.26. From now on, we will assume that p D Γ is endowed with a CAT(´1) metric from [MP21] such that Lemma 2.25 holds.
Moreover, since we are considering a fixed Artin group A Γ , we will from now on simply denote by D and p D the complexes D Γ and p D Γ .
We now describe the stabiliser of the vertices of p D of tree type: Lemma 2.27. The stabiliser of the vertex of tree type v a is exactly the centraliser (and normaliser) of the cyclic subgroup xay. Moreover, this centraliser splits as a direct product of the form xayˆK, where K is a finitely-generated free group. More precisely, the subgroup xay acts trivially on T a , while K acts cocompactly on it with trivial edge stabilisers.
Proof. Let us first note that the normaliser and centraliser of xay coincide. Indeed, if g P A Γ is such that g´1ag P xay, we apply the homomorphism A Γ Ñ Z sending every generator to 1 and deduce that g´1ag " a, hence g centralises xay.
The stabiliser of v a coincides with the global stabiliser of the standard tree T a . Let us show that an element g P A Γ stabilises T a if and only if it normalises xay.
If g stabilises T a , then it sends the vertex xay P T a to the vertex gxay P T a , so in particular we have a¨gxay " gxay, or in other words g´1ag P xay, hence g normalises xay. Conversely, let us assume that g normalises xay, and let x be a point of T a . Let us show that gx P T a . We have a¨gx " gpg´1agqx " gx, the last equality following from the fact that x is fixed by xay by definition of T a . Thus, gx is fixed by a, and it follows that g stabilises T a .
The decomposition of the stabiliser of T a as a direct product as in the statement is a consequence of [MP21,Lemma 4.5]. Observe that Stabpv a q acts cocompactly on T a . Indeed, D contains finitely many A Γ orbits of edges, and no edge is contained in distinct translates of a standard tree by Corollary 2.18.
The quotient K was defined in the proof of [MP21, Lemma 4.5] as the fundamental group of a graph of groups over the graph T a {Stabpv a q, with trivial edge stabilisers and vertex stabilisers that are trivial or infinite cyclic. Since T a {Stabpv a q is a finite graph by the above, it follows that K is finitely generated.
Regarding vertices of dihedral type, we have the following similar result: Lemma 2.28. The stabiliser A ab of the vertex of dihedral type v ab is exactly the centraliser of the cyclic subgroup xz p ab y for all p ‰ 0, which further coincides with the normaliser. Proof. Since the stabiliser of v ab is equal to A ab by construction, and z p ab is central in A ab , it is enough to show that an element in the normaliser of z p ab fixes v ab . Suppose that we have an element h such that h P N pz p ab q. Then z q ab " hz p ab h´1 (for some q ‰ 0) also fixes the vertex hv ab , so z q ab fixes pointwise the unique CAT(0) geodesic of D joining v ab to hv ab . If this geodesic is nontrivial, then z q ab fixes an edge of D containing v ab , since triangles in D have trivial stabilisers. This is impossible, as otherwise z q ab would be contained in an edge stabiliser, hence would be conjugate in A ab to a power of a or b (since the edge in question contains v ab ). Hence z q ab would be equal to a power of a or b since z ab is central in A ab , contradicting Corollary 2.7. So, hv ab " v ab , as required.
We will also mention the following lemma about centralisers, which will be used later in this article (see Lemma 4.5): Lemma 2.29. Let c P A Γ and suppose that either c is a standard generator, or c " z ab for some standard generators a and b generating a dihedral Artin group. Then the centraliser Cpcq satisfies Cpcq " Cpc p q for all p P Z´t0u.
Proof. There are two cases, according to whether c is a standard generator or c " z ab for standard generators a, b.
First consider the case where c " z ab , where a, b are standard generators generating a dihedral type Artin subgroup A ab . Recall that xz ab y fixes a point in p D, namely the vertex v ab . Suppose that, for some p P Z´t0u, we have an element h such that h P Cpz p ab q. By Lemma 2.28, we have that hv ab " v ab , whence h P A ab , and this subgroup is equal to Cpz ab q by Lemma 2.28.
Next consider the case where c is a standard generator c " a, and let e be an edge of the corresponding standard tree T a , which has xay as stabiliser. Suppose that, for some p P Z´t0u, we have an element h such that h P Cpa p q. Then the stabiliser of the edge he is Stabpheq " hxayh´1 Ą xa p y. In particular, the stabilisers of e and he intersect nontrivially, and it follows from Lemma 2.15 that they are in the same standard tree T a . From Corollary 2.18, we get that hT a " T a , so h P Cpaq.
2.5. Links of vertices. The local structure of Deligne complexes will play an important role in this article, so we now describe it in further detail.
Lemma 2.30. Let v a be a vertex of tree type. The link Lk p D pv a q is the standard tree T a , and the action of Stabpv a q on it is cocompact.
Proof. The description of the link follows from the construction of the cone-off, and the cocompactness follows from Lemma 2.27.
Throughout the rest of this subsection, we fix a vertex of D of dihedral type of the form v ab . Before describing the link in the cone-off p D, we start by describing the link in the original Deligne complex D. The link Lk D pv ab q of that vertex has a simple description, which is a direct consequence of the construction of D: Lemma 2.31. The link Lk D pv ab q is A ab -equivariantly isomorphic to the the geometric realisation of the poset of cosets of strict standard parabolic subgroups of A ab . That is, vertices of Lk D pv ab q correspond to cosets of the form gxay, gxby, or gt1u, and for every g P A ab , we add an edge between gt1u and gxay, as well as an edge between gt1u and gxby.
In particular, the action of Stabpv ab q on Lk D pv ab q is cocompact.
This link is a particular case of a general construction that we will use again in Section 5: Definition 2.32 (Graph of orbits). Let G be a graph with an A ab -action, such that the subgroups xay and xby act freely on it. We define a new graph encoding the pattern of intersections of orbits of xay and xby as follows: We put a vertex for every xay-orbit and one vertex for every xby-orbit. If two such orbits have a non-empty intersection, we put an edge between them. The graph of orbits Orbit a,b`G˘i s defined as the first barycentric subdivision of the graph obtained in this way.
Remark 2.33. With that terminology, the link Lk D pv ab q is A ab -equivariantly isomorphic to the graph of orbits Orbit a,b`C ayley a,b pA ab q˘, where Cayley a,b pA ab q denotes the Cayley graph of A ab for the standard generators a, b.
The action of A ab on Lk D pv ab q has been studied by Vaskou in [Vas20]. In particular, the following result, which is a geometric counterpart of Lemma 2.6 will be useful in Section 5: Lemma 2.34 ([Vas20, Proposition 4.7]). Let g be an element of A ab that can be written only with positive letters and that has syllabic length greater than 1. If m ab ě 3, then the xgy-orbit maps to Lk D pv ab q are quasi-isometric embeddings.
Since the Garside element ∆ ab either conjugates the generators a, b (when m ab is odd) or centralises each of them (when m ab is even), the action of x∆ ab y by right multiplication induces an action on the left cosets. Indeed, we have gxay∆ ab " g∆ ab xby and gxby∆ ab " g∆ ab xay if m ab is odd, gxay∆ ab " g∆ ab xay and gxby∆ ab " g∆ ab xby if m ab is even.
We now describe the link of v ab in the coned-off Deligne complex. There is a simple characterisation of the trace of a standard tree on the link of a vertex of dihedral type. The following is a reformulation of [MP21, Lemma 4.3]: Lemma 2.35. Two vertices of the link Lk D pv ab q correspond to edges in the same standard tree of D if and only if the corresponding cosets are in the same x∆ ab y-orbit (for the multiplication on the right).
Moreover, two vertices of the link Lk D pv ab q that are in the same A ab -orbit correspond to edges in the same standard tree of D if and only if the corresponding cosets are in the same xz ab y-orbit (for the multiplication on the right).
This local characterization of standard trees allows us to simply describe the links of vertices in the coned-off space p D: Corollary 2.36. The link Lk p D pv ab q is obtained from Lk D pv ab q by coning-off every x∆ ab y-orbit of vertices corresponding to cosets of the form gxay or gxby.
In particular, the action of Stabpv ab q on Lk p D pv ab q is cocompact. Proof. The description of the link is a consequence of Lemma 2.35. The cocompactness of the action of Stabpv ab q on Lk D pv ab q follows from Lemma 2.31. Moreover, since there are finitely many Stabpv ab q-orbits of edges of D containing v ab , there are in particular finitely many orbits of standard trees containing v ab , and hence finitely many orbits of apices of standard trees containing v ab . The cocompactness of the action of Stabpv ab q on Lk p D pv ab q now follows.

The commutation graph of an Artin group
3.1. The commutation graph. We now construct a complex to which Theorem 1.4 will be applied, and which turns out to be quasi-isometric to the coned-off Deligne complex p D. This complex is obtained via a general construction that encodes the commutation of certain chosen subgroups of a given group that we now describe.
Definition 3.1 (Commutation graph). Let G be a group, and let H be a set of subgroups of G. We define a simplicial graph, called the commutation graph of H and denoted Y H , as follows. The vertex set of Y H is ğ HPH G{N pHq, and we put an edge between gN pHq and hN pH 1 q if gHg´1 and hH 1 h´1 commute, that is, every element of one subgroup commutes with every element of the other subgroup. Note that this is independent of the choice of coset representatives g and h. The group G acts on this graph by left multiplication.
Remark 3.2. Commutation graphs have already been considered in the work of Kim-Koberda on right-angled Artin groups under the name of extension graphs [KK14]. Indeed, for a right-angled Artin group G on generators g 1 , . . . , g n , the extension graph they study is exactly the commutation graph for the family H " xg 1 y,¨¨¨, xg n y ( .
We now discuss the family of subgroups of A Γ that yield the correct commutation graph for our purposes.
First, the motivation. The acylindricity of the action of A Γ on its coned-off Deligne complex and the CAT(0) geometry of this complex can be shown to imply that the maximal subgroups of A Γ that virtually split as products are the dihedral parabolic subgroups (stabilisers of vertices of dihedral type), which are virtual products by Lemma 2.2, and the normalisers of standard generators, which are products by Lemma 2.27. It is an exercise, left to the reader as it is not needed in this article, to identify the minimal infinite subgroups obtained by taking arbitrary intersections of such maximal virtual products. They come in two families: ‚ The cyclic subgroups generated by a conjugate of a standard generator: These subgroups are obtained by taking the intersection of stabilisers of vertices of dihedral type contained in a common standard tree. ‚ The centres of dihedral parabolic subgroups with m ab ě 3: These subgroups are obtained by taking the intersection of the stabilisers of two standard trees that share a vertex.
Remark 3.3. We know from the work of Paris [Par97, Corollary 4.2] that two standard generators are conjugate if and only if there is a path in the presentation graph Γ consisting of edges with odd labels connecting the corresponding vertices. We can thus choose a set of representatives of conjugacy classes of elements of V pΓq, which defines a subset V odd pΓq of V pΓq.
This motivates the following definition: Definition 3.4. We define the following collection of subgroups of the Artin group A Γ : We will simply denote by Y " Y Γ " Y H the commutation graph of H. Proof. We have to show that for two distinct elements x, y of the form z ab or a P V odd pΓq, no non-trivial power of x is conjugate to a non-trivial power of y. We treat several different cases. Let a, b be two standard generators of V odd pΓq. If non-trivial powers a k and b were conjugate, then using the homomorphism A Γ Ñ Z sending every generator to 1 we would obtain that k " . By [Par97, Corollary 5.3] (which says that an element conjugates a k to b k if and only if it conjugates a to b), we get that a and b are conjugate, which is excluded by construction of V odd pΓq.
The centraliser of a standard generator a has the form Kˆxay for some free group K by Lemma 2.27, while the centraliser of an element of the form z a 1 b 1 is the dihedral Artin group A a 1 b 1 by Lemma 2.28. These are not isomorphic for example because the abelianisations can be isomorphic only if K is trivial (consider the cases of m a 1 b 1 odd or even), but A a 1 b 1 is not isomorphic to Z by Lemma 2.3.
Finally, suppose that non-trivial powers of z ab and z a 1 b 1 are conjugate. Then their centralisers are also conjugate. By Lemma 2.28, these centralisers are A ab and A a 1 b 1 . In turn, by Lemma 2.13, these have fixed-point sets respectively consisting of v ab and v a 1 b 1 only, so that these fixed point sets are not translates of each other. Therefore, the centralisers cannot be conjugate, and the powers of z ab and z a 1 b 1 cannot be conjugate.
Notation 3.6. For a standard generator a P Γ, we denote by u a the vertex of Y corresponding to N pxayq. The A Γ -translates of such vertices are said to be of tree type. Analogously, for standard generators a, b spanning an edge of Γ with m ab ě 3, we denote by u ab the vertex of Y corresponding to N pxz ab yq. The A Γ -translates of such vertices are said to be of dihedral type.
In the rest of this section, we construct an equivariant quasi-isometry between the commutation graph Y and the coned-off Deligne complex p D. We start by defining such a map at the level of vertices: Lemma 3.7. We define a map ι : Y p0q Ñ p D p0q as follows: ‚ For a vertex of Y of dihedral type of the form gu ab , we set ιpgu ab q :" gv ab . ‚ For a vertex of Y of tree type of the form gu a , we set ιpgu a q :" gv a . Then ι is well-defined, is injective, and realises bijections between the vertices of Y of dihedral type and the vertices of p D of dihedral type, as well as between the vertices of Y of tree type and the vertices of p D of tree type.
Proof. We have to show the following properties: ‚ if gN pxz ab yq " hN pxz ab yq, then gv ab " hv ab . ‚ if gN pxayq " hN pxayq, where a P V odd pΓq, then gv a " hv a . If gN pxz ab yq " hN pxz ab yq, then g´1h P N pxz ab yq " A ab , the latter equality following from Lemma 2.28, and we know from Lemma 2.28 that N pxz ab yq stabilises v ab . It thus follows that hv ab " gpg´1hqv ab " gv ab , so ι is well defined on vertices of dihedral type. Moreover, it in fact realises a bijection on vertices of dihedral type as we just saw that in both Y and p D these vertices are cosets of the A ab .
If gN pxayq " hN pxayq, then g´1h P N paq, and we know from Lemma 2.27 that N paq stabilises v a . It thus follows that hv a " gpg´1hqv a " gv a , so ι is well defined on vertices of tree type.
Let us show that ι is surjective on vertices of tree type. Let gv a be a vertex of p D of tree type. By construction, there exists a standard generator b P V odd pΓq and an element x P A Γ such that xbx´1 " a. It thus follows that xv b " v a , and in particular gv a " gxv b " ιpgxu b q.
Let us now show that ι is injective on vertices of tree type. Consider a, b P V odd pΓq and g, h P A Γ such that gv a " hv b . We want to show that gu a " hu b . By construction of vertices of tree type, this means that the standard tree of D containing the vertex gxay and the standard tree of D containing the vertex hxby coincide. In particular, the vertices gxay and hxby of D are in the same standard tree, and it follows from Lemma 2.22 that a and b are connected by a path of Γ with odd labels. By definition of V odd pΓq, this implies that a " b. We thus have gv a " hv a , and so g´1h P Fixpv a q " N paq, the latter equality following from Lemma 2.27. We thus have gN paq " hN paq, hence gu a " hu a " hu b , which shows injectivity.
Note that the injectivity of ι : Y p0q Ñ p D p0q is now straightforward, since the sets of vertices of tree type and dihedral type of p D are disjoint.
Lemma 3.8. The map ι is well-defined, is injective, and realises bijections between the vertices of Y of dihedral type and the vertices of p D of dihedral type, as well as between the vertices of Y of tree type and the vertices of p D of tree type. Moreover, two vertices v, v 1 of Y are adjacent if and only if the following occurs: One of them (say v) is of dihedral type, the other (say v 1 ) is of tree type, and ιpvq is contained in the standard tree having ιpv 1 q as apex.
In particular, Y is a bipartite graph with respect to the type of vertices.
Proof. The first statement is exactly Lemma 3.7. Let us now characterise the edges of Y . Consider two vertices v " gN pHq, v 1 " hN pH 1 q of Y that are connected by an edge. The subgroups gHg´1, hH 1 h´1 are infinite cyclic, and we denote by z v , z v 1 the associated generators. The elements z v , z v 1 commute by definition of Y , and they generate a Z 2 subgroup of A Γ by Lemma 3.5. Since the action of A Γ on p D is acylindrical and p D is CAT(0), this subgroup must fix a point. In particular, the fixed-point sets in p D of z v and z v 1 have a nontrivial intersection. We will need the following standard result from group actions on trees, whose proof we omit, to show that certain configurations are impossible: Claim: Let G be a group acting on a simplicial tree T by isometries, let w, w 1 be two distinct vertices of T , and let g, g 1 be two elements of G such that for all non-zero k P Z, we have Fix T pg k q " twu and Fix T ppg 1 q k q " tw 1 u. Then g, g 1 generate a non-abelian free subgroup.
We now consider several cases, depending on the type (dihedral or tree) of the corresponding vertices.
Case 1: Suppose by contradiction that v and v 1 are adjacent vertices of Y of dihedral type. Recall that for an element of the form z ab , we have Fix D pz ab q " tv ab u by Lemma 2.28. Since Fix p D pz v q X Fix p D pz v 1 q is non-empty, these fixed-point sets intersect along at least (at least) one vertex w, which must be of tree type since Fix D pz v q X Fix D pz v 1 q " ∅. This vertex correspond to a standard tree T stabilised by xz v , z v 1 y. Moreover, T contains both vertices ιpvq and ιpv 1 q: Indeed, z v stabilises the unique CAT(0) geodesic between ιpvq and T , and since z v is not conjugate to a power of a standard generator by Lemma 3.5, z v cannot stabilise an edge of D, and it follows that this geodesic is reduced to a point (and similarly for z v 1 ). Thus, xz v , z v 1 y acts on T , and since no non-trivial power of z v or z v 1 is conjugate to a standard generator by Lemma 3.5, it follows that for all non-zero k P Z, we have Fix T pz k v q " tιpvqu and Fix T ppz v 1 q k q " tιpv 1 qu. It now follows from the Claim that z v and z v 1 generate a non-abelian free subgroup, a contradiction.
Case 2: Let us now assume that v, v 1 are two adjacent vertices of Y of tree type. By the above argument, z v and z v 1 fix a vertex w of p D. If w is a vertex of D, then it belongs to Fix D pz v q X Fix D pz v 1 q " T ιpvq X T ιpv 1 q . Since distinct standard trees of D meet in at most one vertex by Corollary 2.18, we can assume that w is that common vertex. Up to conjugation, we can thus assume that z v and z v 1 are two distinct commuting conjugates of standard generators of a dihedral Artin group A ab , which contradicts Lemma 2.8.
Case 3: Finally, we assume that v is of dihedral type and v 1 of tree type. Since Fix Let us show that this is the only possibility. By contradiction, if Fix p D pz v q X Fix p D pz v 1 q does not contain any vertex of D, then these fixed-point sets intersect along (at least) one vertex, which must be of tree type since Fix D pz v q X Fix D pz v 1 q " ∅. This vertex corresponds to a standard tree T . Moreover, the same reasoning as in the previous cases shows that T contains ιpvq and intersects T ιpv 1 q . We thus have that xz v , z v 1 y induces an action on T , z v fixes the vertex w :" ιpvq, z v 1 fixes the vertex w :" T ιpv 1 q X T , w, w 1 are distinct since Fix D pz v q X Fix D pz v 1 q is empty by assumption. Moreover, for every non-zero integer k, we have Fix T pz k v q " twu since no non-trivial power of an element of the form z ab is conjugate to a standard generator by Lemma 3.5, and for every non-zero integer k, we have that Fix T pz k v 1 q " tw 1 u by Lemma 2.15 since T is a standard tree distinct from T ιpv 1 q . It now follows from the Claim that z v and z v 1 generate a non-abelian free subgroup, a contradiction.
The previous lemma is useful in understand the structure of the graph Y . In particular, we have the following results: Lemma 3.9. The graph Y does not contain any triangle or square.
Proof. Since Y is bipartite with respect to the type of vertices by Lemma 3.8, it does not contain triangles.
Let us now show by contradiction that Y does not contain squares. Since Y is bipartite with respect to the type of vertices by Lemma 3.8, such a square would contain exactly two opposite vertices of dihedral type and two opposite vertices of tree type.
In D Γ , this would correspond to two distinct standard trees intersecting in at least two different vertices, which contradicts Corollary 2.18.
Lemma 3.10. If Γ is connected, then the commutation graph Y is connected.
Proof. Let us first show that any two vertices of Y of tree type are connected by a path of Y . Since A Γ is generated by its standard generators, it is enough to show that every pair of vertices of tree type of the form u a and u b are connected by a path of Y . Let a, b be two vertices of Γ. Since Γ is connected, let c 1 " a, . . . , c n " b be a sequence of vertices of Γ such that c i and c i`1 are adjacent for every i. Then the sequence of vertices Moreover, since every vertex of dihedral type of Y is connected by an edge to some vertex of Y of tree type, it follows that Y is connected.
Lemma 3.11. The action of A Γ on Y is cocompact.
Proof. There are finitely many orbits of vertices of Y by construction, since the family H is finite. It follows from Lemma 3.8 and the characterisation of the edges of Y that the set of edges of Y equivariantly embeds into the set of edges of p D. Since the action of A Γ on p D is cocompact, it follows that there are only finitely many A Γ -orbits of edges of Y , hence the result.
Lemma 3.8 allows us to extend the map ι to a map from Y to p D: Definition 3.12. We extend the map ι : Y p0q Ñ p D p0q into a simplicial and A Γ -equivariant map ι : Y Ñ p D as follows: An edge between a vertex u of dihedral type and a vertex u 1 of tree type is sent to the corresponding edge of p D between ιpuq and ιpu 1 q.
Lemma 3.13. Suppose that the graph Γ is connected. Then the map ι : Y Ñ p D is an A Γequivariant quasi-isometry. More precisely, ι embeds Y as a coarsely dense subgraph of p D.
Proof. We construct a quasi-inverse ι : p D Ñ Y as follows. First notice that since p D has finitely many isometry types of simplices, it is enough to define ι at the level of vertices, where the distance between vertices is defined as the length of a minimal path in the 1-skeleton.
By Lemma 3.8, for a vertex v of p D that is either of dihedral or tree type, we define ιpvq to be the unique vertex u P Y such that ιpuq " v. Let us define ι on the remaining vertices. A vertex v of p D corresponding to the coset of a cyclic group generated by a standard generator belongs to a unique standard tree T , and we set ιpvq " ιpv 1 q, where v 1 is the vertex of p D that is the apex corresponding to T . For a vertex v corresponding to a coset of the trivial subgroup, we pick a vertex v 1 of dihedral type of D adjacent to it, and we set ιpvq " ιpv 1 q.
Lemma 3.8 implies that ι˝ι is the identity on the vertices of Y , and that ι˝ι is the identity on the vertices of p D that are either of dihedral type or of tree type. For a vertex v of p D corresponding either to the coset of a trivial, cyclic, or Z 2 -subgroup, the construction implies that ι˝ιpvq and v are at distance 1 in the 1-skeleton of D, which concludes the proof.
Note that if Γ is disconnected and can be written as the disjoint union of two full subgraphs Γ 1 , Γ 2 , then A Γ splits as the free product A Γ 1˚A Γ 2 . In particular, since a free product of hierarchically hyperbolic groups is itself hierarchically hyperbolic (see [BHS19,Corollary 8.24] or [BHS19, Theorem 9.1]), it is enough to consider the case of a connected graph Γ. This motivates the following convention: Convention 3.14. In the rest of this article (except the proof of Theorem A in Section 6.3), we will assume that the underlying presentation graph Γ is connected and not a single vertex.
3.2. The graph of proper irreducible parabolic subgroups of finite type. As an aside, we highlight the connection between the commutation graph Y and the graph of proper irreducible parabolic subgroups of finite type introduced by Morris-Wright [MW21], and generalizing a construction in the spherical type of Cumplido et al. [CGGMW19] . This graph was proposed as an analogue of the curve graph for all Artin groups. We show that this indeed the case for Artin groups of large and hyperbolic type. We emphasise that this subsection is not needed in the rest of the paper and can be omitted in a first reading.
). The graph of irreducible proper parabolic subgroups of finite type P of A Γ is the simplicial graph defined as follows. Vertices correspond to the proper parabolic subgroups of finite type that are irreducible (that is, they do not decompose as a direct product of proper standard parabolic subgroups). Two vertices H, H 1 are connected by an edge when either: ‚ there is a strict inclusion H Ĺ H 1 , or ‚ we have H X H 1 " t1u and H, H 1 commute. The Artin group A Γ acts on P by conjugation.
While this definition makes sense for all Artin groups, we recall that we are only dealing in this article (and in particular in this section) with the case of Artin groups A Γ that are large-type and of hyperbolic type.
Proposition 3.16. Assume that Γ is a connected graph not reduced to a single edge. Then the graph of irreducible parabolic subgroups of finite type of A Γ is equivariantly isomorphic to the commutation graph.
Under such conditions on Γ, the action of A Γ on the coned-off Deligne complex is acylindrical and universal by [MP21, Theorem A]. The following is thus a direct consequence of Lemma 3.13: Corollary 3.17. Assume that Γ is a connected graph not reduced to a single edge. Then the graph of irreducible proper parabolic subgroups of finite type of A Γ is hyperbolic of infinite diameter, and the action of A Γ on it is acylindrical and universal.
In order to prove Proposition 3.16, we will need the following result: Lemma 3.18. Assume that Γ is a connected graph not reduced to a single edge. Then distinct elements of H have normalisers that are in different conjugacy classes.
Proof. The normaliser of a standard generator is of the form ZˆF k for F k a finitely-generated free group by Lemma 2.27, while the normaliser of an element of the form z ab is equal to the dihedral Artin group A ab by Lemma 2.28. We show that these groups are non-isomorphic. This is clear if k " 1 as a dihedral Artin group with m ab ě 3 is not isomorphic to Z 2 since it virtually contains a non-abelian free group by Lemma 2.3. For k ě 2, this follows for instance from the description of their abelianisations: The abelianisation of ZˆF k is a free abelian group of rank k`1 ě 3, while the abelianisation of A ab is generated by two elements since A ab is. Thus, the normaliser of a standard generator and the normaliser of an element of the form z ab are not conjugated.
Let z ab , z a 1 b 1 be two elements associated to two distinct edges of Γ, and let g P A Γ . By Lemma 2.28, the normalisers of z ab and gz a 1 b 1 g´1 are the dihedral Artin groups A ab and gA a 1 b 1 g´1 and their fixed-point sets in D are v ab and gv a 1 b 1 respectively. As these points are in different A Γ -orbits, it follows that N pz ab q and N pz a 1 b 1 q are not conjugate.
Finally, let a, b two distinct standard generators in V odd pΓq, and suppose by contradiction that N paq " gN pbqg´1 for some g P A Γ . By Lemma 2.27, the normalisers a and b have the form xayˆF k and xbyˆF k respectively (note that the rank of the free group factors need to coincide since the subgroups are conjugated). If k ě 2, then since N paq and N pbq are conjugated, so are their centres xay and xby. Using the homomorphism A Γ Ñ Z sending each standard generator to 1, it follows that a and b are conjugated. By construction of V odd pΓq, it follows that a " b, a contradiction.
Assume now that k ď 1. We thus have two standard generators a, b such that N paq, N pbq are isomorphic to Z or Z 2 . We will use the following general claim: Claim: Let x be a standard generator. If the normaliser N pxq is isomorphic to Z, then x corresponds to an isolated vertex of Γ. If the normaliser N pxq is isomorphic to Z 2 , then x corresponds to a leaf of Γ.
Let us prove the Claim. By Lemma 2.27, the normaliser N pxq is of the form xxyˆF for some finitely-generated group F . An explicit basis B of F was given in [MP21, Remark 4.6]. In particular, the following holds: (i) If x is an isolated vertex of Γ, then F is trivial.
(ii) The basis B contains the element z xy for every vertex y of Γ adjacent to x.
(iii) Suppose that y is adjacent to x and the edge of Γ between them has an odd label. If w is a vertex of Γ adjacent to y and distinct from x, then B contains a suitable conjugate of z yw . We note that such an element cannot be equal to an element described in piiq by Lemma 3.5. If N pxq is isomorphic to Z, then F is trivial, and by item piiq in the above description of a basis of F , x must be an isolated vertex. If N pxq is isomorphic to Z 2 , then F is isomorphic to Z, and by item piiq in the above can have at most one neighbour in Γ. It cannot be an isolated vertex for otherwise F would be trivial by item piq above. Thus x is a leaf of Γ. This proves the Claim.
Since N paq, N pbq are isomorphic to Z or Z 2 , it follows from the above Claim that a and b are leaves of Γ (since Γ is connected by assumption). Moreover, since N paq, N pbq -Z 2 and Γ is not reduced to a single edge, item piiiq above implies that the corresponding edges of Γ containing a and b have even labels.
By Lemma 2.27, N paq " gN pbqg´1 stabilises the standard tree containing the vertex xay and the standard tree containing gxby. Because a, b are leaves of Γ with corresponding edges having even labels, the corresponding standard trees T a and gT b have a very simple structure, which is a direct consequence of [MP21, Lemma 4.3] (see also [MP21,Example 4.7] for a concrete example): Let a 1 be the unique neighbour of a in Γ, and let e be the edge of D between the parabolic subgroups xay and A aa 1 . Then T a is the union of the xz aa 1 y-translates of e (such a description was for instance given in [MP21, Example 4.7]). In particular, T a (and similarly for gT b ) is isomorphic to a cone over a countably infinite discrete set.
Let us show that the standard trees T a and gT b intersect. If these two convex trees were disjoint, then N paq " gN pbqg´1 would also stabilise the unique CAT(´1) geodesic of D between them, and so both normalisers would be contained in an edge-stabiliser or trianglestabiliser. This is impossible as such stabilisers do not contain Z 2 , as they are infinite cyclic or trivial respectively.
Thus these two standard trees meet. The vertices in T a that are not the central vertex of dihedral type correspond to cosets of xay (and we have an analogous result for b). Since xay ‰ xby by assumption, the trees T a and gT b necessarily meet at the central vertex of dihedral type. Let a 1 (respectively b 1 ) be the unique neighbour of a (respectively b) in Γ. The unique vertex of T a of dihedral type corresponds to a coset of A aa 1 . Similarly, the unique vertex of gT b of dihedral type corresponds to a coset of A bb 1 . As these vertices agree, it follows that ta, a 1 u " tb, b 1 u, and since a ‰ b this implies a and b are adjacent in Γ. Since a and b are leaves of Γ by the above, it follows that Γ is a single edge, contradicting our assumption on Γ.
Lemma 3.19. Assume that Γ is a connected graph not reduced to a single edge. Consider the map ϕ : Y p0q Ñ P p0q defined as follows: ‚ For a vertex gA ab of Y of dihedral type, we set ϕpgA ab q " gA ab g´1. ‚ For a vertex gN paq of Y of tree type, we set ϕpgN paqq " gxayg´1. Then ϕ is well-defined and realises an A Γ -equivariant bijection between the vertex sets of Y and P .
Proof. The map ϕ is well-defined on vertices of dihedral type, but we need to check that the definition makes sense for vertices of tree type. It is enough to show that if a, b are standard generators in V odd pΓq with N paq " N pbq, then a " b. This is a direct consequence of Lemma 3.18.
Let us show that ϕ is surjective. It is clear from the construction that every conjugate of the form gA ab g´1 " ϕpgA ab q is in the image of ϕ. Let a be a standard generator of A Γ . By construction of V odd pΓq, there exists a standard generator b P V odd pΓq and an element h P A Γ such that a " hbh´1. Thus, any conjugate of the form gxayg´1 can be written as gxayg´1 " pghqxbypghq´1 " ϕpghN pbqq. Thus, ϕ is surjective.
Let us show that ϕ is injective. For a given standard generator a P V pΓq, the map ϕ induces a bijection between the conjugates of xay and the cosets of N paq. Similarly, since dihedral parabolic subgroups of A Γ are self-normalizing by Lemma 2.13, we have that for a given edge a, b of Γ with m ab ě 3, the map ϕ induces a bijection between the conjugates of A ab and the cosets of A ab . To show that ϕ is injective, it remains to show that for H, H 1 different elements of H, we cannot have an equality of the form ϕpgN pHqq " ϕpg 1 N pH 1 qq for g, g 1 P A Γ . Such an equality would amount to an equality between parabolic subgroups. Note that a conjugate of the form gxayg´1, which is isomorphic to Z, cannot be isomorphic to a conjugate of a dihedral parabolic subgroup by Lemma 2.3. There are thus two cases to consider: If H, H 1 correspond to cyclic subgroups of the form xay, xa 1 y respectively, then the equality ϕpgN pHqq " ϕpg 1 N pH 1 qq yields the equality gxayg´1 " g 1 xa 1 ypg 1 q´1. Since a ‰ a 1 , this equality is impossible by Lemma 3.5.
If H, H 1 correspond to cyclic subgroups of the form xz ab y, xz a 1 b 1 y respectively, then the equality ϕpgN pHqq " ϕpg 1 N pH 1 qq and the fact that A ab " N pz ab q by Lemma 2.28 yield the equality gN pz ab qg´1 " g 1 N pz a 1 b 1 qpg 1 q´1. Since H ‰ H 1 , this equality is impossible by Lemma 3.18.
Proof of Proposition 3.16. We know from Lemma 3.19 that the map ϕ realises an A Γ -equivariant bijection between the vertex sets of Y and P . Let us show that this bijection extends to an equivariant isomorphism between P and Y .
First, it follows from Lemma 3.8 that two vertices that are adjacent in Y are also adjacent in P , and we now show the converse.
Let Q and Q 1 be two proper irreducible parabolic subgroups of finite type, i.e. parabolic subgroups on one or two generators with m ab ě 3, and assume that Q and Q 1 are connected in P . There are two cases to consider, depending on the two types of edges of P : Let us first assume that Q, Q 1 intersect trivially and commute. Since A Γ is of cohomological dimension 2 by [CD95] and dihedral parabolic subgroups contain a copy of Z 2 , this can only happen if both Q and Q 1 are infinite cyclic, in which case there is a Y -edge between them by construction of Y .
Suppose now that Q Ĺ Q 1 . First notice that Q and Q 1 cannot be both dihedral parabolic subgroups by Lemma 2.13. We can thus assume that Q is infinite cyclic. Let e be an edge in the standard tree T associated to Q, and let σ 1 be a maximal simplex of FixpQ 1 q (i.e. σ 1 is a vertex or an edge depending on whether Q 1 is of dihedral type or infinite cyclic respectively). Since Q fixes both v and σ 1 by assumption, Lemma 2.17 implies that v and σ 1 are contained in a common standard tree, namely the standard tree T .
Let us show by contradiction that Q 1 cannot be infinite cyclic. Let T 1 be the standard tree corresponding to Q 1 . Then since T and T 1 both contain e, it follows from Corollary 2.18 that T " T 1 . Since Q, Q 1 is the pointwise stabiliser of T, T 1 respectively by construction, we get Q " Q 1 , contradicting the assumption that Q Ĺ Q 1 .
Thus, Q 1 is a parabolic subgroup of dihedral type. We thus have that the vertex of D corresponding to Q 1 is contained in the standard tree associated to Q. By Lemma 3.8, this implies that Q and Q 1 are connected by an edge of Y . 3.3. Adjacency in the commutation graph and intersection of neighbourhoods of cosets. In this section, we study the interactions between cosets associated to adjacent vertices of the commutation graph Y . Namely, we show that two such adjacent cosets have neighbourhoods that intersect along a quasi-flat (see Lemma 3.23). We start with the following result: Lemma 3.21. Let H 1 , H 2 P H and suppose that gN pH 1 q, hN pH 2 q are adjacent vertices of Y for some g, h P A Γ . Then we have gN pH 1 qg´1 X hN pH 2 qh´1 " xgH 1 g´1, hH 2 h´1y.
Proof. By Lemma 3.8, we can assume that gN pH 1 q is mapped under ι to the apex of a standard tree T 1 and hN pH 2 q is mapped under ι to a vertex of dihedral type contained in T 1 . Notice that, up to a conjugation, it is enough to prove the desired equality when h is trivial, so we will assume that this is the case in the rest of the proof. The subgroup H 2 P H is of the form xz ab y for some standard generators a, b, and so the vertex v corresponds to the trivial coset A ab by construction of ι. Since the standard tree T 1 contains the vertex A ab , it also contains a vertex of the form xxay or xxby for some x P A ab . Without loss of generality, we can assume that T 1 contains a vertex of the form xxay for some chosen x P A ab .
We have hN pH 2 qh´1 " N pz ab q " A ab by Lemma 2.28. Moreover, since two edges of T 1 have the same stabiliser by construction, we have gH 1 g´1 " xxayx´1, hence gN pH 1 qg´1 " xN paqx´1 " xCpaqx´1, the latter equality following from Lemma 2.27. Thus, the intersection gN pH 1 qg´1 X hN pH 2 qh´1 can be identified with the centraliser of the element xax´1 in A ab . By [Cri05, Lemma 7.piiq], this is equal to xxax´1, z ab y " xgH 1 g´1, hH 2 h´1y.
For the "moreover" part, note that the intersection is naturally isomorphic to the product by definition of edges of Y and Lemma 3.5. We fix a constant r ě 0 so that whenever a, b are conjugate standard generators there exists x P Bpe, rq so that xax´1 " b. The r-neighbourhood N pHq`r of N pHq is defined as N pHq`r :" ď cPBpe,rq N pHqc.
Lemma 3.23. There exists a constant ě 0 such that the following holds: Let H 1 , H 2 P H and suppose that gN pH 1 q, hN pH 2 q are adjacent vertices of Y for some g, h P A Γ .
Then gN pH 1 q`r X hN pH 2 q`r is nonempty. Also, the subgroup gN pH 1 qg´1 X hN pH 2 qh´1 (which is isomorphic to Z 2 by Lemma 3.21) acts coboundedly on the intersection gN pH 1 q`r X hN pH 2 q`r, and the quotient of this action has diameter at most .
Remark 3.24. We advise the reader to focus on the case of even Artin groups (i.e. m ab is even for all standard generators) in a first reading. In that particular case, we have V odd pΓq " V pΓq and we can take r " " 0 in the above statement. In particular, we are simply looking at intersections of cosets.
Let us explain briefly the need to consider neighbourhoods of cosets in the general case. If x and y are two standard generators connected by an edge of Γ with an odd label, then we have ∆ xy x∆´1 xy " y, where ∆ xy is the Garside element of A xy . In particular, we have g∆ xy N pxq∆´1 xy " gN pyq, and so the cosets g∆ xy N pxq and gN pyq are at bounded Hausdorff distance of one another (they are in a sense "parallel"). Let us assume that we are looking at vertices gN pHq and hN pH 1 q where H, H 1 correspond to standard generators a, b. As we saw in the proof of Lemma 3.21, the intersection gN pHqg´1 X hN pH 1 qh´1 stabilises a vertex of dihedral type. However, while H, H 1 correspond to standard generators a, b in V odd pΓq, it may happen that this vertex of dihedral type correspond to two different standard generators a 1 , b 1 that are conjugated to a, b respectively. Thus, while it may not be clear how to study the intersection gN paq X hN pbq, it is much simpler to study the corresponding intersection of parallel copies corresponding to cosets of N pa 1 q and N pb 1 q. This is why we do not simply consider the cosets gN paq, hN pbq but consider suitable neighbourhoods that contain those parallel copies corresponding to cosets of N pa 1 q and N pb 1 q.
Proof of Lemma 3.23. Let H 1 , H 2 P H and let gN pH 1 q, hN pH 2 q be cosets corresponding to adjacent vertices of Y .
By Lemma 3.8, we can assume that H 1 correspond to a standard generator a 1 , H 2 corresponds to an element of the form z b 1 b 2 , and the standard tree T 1 with apex ιpgN pH 1 qq contains the vertex of dihedral type ιphN pH 2 qq. This vertex corresponds to a coset of the form hA b 1 b 2 . Since the tree T 1 contains the vertex hA b 1 b 2 , it also contains a vertex corresponding to a coset of the form hxxb 1 y or hxxb 2 y for some x P A b 1 b 2 , so without loss of generality we will assume that it contains a vertex corresponding to a coset of the form hxxb 1 y. Since N pH 2 q " A b 1 b 2 by Lemma 2.28, we have that hxN pH 2 q " hN pH 2 q, so without loss of generality we will assume, up to replacing the element h by hx, that T 1 contains the vertex hxb 1 y. Moreover, since T 1 contains vertices corresponding to cosets of xa 1 y and xb 1 y, it follows from Lemma 2.22 that a 1 and b 1 are conjugated: By construction of r, we choose an element x 1 P Bpe, rq so that x 1 a 1 x´1 1 " b 1 .
By construction of x 1 , we have hx 1 v a 1 " hv b 1 . Thus, the apex of the standard tree T 1 is the coset gN pa 1 q " hx 1 N pa 1 q, so we can assume without loss of generality that g " hx 1 . We thus have gN pH 1 q " hx 1 N pa 1 q and hN pH 2 q " hA b 1 b 2 . Since x 1 P Bpe, rq, the intersection gN pH 1 q`r X hN pH 2 q`r " hx 1 N pa 1 q`r X hN pz b 1 b 2 q`r contains the intersection hx 1 N pa 1 qx´1 1 X hN pz b 1 b 2 q " hN pb 1 q X hN pz b 1 b 2 q " hxb 1 , z b 1 b 2 y, the latter equality following from Lemma 3.21.
Since Bpe, rq contains only finitely many elements of A Γ and H is finite, it follows from [Hru10, Proposition 9.4] that there exists a constant ą 0 (that does not depend on the choice of standard generators a 1 , b 1 , b 2 and element x 1 ) such that and in particular gN pH 1 q`r X hN pH 2 q`r " hx 1 N pa 1 q`r X hN pz b 1 b 2 q`r Ă hxb 1 , z b 1 b 2 y` .
Thus, the intersection gN pH 1 q`r X hN pH 2 q`r is at Hausdorff distance at most from hxb 1 , z b 1 b 2 y.
The conjugate hxb 1 , z b 1 b 2 yh´1 is equal to xga 1 g´1, hz b 1 b 2 h´1y " gN pH 1 qg´1 X hN pH 2 qh´1 by Lemma 3.21. Since kxb 1 , z b 1 b 2 yk´1 acts coboundedly on kxb 1 , z b 1 b 2 y and gN pH 1 q`r X hN pH 2 q`r is at Hausdorff distance at most from kxb 1 , z b 1 b 2 y, it follows that gN pH 1 qg´1 X hN pH 2 qh´1 acts coboundedly on gN pH 1 q`r XhN pH 2 q`r and the quotient space has diameter at most .
Definition 3.25. Let gN pHq be a vertex of the commutation graph Y . For each Y -adjacent coset hN pH 1 q, the hH 1 h´1-orbit of a point of gN pHq`r X hN pH 1 q`r is called an incident direction at gN pHq.
Fixing gN pHq and letting hN pH 1 q vary over the vertices of Y that are adjacent to gN pHq, we denote by IpgN pHqq the set of all such incident directions.
We advise the reader to look at Figure 3 for an illustration of this notion. For later use, we observe: Lemma 3.26. For each vertex gN pHq of the commutation graph Y , there are finitely many gN pHqg´1-orbits of incident directions at gN pHq.
Proof. By Lemma 2.30 and Corollary 2.36, there are finitely many gN pHqg´1-orbits of vertices of Y adjacent to the vertex gN pHq. Thus, it is enough to show that for a fixed vertex hN pH 1 q adjacent to gN pHq, there are only finitely many gN pHqg´1-orbits of incident directions corresponding to hH 1 h´1-orbits. This is indeed the case, since gN pHqg´1 contains gN pHqg´1 X hN pH 1 qh´1, which acts coboundedly, hence cofinitely (since the Cayley graph of A Γ is locally finite) on the intersection gN pHq`r X hN pH 1 q`r by Lemma 3.23.
For later purposes we also note the following consequence: Corollary 3.27. There are finitely many orbits of edges in Y .

The blown-up commutation graph
We now introduce a variant of the commutation graph of a group. This is this complex X to which we apply Theorem 1.4 for Artin groups of large and hyperbolic type.
As per Convention 3.14, we are considering an Artin group of large and hyperbolic type A Γ with Γ connected, and let H be as in Definition 3.4. For each H P H, let N pHq denote its normaliser. Recall that Ů HPH A Γ {N pHq is the vertex set of the commutation graph Y . Our goal is to produce X by modifying Y . Roughly speaking, each vertex of Y will be "blown up" to a cone on a certain discrete space quasi-isometric to Z, and each edge of Y will be blown up to the simplicial join of the blow-ups of its vertices. Figure 4 and the discussion following Definition 4.7 may help the reader visualise the construction.
In what follows, we equip A Γ with the word metric d A Γ coming from the standard generating set given by the vertices of Γ. When we refer to distances in subspaces of A Γ (e.g. cosets of subgroups), we are using the subspace metric inherited from pA Γ , d A Γ q.

4.1.
Blow-up data. The action of A Γ on the commutation graph Y hides too much information to use it as the underlying simplicial complex in a combinatorial HHS structure. For example, subgroups of the form N pHq, H P H, fix vertices in the commutation graph, so by Lemma 3.21, edges have Z 2 stabilisers. By Lemma 3.9, edges are maximal simplices. Thus any Y -graph (see Definition 1.1) has Z 2 vertex stabilisers, and so it is not a quasi-isometry model for A Γ . We remedy this by replacing the commutation graph by a complex X in which the vertices of the commutation graph have been "blown up".
We now describe the geometric data necessary to perform such a blow-up construction and show that it exists in the case of Artin groups of large and hyperbolic type. Some intuition for the definition of blow-up data is provided by Figure 3.
N pH 2 q xH, H 1 y xH, H 2 y Figure 3. A simplified picture illustrating two adjacent edges of Y , with above each vertex a depiction of the corresponding cosets N pHq, N pH 1 q, N pH 2 q. Each such coset is quasi-isometric to a product of a tree with a line, and the directions H, H 1 , H 2 are represented vertically in each coset. Two cosets N pHq and N pH 1 q that are adjacent in Y have uniform neighbourhoods that meet along a subset with a cobounded action of the Z 2 -subgroup HˆH 1 (shaded regions in the picture). The H 1 -and H 2 -orbits in the corresponding shaded regions have a certain "slope" when seen inside N pHq. The action of N pHq on the quasiline Λ N pHq from the blow-up data is chosen so that H acts with unbounded orbits on it, but H 1 and H 2 act with bounded orbits on Λ N pHq .
The reader may want to read the statement of Lemma 4.19 already, in order to understand the importance of the previous conditions, especially condition (C). While the equivariance condition (A) will be necessary for our constructions, it is slightly redundant in the previous definition, as we can start from a "non-equivariant" blow-up data and extend it equivariantly: Lemma 4.2. Suppose that for every H P H, we have: ‚ a discrete metric space Λ N pHq that is pB 0 , B 0 q-quasi-isometric to Z, with an action of N pHq on it by isometries with at most B 0 orbits of points and which fixes the Gromov boundary pointwise. ‚ a N pHq-equivariant, pL 0 , L 0 q-coarsely lipschitz map Then it is possible to extend equivariantly this data into a blow-up data for Y .
Proof. Fix a favourite representative g of the coset gN pHq. Let Λ gN pHq be a copy of Λ N pHq . We define φ gN pHq pgxq " φpxq for x P N pHq`r, and the action of gN pHqg´1 on Λ gN pHq is defined as gng´1¨λ " nλ for n P N pHq and λ P Λ gN pHq . (Note that the coset representative g is fixed throughout.) We have that Λ gN pHq is pB 0 , B 0 q-quasi-isometric to Z, gN pHqg´1 acts by isometries, it has at most B 0 orbits, and the orbits of gHg´1 are unbounded. The fact that φ gN pHq is coarsely Lipschitz is also straightforward.
Let us check equivariance (which is not immediate as we chose coset representatives to define the φ maps). Let g P A Γ , and consider a coset hN pHq. We can assume that h is the favourite representative of its coset, and let ghn be the favourite representative of ghN pHq, where n P N pHq. The isometry g : Λ hN pHq Ñ Λ ghN pHq is defined as gpxq " n´1x (so that we are using the action of N pHq on Λ N pHq ). For x P Λ hN pHq we have φ ghN pHq pgxq " φ ghN pHq pghn n´1h´1xq To show C, we note that by equivariance, for x P gN pHq`r and hN pH 1 q adjacent to gN pHq in Y , we have φ gN pHq phH 1 h´1¨xq " gφ N pHq pg´1phH 1 h´1¨xqq.
Note that g´1phH 1 h´1¨xq " g´1hH 1 h´1g¨pg´1xq P N pHq`r and g´1hH 1 h´1g commutes with H, so that N pHq is adjacent to g´1hN pH 1 q. In particular g´1hH 1 h´1¨x is in IpN pHqq, so that φ N pHq pg´1phH 1 h´1¨xqq has diameter at most B 0 . Since g is an isometry, the same holds for φ gN pHq phH 1 h´1¨xq, as required.
Existence of blow-up data. We now verify that blow-up data exists for A Γ and Y . The rest of this subsection is independent of the rest of the article, as the arguments in the next sections rely on the existence of blow-up data, but not on the specific choice of blow-up data. The reader may thus want to skip the rest of this subsection in a first reading.
As we will see, the key idea is to transform quasimorphisms into actions on quasilines. We use a lemma from [ABO19] to do so (the idea behind it can be traced further back, see e.g. [Man05, Proposition 4.4]). We need the following two general lemmas, which also underpin the strategy followed in the forthcoming paper [HRSS21]. Proof. Since the central extension corresponds to a bounded class, there is a set-theoretic section s : F Ñ G of π such that the element spf 1 qspf 2 qspf 1 f 2 q´1 takes finitely many values as f 1 , f 2 vary in F . Since s is a section of π, we have spf 1 qspf 2 qspf 1 f 2 q´1 P kerpπq " ιpZq for all f 1 , f 2 P F , so for each f 1 , f 2 P F we can choose an integer cpf 1 , f 2 q such that ιpcpf 1 , f 2 qq " spf 1 qspf 2 qspf 1 f 2 q´1. Note that |cpf 1 , f 2 q| is bounded independently of f 1 , f 2 , because spf 1 qspf 2 qspf 1 f 2 q´1 takes only finitely many possible values.
We now define φ. Let x P G. Then there exist unique f x P F, t x P Z such that x " spf x qιpt x q. We set φpxq " t x . We first verify that φ is a quasimorphism. Indeed, let x, y P G. Then, since ιpZq is central, we have xy " spf x qιpt x qspf y qιpt y q " spf x qspf y qιpt x`ty q " spf x t y qιpcpf x , f y q`t x`ty q.

So
φpxyq " t x`ty`c pf x , f y q " φpxq`φpyq`cpf x , f y q, whence φ is a quasimorphism because of the uniform bound on |cpf x , f y q|. Finally, observe that for all t P Z, we have φpιptqq " t, proving the last part of the lemma.
Lemma 4.4 (Action on a quasiline). Let 1 Ñ Z ι ÝÑ G π ÝÑ F Ñ 1 be a central extension with F a hyperbolic group. Let tC α u αPI be a finite collection of infinite cyclic subgroups of G such that tπpC α qu αPI is a malnormal collection of infinite cyclic subgroups of F . Then G has an (infinite) generating set S such that: ‚ CaypG, Sq is quasi-isometric to Z; ‚ ιpZq is unbounded in CaypG, Sq; ‚ there exists D ě 0 so that for each x, g P G and each α the diameter of gC α g´1x is bounded by D.
Proof. By Lemma 4.15 in [ABO19], it suffices to produce a quasimorphism ψ : G Ñ R such that ψ is unbounded on ιpZq, and each ψpgC α g´1xq is contained in an interval of uniformly bounded size. For each α P I, let a α generate the infinite cyclic group πpC α q. So, txa α yu αPI is a malnormal collection in F , and each a α has infinite order.
We now construct a collection of quasimorphisms; the quasimorphism ψ will be a linear combination of these.
First, let φ : G Ñ Z be the quasimorphism provided by Lemma 4.3, which applies since F is hyperbolic and therefore every cohomology class is bounded [Min01,Theorem 15]. Let ψ 0 be the homogenisation of φ.
Then we claim that ψ is a quasimorphism on G with the required properties. Indeed, ψ| ιpZq " ψ 0 | ιpZq , so ψ is unbounded on ιpZq. Next, note that ψpC α q " t0u, so that for any c P C α and g, x P G we have that |ψpgcg´1xq´ψpxq| is bounded by 3 times the defect of ψ, showing that ψpgC α g´1xq is contained in an interval of uniformly bounded size.
We now use the two preceding lemmas to produce blow-up data. Choose one element of each N pHq-orbit in IpN pHqq; there are finitely many by Lemma 3.26. Each of these is an orbit g i H i g´1 i¨x i in N pHq`r.
Each g i H i g´1 i is infinite cyclic. Moreover, g i H i g´1 i XH " t1u by Lemma 3.21, so πpg i H i g´1 i q is again infinite cyclic.
Observe that tπpg i H i g´1 i q : α P Iu is a malnormal collection of subgroups of F . Indeed, suppose not, so that for distinct i, j, the following holds. Choose elements c, c i , c j (either standard generators or elements of the form z ab ) generating H, H i , H j respectively. Then, for some k P N pHq, there exist positive integers p, q and an integer r such that kc p j k´1 " c q i c r . Now, c q i c r centralises c i , whence kc p j k´1 centralises c i . So, c i centralises kc j k´1, by Lemma 2.29. Hence c i , kc j k´1, and c pairwise commute, contradicting Lemma 3.9.
So, Lemma 4.4 yields a generating set S of N pHq. Recall that N pHq`r is a finite disjoint union of right cosets N pHqc i of N pHq. We can then consider the graph with vertex set N pHq`r with vertices g, h connected by an edge if g´1h or h´1g belong to S Y tc i u. By [CC07, Theorem 5.1] (or a direct argument) the vertex set Λ N pHq endowed with the induced metric is N pHq-equivariantly quasi-isometric to to the quasiline CaypG, Sq.
We can define φ N pHq : N pHq Ñ Λ N pHq to be the identity map, and we observe that Λ N pHq and φ N pHq satisfies the conditions in Lemma 4.2.
Since there are finitely many subgroups H P H, we can choose constants B 0 , L 0 such that each Λ N pHq is pB 0 , B 0 q-quasi-isometric to Z, the maps Φ N pHq are pL 0 , L 0 q-coarsely lipschitz (where N pHq carries the word metric from the fixed finite generating set of A Γ ), and each element of IpN pHqq maps to a set in Λ N pHq of diameter at most B 0 , for all H. By construction, there are boundedly many N pHq-orbits in Λ N pHq .
We thus have that tpΛ N pHq , φ N pHq u satisfies conditions (B') and (C') from Lemma 4.2. By Lemma 4.2, it possible to extend the construction equivariantly to obtain condition (A).

4.2.
The blown-up commutation graph. We now define the simplicial complex X. Fix a blow-up data as in Definition 4.1.
Definition 4.6 (Blown-up commutation graph). We define a simplicial graph, called the blown-up commutation graph, as follows: ‚ The graph has a vertex for each element of ğ HPH G{N pHq, and for each coset gN pHq in the above set, we add Λ gN pHq to the vertex set. ‚ For every g P G and H P H, we put an edge between gN pHq and each vertex of Λ gN pHq . Moreover, whenever g, h P G and H, H 1 P H are such that gHg´1 and hH 1 h´1 are distinct and commute (i.e. when gN pHq and hN pH 1 q are Y -adjacent), we join every vertex in tgN pHqu Y Λ gN pHq to every vertex of thN pH 1 qu Y Λ hN pH 1 q . We denote by X the flag completion of this graph.
Observe that A Γ acts on X by simplicial automorphisms, because of Definition 4.1.(A).
Definition 4.7 (Projection, fibre, roots and leaves). There is a natural A Γ -equivariant simplicial projection p : X Ñ Y obtained as follows. At the level of vertices, for H P H and g P G, we set ppgN pHqq " gN pHq, ppΛ gN pHq q " gN pHq. Note that two adjacent vertices of X are either sent to the same vertex or to two adjacent vertices of Y . Thus, this definition extends to a map from X to Y . For a simplex ∆ of X, we denote by ∆ the image simplex pp∆q Ă Y .
Note that for each vertex v :" gN pHq P Y , the preimage X v :" p´1pvq, called the fibre of v, is a tree that is the simplicial cone over Λ gN pHq : where "˚" denotes the simplicial join between two sub-complexes. The root of X v will be the vertex gN pHq, while the other vertices of X v will be called leaves.
Thus, X can be thought of as being obtained from Y by blowing-up each vertex gN pHq of Y into a cone over a quasiline, in such a way that fibres X v and X v 1 over adjacent vertices v, v 1 of Y span a simplicial join X v˚Xv 1 in X. Figure 4. A portion of the blow-up X over an edge of Y . The fibres X v and X v 1 span a simplicial join in X. The figure illustrates how an edge of X v and an edge of X v 1 span a tetrahedron of X.
Definition 4.8. We define the map q : X Ñ p D as q :" ι˝p, where ι : Y Ñ p D is the map to the coned-off Deligne complex defined in Lemma 3.7 and Definition 3.12.
Lemma 4.9. The map q : X Ñ p D is an A Γ -equivariant simplicial quasi-isometry.
Proof. The map q is simplicial, being a composition of simplicial maps, and is equivariant by construction. We have that p is a surjective simplicial whose fibres have uniformly bounded diameter (the bound being 2), so that p is a quasi-isometry. Also, ι is a quasi-isometry by Lemma 3.13, and hence q is also a quasi-isometry.

4.3.
Simplices of the blow-up and their links. In this subsection, we describe the simplices of X and their links, and prove that X satisfies several of the properties from Definition 1.3 and conditions from Theorem 1.4.
Definition 4.10 (Join decomposition of simplices). A simplex ∆ of X naturally decomposes as a join as follows. For each vertex v P ∆, we define the fibre of ∆ at v by ∆ v :" ∆XX v Ă ∆, which is either a single vertex or a single edge of ∆. Then ∆ is naturally the join of all its fibres: Lemma 4.11. Let ∆ be a simplex of X. Then its projection ∆ is either a vertex or an edge. In particular, X is a 3-dimensional simplicial complex.
Proof. Note that the projection ∆ is a simplex of Y . It thus follows from Lemma 3.9 that ∆ is either a vertex or an edge. Since the simplices of X v are either vertices or edges, the result follows.
Lemma 4.12. Let ∆ be a simplex of X. The link Lk X p∆q decomposes as the following simplicial join: Proof. A vertex in Lk X p∆q is adjacent to every vertex of ∆ by definition. Since edges of X either project to vertices or edges of Y , we have that p`Lk X p∆q˘Ă St Y p∆q. For two adjacent vertices v, v 1 of Y , every vertex of X v is adjacent to every vertex of X v 1 . It follows that for v P St Y p∆q, a vertex w P X v is in Lk X p∆q if and only if it is in Lk Xv p∆ X X v q. The result follows.
Theorem 1.4 requires that the action of A Γ on X has finitely many orbits of links, which we now verify.
Lemma 4.13 (Finite index set). The action of A Γ on X has finitely many orbits of subcomplexes of the form Lk X p∆q, where ∆ is a simplex of X.
In the proof of the lemma we will give a classification of links which will also be useful later on.
Proof. Let ∆ be a non-maximal simplex of X and let∆ be the simplex of Y to which ∆ projects. We divide into cases according to∆ and the fibers in ∆ over the vertices of∆.
(1)∆ is a vertex gN pHq, and ∆ is the single vertex gN pHq. There are finitely many A Γ orbits of such ∆, since H is finite. Hence there are finitely many orbits of links of such ∆.
(2)∆ is a vertex gN pHq, and ∆ is a vertex λ P Λ gN pHq . There are boundedly many gN pHqg´1 orbits of λ for the given gN pHq, by Definition 4.1, so there are finitely many orbits of such ∆. (3)∆ is a vertex gN pHq, and ∆ is spanned by tgN pHq, λu for some λ P Λ gN pHq . Since gN pHqg´1 acts on Λ gN pHq with boundedly many orbits, and there are finitely many orbits of vertices gN pHq, there are finitely many orbits of such simplices ∆. (4)∆ is an edge joining gN pHq to hN pH 1 q, and ∆ is the 1-simplex spanned by tgN pHq, hN pH 1 qu.
By Corollary 3.27, there are finitely many orbits of such simplices. (5)∆ is an edge joining gN pHq to hN pH 1 q, and ∆ joins gN pHq to some µ P Λ hN pH 1 q .
Observe that all simplices that join gN pHq to a point in Λ hN pH 1 q have the same link, so there are at most as many A Γ orbits of links of such ∆ as there are orbits of edges in Y , i.e. finitely many by Corollary 3.27. (6)∆ is an edge joining gN pHq to hN pH 1 q, and ∆ joins λ P Λ gN pHq to µ P Λ hN pH 1 q .
Then Lk X p∆q is the edge joining gN pHq, hN pH 1 q; there are finitely many orbits of these. (7)∆ is an edge joining gN pHq, hN pH 1 q and ∆ is spanned by tgN pHq, λ, hN pH 1 qu, where λ P Λ gN pHq . Then Lk X p∆q " Λ hN pH 1 q , and there are finitely many orbits of such subcomplexes since there are finitely many orbits of vertices in Y . (8)∆ is an edge joining gN pHq, hN pH 1 q, and ∆ is spanned by tgN pHq, λ, µu, where λ P Λ gN pHq and µ P Λ hN pH 1 q . Then Lk X p∆q " hN pH 1 q, and there are finitely many orbits of such links because Y has finitely many orbits of vertices. (9) ∆ " H, in which case Lk X p∆q " X, and there is a single orbit. This exhausts all the cases.
The list in the previous proof allows to prove that the blow-up X satisfies the finite complexity condition from Definition 1.3.
Proposition 4.14 (Finite complexity). There exists N such that the following holds. Let ∆ 1 , . . . , ∆ n be non-maximal simplices of X such that Lk X p∆ i q Ĺ Lk X p∆ i`1 q for 1 ď i ď n´1. Then n ď N .
Proof. Suppose that Σ, ∆ are non-maximal simplices with Lk X pΣq Ĺ Lk X p∆q. Considering the nine types of simplex explained in the proof of Lemma 4.13, we see that links of the same type cannot be properly contained in each other. Hence the length of a chain as in the statement is at most 9.
In the rest of this subsection is to prove the following: For all simplices ∆ 1 , ∆ 2 of X such that there exists a simplex Σ of X with unbounded link and with the property that Lk X pΣq Ă Lk X p∆ 1 q X Lk X p∆ 2 q the following holds. There exists a simplex ∆ of X containing ∆ 1 , and such that for every simplex Σ of X with unbounded link and such that Lk X pΣq Ă Lk X p∆ 1 q X Lk X p∆ 2 q, we have Lk X pΣq Ă Lk X p∆q.
Proof. The finite height condition follows from Proposition 4.14.
Note that LkpΣq is unbounded if and only if Σ is either the empty simplex, an edge contained in a fibre of X, or a triangle of X containing exactly two roots, by the classification given in the proof of Lemma 4.13.
The case Σ " ∅ is immediate since in that case Lk X pΣq " Lk X p∆ 1 q " Lk X p∆ 2 q " X, and we can take ∆ " ∅. We thus focus on the other two cases.
Case 1: Suppose that there exists such a simplex Σ such that Lk X pΣq Ă Lk X p∆ 1 q X Lk X p∆ 2 q, and such that Σ is an edge contained in a fibre of X. Then its projection is a single vertex v P Y , and by Lemma 4.12 we have Lk X pΣq " p´1`Lk Y pvq˘. Since Lk X pΣq is contained in Lk X p∆ 1 q X Lk X p∆ 2 q, it follows from Lemma 3.9 that ∆ 1 , ∆ 2 are also contained in X v . Since ∆ 1 is either a vertex or an edge of X v , we can choose an edge ∆ contained in St Xv p∆ 1 q. We thus have Lk X p∆q " p´1`Lk Y pvq˘by Lemma 4.12, and Lk X p∆q satisfies the required property. Indeed, any link contained in Lk X p∆ 1 q X Lk X p∆ 2 q and strictly larger than Lk X p∆q must contain a vertex of X v , and so this link is a simplicial join, hence bounded.
Case 2: Suppose that for every simplex Σ such that Lk X pΣq Ă Lk X p∆ 1 q X Lk X p∆ 2 q, Σ is a triangle of X with exactly two roots. Choose such a simplex Σ. Then there exists a vertex v P Y such that Lk X pΣq consists of all the leaves of X v . Moreover, it follows from Lemma 3.9 that for every other simplex Σ 1 with unbounded augmented link such that Lk X p∆ 1 q X Lk X p∆ 2 q, we have Lk X pΣ 1 q Ă X v , and in particular Lk X pΣ 1 q " Lk X pΣq. Now since Lk X pΣq Ă Lk X p∆ 1 q, then ∆ 1 is either a vertex or an edge of St Y pvq.
If ∆ 1 " tvu, then we choose a vertex u adjacent to v, an edge f u of X u , and we define ∆ as the triangle of X spanned by f u and the root of X v . We thus have ∆ Ă St X p∆ 1 q, and ∆ satisfies the required conditions.
If ∆ 1 is an edge containing v, we denote by u the other vertex of ∆ 1 . We choose an edge f u of X u that belongs to St Xu pp∆ 1 q u q, and we define ∆ as the triangle of X spanned by f u and the root of X v . We thus have ∆ Ă St X p∆ 1 q, and ∆ satisfies the required conditions. 4.4. From maximal simplices to elements of A Γ . In this subsection, we associate to each maximal simplex of X a uniformly bounded subset of A Γ . We first describe the maximal simplices of X: Remark 4.16 (Description of maximal simplices). Let ∆ be a maximal simplex of X. Then ∆ has the following form. First, ∆ is an edge of Y joining a vertex gN pHq to a vertex hN pH 1 q. Then, there are vertices λ P Λ gN pHq and µ P Λ hN pH 1 q such that ∆ is the join of the edges tgN pHq, λu and thN pH 1 q, µu.
Definition 4.17. For simplices σ 1 , σ 2 of X projecting to vertices of Y and spanning a simplex of X, we denote that simplex by ∆pσ 1 , σ 2 q.
Recall that one of our goals is to define an X-graph W quasi-isometric to A Γ , and whose vertex set is the set of maximal simplices of X, see Definition 1.3. We start by defining the vertices of this graph and we construct a map to A Γ . The edges of W and the quasi-isometry W Ñ A Γ will be constructed in Section 5.
Definition 4.18. We denote by W p0q the set of maximal simplices of X.
The blow-up data was constructed to obtain the following crucial lemma: Lemma 4.19 (Bounded sets in A Γ ). There exists B 1 ě 0 such that the following holds for all B ě B 1 . Let gN pHq, hN pH 1 q be Y -adjacent cosets. Let λ P Λ gN pHq , µ P Λ hN pH 1 q . Then the subset φ´1 gN pHq pN B pλqq X φ´1 hN pH 1 q pN B pµqq of A Γ is nonempty and has diameter bounded in terms of B. Moreover, φ gN pHq pgN pHq`r X hN pH 1 q`rq is B 1 -dense.
We prove the lemma after the following two auxiliary lemmas (coboundedness in the first lemma will only be needed in the next section).
Lemma 4.20. Suppose C 0 , C 1 are infinite cyclic groups and Λ 0 and Λ 1 are quasilines. Suppose that C 0ˆC1 acts on both Λ i with the property that C i acts with unbounded orbits on Λ i and with bounded orbits on Λ i`1 (where the action of C i is the restriction of the action to the subgroup C i of C 0ˆC1 ). Then the diagonal action of C 0ˆC1 on Λ 0ˆΛ1 is proper and cobounded (where the product is given the 1 metric).
Let B be the bound on C i orbits in Λ i`1 . Then d Λ 0 ppc 0 , c 1 q¨µ 0 , pc 0 , 1q¨µ 0 q ď B and d Λ 1 ppc 0 , c 1 q¨µ 1 , p1, c 1 q¨µ 1 q ď B. So and d Λ 0 pp1, c 1 q¨µ 1 , µ 1 q ď L`B. Since C i has unbounded orbits on the quasiline Λ i , there are finitely many such c i , for i P t0, 1u. So pc 0 , c 1 q is one of finitely many elements of C 0ˆC1 , as required.
Regarding coboundedness, the action of the cyclic group C i on the quasiline Λ i is, say, B icobounded, which easily implies that the action of C 0ˆC1 is pB 0`B1`2 Bq-cobounded.
Lemma 4.21. Let X and Y be metric spaces, and let f : X Ñ Y be a coarsely Lipschitz map. Suppose that a group G acts metrically properly on Y and coboundedly on X, and that f is G-equivariant. Then f is uniformly metrically proper, that is, for every L there exists D such that the preimage under f of any ball of radius L in Y has diameter at most D.
Proof. Let K be such that f is pK, Kq-coarsely lipschitz. Fix x P X and R ě 0 such that G¨B X R pxq " X. Let L ě 0. By properness, there exists a finite set F L Ă G such that d Y phf pxq, f pxqq ď L`2pKR`Kq implies h P F L .
We will show that if two points of X map L-close in Y then they are DpLq-close in X; this is easily seen to imply the lemma (after increasing D).
Suppose x 0 , x 1 P X satisfy d Y pf px 0 q, f px 1 qq ď L. Choose g 0 , g 1 P G such that d X px i , g i xq ď R for i P t0, 1u. Then d Y pg 0 f pxq, g 1 f pxqq ď L`2pKR`Kq, so d Y pg´1 1 g 0 f pxq, f pxqq ď L`2pKR`Kq. Thus g´1 1 g 0 P F L , so as required.
Proof of Lemma 4.19. Fix Y -adjacent cosets gN pHq, hN pH 1 q. Recall from Lemma 3.21 that xgHg´1, hH 1 h´1y is naturally isomorphic to gHg´1ˆhH 1 h´1. By definition of the blow-up data, we have a gHg´1ˆhH 1 h´1-equivariant map φ gN pHqˆφhN pH 1 q : gN pHq`r X hN pH 1 q`r Ñ Λ gN pHqˆΛhN pH 1 q .
Equipping gN pHq`r X hN pH 1 q`r with the word metric and Λ gN pHqˆΛhN pHq with the 1metric, we see that φ gN pHqˆφhN pH 1 q is p2L 0 , 2L 0 q-coarsely lipschitz.
Since gHg´1 stabilises hN pH 1 q`r (because it is contained in hN pH 1 qh´1 by definition of the edges of Y ), we have that it also acts on gN pHq`r X hN pH 1 q`r. Since gHg´1 has unbounded orbits in Λ gN pHq by Condition (B), we have that φ gN pHq restricts to a B 1 -coarsely surjective map on gN pHq`r X hN pH 1 q`r for some B 1 (proving the moreover part). By equivariance (Condition (A) ) and the fact that there are finitely many orbits of edges of Y (Corollary 3.27), the constant B 1 can be chosen independently of the cosets in question.
Let us prove the "nonempty" part. Fix λ P Λ gN pHq and µ P Λ hN pH 1 q . By coarse surjectivity, there exists x P gN pHq`r X hN pH 1 q`r such that d Λ gN pHq pφ gN pHq pxq, λq ď B 1 . In view of Condition (C), the hH 1 h´1-orbit of φ gN pHq pxq is contained in the pB 0`B1 q-neighbourhood of λ. Since the action of hH 1 h´1 on Λ hN pHq is cobounded (with uniform constant), up to enlarging B 1 we can find an element in hH 1 h´1-orbit of x in gN pHq`r X hN pH 1 q`r that maps B 1 -close to both λ and µ, as required.
We now make the following claim, which will also be used later.
Claim 4.22. Fix Y -adjacent cosets gN pHq, hN pH 1 q. Then the map φ gN pHqˆφhN pH 1 q : gN pHq`r X hN pH 1 q`r Ñ Λ gN pHqˆΛhN pH 1 q is proper. More precisely, there exists f 0 : R ě0 Ñ R ě0 independent of gN pHq, hN pH 1 q such that, for all s P R ě0 , the preimage under φ gN pHqˆφhN pH 1 q of any s-ball in Λ gN pHqˆΛhN pH 1 q has diameter at most f 0 psq. Moreover, the action of xgHg´1, hH 1 h´1y on Λ gN pHqˆΛhN pH 1 q is cobounded.
Proof. By Lemma 4.20 and Conditions (B) and (C), the action of gHg´1ˆhH 1 h´1 on Λ gN pHqˆΛhN pH 1 q is proper and cobounded. Then, by Lemma 4.21 (with f " φ gN pHqφ hN pH 1 q ) and Lemma 3.23 we have there exists f 0 : R ě0 Ñ R ě0 such that, for all s P R ě0 , the preimage under φ gN pHqˆφhN pH 1 q of any s-ball in Λ gN pHqˆΛhN pH 1 q has diameter at most f 0 psq. Finiteness of H and condition (A) imply that this f 0 can be chosen independently of the cosets gN pHq, hN pH 1 q. Now, if x, y P gN pHq`r X hN pH 1 q`r map B-close to the image of x under both φ gN pHq and φ hN pH 1 q , then d A Γ px, yq ď f 0 p2Bq, yielding the required bound on the diameter of φ´1 gN pHq pN B pλqq X φ´1 hN pH 1 q pN B pµqq.
We are almost ready to fix constants for the rest of the paper, but we first need lemma about the geometry of quasilines.  Fix gN pHq. Recall that any kH 1 k´1¨x P IpgN pHqq has image in Λ gN pHq of diameter bounded by B 0 (and B 0 does not depend on H, H 1 , g, k, x), by Definition 4.1.(C).
Let δ P W p0q be a maximal simplex spanned by vertices gN pHq, hN pH 1 q and λ P Λ gN pHq , µ P Λ hN pH 1 q . Recall from Lemma 4.19 that φ´1 gN pHq pN B 1 pλqq X φ´1 hN pH 1 q pN B 1 pµqq is a nonempty subset of A Γ , and the corresponding subset with B 1 replaced by B 1`B0 has diameter (in the word metric) at most B 2 .
This defines a coarse map w : W p0q Ñ A Γ .
Lemma 4.26 (Basic properties of w). The coarse map w is A Γ -equivariant and sends each maximal simplex to a nonempty subset of A Γ of diameter at most B 2 . Moreover, if δ P W p0q projects to the edge of Y joining cosets gN pHq, hN pH 1 q, then wpδq Ă gN pHq`r X hN pH 1 q`r.
Proof. Let δ be a maximal simplex, as above. By construction, wpδq is which is nonempty by our choice of B 1 (which we made in Convention 4.24). Also, by the choice of B 2 , wpδq has diameter at most B 2 . The "moreover" assertion holds by construction, as does A Γ -equivariance (see Condition A).
We also have: Proof. This follows since w is A Γ -equivariant and the action of A Γ on itself has one orbit.

The augmented complex
As per Convention 3.14, we are considering an Artin group of large and hyperbolic type A Γ with Γ connected, and we work with the commutation graph Y and blow-up X constructed in the preceding sections. In this section we construct a combinatorial HHS with underlying simplicial complex X.

5.1.
Construction of the augmented complex. Let r 1 , r 2 ě 0 be constants to be determined. The first step is to define a graph W r 1 ,r 2 whose vertex set is the set W p0q of maximal simplices of X; recall that Remark 4.16 and Definition 4.17 describe maximal simplices. The edges of W r 1 ,r 2 are as follows.
Definition 5.1 (W r 1 ,r 2 -edges). Let δ 0 , δ 1 be maximal simplices of X, where δ i " ∆pα i , β i q. Here, α i is an edge joining g i N pH i q to λ i P Λ g i N pH i q and β i is an edge joining h i N pH 1 i q to ‚ W -edges of type 1: Suppose g 0 N pH 0 q " g 1 N pH 1 q and h 0 N pH 1 0 q " h 1 N pH 1 1 q, and µ 0 " µ 1 . We declare δ 0 to be W r 1 ,r 2 -adjacent to δ 1 if d Λ g 0 N pH 0 q pλ 0 , λ 1 q ď r 1 .
Note that simplices joined by an edge of type 1 differ on a single vertex and project to the same edge of Y . ‚ W -edges of type 2: In this case, h 0 N pH 1 0 q " h 1 N pH 1 1 q and µ 0 " µ 1 , while g 0 N pH 0 q ‰ g 1 N pH 1 q. (So, ppδ 0 Y δ 1 q is the union of two distinct edges of Y that share a vertex.) In this case, we declare δ 0 and δ 1 to be W r 1 ,r 2 -adjacent if d A Γ pσpλ 0 q, σpλ 1 qq ď r 2 .
Note that simplices joined by an edge of type 2 intersect in an edge of the fibre over h 0 N pH 1 0 q " h 1 N pH 1 1 q. The action of A Γ on W p0q extends to an action of A Γ on W by graph automorphisms. The (finitely many) conditions needed on r 1 , r 2 for our constructions to work appear in the proofs of the various results below.
Morally, the definition of W -edges of type 2 should be the inequality in the following lemma. However, we need the definition exactly as stated to show fullness of links (see Proposition 5.6).
Lemma 5.2. For every r 2 there exists r 3 with the following property. Fix the notation of the "W -edges of type 2" part of Definition 5.1. Then Proof. We first make two preliminary claims.
Claim 5.3. There exists a function s such that the following holds. Given cosets g 0 N pH 0 q, hN pHq, and g 1 N pH 1 q that, as vertices of Y , form a path, and x i P g i N pH i q`r we have d A Γ px 0 , g 1 N pH 1 q`r X hN pHq`rq ď s´d A Γ px 0 , x 1 q¯.
Proof. Set d " d A Γ px 0 , x 1 q. Up to multiplying all objects on the left by x´1 0 , we can assume that x 0 is the identity. Note that there are N " N pdq ă`8 cosets gN pH 1 q with H 1 P H within distance d`r of x 0 . Therefore, it suffices to consider fixed cosets g 0 N pH 0 q, g 1 N pH 1 q as in the statement, and prove the claimed inequality for x i in those particular g i N pH i q`r. Note that since Y does not contain triangles (Lemma 3.9), for each pair g 0 N pH 0 q, g 1 N pH 1 q there is at most one hN pHq at distance 1 in Y from both. In particular, there are at most finitely many hN pHq that can occur, so that once again we can fix one. But at this point, the left hand side is a number, so we are done.
Claim 5.4. There exists a function t such that the following holds. Suppose that gN pH 1 q, hN pHq are adjacent vertices of Y and λ P Λ gN pH 1 q . Given x P σpλq there exists y P σpλq X hN pHq`r such that d A Γ px, yq ď tpd A Γ px, gN pH 1 q`r X hN pHq`rqq.
Proof. Similarly to the proof of the previous claim, we can assume that x is the identity and that gN pH 1 q, hN pHq are fixed. However, λ is not; there are infinitely many possible ones. To overcome this, we will find two candidates for y, and we will see that at least one of them is in σpλq X hN pHq`r, which suffices.
Towards this, recall the order ă from Lemma 4.23. From Convention 4.24 we know that the lemma applies with M " B 1 {10, and that φ gN pH 1 q pgN pH 1 q`r X hN pHq`rq is B 1 {100-dense.
Hence, we can find y 1 , y 2 P gN pH 1 q`r XhN pHq`r such that z i " φ gN pH 1 q py i q P Λ gN pH 1 q satisfy the following: There are now two cases. If d Λ gN pH 1 q pz, λq ď B 1 {2, then either z i lies in N B 1 pλq, that is, either y i lies in σpλq, and we are done. If not, any two points λ, z, z 1 , z 2 are either ăcomparable by construction, or at distance at least B 1 {10, whence, again, ă-comparable. If λ ă z 1 , then it is readily seen from the last item of Lemma 4.23 that z 1 P N B 1 pλq, since z lies in said ball and a geodesic from λ to z passes within distance B 1 {100 of z 1 , so that Similarly, if z 2 ă λ then z 2 P N B 1 pλq. On the other hand, we cannot have z 1 ă λ ă z or z ă λ ă z 2 , for otherwise we would have d Λ gN pH 1 q pz, λq ă B 1 {2. We now covered all cases.
We now argue that for every r 2 there exists r 2.5 so that if d A Γ pσpλ 0 q, σpλ 1 qq ď r 2 then d A Γ pσpλ 0 q X h 0 N pH 1 0 q`r, σpλ 1 q X h 0 N pH 1 0 q`rq ď r 2.5 (That is, if the σ come close, then they come close in the subspace of A Γ corresponding to the middle vertex h 0 N pH 1 0 q " ppδ 0 q X ppδ 1 q.) Indeed, for x i P σpλ i q such that d A Γ px 0 , x 1 q ď r 2 , we can find y i as in the second claim above, and in view of both claims we have d A Γ py 1 , y 2 q ď 2tpspr 2 qq`r 2 .
Pick p P σpλ 0 q X h 0 N pH 1 0 q`r and q P σpλ 1 q X h 0 N pH 1 0 q`rq with d A Γ pp, qq ď r 2.5 . By translating the pair p, q by an element h 0 H 1 0 h´1, we find points p 1 , q 1 so that ‚ d A Γ pp 1 , q 1 q ď r 2.5 (since we multiplied by an element of A Γ , ‚ d Λ g 0 N pH 0 q pφ g 0 N pH 0 q ppq, φ g 0 N pH 0 q pp 1 qq ď B 0 and d Λ g 1 N pH 1 q pφ g 1 N pH 1 q pqq, φ g 1 N pH 1 q pq 1 qq ď B 0 (Condition C), ‚ d Λ h 0 N pH 1 0 q pµ 0 , φ h 0 N pH 1 0 q pp 1 qq ď B 1 (Convention 4.24, coboundedness property of B 1 ). Furthermore, by Definition 4.1 (blow-up data), the first and third items yield d Λ h 0 N pH 1 0 q pµ 0 , φ h 0 N pH 1 0 q pqqq ď L 0 r 2.5`L0`B1 . Pick p 2 P wpδ 0 q and q 2 P wpδ 1 q. The we see that p 1 and p 2 are both points in g 0 N pH 0 q`r X h 0 N pH 1 0 q`r that map within bounded distance of pλ 0 , µ 0 q P Λ g 0 N pH 0 qˆΛh 0 N pH 1 0 q under φ g 0 N pH 0 qˆφh 0 N pH 1 0 q . By the properness part of Claim 4.22, there is a bound on d A Γ pp 2 , p 1 q. A similar argument applies to q 2 , and therefore we get a bound on d A Γ pp 2 , q 2 q, as required.
To use Theorem 1.4, we will need the following: Proposition 5.5. There exist R 1 , R 2 ě 0 such that the following holds for all r 1 ě R 1 , r 2 ě R 2 . Let W " W r 1 ,r 2 . Then W is connected and the action of A Γ on W is proper and cobounded. In particular, any orbit map A Γ Ñ W is a quasi-isometry.
Proof. The action is proper. Recall the A Γ -equivariant coarse map w : W p0q Ñ A Γ from Definition 4.25. Using Lemma 4.27, choose δ 0 P W p0q such that 1 P wpδ 0 q. We show that w is coarsely Lipschitz, for any r 1 , r 2 , which proves properness of the action since it provides a bound on d A Γ pwpγ 1 δ 0 q, wpγ 2 δ 0 qq ě d A Γ pγ 1 , γ 2 q´2B 2 when d W pγ 1 δ 0 , γ 2 δ 0 q is bounded.
First suppose that δ 0 , δ 1 are joined by a W -edge of type 2. Then, by Lemma 5.2, we have d A Γ pwpδ 0 q, wpδ 1 qq ď r 3 . Second, suppose that δ 0 , δ 1 are joined by a W -edge of type 1. Then, by definition, we have the following: associated to the coset h 0 N pH 1 0 q " h 1 N pH 1 1 q, we have µ 0 " µ 1 , and associated to the coset g 0 N pH 0 q " g 1 N pH 1 q, we have λ 0 , λ 1 P Λ gnN pH 0 q with d Λ g 0 N pH 0 q pλ 0 , λ 1 q ď r 1 .
Consider the map φ g 0 N pH 0 qˆφh 0 N pH 1 0 q : g 0 N pH 0 q X h 0 N pH 1 0 q Ñ Λ g 0 N pH 0 qˆΛh 0 N pH 1 0 q , and let f 0 : R ě0 Ñ R ě0 be the (properness) function as in Claim 4.22 (which says that φ g 0 N pH 0 qˆφh 0 N pH 1 0 q is proper). Letting x P wpδ 0 q and y P wpδ 1 q, we have by Lemma 4.26: In Λ h 0 N pH 1 0 q , the images of x, y are both pB 1`B0 q-close to µ 0 " µ 1 . So, d A Γ px, yq ď f 0 p4B 1`4 B 0`r1 q. This completes the proof that w is coarsely lipschitz. As a side remark, the constants depend on A Γ , the blow-up data, and the choice of r 1 , r 2 , but we have not yet needed to impose any restriction on r 1 , r 2 .
Connectedness and coboundedness. We will show at the same time that, for sufficiently large r 1 , r 2 , W r 1 ,r 2 is connected and the action of A Γ is cobounded. Note that a set of representatives for the A Γ -orbits can be taken to be tδ 1 i " ∆pα i , β i qu where ‚ the set tppδ 1 i qu of edges of Y is finite (we can arrange this since there are finitely many orbits of edges of Y by Corollary 3.27), and ‚ there exists a constant C so that if ppδ 1 i q " ppδ 1 j q then the following holds. Suppose that the endpoints of ppδ 1 i q are gN pHq and hN pH 1 q then d Λ gN pHq pλ i , λ j q ď C, where α k " tgN pHq, λ k u, and similarly in Λ hN pH 1 q .
The second item can be arranged because the action of xgHg´1, hH 1 h´1y on Λ gN pHqΛ h 1 N pH 1 q is cobounded by Claim 4.22 ("moreover" part). Since Y is connected by Lemma 3.10 (recalling that we are assuming that Γ is connected throughout), to show connectedness and coboundedness it suffices to show that there is a path in W r 1 ,r 2 connecting δ 0 to δ 1 when either (1) δ 0 , δ 1 are maximal simplices with ppδ 0 q " ppδ 1 q, or (2) δ 0 , δ 1 are maximal simplices such that ppδ 0 q intersects ppδ 1 q at a vertex gN pHq, and that moreover in the first case the length of the path is bounded if we are in the situation of the second bullet point above.
In the first case, it is easily seen that we can in fact connect the maximal simplices only using edges of type 1 (roughly, changing one vertex at a time moving a bounded amount in one of the relevant quasilines). This requires r 1 to be sufficiently large to move in the quasilines.
Since there are only finitely many gN pHqg´1-orbits of edges of Y emanating from gN pHq and N pHq is finitely generated, there is a sequence of edges ppδ 0 q " e 0 , . . . , e n " ppδ n q emanating from gN pHq so that the possible pairs pe i , e i`1 q belong to a fixed set of A Γ -orbits of pairs of edges of Y . From here, we see that we can change the orders of quantifiers, meaning that it suffices to show that given maximal simplices δ 0 , δ 1 of X as in case 2, there exist r 1 and r 2 such that δ 0 and δ 1 are W r 1 ,r 2 -adjacent.
To show that this holds, denote h i N pH 1 i q the other endpoint of ppδ i q, and observe that since the distance between the sets gN pHq`r X h i N pH 1 i q`r is at most some R 2 , for any r 2 ě R 2 there is a W -edge of type 2 connecting maximal simplices δ 1 0 , δ 1 1 with ppδ i q " δ 1 i . We know already that δ i can be connected to δ 1 i , so we are done.
To save notation, we often denote W r 1 ,r 2 by W , even though we will not fix r 1 , r 2 until later.
5.2. Fullness of links. On our way to proving that pX, W q is a combinatorial HHS, so as to apply Theorem 1.4, we need to verify condition (III) from Definition 1.3, which we do in the following proposition.
Proposition 5.6 (W -fullness of links). Let ∆ be a non-maximal simplex of X, and let v 1 , v 2 be two distinct non-adjacent vertices of Lk X p∆q that are contained in W -adjacent maximal simplices. Then v 1 , v 2 are contained in W -adjacent maximal simplices of St X p∆q.
Since W -edges come in two types, we split Proposition 5.6 in two.
Lemma 5.7 (Type 1 fullness). Let ∆ be a non-maximal simplex of X, and let v 1 , v 2 be two distinct non-adjacent vertices of Lk X p∆q. Suppose that there exist maximal simplices ∆ 1 , ∆ 2 of X such that v 1 P ∆ 1 , v 2 P ∆ 2 , and ∆ 1 , ∆ 2 are connected by a W -edge of type 1. Then v 1 , v 2 are contained in W -adjacent maximal simplices of St X p∆q.
Proof. Recall that W -edges of type 1 connect maximal simplices with the same projection in Y . Let e "∆ 1 "∆ 2 be the edge of Y to which ∆ 1 , ∆ 2 project, and denote the endpoints of e by u and v. Then ∆ i " ∆pf i u , f i v q, where f i u is an edge with an endpoint λ 1 u P Λ u , and similarly for v. Also, we have say, f 1 u " f 2 u and d Λv pλ 1 v , λ 2 v q ď r 1 . Since v 1 , v 2 are not adjacent in X, we have v 1 " λ 1 v and v 2 " λ 2 v , since those are the only non-common vertices of ∆ 1 and ∆ 2 . Since v 1 , v 2 P Lk X p∆q, Lemma 4.12 implies that there exist a vertex u 1 of Y adjacent to v and an edge f u 1 of X u 1 such that f u 1 is contained in St X p∆q. By Definition 5.1, the maximal simplices ∆pf 1 v , f u 1 q and ∆pf 2 v , f u 1 q are joined by a type 1 edge in W . These simplices respectively contain v 1 , v 2 and lie in the star of ∆, as required.
Lemma 5.8 (Type 2 fullness). Let ∆ be a non-maximal simplex of X, and let v 1 , v 2 be two distinct non-adjacent vertices of Lk X p∆q. Suppose that there exist maximal simplices ∆ 1 , ∆ 2 of X such that v 1 P ∆ 1 , v 2 P ∆ 2 , and ∆ 1 , ∆ 2 are connected by a W -edge of type 2. Then v 1 , v 2 are contained in W -adjacent maximal simplices of St X p∆q.
Proof. Since ∆ 1 and ∆ 2 are connected by a W -edge of type 2, the edges ∆ 1 and ∆ 2 share exactly one vertex u of Y . Let us denote by u 1 , u 2 the vertices of ∆ 1 , ∆ 2 , respectively, that differ from u.
By the definition of W -edges of type 2, there is an edge f u of X u and edges f u 1 , f u 2 of X u 1 , X u 2 such that ∆ i " ∆pf u i , f u q, for i " 1, 2. Letting λ i be the endpoint of f u i which is not u i , we also have d A Γ pσpλ 1 q, σpλ 2 qq ď r 2 .
Candidate simplices: Now, since v 1 , v 2 are not adjacent in X, we have v i P f u i for i " 1, 2. Since Y does not contain triangles, by Lemma 3.9, the vertices u 1 , u 2 are at distance 2 in Y . Since v 1 , v 2 P Lk X p∆q, we therefore have by Lemma 4.12 that ∆ is a vertex. Since, by Lemma 3.9, Y does not contain squares, it follows that ∆ " u.
Let f 1 u be an edge of X u containing ∆. Let δ i " ∆pf 1 u , f u i q for i " 1, 2. So, δ 1 , δ 2 are maximal simplices that share the edge f 1 u of X u and project to∆ 1 ,∆ 2 respectively. Note that δ 1 and δ 2 contain f 1 u and hence contain ∆. Thus δ 1 , δ 2 Ă St X p∆q.
W -adjacency: We are only left to observe that there is a W -edge of type 2 connecting δ 1 and δ 2 , since the σ sets associated to δ i and ∆ i coincide. (That is, roughly, in the definition of edges of type 2 the "middle" edge does not play a role.) Now the proposition is immediate: Proof of Proposition 5.6. Combine Lemma 5.7 and Lemma 5.8. 5.3. Maps between augmented complexes. Recall that the augmented complex X`W was defined in Definition 1.1. We will also use a corresponding construction in Y : Definition 5.9. We define the augmented graph Y`W which is obtained from Y by adding a W -edge between two vertices v, v 1 of Y whenever their fibres X v , X v 1 contain vertices that are connected by a W -edge. In that case, we say that v, v 1 are connected by a W -edge. Similarly, for a full subgraph Y 0 of Y , we denote by Y`W 0 the full subgraph of Y`W induced by Y 0 .
Remark 5.10. Note that vertices of Y connected by a W -edge lie at distance at most 2 from each other, by definition of the W -edges.
The simplicial map p mentioned below is defined in Definition 4.7, while ι is defined in Lemma 3.7 and Definition 3.12. Recall from Lemma 3.13 that ι is a simplicial quasi-isometry, and recall also that q is just the composition of p and ι.
Definition 5.11. The simplicial maps extend to maps that we still denote by The extended map p : X`W Ñ Y`W is still simplicial, while the other two can be taken to 2-Lipschitz by Remark 5.10 Again by Remark 5.10 the inclusion Y ãÑ Y`W is an A Γ -equivariant quasi-isometry, so ι : Y`W Ñ p D is an A Γ -equivariant quasi-isometry. Similarly, p : X`W Ñ Y`W is an A Γ -equivariant quasi-isometry. Hence, so is q : X`W Ñ p D.
We will need the following preliminary lemma, in a similar spirit to Proposition 5.5.
Lemma 5.12. For any sufficiently large r 2 the following holds. Let v be a vertex of Y and let ∆ be an edge of X contained in a fibre X v . Then the augmented links Lk Y pvq`W and Lk X p∆q`W are connected, and p restricts to a quasi-isometry of said augmented links.
Proof. Connectedness: We start by showing that Lk Y pvq`W is connected.
Set v 1 :" ιpvq. By Lemmas 2.30 and 2.31 , the action of Stabpv 1 q on Lk p D pv 1 q is cocompact. We choose a finite connected subcomplex K of Lk p D pv 1 q such that its Stabpv 1 q-orbits cover Lk p D pv 1 q. We can further assume that K satisfies the following additional property: Whenever K contains a vertex w that is not of tree or dihedral type, then it also contains every vertex of tree or dihedral type adjacent to it. Indeed, vertices of p D that are not of dihedral or tree type are cosets of the form gt1u or gxay, and they have only finitely many neighbours in D that are vertices of dihedral type by construction of the modified Deligne complex. Moreover, the cone-off procedure adds no additional neighbour for vertices of the form gt1u and add only one additional neighbour for vertices of the form gxay (namely, the apex of the standard tree gT a ). Without loss of generality, we can thus assume that K contains all such neighbours.
Let S be the set of vertices of K that are of dihedral or tree type. The set ι´1pSq forms a finite set of vertices of Y that are adjacent to v by Lemma 3.8. Let us choose an edge e of X v . For each w P ι´1pSq, we choose a maximal simplex ∆ w of X above the edge vw of Y and whose fibre over v is the edge e. Since the action of A Γ on Y is cocompact by Lemma 3.11, we can now require the constant r 2 in the definition of W -edges (see Definition 5.1) to be large enough so that for every distinct w, w 1 in ι´1pSq, we have that ∆ w and ∆ w 1 are connected by a W -edge of type 2. In particular, all pairs of vertices of ι´1pSq are connected by an edge of Y`W .
Let us now show that Lk Y pvq`W is connected. Indeed, if x, x 1 are two vertices of Lk Y pvq`W , then by definition of the subcomplex K we can find a finite sequence of elements g 1 , . . . , g n P A Γ such that ιpxq P g 1 K, ιpx 1 q P g n K and for every 1 ď i ă n, g i K X g i`1 K ‰ ∅. By the additional property imposed on K, it follows that each such g i K X g i`1 K contains a vertex v 1 i of p D that is either of tree or dihedral type. By Lemma 3.8, we can define v i :" ι´1pv 1 i q. We thus have a sequence of vertices v 0 :" x, v 1 , . . . , v n´1 , v n :" x 1 and by construction, for every 0 ď i ă n we have that g´1 i`1 v i and g´1 i`1 v i`1 are in ι´1pSq, hence v i and v i`1 are either equal or connected by an edge of Y`W . This show that Lk Y pvq`W is connected.
Let us now show that Lk X p∆q`W is connected. Since ∆ is an edge contained in the fibre X v , it follows from Lemma 4.12 that Lk X p∆q is the disjoint union of all the fibres X w for w P Lk Y pvq. For every vertex w of Lk Y pvq, the fibre X w is connected by construction. Since in addition Lk Y pvq`W is connected by the above, it is now straightforward to check that Lk X p∆q`W is also connected.
Quasi-isometry: By construction of Y`W , for every e edge of Lk Y pvq`W , there exists an edge e 1 in Lk X p∆q`W such that ppe 1 q " e. Thus, we can construct a set-theoretic section p : Lk Y pvq`W Ñ Lk X p∆q`W by choosing for every point x P Lk Y pvq`W a point in p´1pxq. Let us show that p is a quasi-inverse of p.
We have p˝p " Id by construction. Since all fibres of the form X w are connected of diameter 2 by construction, it follows that for every vertex w P Lk Y pvq and every point x P X w , we have dpx, p˝ppxqq ď 2. Moreover, for every other point x P Lk X p∆q`W , x is thus contained in an edge with endpoints in two distinct fibres X w and X w 1 , so we have that p˝ppxq is contained in a possibly different edge between X w and X w 1 . It follows that dpx, p˝ppxqq ď 2`diampX w q ď 4. Moreover, both p and p are coarsely Lipschitz. That p is coarsely Lipschitz follows from the fact that it sends an edge of Lk X p∆q`W to an edge or a vertex of Lk Y pvq`W . A similar argument as above shows that if x, y are two points belonging to some edge of Y , then dpppxq, ppyqq ď 4. Thus, p is coarsely Lipschitz, and it then follows that p is a quasi-isometry.
The following lemma is crucial as it allows us to compare links in X (which are the ones we are interested in) to links in p D (which are the ones we understand).
Lemma 5.13. For any sufficiently large r 1 , r 2 the following holds for W " W r 1 ,r 2 : Let v be a vertex of Y . Then the quasi-isometry induces a quasi-isometry between Lk Y pvq`W and Lk p D pιpvqq.
Moreover, let ∆ be an edge of X contained in a fibre X v , for some vertex v of Y . Then the quasi-isometry q : X`W Ñ p D restricts to a quasi-isometry q : pLk X p∆q`W q p0q Ñ Lk p D pιpvqq. Proof. Using the actions of A Γ , we can and will assume that v " N pHq for some H P H. We first notice that the map ι induces a Stabpvq-equivariant bijection between the vertices of Lk Y pvq`W and the vertices of Lk p D pιpvqq of tree or dihedral type. Indeed, this follows from Lemma 3.8, the fact that the bijection between tree-type/dihedral-type sets of vertices of Y and p D stated in Lemma 3.8 passes to links is a direct consequence of the characterisation of edges of Y therein.
We remark that Lk p D pιpvqq is connected, since it is either a tree when v is of tree type or we can use Remark 2.33 and Corollary 2.36. We can thus extend the restriction ι : Lk Y pvq Ñ Lk p D pιpvqq into a map ι : Lk Y pvq`W Ñ Lk p D pιpvqq by sending a W -edge between two distinct vertices w, w 1 P Lk Y pvq to a geodesic path between ιpwq and ιpw 1 q. Let us show that this extension is coarsely Lipschitz.
Since ι is Stabpvq-equivariant, it suffices to show that there are finitely many Stabpvq-orbits of W -edges in Lk Y pvq`W (note that a G-equivariant map between graphs Γ 1 Ñ Γ 2 is coarsely Lipschitz provided that Γ 2 is connected and there are finitely many G-orbits of edges in Γ 1 ). Since there are finitely many Stabpvq-orbits of vertices in Lk Y pvq (Lemma 3.26), it suffices to fix a vertex z " h 1 N pH 1 q P Lk Y pvq and show that there are finitely many pStabpvqXStabpzqqorbits of W -edges in Lk Y pvq`W containing z. Consider some vertex z 1 " h 2 N pH 2 q P Lk Y pvq connected by a W -edge to z, and notice that the W -edge needs to be of type 2 (since W -edges of type 1 connect maximal simplices that project to the same edge in Y ). By definition of W -edges, we must have d A Γ pN pHq`r X h 1 N pH 1 q`r, N pHq`r X h 2 N pH 2 q`rq ď r 2 . p˚q Since K " Stabpvq X Stabpzq " N pHq X h 1 N pH 1 qh´1 1 acts coboundedly on N pHq`r X h 1 N pH 1 q`r (Lemma 3.23) and H is finite, there are finitely many K-orbits of cosets h 2 N pH 2 q satisfying p˚q, there must be finitely many K-orbits of vertices z 1 as required.
Let us now construct a coarsely Lipschitz quasi-inverse. By Lemma 2.31 or Lemma 2.30, the action of Stabpιpvqq on Lk p D pιpvqq has finitely many orbits of edges and Lk Y pvq is connected by Lemma 5.12, and as above this suffices.
It thus follows that ι : Lk Y pvq`W Ñ Lk p D pιpvqq is a quasi-isometry. Regarding the moreover clause, in view of Lemma 5.12, q " ι˝p restricts to a composition of quasi-isometries of the relevant links.
In this section we will often make the following abuse of notation: Convention 5.14. Let A be a subcomplex of a simplicial complex B. Then by B´A we mean the full subcomplex of B with vertex set B p0q´Ap0q . Note for example that with this abuse we have`X´Sat X p∆q˘`W "`X p0q´S at X p∆q˘`W .
We also mention the following related result that will be needed in the Section 6.2: Lemma 5.15. For any sufficiently large r 1 , r 2 the following holds. Let ∆ be an edge of X contained in a fibre X v , for some vertex v of Y . If the presentation graph Γ is a single edge, we have X´Sat X p∆q " Lk X p∆q. Otherwise, X´Sat X p∆q is the full subcomplex of X spanned by X´X v , and the restriction q : X´Sat X p∆q Ñ p D´ιpvq is well-defined and extends to a coarsely Lipschitz map q : pX´Sat X p∆qq`W Ñ p D´ιpvq.
Proof. Case where Γ is a single edge: Let us first consider the case where Γ is a single edge between the two standard generators a, b. Then by construction, Y contains a unique vertex of dihedral type, namely u ab . Let us first assume that ∆ is contained in the fibre of a vertex of tree type. Since Y is bipartite with respect to the vertex type (tree or dihedral) by Lemma 3.8, it follows that for every vertex u of Y of tree type, its link consists of the single vertex u ab . It now follows from Lemma 4.12 that the link of any edge contained in the fibre of a vertex of tree type is equal to the fibre X u ab . Moreover, no vertex of X u ab has X u ab in its link, so we get that X´Sat X p∆q " X u ab " Lk X p∆q.
Let us now assume that ∆ is contained in the fibre X u ab . It follows from Lemma 4.12 that the link of any edge of X u ab is exactly X´X u ab . It also follows from the above discussion that no other simplex of X has link X´X u ab , hence X´Sat X p∆q " X´X u ab " Lk X p∆q.
General case: Let us now consider the case where Γ is not a single edge. Let us first show that no other vertex of Y has the same link as v. Assume by contradiction that there exists another such vertex v 1 . Since Y does not contain squares by Lemma 3.9, we have that Lk Y pvq " Lk Y pv 1 q is a single vertex. Since a vertex of Y of dihedral type gv ab contains at least two vertices in its link, namely gv a and gv b by Lemma 3.8, it follows that v and v 1 are of tree type, say v " gv a and v 1 " hv b . This now forces a and b to be leaves of Γ: Indeed, if a had at least two neighbours c, d in Γ, then Lk Y pgv a q would contain the two distinct vertices gv ac and gv ad . Thus, a and b are leaves of Γ. Moreover, since gv a and hv b have a common neighbour that is of dihedral type, this implies that this common neighbour is of the form kv ab , and in particular a and b are adjacent in Γ. Since Γ is connected and a and b are two adjacent leaves of Γ, it follows that Γ is a single edge, a contradiction. Thus, no other vertex of Y has the same link as v.
By Lemma 4.12, the link of ∆ is the disjoint union of all the fibres of the form X w for w a vertex of Lk Y pvq. Moreover, since distinct vertices of Y do not have the same link by the above, a simplex of X has the same link as ∆ if and only if it is an edge of X v . Thus, X´Sat X p∆q is the full subcomplex of X obtained by removing the fibre X v .
It follows from the above description of X´Sat X p∆q that the restriction map q : X´Sat X p∆q Ñ p D´ιpvq is well-defined (recall that q is simplicial on X, so it is enough to check that the restriction is well-defined on the 0-skeleton). Since p D´ιpvq is connected (since the link of ιpvq is connected as explained in the proof of Lemma 5.13), we extend this map to a map q : pX´Sat X p∆qq`W Ñ p D´ιpvq by sending edges to geodesics in p D´ιpvq. To show that this map is coarsely Lipschitz, it is enough to find a uniform constant C such that for two distinct vertices w, w 1 of Y´v connected by a W -edge, we have d p D´ιpvq pιpwq, ιpw 1 qq ď C. Note that since w and w 1 are distinct, this W -edge must be of type 2, and in particular w and w 1 are at distance at most 2 in Y . Since Y does not contain triangles and squares by Lemma 3.9, this distance is exactly 2 and we can consider the midpoint w 2 .
If w 2 P Y´v, then ιpw 2 q P p D´ιpvq since ι is injective by Lemma 3.8, and we thus have d p D´ιpvq pιpwq, ιpw 1 qq ď 2.
Otherwise, w 2 " v and so w, w 1 are adjacent vertices of Lk Y pvq`W . It follows from Lemma 5.13 that the maps p : Lk X p∆q`W Ñ Lk Y pvq`W and q : Lk X p∆q`W Ñ Lk p D pιpvqq are quasi-isometries. Since w, w 1 are adjacent vertices of Lk Y pvq`W , it follows that there exists a constant C (depending only on A Γ ) such that d Lk x D pιpvqq pιpwq, ιpw 1 qq ď C, and hence d p D´ιpvq pιpwq, ιpw 1 qq ď C. This shows that the mapq : pX´Sat X p∆qq`W Ñ p D´ιpvq is coarsely Lipschitz.
We will later prove a similar statement for simplices ∆ that are triangles of X, see Lemma 6.11.

The geometry of the augmented complex
In this section we finish the proof of Theorem A, by verifying that all conditions of Theorem 1.4 apply to the pair pX, W q where X is the blown-up commutation graph (Definition 4.6) and W " W r 1 ,r 2 is described in Subsection 5.1. Up until Subsection 6.3 (the final argument) we work under Convention 3.14, that is, we work with an Artin group of large and hyperbolic type A Γ with Γ connected and not a single vertex.
6.1. Hyperbolicity of the augmented links. The goal of this section is to prove the following proposition, verifying part of condition (II) from Definition 1.3 for the candidate combinatorial HHS pX, W q: Proposition 6.1. For all sufficiently large r 1 , r 2 the following holds. The augmented links Lk X p∆q`W of all non-maximal simplices ∆ of X are connected and hyperbolic.
Before the proof, we assemble the ingredients. Let ∆ be a simplex of ∆. If Lk X p∆q is connected and of bounded diameter, so is its augmented link and there is nothing to prove. Thus, we only need focus on unbounded links. Lemma 6.2. Let ∆ be a non-maximal simplex of X. Then its link (and hence its augmented link) is connected and of bounded diameter, except possibly when ∆ is one of the following: ‚ the empty simplex, ‚ an edge of X contained in a fibre, ‚ a triangle of X containing exactly two roots.
Proof. Let ∆ be a non-empty simplex of X. Note that if a link decomposes as a non-trivial join, it is automatically connected and bounded. In view of Lemma 4.12, Lk X p∆q is connected and bounded, by virtue of decomposing non-trivially as a join, except possibly in one of the following two situations. Case 1: Lk Y p∆q ‰ ∅ and all the links of fibres are empty. In that case, ∆ is a single vertex v of Y , and its associated fibre ∆ v " ∆ is an edge (and this case is listed in the statement).
Case 2: Lk Y p∆q " ∅ and exactly one of the fibres of ∆ has an empty link. In that case, ∆ is an edge, and ∆ is a triangle (since it is non-maximal). If ∆ contains only one root, then Lk X p∆q is a single vertex, so it is bounded. (The remaining case of two root is listed in the statement.) We now treat the remaining cases separately.
Lemma 6.3. The augmented link of the empty simplex of X is equivariantly quasi-isometric to p D, and in particular is hyperbolic.
Corollary 6.6. The simplices of X that have an unbounded augmented link are exactly: ‚ the empty simplex of X, ‚ the triangles of X containing exactly two roots. ‚ the edges contained in a fibre X v of X, except when v corresponds to a standard generator of A Γ that comes from a leaf of Γ contained in an edge of Γ with even label. All other simplices of X have an augmented link of diameter at most 3.
6.2. Quasi-isometric embeddings of augmented links. It remains to verify that pX, W q satisfies the quasi-isometric embedding part of condition (II) from Definition 1.3. This is achieved by the following proposition: Proposition 6.7. For every non-maximal simplex ∆ of X, the augmented link Lk X p∆q`W is quasi-isometrically embedded in the augmented complex`X p0q´S at X p∆q˘`W .
The proposition is clear when the augmented link is bounded. We treat the remaining cases, listed in Corollary 6.6, separately. Note that there is nothing to do in the case of the empty simplex, as the inclusion Lk X pHq`W ãÑ`X´Sat X pHq˘`W is the identity map of X`W .
Overview. Let us give an overview of the general strategy. We will use the quasiisometries between X`W , Y`W and p D (see Lemma 3.13 and Definition 5.11) to construct models for Lk X p∆q`W and`X´Sat X p∆q˘`W that are easier to work with. More precisely, we will construct a diagram of the form ‚ the complexes U ∆ , V ∆ are built out of subcomplexes of p D, ‚ the top horizontal map is a quasi-isometry (in the spirit of Lemma 5.13), ‚ the bottom horizontal map is coarsely lipschitz (in the spirit of Lemma 5.15). ‚ both horizontal maps coincide with the map q when restricted to vertex sets, ‚ the diagram commutes up to bounded error. This is due to some bounded choices appearing when constructing the various horizontal arrows (and more precisely extending the map q to W -edges), see the proof of Lemma 5.13, 5.15, and 6.11. Then, it will be enough to show that U ∆ ãÑ V ∆ is a quasi-isometric embedding. This in turn will be done using either the CAT(0) geometry of p D to construct a coarsely Lipschitz retraction V ∆ Ñ U ∆ (in Lemma 6.10), or by using additional properties of dihedral Artin groups when the complexes U ∆ , V ∆ are defined using neighbourhoods of vertices of dihedral type (in Lemma 6.12 and 6.13).
We start by introducing the main tool for constructing coarsely Lipschitz retractions in p D, and more generally in CAT(0) complexes: Definition 6.8 (Combinatorial neighbourhood). Let C be a subcomplex of a complex X . The combinatorial neighbourhood of C, denoted N pCq, is the subcomplex consisting of the union of all the simplices of X intersecting C. The combinatorial sphere around C, denoted BN pCq, is the full subcomplex with vertex set N pCq p0q´Cp0q . Finally, we let N pCq " N pCq´BN pCq.
For C a subcomplex of a metric complex X , note that X´N pCq is the full subcomplex spanned by X´C, meaning the full subcomplex with vertex set the vertices of X that are not contained in C.
In the statement below we regard X´N pCq as endowed with its intrinsic path metric, and similarly for BN pCq.
Lemma 6.9. Let X be a CAT(0) simplicial complex with finitely many isometry types of simplices, and let C be a convex subcomplex of X . Then there exists a coarsely Lipschitz retraction from X´N pCq to BN pCq. In particular, BN pCq is quasi-isometrically embedded in X´N pCq.
Proof. We construct a projection from X´N pCq to BN pCq as follows. Since the CAT(0) space X has finitely many isometry types of simplices, we can choose a number ε ą 0 such that the closed metric neighbourhood N pC, εq (for the CAT(0) metric) is contained in N pCq. Since C, whence N pC, εq, is convex for the CAT(0) metric, the closest-point projection π : X´N pCq Ñ N pC, εq is well-defined and 1-Lipschitz. Given a point x P X´N pCq, we first compute the projection πpxq, we choose a simplex ∆ of N pCq containing πpxq (such a simplex cannot be contained in C), and we define π 1 pxq to be any vertex of ∆ not in C. This defines the map π 1 : X´N pCq Ñ BN pCq.
Let x, y P X´N pCq be two points at distance at most ε{3 in X´N pCq, and therefore in X . Their closest-point projections πpxq, πpyq are also at distance at most ε{3. Thus, the CAT(0) geodesic γ between πpxq and πpyq is of length at most ε{3, and in particular γ does not intersect C since its endpoints are at distance ε from C. Also, γ is contained in the combinatorial neighbourhood N pCq since it is even contained in the convex subspace N pC, εq. By [BH99, Corollary 7.30], there is a uniform bound k depending only on X and ε but not on x, y such that γ intersects at most k simplices of N pCq (none of them contained in C). This now implies that there is a path in BN pCq connecting π 1 pxq and π 1 pyq consisting of a concatenation of at most k paths each contained in a simplex of BN pCq (indeed, note that simplices ∆ 1 , ∆ 2 of N pCq that intersect outside of C also satisfy ∆ 1 X ∆ 2 X BN pCq ‰ H). Since X has finitely many isometry types of simplices, the k paths can be taken to have uniformly bounded length, and this implies that the map π 1 : X´N pCq Ñ BN pCq is coarsely Lipschitz.
Moreover, again since X has finitely many isometry types of simplices, a similar reasoning implies that there exists a constant k 1 such that for any vertex v of BN pCq, the CAT(0) geodesic between v and πpvq meets at most k 1 simplices, which implies that v and π 1 pvq lie within bounded distance in X´N pCq. Thus, π 1 is a coarse retraction.
Lemma 6.10. Let ∆ be an edge of X contained in the fibre of a vertex of Y . Then the augmented link Lk X p∆q`W is quasi-isometrically embedded in`X´Sat X p∆q˘`W .
Proof. If Γ is a single edge, then we have Lk X p∆q " X´Sat X p∆q by Lemma 5.15, hence Lk X p∆q`W "`X´Sat X p∆q˘`W , and the inclusion is a quasi-isometric embedding.
Let us now consider the case where Γ is not reduced to a single edge. The edge ∆ is contained in the fibre over a vertex that we call u P Y , and we set v :" ιpuq P p D. It follows from Lemma 5.15 that X´Sat X p∆q is the full subcomplex of X obtained by removing the fibre X u . Recall also from the same lemma the mapq : pX´Satp∆qq`W Ñ p D´tvu. Also, Lemma 5.13 gives us a quasi-isometry q 1 : Lk X p∆q`W Ñ Lk p D pvq (note that, in the statement of the lemma, the map q is restricted to the 0-skeleton of Lk X p∆q`W , and q 1 here is an extension of q across all edges).
Consider now the following diagram: Note that the diagram commutes at the level of vertices, so that it commutes up to bounded arrow since all maps are coarsely Lipschitz.
By Lemma 6.9, there exists a coarsely Lipschitz retraction π : p D´tvu Ñ BN pvq. Thus, the inclusion BN pvq ãÑ p D´tvu is a quasi-isometric embedding. Since the top and right arrows are quasi-isometric embeddings, so is their composition. Since the left arrow is coarsely Lipschitz, and so is the bottom arrow, the left arrow is a quasi-isometric embedding because the diagram commutes up to bounded error. In other words, Lk X p∆q`W ãÑ`X´Sat X p∆q˘`W is also a quasi-isometric embedding.
Before considering the final two cases where ∆ is a triangle of X, we prove the following result that will allow us to show that the bottom map of our (almost) commutative diagram is coarsely Lipschitz. Recall that we have simplicial maps p : X Ñ Y , and ι : Y Ñ p D, and we set q " ι˝p.
Lemma 6.11. Let ∆ be a triangle of X that contains an edge in the fibre of a vertex of Y and an additional root u. Then we have Moreover, let Z " Z u be the graph obtained from the disjoint unioǹ p D´St p D pιpuqq˘\ Λ`W u as follows: We add an edge between a vertex λ P Λ u and a vertex v P p D´St p D pιpuqq if there exists a W -edge of X`W between λ and a vertex of X v .
Then the restriction q : X´Sat X p∆q Ñ Z is well-defined and extends to a coarsely Lipschitz map Proof. It follows from Lemma 4.12 that Sat X p∆q consists of all the triangles ∆ 1 of X such that p∆ 1 q u " ∆ u (that is, ∆ 1 is the join of the root u and some edge). The structure of X´Sat X p∆q follows immediately. Since ι is injective by Lemma 3.8, as well as simplicial, we have ι´1pSt p D pιpuqq Ď St Y puq, and therefore q´1pSt p D pιpuqq Ď p´1pSt Y puqq. Therefore, the restriction of q as in the statement is well-defined.
The required coarsely Lipschitz extension exists provided that any two distinct vertices v, v 1 P X´Sat X p∆q joined by an edge get mapped to vertices of Z that lie within uniformly bounded (in particular, finite) distance. First notice that, by construction, edges that are not W -edges between two distinct vertices of X´Sat X p∆q are either mapped to an edge of Z or collapsed to a point. Moreover, if v P X´Sat X p∆q´Λ u and a point λ P Λ u are joined by a W -edge, then their images under q are joined by an edge in Z by definition of Z. Thus, it is enough to show that there exists a uniform constant C such that if v, v 1 P`X´Sat X p∆q˘´Λ u are distinct vertices joined by a W -edge, then d Z pqpvq, qpv 1 qq ď C.
Let v, v 1 P`X´Sat X p∆q˘´Λ u be two distinct vertices joined by a W -edge. First notice that ppvq, ppv 1 q belong to Y´St Y puq by the first part of the statement. It is enough to consider the case where ppvq and ppv 1 q are not connected by an edge in Y . Since ppvq and ppv 1 q are connected by a W -edge, they are either equal (for W -edges of type 1) or at distance 2 in Y (for W -edges of type 2). It is enough to consider the second case, so let δ, δ 1 be maximal simplices of X, containing v, v 1 respectively, that are connected by a W -edge of type 2.
Let u 1 be the midpoint between ppvq and ppv 1 q. If u 1 P Y´St Y puq, then qpvq, ιpu 1 q, qpv 1 q is a combinatorial path of length 2 in Z, so d Z pqpvq, qpv 1 qq ď 2.
Otherwise, u 1 must be a vertex of Lk Y puq (since u 1 ‰ u for otherwise v and v 1 would lie in Satp∆q). Up to the action of A Γ , we can assume that u, u 1 are of the form u a , u ab (up to permutation) and in particular there are finitely many possible pairs to consider. Hence, it suffices to fix u, u 1 and produce a constant for those. By construction, we have that wpδq, wpδ 1 q are contained in the "thickened" coset N pz u 1 q`r. Moreover, since δ and δ 1 are connected by a W -edge, it follows from Lemma 5.2 that d A Γ pwpδq, wpδ 1 qq ď r 3 .
To show that qpvq, qpv 1 q are at uniformly bounded distance in Z, it will be enough to show that they are at uniformly bounded distance in the full subgraph Z 1 of Z spanned by pLk We modify the complex Z 1 in an equivariant way to obtain a graph on which N pz u 1 q acts naturally. Let Z 2 be the full subcomplex of X`W spanned by the quasi-lines Λ`W w for w P Lk Y pu 1 q. By construction of the augmented link Lk Y pu 1 q`W , the projection p : X`W Ñ Y`W induces a surjective simplical map p : Z 2 Ñ Lk Y pu 1 q`W . Moreover, since the link Lk Y pu 1 q`W is connected by Lemma 5.12 and the quasi-lines Λ`W w (that is, the fibers of p over the vertices) are connected by Lemma 6.4, it follows that Z 2 is connected. Since Stab Y pu 1 q " N pz u 1 q stabilises Lk Y pu 1 q`W and preserves the family of quasi-lines Λ w by construction of the action A Γ ñ X`W , it follows that there is an induced action of N pz u 1 q on the connected graph Z 2 .
In particular, for x P Z 2 , the orbit map is coarsely Lipschitz (with respect to any given word metric on N pz u 1 q), and the Lipschitz constant does not depend on x P Z 2 since the action is cobounded, see Lemmas 3.26 and 4.5. We can extend ω x to a mapω x : N pz u 1 q`2 r Ñ Z 2 which is still N pz u 1 q-equivariant and coarsely Lipschitz when N pz u 1 q`2 r is endowed with the metric inherited from A Γ .
Moreover, by construction of Z, the projection q : X`W Ñ p D`W induces a Lipschitz map q : Z 2 Ñ Z 1 obtained by collapsing each quasi-line Λ`W w other than Λ u to the corresponding vertex ιpwq. Thus, the composition map N pz u 1 q ωx ÝÑ Z 2 q ÝÑ Z 1 is coarsely Lipschitz, where the coarse Lipschitz constant can be chosen to be uniform.
Let us now bound above the distance d Z 1 pqpvq, qpv 1 qq. After perturbing wpδq up to distance r, we can assume that it is contained in yN pzq X N pz u 1 q`2 r , where yN pzq " qpvq for some element z belonging to the finite set of elements of the form a or z ab (for some a, b P Γ). Similarly, we perturb wpδ 1 q and define y 1 , z 1 analogously for δ 1 . We pick as basepoint for the orbit map the point x :" N pzq. The coarsely Lipschitz map q˝ω x sends wpδq to a set (uniformly bounded by 4.26) containing yN pzq " qpvq. Moreover, if we denote by k a uniform upper bound on the distance in Lk Y pu 1 q between vertices of the form N pz 1 q and N pz 2 q, for z 1 , z 2 of the form a or z ab (a, b P Γ), then the map q˝ω x sends wpδ 1 q to a set (again uniformly bounded by 4.26) containing y 1 N pzq, which is at distance at most k from y 1 N pz 1 q " qpv 1 q. Moreover, since q˝ω x is coarsely Lipschitz and d A Γ pwpδq, wpδ 1 qq ď r 3`4 r (recall the rperturbation), it follows that there exists a constant C 1 such that d Z 1 pqpvq, qpv 1 qq ď C 1 , hence d Z pqpvq, qpv 1 qq ď C 1 . This constant C 1 depends on k, r, r 3 and the coarse Lipschitz constant forω x (in particular, it is independent of v, v 1 , u 1 ).
We thus have that d Z pqpvq, qpv 1 qq is uniformly bounded above, and it follows that the map q 1 is coarsely Lipschitz.
Lemma 6.12. Let ∆ be a triangle of X that contains two roots, and an edge in the fibre of a vertex of tree type of Y . Then the augmented link Lk X p∆q`W is quasi-isometrically embedded in the augmented complex`X´Sat X p∆q˘`W .
Proof. Up to the action of A Γ , we can assume that there exist adjacent generators a, b P Γ such that ∆ is the edge between u a and u ab . It follows from Lemma 4.12 that Sat X p∆q consists of all the triangles ∆ 1 of X such that p∆ 1 q u ab " ∆ u ab (that is, ∆ 1 is a join of u ab and some edge). Moreover, by Proposition 5.6, the augmented link Lk X p∆q`W consists of the quasi-line Λ u ab , equipped with its W -edges (this augmented link is still quasi-isometric to Z by Lemma 6.4).
Let q 1 : pX´Sat X p∆qq`W Ñ Z be the coarsely Lipschitz map constructed in Lemma 6.11. We have the following diagram that commutes up to bounded error (as it commutes exactly on vertices and all maps are coarsely Lipschitz): In particular, to show that Lk X p∆q`W ãÑ`X´Sat X p∆q˘`W is a quasi-isometric embedding, it is enough to show that the inclusion Λ u ab ãÑ Z is a quasi-isometric embedding. By construction of the W -edges, the inclusion Λ u ab ãÑ Z is a quasi-isometric embedding provided the xz ab y-orbits are quasi-isometrically embedded in Z.
Let Z 1 be the full subcomplex of p D generated by Z Y tv ab u. By construction of Z (see Lemma 6.11), the subcomplex Z 1 is obtained from p D by removing the apices of all the standard trees containing v ab . Since Z 1 is obtained from p D by removing a family of open cones, it is still CAT(0) for the induced metric by Lemma 2.24. In particular, it follows from Lemma 6.9 that there is a coarsely Lipschitz retraction Thus, it is enough to show that the xz ab y-orbits are quasi-isometrically embedded in Lk Z 1 pv ab q. But by construction of Z 1 , Lk Z 1 pv ab q is A ab -equivariantly isomorphic to Lk D pv ab q. Thus, it is enough to show that the xz ab y-orbits are quasi-isometrically embedded in Lk D pv ab q, where D is the original Deligne complex. Since A ab is of large type, this now follows from Lemma 2.34 applied to g " z ab . Lemma 6.13. Let ∆ be a triangle of X that contains two roots, and an edge in the fibre of a dihedral vertex of Y . Then the augmented link Lk X p∆q`W is quasi-isometrically embedded in the augmented complex`X´Sat X p∆q˘`W .
The proof of this lemma is rather long, but follows a similar strategy of reduction. The rest of this section will be devoted to it.
Up to the action of the group, we can assume that the projection ∆ is an edge of Y between the vertices u a and u ab . Since the fibre of ∆ ua is the apex of X ua , the saturation Sat X p∆q consists of the union of all the triangles of X that have the same fibre at u a as ∆.
In particular, we have X´Satp∆q " Λ ua \ q´1`p D´p T a˘, and`X´Satp∆q˘`W is obtained from the previous graph by adding all the W -edges whose endpoints are in X´Satp∆q.
Step 1: Let q 1 " π 1 :`X´Sat X p∆q˘`W Ñ Z be the coarsely Lipschitz map constructed in Lemma 6.11. The the following diagram commutes up to bounded error: In particular, to show that the inclusion is a quasi-isometric embedding, it is enough to show that the inclusion Λ a ãÑ Z is a quasi-isometric embedding.
We reduce the problem further, by using the CAT(0) geometry of p D to show that Z coarsely retracts onto a neighbourhood of p T a . Consider the combinatorial neighbourhood BN p p T a q " N pT a q´v a . We define the complex Z a as the full subcomplex of Z spanned by Λ a \ BN p p T a q.
Step 2: There exists a coarsely Lipschitz retraction π 2 : Z Ñ Z a such that the following diagram commutes: In particular, to show that the inclusion Λ a ãÑ Z is a quasi-isometric embedding, it is enough to show that the inclusion Λ a ãÑ Z a is a quasi-isometric embedding.

Proof of
Step 2: Since T a is convex in p D´v a by Lemma 2.25, there exists a coarse retraction of p D´p T a on BN p p T a q by Lemma 6.9. We define the map π 2 : Z Ñ Z a as follows: π 2 is the identity on Z a and is the coarse retraction p D´p T a Ñ BN p p T a q on Z´Λ a . This definition makes sense as Z a X pZ´Λ a q " BN p p T a q and π 2 is defined as the identity on BN p p T a q in both cases. To conclude that this defines a coarse retraction Z Ñ Z a , let us check that an edge between a vertex x of Z´Λ a and a vertex y of Z a is either contained in Z´Λ a or in Z a . If y R Λ a , then x, y belong to Z´Λ a , and so does the edge between them. If y P Λ a , then by construction the edge e between x and y comes from a W -edge of type 2 of X`W . But by construction of W -edges of type 2, this forces x to belong to BN pT a q, and thus x, y, and e belong to Z a .
We now want to show that the inclusion Λ a ãÑ Z a is a quasi-isometric embedding. For a vertex v of T a , we denote by Z v the subgraph of Z a spanned by Λ a and BN p p T a q X N pvq. In particular, Z a is the union of all the Z v , where v runs among the vertices of T a . For a generator b P Γ adjacent to a, we simply denote by Z ab the graph Z v ab . To show that the inclusion Λ a ãÑ Z a is a quasi-isometric embedding, we will first show that each inclusion Λ a ãÑ Z v is a quasi-isometric embedding (Steps 3 and 4) and then combine these quasiisometric embeddings to show that Λ a ãÑ Z a is a quasi-isometric embedding (Step 5).
Before reducing the situation further, we need further information about the graph of orbits introduced in Definition 2.32 and some variants. As the graphs we are looking at have been obtained from Lk D pv ab q by removing the vertices corresponding to the tree T a , we introduce a variant of the graph of orbits: Definition 6.14. Let G be either Cayley a,b pA a,b q or Cayley a,b pA a,b q{x∆ ab y (where as usual, x∆ ab y acts by multiplication on the right). We define a new graph, called the blown-up graph of orbits and denoted Orbitå ,b`G˘a s follows.
We denote by L a the simplicial line in G spanned by the xay-orbit of the identity element. We put a vertex for every xay-orbit of G except the one corresponding to L a , one vertex for every xby-orbit in G, and one vertex for each vertex of L a . If two such subsets of G have a non-empty intersection, we put an edge between them. Moreover, for every vertex v of L a , we also put an edge between v and av.
This construction can be thought of as obtained from the original graph of orbits Orbitå ,b`Gb y blowing up the vertex corresponding to the coset xay, and replacing it with a copy of L a .
Step 3: There exists a coarsely Lipschitz map π 3 : Z ab Ñ Orbitå ,b`C ayley a,b pA a,b q˘{ x∆ ab y and a quasi-isometry Λ a Ñ L a such that the following diagram commutes up to bounded error and the inclusion Λ a ãÑ Z ab coarsely Lipschitz: In particular, to show that the inclusion Λ a ãÑ Z ab is a quasi-isometric embedding, it is enough to show that the inclusion L a ãÑ Orbitå ,b`C ayley a,b pA a,b q˘{x∆ ab y is a quasi-isometric embedding.

Proof of
Step 3. First notice that one can identify the subgraph Z ab´Λa with the subgraph Lk p D pv ab q´Lk p DXTa pv ab q. Now consider the (coarsely Lipschitz) quotient map Z ab Ñ Z ab {x∆ ab y for the action of x∆ ab y on the right. The image in that quotient of the apex of a standard tree is a vertex of valence 1. The graph Z ab {x∆ ab y thus retracts via a Lipschitz map onto the subgraph obtained by removing all these valence 1 vertices. Using the identification of Lk D pv ab q with the graph of orbits Orbit a,b`C ayleypA ab q˘, one sees that the graph obtained is the blown-up graph of orbits Orbitå ,b`C ayley a,b pA a,b q˘{x∆ ab y. The composition π 3 : Z ab Ñ Z ab {x∆ ab y Ñ Orbitå ,b`C ayley a,b pA a,b q˘{x∆ ab y satisfies the required property. Moreover, π 3 is xay-equivariant.
Pick a bounded set B whose xay-translates cover Λ a containing exactly one element of each orbit. There is a unique xay-equivariant map Λ a Ñ L a that sends B to the identity element. This is a quasi-isometry since xay acts coboundedly on both Λ a and L a . We use this as the top arrow in the diagram. Since the other three maps are also xay-equivariant, the diagram commutes up to bounded error.
Note that the map Λ a ãÑ Z ab is simplicial, hence coarsely Lipschitz.
Step 4: The inclusion L a ãÑ Orbitå ,b`C ayley a,b pA a,b q˘{x∆ ab y is a quasi-isometric embedding.
Proof of Step 4. Let us start with a preliminary remark. Starting from the Cayley graph of A ab , there are two operations one can perform: We can mod out by the right-action of x∆ ab y, or we can construct the aforementioned blown-up graph of orbits Orbitå ,b`C ayley a,b pA a,b q˘. These two constructions commute. More precisely, there exists an isomorphism (in dotted lines below) that makes the following diagram commute: Orbitå ,b`C ayley a,b pA a,b q{x∆ ab y-L a " v ) ) @ Ø

5
Orbitå ,b`C ayley a,b pA ab q˘{x∆ ab y Thus, it is enough to show that the inclusion L a ãÑ Orbitå ,b`C ayley a,b pA a,b q{x∆ ab yȋ s a quasi-isometric embedding. The graph Cayley a,b pA a,b q{x∆ ab y (for the action of x∆ ab y on the right) is a tree, as it is obtained from the quasi-tree T ab defined in Section 2.2 by removing the edges of T ab labelled by an atom on more than one generator. Moreover, the various gL a and gL b are quasi-lines in the tree Cayley a,b pA a,b q{x∆ ab y that are gxayg´1invariant and gxbyg´1-invariant respectively. Moreover, for any r ě 0 there is a constant f prq such that the intersection of the r-neighbourhoods of any two such distinct quasi-lines has diameter at most f prq. The fact that L a ãÑ Orbitå ,b`C ayley a,b pA a,b q{x∆ ab y˘is a quasiisometric embedding is now a consequence of Step 3 and the following Claim, applied to the case Z " Cayley a,b pA a,b q{x∆ ab y, L i 0 " L a , and the L i are the gL b and the gL a other than L a .
Claim. For all δ, κ and functions f : R`Ñ R`, there exists D ě 0 such that the following holds. Let Z be a δ-hyperbolic space and let tL i u be a family of pκ, κq-quasilines in Z such that for all distinct i, j and all r, the intersection of the r-neighbourhoods of L i and L j has diameter at most f prq. Then the following holds for all L i 0 . Let p Z be obtained from Z by coning off each L i , i ‰ i 0 . Then the image of L i 0 under the inclusion Z Ñ p Z is a pD, Dq-quasi-line.
Proof of Claim. The closest-point projection proj : Z Ñ L i 0 has the property that the image of all L i , i ‰ i 0 is uniformly bounded. Therefore, proj gives a coarsely Lipschitz coarse retraction of p Z onto the image of L i 0 in p Z. The existence of a coarse retraction implies that L i 0 is quasi-isometrically embedded in p Z, as required.
This concludes the proof of Step 4.
It follows from Steps 3 and 4 that the various inclusions Λ a ãÑ Z v are quasi-isometric embeddings, for vertices v of T a . Note that because the action of StabpT a q on T a is cocompact by Lemma 2.30, we can choose a constant K ě 0 such that all such inclusions Λ a ãÑ Z v are pK, Kq-quasi-isometric embeddings. We now want to combine these various quasi-isometric embeddings into a global quasi-isometric embedding Λ a ãÑ Z a .
Step 5: The inclusion Λ a ãÑ Z a is a quasi-isometric embedding.
Proof of Step 5. For every distinct vertices v, w of T a , notice that vertices in Z v X Z w are at (combinatorial) distance at most 1 from Λ a . In particular, given a finite combinatorial geodesic γ in Z a between two points x, x 1 of Λ a , there exist points x 0 :" x, x 1 , . . . , x n :" x 1 of Λ a , and distinct points y 0 :" x, y 1 , . . . , y n :" x 1 of γ Ă Z a such that the portion of γ between y i and y i`1 is contained in some Z v , and each y i is either equal or adjacent to x i . From Steps 3 and 4 and the definition of the x i and y i , it follows that for each i, we get the following inequality between combinatorial length: d Λa px i , x i`1 q ď Kd Zv py i , y i`1 q`K`2 ď p2K`2qd Zv py i , y i`1 q, the last inequality following from the fact that d Zv py i , y i`1 q ě 1. We thus have d Λa px, x 1 q ď ÿ i d Λa px i , x i`1 q ď p2K`2q ÿ i d Zv py i , y i`1 q " p2K`2qd Za px, x 1 q.
This shows that the inclusion Λ a ãÑ Z a is a quasi-isometric embedding.
Combining all the previous steps, it follows that the inclusion Lk X p∆q`W ãÑ`XŚ atp∆q˘`W is a quasi-isometric embedding, which concludes the proof of Lemma 6.13.
Proof of Proposition 6.7. The preceding sequence of Lemmas 6.10, 6.12, 6.13, together with Corollary 6.6, imply the proposition. 6.3. Obtaining hierarchical hyperbolicity. We are now ready to prove the main theorem of this article, Theorem A, which says that Artin groups of large and hyperbolic type are hierarchically hyperbolic. First, we just prove that a structure exists, and later on we give a more detailed description for the interested reader.
Proof of Theorem A. There are three cases to consider. Single vertex: If Γ is a single vertex, then A Γ is isomorphic to Z, which is an HHG. Connected case: We consider the case where Γ is connected and contains at least two vertices, so that we are in the case covered by Convention 3.14. Let us first check that pX, W q satisfy the conditions of Definition 1.3: It satisfies (I) by Proposition 4.14, condition (II) by Propositions 6.1 and 6.7, condition (III) by Proposition 5.6, and condition (IV) by Proposition 4.15 and Corollary 6.6. Moreover, the action of A Γ is proper and cobounded by Proposition 5.5 and there are finitely many orbits of links by Proposition 4.13. It thus follows from Theorem 1.4 that A Γ is hierarchically hyperbolic. General case: In general, A Γ is the free product of Artin groups on connected graphs, which are hierarchically hyperbolic by the above. Taking free products preserves the property of being a hierarchically hyperbolic group, by [BHS19, Corollary 8.24] or [BHS19, Theorem 9.1].
Description of the HHG structure. We will now describe the HHG structure, and we introduce some notation to this end; we will use standard HHS terminology from [BHS19, Definition 1.1].
Given an HHS pX, Sq, we can write S " S f in \ S 8 , where S 8 is the set of indices for which the corresponding hyperbolic space is unbounded. We typically have various bounded hyperbolic spaces in an HHS structure which need to be there because of the orthogonality axiom, but do not otherwise play a major role. The interesting part of the structure is S 8 .
Recall from Definition 3.4 that given an Artin group A Γ we denote H :" xay | a a standard generator of A Γ ( Y xz ab y | a, b span an edge of Γ with m ab ě 3 ( .
We denote two distinct copies of Ů HPH G{N pHq by S line and S tree . We denote the copy of gN pHq in S line by gN pHq line , and similarly for S tree .
Finally, we denote by S leaf the subset of S tree consisting of all cosets of the form gN pxayq tree , where a is a standard generator corresponding to a valence-1 vertex of Γ whose incident edge has even label.
Theorem 6.15. Let A Γ be an Artin group of large and hyperbolic type, with Γ connected and not a single vertex. Then there is an HHG structure S on A Γ with the following properties. We have S 8 " tSu \ S line \`S tree´Sleaf˘, and all of the following hold: (1) S is the Ď-maximal element of S and CpSq can be taken to be either the coned-off Deligne complex or the complex of irreducible parabolics of finite type.
(2) For each gN pHq P S line , CpgN pHqq is a quasiline on which gN pHqg´1 acts, with gHg´1 having unbounded orbits. Proof. The HHG structure can be obtained from the description of the HHS structure for a combinatorial HHS [BHMS20, Theorem 1.18, Remark 1.19]. The index set of the HHS Ď-maximal hyperbolic space constructed in [ABD21] is A Γ -equivariantly quasi-isometric to p D, which proves the assertion.
In [ABD21], Theorem B follows combining [ABD21, Corollary 6.2], which characterises stability for any HHS satisfying an additional condition (having unbounded products), and [ABD21, Theorem 3.7], which says that says that one can change an HHG structure into one satisfying the additional condition.
What we used in the proof of Corollary B is that this change does not affect the equivariant quasi-isometry type of the Ď-maximal hyperbolic space, as we now justify.
Lemma 6.18. Let pG, Sq be an HHG with the property that for each non-Ď-maximal V P S with unbounded CV there exists U P S with U KV and unbounded CU . Then G admits an HHG structure pG, Tq with unbounded products where the Ď-maximal hyperbolic space is Gequivariantly quasi-isometric to that of pG, Sq.
Proof. We claim that the HHG structure pG, Tq constructed in the second paragraph of the proof of [ABD21, Theorem 3.7] has Ď-maximal hyperbolic space T S with the required property; we now recall the construction. First, for a constant M let S M be the set of all U P S for which there exist U Ď V and V KW with diam CV ą M and diam CW ą M . We can choose M such that diam CV ą M is equivalent to diam CV " 8 (because the G-action on S is cofinite by definition of HHG). Then T S is defined as the factored spaceĜ S M . As a set, this is G, however it is endowed with a different metric. Roughly, we start with G and cone-off certain product regions corresponding to the elements of S M , but we will not need the exact definition here.
By [BHS17a, Proposition 2.4],Ĝ S M is an HHS with index set S´S M , and with the same associated hyperbolic spaces and projections as in pG, Sq. By the distance formula for HHS [BHS19, Theorem 4.5], for any suitably large constant L there are constants K, C so that for all x, y PĜ S M we have.
where « K,C denotes quantities that differ by multiplicative constant at most K and additive constant at most C, and rAs L denotes A if A ě L and 0 otherwise. Now it is time to use our hypothesis. In our case S M contains all U P S with diamCU ą M except the Ď-maximal element S. If we take the threshold L to be larger than M , the distance formula states that π S is a quasi-isometry from T S "Ĝ S M to CS. Since π S is G-equivariant by definition of HHG, we are done.