A Deligne complex for Artin Monoids
aa r X i v : . [ m a t h . G R ] J u l A DELIGNE COMPLEX FOR ARTIN MONOIDS
RACHAEL BOYD, RUTH CHARNEY AND ROSE MORRIS-WRIGHT
Dedicated to the memory of Patrick Dehornoy.
Abstract.
In this paper we introduce and study some geometric objects associated to Artinmonoids. The Deligne complex for an Artin group is a cube complex that was introduced bythe second author and Davis [CD95a] to study the K ( π,
1) conjecture for these groups. Using anotion of Artin monoid cosets introduced by the first author in [Boyd20], we construct a versionof the Deligne complex for Artin monoids.We show that for any Artin monoid this cube complex is contractible. Furthermore, we studythe embedding of the monoid Deligne complex into the Deligne complex for the correspondingArtin group. We show that for any Artin group this is a locally isometric embedding. In thecase of FC-type Artin groups this result can be strengthened to a globally isometric embedding,and it follows that the monoid Deligne complex is CAT(0) and its image in the Deligne complexis convex. We also consider the Cayley graph of an Artin group, and investigate properties ofthe subgraph spanned by elements of the Artin monoid. Our final results show that for a finitetype Artin group, the monoid Cayley graph embeds isometrically, but not quasi-convexly, intothe group Cayley graph. Introduction
Artin groups, also known are Artin-Tits groups, are a broad class of groups whose presentationsare encoded by labelled graphs. Given a simple graph Γ with a finite vertex set S and edge set E ,such that each edge ( s, t ) ∈ E is labelled by an integer m st ≥
2, we define the
Artin group , A Γ , tobe the group with presentation A Γ = h S | sts . . . | {z } length m st = tst . . . | {z } length m st ∀ ( s, t ) ∈ E i . If there is no edge in Γ between s and t in S , we say that m st = ∞ and there is no relation between s and t in the presentation.Artin groups are closely related to Coxeter groups. Given a graph Γ as above, the Coxeter group W Γ is the group whose presentation is the same as A Γ with the added relations s = e for all s ∈ S , where e is the identity element. One important example of a Coxeter group is the symmetricgroup S n . The symmetric group acts on R n by permuting the coordinates. Each transposition s i exchanging the i th coordinate with the i + 1 th coordinate in S n is an involution, or reflection. Thepresentation of S n generated by S = { s , . . . , s n − } is that of a Coxeter group. The correspondingArtin group is the braid group on n strands. More generally, any Coxeter group can be realized as Mathematics Subject Classification.
Key words and phrases.
Artin monoids, Artin groups, K ( π,
1) conjecture.Charney was partially supported by NSF grant DMS-1607616.Boyd was partially supported by the London Mathematical Society Cecil King Scholarship. a discrete group generated by reflections on a finite dimensional vector space with respect to someinner product. The Coxeter group is finite precisely when this inner product is positive definite.The braid group is the prototypical example of a finite type
Artin group, an Artin group whosecorresponding Coxeter group is a finite group. The combinatorial structure of these groups wasfirst studied by Garside [Gar69] who found a particularly nice solution to the word problem forthese groups which has played a major role in the study of finite type Artin groups. The notion ofa
Garside group was later introduced by Dehornoy and Paris [DP99] to include other groups witha similar combinatorial structure.However, the definition of Artin groups encompasses a class of groups much larger than only thefinite type groups, and as a whole, this class is very poorly understood. Particular types of Artingroups are well studied, including finite type, right-angled and FC-type Artin groups. Howeverfor general Artin groups, many basic questions remain unanswered. For example, it is unknownwhether they are torsion-free and whether they have solvable word problem, as well as many otherproperties.Questions that seem intractable for Artin groups are often easier to solve in the monoid case.Given a labelled graph Γ as above, the
Artin monoid A Γ+ is defined to be the monoid given by thepositive presentation of the Artin group: i.e. elements in A Γ+ are represented by words which useonly positive powers of the generating set S . A +Γ = h S | sts . . . | {z } length m st = tst . . . | {z } length m st ∀ ( s, t ) ∈ E i + . Some properties of the monoid are immediate, and solve questions that are still unknown for thegroup. For example, because all of the relations in the above presentation preserve the length of aword in terms of the standard generating set, there is a well defined length function for any givenmonoid element. Thus the word problem in the monoid is easily solved.The monoid also has the structure of a partially ordered set, with a relation determined by aprefix (or suffix) order on the elements. This poset is especially useful in the finite type case, whenit turns out the poset is a lattice, yet it can also be useful in the more general case. One goal ofthis work is to use the monoid and this poset to study the group in as much generality as possible.One challenge in using the Artin monoid to study the Artin group is that very little is knownabout the relationship between the two. In [Par02], Paris shows that the natural map A Γ+ → A Γ is an injection, and even this result is highly non-trivial. In the current paper, we study geometricrelationships between monoids and groups with the hope of building new tools to study both themonoid and the group.1.1. The monoid Deligne complex.
To date, the most effective approaches to studying infinitetype (non-finite type) Artin groups have been geometric. In particular, CAT(0) cube complexeshave been a primary tool in the study of right-angled Artin groups and, more generally, FC-typeArtin groups.The subgroup A T generated by a subset T ⊆ S is called a special subgroup of A Γ . By a theoremof van der Lek [vdL83], A T is isomorphic to the Artin groups associated to the (full) subgraph of Γspanned by T . An Artin group is called FC-type if any subset T that spans a clique in Γ, generatesa finite type Artin group. FC-type Artin groups were originally defined by the second author andDavis in [CD95a]. Building on work of Deligne, they define a cube complex whose vertices are givenby cosets of finite type special subgroups of A Γ . Charney and Davis call this complex the modifiedDeligne complex, and it has since become known as the Deligne complex. DELIGNE COMPLEX FOR ARTIN MONOIDS 3
Definition 1.1.
Let A Γ be an Artin group. The Deligne complex D Γ is the cube complex withvertex set all cosets gA T , such that A T is finite type. For any pair of vertices gA T ⊂ gA T ′ , theinterval [ gA T , gA T ′ ] spans a cube of dimension | T ′ r T | .The action of A Γ on its cosets induces a cocompact action by isometries of A Γ on the Delignecomplex D Γ . Note, however, that this action is not proper as the stabilizer of a vertex gA T is thesubgroup gA T g − .FC-type Artin groups are precisely those for which the standard cubical metric on this complex isCAT(0). The “FC” stands for Flag Complex, which comes from the flag condition required to showthat a cube complex is CAT(0). The Deligne complex has been used to show that FC-type Artingroups have many desirable properties. It was originally introduced to prove the K ( π,
1) conjecture,which we discuss below. In addition, it has been used to show that FC-type Artin groups havesolvable word problem, are torsion-free and have finite virtual cohomological dimension, amongother properties [Alt98, CD95a, God07].In this paper, we use the Deligne complex to study the relation between the Artin monoid andthe Artin group. The Deligne complex D Γ is built from cosets of finite type special subgroups of A Γ . In the first author’s work [Boyd20], an analogue to these cosets for the Artin monoid is definedand studied, in the setting of homological stability. This has inspired the current work, in whichwe imitate the construction of the Deligne complex for the Artin monoid to produce a new cubecomplex, D +Γ , which we call the monoid Deligne complex .First, we show that for any Artin monoid A Γ+ , the complex D +Γ is contractible. The analogousresult for D Γ is known only for certain restricted classes of Artin groups. (Indeed, this is one of themajor open problems for a general Artin group.) Theorem 4.1.
Let A +Γ be an arbitrary Artin monoid. Then the cube complex D +Γ is contractible. We then compare the geometry of this new complex D +Γ to the full Deligne complex D Γ . Ourmain results are the following. Theorem 5.1.
Let A Γ be any Artin group. Then D +Γ embeds as a subcomplex of D Γ and theinclusion map ι : D +Γ → D Γ is a locally isometric embedding. This theorem applies to all Artin groups, but has important consequences when restricted to theFC-type case.
Corollary 5.2. If A Γ is an FC-type Artin group, then the inclusion map ι : D +Γ → D Γ is anisometric embedding, hence D +Γ is CAT(0) and its image is convex in D Γ . The monoid Cayley graph.
The Deligne complex considers group elements ‘up to finitetype cosets’, which in many cases reduces problems to the well-studied finite type case. We canhowever, consider the whole set of group elements by studying the Cayley graph of the group. Weconsider the Cayley graph of a given Artin group and study the properties of the subgraph spannedby elements in the monoid, which we call the monoid Cayley graph.Identifying elements of the group and monoid with vertices in the Cayley graph and monoidCayley graph, we get a metric on A Γ and A Γ+ given by minimal path lengths in the correspondinggraph. Using the Cayley graph of A Γ defined with respect to a particular finite generating set called M , the set of minimal elements, we show the following. Proposition 6.3.
Suppose A Γ is a finite type Artin group, then with respect to the generating set M , the associated monoid A Γ+ embeds isometrically in the group A Γ . DELIGNE COMPLEX FOR ARTIN MONOIDS 4
On the other hand, we give an example showing that this inclusion is not in general convex oreven quasi-convex. That is, geodesics in the Cayley graph of A Γ connecting two monoid vertices,need not stay uniformly bounded distance from the monoid subgraph.1.3. Motivation: the K ( π, conjecture. Much of this paper focuses on establishing a geometricrelationship between an Artin group and its corresponding monoid, but Artin monoids and theirgeometry are also interesting to study in their own right.Much of the early work on Artin groups focussed on solving a conjecture formulated in its currentform by Arnol’d, Brieskorn, Pham and Thom. First we consider the case of finite type Artin groups.In this case, given a defining graph Γ, one can associate to the (complexified) action of the Coxetergroup W Γ on C n a hyperplane complement obtained by removing all of the hyperplanes fixed bysome reflection. We will denote this hyperplane complement by H Γ . The group W Γ acts freelyon H Γ and the corresponding quotient H Γ /W Γ has as its fundamental group the Artin group A Γ .In fact, this was the original motivation for the definition of Artin groups by Brieskorn [Bri71].For example, in the case that W Γ is the symmetric group on n letters, H Γ /W Γ is the configurationspace of n (unordered) distinct points in the complex plane, and the braid group can be naturallyidentified with the fundamental group of this space.In work of Deligne [Del72], the universal cover of H Γ (and hence also of H Γ /W Γ ) is shown tobe contractible and it follows that H Γ /W Γ is an Eilenberg-Maclane space for A Γ , otherwise knownas a K ( A Γ ,
1) space or classifying space BA Γ . His proof, however, applies only to the finite typecase. For infinite type Artin groups, there is an analogue of the hyperplane complement formulatedby Vinberg [Vin71] by restricting to an open cone in C n . (For a more detailed description seeDavis [Dav08], notes by Paris [Par14] and the introduction of [Cha07].) We again denote thishyperplane complement by H Γ . Van der Lek [vdL83] showed that the fundamental group of H Γ /W Γ is isomorphic to the Artin group A Γ for any G . The K ( π, conjecture states that an analogueof Deligne’s theorem holds for all Artin groups, that is, the universal cover of H Γ is contractible.This conjecture is open in general, but is known to hold for many classes of Artin groups, includingfinite type [Del72], FC-type [CD95a], affine type [PS19], and 2-dimensional Artin groups [Hen85]The K ( π,
1) conjecture has been rephrased in many ways. Charney and Davis [CD95a] showedthat the Deligne complex of Definition 1.1 is homotopy equivalent to the universal cover of H Γ forany Artin group and thus contractibility of the Deligne complex would prove the K ( π,
1) conjecture.Most of the cases for which the conjecture is known were proved in this way.There are also several other approaches. Notably, Salvetti constructed a finite dimensional CWcomplex Sal Γ homotopy equivalent to H Γ [Sal94] (see also [CD95b]), so proving that this complexis aspherical would also prove the K ( π,
1) conjecture.In 2006 Dobrinskaya proved that the quotient H Γ /W Γ has the same homotopy type as BA +Γ ,the classifying space of the Artin monoid [Dob06]. This was later reproven by Ozornova [Ozo17]and Paolini [Pao17]. Recall that, contrary to classifying spaces of groups, the classifying space of amonoid can exhibit any connected homotopy type [McD79]. It follows that the K ( π,
1) conjectureis true for A Γ , if and only if the natural map BA +Γ → BA Γ is a homotopy equivalence. We give adiagrammatic depiction of some of the known K ( π,
1) conjecture equivalences in the digram below.Finding a proof which confirms any question mark shown in the diagram would in turn provethe K ( π,
1) conjecture. Some of our results (Theorem 4.1 and part of Theorem 5.1) on the monoid
DELIGNE COMPLEX FOR ARTIN MONOIDS 5
Deligne complex are shown in red. D +Γ ∼ = ∗ (cid:127) _ (cid:15) (cid:15) ^ H Γ /W Γ ∼ = D Γ ? ∼ = ∗ (cid:15) (cid:15) Sal Γ /W Γ ≃ / / H Γ /W Γ BA Γ K ( π,
1) conj. ≃ ? o o BA +Γ ≃ O O ≃ ? The initial definition of Artin groups arose from their relation to hyperplane complements: thegroup encodes some of the homotopy information about the hyperplane complement, namely thefundamental group. The K ( π,
1) conjecture then implies that the Artin group in fact encodes all of the homotopy information, and this has been shown to be true in many cases. However, fromDobrinskaya’s work, we know that the Artin monoid encodes all of the homotopy information forany Γ. This has motivated our study of these monoids and the related construction of a monoidDeligne complex.1.4.
Discussion and further questions.
We believe that this work opens the door to manyfurther questions on the geometry of Artin monoids and the relationship with their correspondingArtin group. Readers who have experience with Artin groups or CAT(0) geometry will no doubtbe able to think of further questions in this direction, we will discuss a few below that seem naturalto us.There are a number of other geometric structures that have been used to study different classesof Artin groups. For example, in [CMW19], the second and third authors use another cube complex,called the clique cube complex, to show that most irreducible infinite type Artin groups have trivialcenter and are acylindrically hyperbolic. The clique cube complex is defined using cosets of specialsubgroups corresponding to cliques in the defining graph. Unlike the Deligne complex, the cliquecube complex is CAT(0) for all Artin groups. This complex has also been used to show that manyquestions about general Artin groups can be reduced to the case of Artin groups whose defininggraph is a single clique [GP12, CMW19]. One can define a monoid clique cube complex , in the samevein as the monoid Deligne complex, using the monoid cosets of special subgroups correspondingto cliques. This inspires the next question, which we leave broad.
Question.
Which of our results for the monoid Deligne complex hold true for the monoid cliquecube complex?Our results are particularly strong in the FC-type case because in that case, the cubical metricon the Deligne complex is CAT(0). There is another metric on the Deligne complex, known as theMoussong metric, that is conjectured to be CAT(0) for all Artin groups. This is known to be thecase when D Γ is 2-dimensional [CD95a]. Question.
Is the embedding D Γ+ → D Γ locally isometric with respect to the Moussong metric?In the 2-dimensional case, this would imply that D Γ+ is convex and hence also CAT(0). DELIGNE COMPLEX FOR ARTIN MONOIDS 6
Considering our results on the monoid Cayley graph, there are many potential strengtheningsone might desire. The following question seems reasonable to us, in light of the fact that in theFC-type case, we have shown that D +Γ embeds isometrically into D Γ . Question.
In the FC-type case, does the monoid Cayley graph isometrically embed into the Cayleygraph of the full group with respect to either the standard generating set S or the generating setof minimal elements M ?We finish with a very ambitious question related to our main motivating problem, the K ( π, D +Γ in D Γ , given by the actionof the Artin group on D Γ . Taking ‘enough’ of these translates will cover D Γ . Question.
Can one use the covering of D Γ by translates of D +Γ to prove contractibility of D Γ , andhence prove the K ( π,
1) conjecture, for some new classes of Artin groups?One might begin by considering this question in the FC-type case – where the contractibilityof D Γ is known – as this may provide further insight into the question for more general D Γ .1.5. Outline.
In Section 2 we review background material on Artin groups and their monoids aswell as the geometry of CAT(0) cube complexes. In Section 3, we define the monoid analogue ofcosets, as originally described in [Boyd20], and use these cosets to define a monoid version of theDeligne complex, denoted D +Γ . In Section 4, we prove that the monoid Deligne complex D +Γ isalways contractible. In Section 5, we investigate the geometric properties of the embedding of D +Γ into the Deligne complex D Γ . In Section 6, we consider the Cayley graph of an Artin group andthe subgraph spanned by monoid elements. Specifically we consider questions about the convexityof this subgraph.1.6. Acknowledgements.
The first author would like to thank the Max Plank Institute for Math-ematics in Bonn for its support and hospitality.This project began during a two-week summer school at the Institut des Hautes ´Etudes Scien-tifiques. All three authors would like to thank IHES for their hospitality.2.
Background
In this section we collect basic facts and lemmas that we require in the rest of the paper.2.1.
Artin groups and monoids.
In this section we recall some basic definitions and propertiesof the groups and monoids we work with. General references for readers are Paris [Par14], [Mic99]and Brieskorn and Saito [BS72] (an English translation of this paper also exists [CCC + S and a finite simple graph Γ with vertex set S and edge set, E , where eachedge ( s, t ) ∈ E is labelled by an integer greater m st ≥
2. If there is no edge between s and t we set m st = ∞ . Definition 2.1.
Given a finite generating set S and corresponding graph Γ as above, we define the Artin group A Γ to be the group with presentation A Γ = h S | sts . . . | {z } length m st = tst . . . | {z } length m st ∀ ( s, t ) ∈ E i . For each graph Γ there also exists a corresponding Coxeter group W Γ , given by adding therelations s = e for all s ∈ S to the presentation for A Γ . The finite Coxeter groups were classifiedby Coxeter [Cox33], and if W Γ is finite, we say that A Γ is a finite type Artin group (note since allgenerators have infinite order an Artin group is never itself finite).
DELIGNE COMPLEX FOR ARTIN MONOIDS 7
Definition 2.2.
Given a finite generating set S and corresponding graph Γ, the Artin monoid A Γ+ is defined to be the monoid given by the positive presentation of the Artin group: i.e. elementsin A Γ+ are represented by words which use only positive powers of the generating set S . A +Γ = h S | sts . . . | {z } length m st = tst . . . | {z } length m st ∀ ( s, t ) ∈ E i + . Note here that A Γ is the group completion of A +Γ . Artin monoids appear in much of the work onArtin groups. For instance in the seminal work of Deligne [Del72] and Brieskorn and Saito [BS72]properties of Artin monoids, and the relationship between the monoids and the groups, play a hugerole. In particular, a key property of finite type Artin groups is that every element w can be writtenas w = ab − for a and b in the corresponding Artin monoid [Gar69, Del72]. Definition 2.3.
Given a subset T ⊆ S , the full subgraph of Γ spanned by the vertex subset T defines an Artin group in its own right, which is a subgroup of A Γ . We denote this subgroup A T ,and call such subgroups special subgroups . If a special subgroup A T is a finite type Artin group,we call it a spherical or finite type subgroup. In this setting it is useful to keep track of whichsubsets T ⊆ S give rise to spherical subgroups. We define S f = { T ⊆ S | A T is finite type } . We similarly define A + T to be the submonoid of A +Γ corresponding to the subgraph of Γ spannedby T .Many of the technical lemmas in this work involve manipulation of words or elements in a givenArtin group or monoid. The following definitions and lemmas provide us with a tool-kit with whichto compare or manipulate these elements. Definition 2.4.
Let α be an element in A +Γ . We define the length of α with respect to the standardgenerating set S by l ( α ) = k if α can be written as the word α = s . . . s k for s i ∈ S. The length function l : A +Γ → N is a well-defined monoid homomorphism. It is independent ofthe word chosen to represent the element α , since the relations in the Artin monoid equate wordsof the same length, and it is a homomorphism since multiplication in the monoid corresponds toaddition of lengths. Lemma 2.5 ([Mic99]) . Artin monoids satisfy right cancellation: that is if a, b, c ∈ A +Γ satisfy ac = bc then it follows that a = b . Likewise for left cancellation. There are two partial orderings on the Artin monoid that will play a key role in the forthcomingarguments.
Definition 2.6.
For a, b ∈ A +Γ , let (cid:22) L denote the partial ordering on A Γ+ defined by a (cid:22) L b if b = ac for some c ∈ A Γ+ . In this case we say a is a left divisor of b , and b is a left multiple of a .Similarly, let (cid:23) R denote the partial ordering on A Γ+ defined by b (cid:23) R a if b = ca for some c ∈ A Γ+ .In this case we say a is a right divisor of b , and b is a right multiple of a .Note that in the definitions above, ‘left’ and ‘right’ refers to the choice of ordering, (cid:22) L or (cid:23) R .We caution that some authors use these terms differently. Definition 2.7.
Given a subset X of an Artin monoid A +Γ , we say that b ∈ A +Γ is a common leftmultiple for X if x (cid:22) L b for all x ∈ X . We say that a ∈ A +Γ is a common left divisor for X if a (cid:22) L x for all x ∈ X . Similarly for common right multiples and divisors. DELIGNE COMPLEX FOR ARTIN MONOIDS 8
Lemma 2.8 ([BS72, Proposition 4.1]) . If a finite subset X of an Artin monoid has a commonleft/right multiple, then it has a (unique) least common left/right multiple, which we denote by lcm L ( X ) , or lcm R ( X ) . Lemma 2.9 ([BS72, Proposition 4.2]) . Any finite subset X of an Artin monoid has a (unique)greatest common left/right divisor, which we denote by gcd L ( X ) , or gcd R ( X ) . A key property of finite type Artin groups is that the generating set S has a common left multipleand a common right multiple and these common multiples are equal. Moreover, conjugation bythis common multiple permutes the elements of S . Definition 2.10.
Let A Γ be a finite type Artin group. The Garside element ∆ S ∈ A Γ+ is theunique element satisfying ∆ S = lcm L ( S ) lcm R ( S ).Gaussian and Garside monoids and groups were introduced by Dehornoy and Paris in 1999,as generalisations of finite type Artin monoids and groups [DP99]. Finite type Artin groups areexamples of Garside groups, and thus they have a Garside structure , based on the existence of theGarside element, which we exploit throughout this work. In particular, the Garside element allowsfor a direct correspondence between group elements and monoid elements.
Lemma 2.11.
Given a finite type Artin group A Γ and an element α ∈ A Γ , there exists m ∈ A +Γ and k ∈ Z such that α = m ∆ kS . Moreover, this decomposition is unique if we require that m (cid:15) R ∆ S . We emphasize that Garside elements exist only for finite type Artin groups. Indeed, Brieskornand Saito [BS72] prove that for A Γ of infinite type, no element in the monoid can be a commonright or left multiple of S . Definition 2.12.
Let α ∈ A +Γ . Define the subset T α ⊆ S to be the set of all generators s ∈ S suchthat α (cid:23) R s , that is s ∈ T α ⇐⇒ α = βs for some β ∈ A +Γ . Lemma 2.13 ([BS72]) . For any α ∈ A +Γ , the subset T α ⊆ S is finite type, i.e. T α ∈ S f . CAT(0) cube complexes.
In addition to the combinatorial structure of the Artin monoidand Artin group, we will be interested in the geometric structure of their associated Deligne com-plexes. In this section we review some geometric notions that will be used in later sections. Formore details and proofs, see [BH11].Let (
X, d ) be a geodesic metric space, that is, a metric space in which any two points x, y areconnected by a path of length d ( x, y ). Such a path is called a geodesic from x to y . Let T ( a, b, c )denote a triangle in X with vertices a, b, c and geodesic edges [ a, b ] , [ b, c ] , [ c, a ]. A comparison triangle is a triangle T ( a ′ , b ′ , c ′ ) in the Euclidean plane with the same edge lengths. The CAT(0) conditionstates that triangles in X are “at least as thin” as their comparison triangles in E . More precisely, Definition 2.14.
A geodesic metric space X is CAT(0) if for any geodesic triangle T ( a, b, c ) in X and any points x ∈ [ a, b ], y ∈ [ b, c ], the corresponding points in a comparison triangle x ′ ∈ [ a ′ , b ′ ], y ′ ∈ [ b ′ , c ′ ] satisfy d X ( x, y ) ≤ d E ( x ′ , y ′ ) . We say X is locally CAT(0) if every point in X has a neighbourhood which is CAT(0).CAT(0) spaces satisfy many nice properties. Here are a few well-known facts. DELIGNE COMPLEX FOR ARTIN MONOIDS 9 • If X is CAT(0), then any two points in X are connected by a unique geodesic. • If X is CAT(0), then it is contractible. • If X is locally CAT(0) and simply connected then it is (globally) CAT(0). • Any locally geodesic path in a CAT(0) space is a geodesic.A particularly useful class of CAT(0) spaces are CAT(0) cube complexes (CCCs), both becausethey are easy to construct and because they come with a nice combinatorial structure in addition totheir geometric structure. We recall a few basics of CCCs here and refer the reader to [HW08, Sag12]for more details.A cube complex is a space obtained by gluing together a collection of standard Euclidean cubes,[0 , n , of varying dimensions, via isometries of faces. Let X be such a complex with the inducedpath metric d . The link of a vertex v in X , denoted lk X ( v ), (or simply lk ( v ) when X is understood)is the simplicial complex with a k -simplex for each k + 1 cube C containing v . Viewing this simplexas the unit tangent space of v in C , we can identify this simplex with a quadrant in the unit sphere S k . This gives rise to a natural piece-wise spherical metric on lk ( v ) with all edges of length π .Identifying lk ( v ) with the unit tangent space of v in X , distances in lk ( v ) correspond to anglesbetween tangent vectors. In particular, a path p passing through v is locally geodesic at v if andonly if its incoming and outgoing tangent vectors have distance at least π in lk ( v ). A similarpiece-wise spherical metric can be put on the link of a point x in a higher dimensional face andonce again, a path in X passing through x is locally geodesic at x if and only if its incoming andoutgoing tangent vectors have distance at least π in this lk ( x ).Using this fact, Gromov showed that a certain combinatorial condition on these links was suffi-cient to determine whether X is locally CAT(0). Definition 2.15.
A simplicial complex L is a flag complex if every set of k vertices in L thatare pairwise joined by edges, span a k + 1 simplex. In particular, a flag complex is completelydetermined by its one-skeleton.Gromov’s condition states, Theorem 2.16.
A cube complex X is locally CAT(0) if and only if for every vertex v in X , lk X ( v ) is a flag complex. In addition to their geometry, CAT(0) cube complexes come with a combinatorial structure givenby hyperplanes . These are codimension-one subspaces, made up of midplanes of cubes, that dividethe complex into two components. Much of the theory of CCCs depends on understanding theinterplay between hyperplanes. Moreover, in addition to the CAT(0) metric on a CCC X , there isanother metric known as the ℓ (1) -metric. This is usually only applied to the 1-skeleton of X andit is defined to be the minimal length of an edge path between two points in X (1) . This metric isnot CAT(0) and there may be many minimal length edge paths between two points. Nevertheless,the CAT(0) geodesic and the ℓ (1) -geodesics between two point x, y are related by the fact that theycross exactly the same hyperplanes, namely the hyperplanes that separate x from y . Definition 2.17.
Let X be a CAT(0) cube complex. For two vertices v, w in X , the subcomplexspanned by the ℓ (1) -geodesics from v to w is called the cubical convex hull of x, y . With respect tothe CAT(0) metric, it is the smallest convex subcomplex in X containing the geodesic from v to w .For example, dividing the Euclidean plane R into unit squares with vertices in Z , for any twovertices x, y the CAT(0) geodesic is the straight line connecting them and their cubical convex hullis the rectangle with this line as its diagonal. DELIGNE COMPLEX FOR ARTIN MONOIDS 10 Definition of D +Γ Recall from Definition 1.1 that the Deligne complex D Γ is the cube complex associated to thepartially ordered set of cosets aA T , T ∈ S f . To define the monoid Deligne complex D +Γ , we willneed to define and understand the properties of monoid cosets. Definition 3.1.
Given A + T a submonoid of A + , consider the relation ∼ T on A Γ+ given by α ∼ T β ⇐⇒ αt = βt ′ for some t and t ′ in A + T . The relation ∼ T is symmetric and reflexive. Let ≈ T be the transitive closure of ∼ T . That is, α ≈ T β if there is a chain of elements τ i in A Γ+ such that: α ∼ T τ ∼ T τ ∼ T · · · ∼ T τ k ∼ T β for some k . Denote the equivalence class of α under the relation ≈ T as [ α ] T .It is shown in [Boyd20] that for any α ∈ A Γ+ , the set of right divisors of α that lie in thesubmonoid A + T has a unique maximal element which we denote by end T ( α ), and we can factor α as α = α T · end T ( α ) . Lemma 3.2.
Let α ∈ A + . Then for any β ≈ T α , β T = α T so [ α ] T = { β ∈ A + | β = α T t for some t ∈ A + T } . Moreover, α T is the unique minimal length representative of [ α ] T .Proof. The first statement is Lemma 5.21 in [Boyd20]. By definition, α T is the smallest left divisorof α lying in [ α ] T . Since these minimal left divisors are the same for every β ∈ [ α ] T , the secondstatement follows. (cid:3) We now establish some useful properties of these monoid cosets.
Lemma 3.3.
Let α ∈ A + and T , T ⊆ S . Then(1) [ α ] T ⊂ [ α ] T if and only if T ⊂ T .(2) [ α ] T ∩ [ α ] T = [ α ] T ∩ T .Proof. (1) T ⊂ T implies [ α ] T ⊂ [ α ] T is clear from the definition of the equivalence relation.Conversely, suppose [ α ] T ⊂ [ α ] T . Then α T = α T t for some t ∈ A + T . So for any s ∈ T , theelement β = α T s = α T ts lies in [ α ] T ⊂ [ α ] T . Thus we can also write β = α T t ′ for some t ′ ∈ A + T .Cancelling α T we see that ts = t ′ ∈ T , and conclude that s ∈ T .(2) The inclusion [ α ] T ∩ [ α ] T ⊇ [ α ] T ∩ T follows from part (1), so it remains to show that[ α ] T ∩ [ α ] T ⊆ [ α ] T ∩ T . That is, given β such that β T = α T and β T = α T we wish toshow β T ∩ T = α T ∩ T . Claim : For any word γ in A + , γ T ∩ T is the least common left-multiple of γ T and γ T .Given the claim, it follows that if β T = α T and β T = α T then their least common left-multiplesare also equal, that is β T ∩ T = α T ∩ T .To prove the claim, recall that for any T , γ = γ T end T ( γ ) where end T ( γ ) is the greatest rightdivisor of γ contained in A + T . Thus, γ T ∩ T is the least common left-multiple of γ T and γ T if andonly if end T ∩ T ( γ ) is the greatest common right divisor of end T ( γ ) and end T ( γ ). Denote thisgreatest common right divisor by g . DELIGNE COMPLEX FOR ARTIN MONOIDS 11
Γ : ts ur [ e ] ∅ [ e ] { r } [ e ] { s } [ e ] { t } [ e ] { u } [ e ] { r,s } [ e ] { r,t } [ e ] { s,t } [ e ] { t,u } [ e ] { r,s,t } lk ([ e ] ∅ ) Figure 1.
The fundamental domain for the complex D +Γ where Γ is as shownwith labels such that the A { r,s,t } is finite type. Note the 3-cube is filled in. Thelink of [ e ] ∅ is shown in blue.Since g is a right divisor of end T ( γ ) ∈ A + T i for i = 1 ,
2, it is a right divisor of γ that lies in A + T ∩ T . So by definition, end T ∩ T ( γ ) (cid:23) R g . Conversely, any right divisor of γ in A + T ∩ T is also aright divisor in A + T i , so end T ∩ T ( γ ) = end T ∩ T (end T i ( γ )) . It follows that end T i ( γ ) (cid:23) R end T ∩ T ( γ ) for both i = 1 ,
2, so g (cid:23) R end T ∩ T ( γ ) since g is thegreatest common right divisor. This proves the claim. (cid:3) In the discussion that follows, we will mostly be interested in cosets associated to subsets T ∈ S f .In this case, the monoid cosets are closely related to the groups cosets. Lemma 3.4.
Let α ∈ A +Γ . If T ∈ S f , then [ α ] T = αA T ∩ A +Γ .Proof. It is clear from the definition that [ α ] T ⊆ αA T ∩ A +Γ . For the reverse inclusion, let β ∈ αA T ∩ A +Γ , so β = αg for some g ∈ A T . Since T ∈ S f , g can be written in the form g = cd − where c, d ∈ A + T . Thus βd = αc in the monoid, and we conclude that β ∈ [ α ] T . (cid:3) Remark 3.5.
In the the case that T and T are in S f , there is an alternate proof of Lemma3.3(2). Namely, it follows from [vdL83] that for any T , T , the corresponding special subgroupssatisfy A T ∩ A T = A T ∩ T and hence also αA T ∩ αA T = αA T ∩ T . Intersecting both sides with A + gives the desired equality. Definition 3.6.
We define the monoid Deligne complex D +Γ to be the cube complex with verticesgiven by [ α ] T for α ∈ A + and T ∈ S f . For T ⊂ T , let the interval [[ α ] T , [ α ] T ] spans a cube ofdimension | T r T | .The Artin monoid A Γ+ acts on D +Γ by left multiplication of cosets, that is, β · [ α ] T = [ βα ] T .This clearly preserves inclusions, and hence maps cubes to cubes.The vertices of the form [ e ] T for some T ∈ S f span a finite subcomplex, denoted F . The entirecomplex D +Γ can be built by taking translates of this subcomplex by elements in the monoid andidentifying vertices of F with vertices of αF when the corresponding cosets are equal. For thisreason, we call F the fundamental domain of D +Γ . See Figure 1 for an example of the fundamentaldomain. DELIGNE COMPLEX FOR ARTIN MONOIDS 12
The lemma below shows that we can view the complex D +Γ as a subcomplex of D Γ . Lemma 3.7.
The map ι : D +Γ → D Γ taking [ α ] T to αA T is injective and two vertices in D +Γ areconnected by an edge if and only if their image in D Γ is connected by an edge.Proof. For
T, R ∈ S f and α, β ∈ A +Γ , it follows from Lemmas 3.3 and 3.4 that [ α ] T ⊆ [ β ] R ifand only if αA T ⊆ βA R . Two such cosets span an edge in D +Γ (respectively D Γ ) precisely when | R r T | = 1. (cid:3) Contractibility for arbitrary Artin groups
In this section we prove that the monoid Deligne complex is contractible for any
Artin monoid.
Theorem 4.1.
Let A +Γ be an arbitrary Artin monoid. Then the cube complex D +Γ is contractible. Definition 4.2.
We define the fundamental domain of D +Γ to be the subcomplex F consisting ofall the cubes spanned by the cosets of the form [ e ] T for some T ∈ S f .To show that D Γ+ is contractible, we construct it inductively, starting with the fundamentaldomain and adding translates of F by monoid elements of specific lengths. At each stage we provecontractibility. We make this inductive procedure precise below. Definition 4.3.
Let α ∈ A +Γ , and denote by αF the translate of the fundamental domain F underleft multiplication by α , that is, αF consists of all the cubes spanned by cosets of the form [ α ] T for some T ∈ S f . For k ≥
0, we define D + k to be the union of the subcomplexes αF for all α withlength l ( α ) ≤ k . D + k = [ l ( α ) ≤ k αF . We state the following proposition, and prove Theorem 4.1 assuming this proposition to be true.We then finish this section by proving the proposition.
Proposition 4.4.
Let α ∈ A Γ+ with l ( α ) = k . Then(a) αF ∩ D + k − is non-empty and contractible, and(b) for any β = α ∈ A +Γ with l ( β ) = k , αF ∩ βF is contained in D + k − .Proof of Theorem 4.1. The proof is by induction on k , noting that D +Γ = lim k →∞ D + k . For the basecase k = 0, we note that D +0 = F so we must prove the fundamental domain is contractible. F hasa cone point, since [ e ] ∅ is a subset of every other coset in F . Therefore F is contractible, and thisproves the base case. We assume, for our inductive hypothesis, that D + k − ≃ ∗ . We consider D + k andshow that, assuming Proposition 4.4, this space is also contractible. Let α ∈ A +Γ satisfy l ( α ) = k .Then, using a similar argument as in the base case, the translate of the fundamental domain αF is contractible. It follows from Proposition 4.4 (a) that D + k − ∪ αF ∼ = ∗ since this is the union oftwo contractible subcomplexes with (non-empty) contractible intersection.Suppose β = α ∈ A +Γ with l ( β ) = k . Then by Proposition 4.4 (b), the intersection αF ∩ βF lies in D + k − , so ( D + k − ∪ αF ) ∩ βF = D + k − ∩ βF which is contractible by part (a). Thusas before, ( D + k − ∪ αF ) and βF are two contractible spaces with contractible intersection, so( D + k − ∪ αF ) ∪ βF ∼ = ∗ . Iterating this argument over all γ ∈ A +Γ with l ( γ ) = k , it follows that D + k = D + k − [ l ( γ )= k γF is contractible, as required. This completes the induction, and thus the proof. (cid:3) DELIGNE COMPLEX FOR ARTIN MONOIDS 13 [ α ] ∅ [ α ] R [ α ] R [ α ] R [ α ] R ∩ T α [ α ] R ∩ T α [ α ] T α Figure 2.
An example of the deformation retraction described in the proof ofProposition 4.4. In this example we assume that T α ⊂ R . The subcomplex Y ,shown in green, first retracts along the arrows to the subset Y in blue, and thenthe Y retracts to the cone point [ α ] T α . Proof of Proposition 4.4.
We first prove statement (b). If αF ∩ βF = ∅ then there exist T and T in S f such that [ α ] T = [ β ] T . Applying Lemma 3 . T = T = T .It is shown in Lemma 3.2 that any coset [ α ] T has a unique shortest element, namely ¯ α T . So if[ α ] T = [ β ] T and l ( α ) = l ( β ) = k , then either α = β (a contradiction) or the length of ¯ α T is strictlyless than k . In the latter case it follows that [ α ] T = [¯ α ] T ∈ D + k − .For statement (a), note that a vertex corresponding to the coset [ α ] T lies in αF ∩ D + k − if andonly if ¯ α T has length less that k , in which case α = ¯ α T ρ for some non-trivial element ρ ∈ A + T . Inparticular, for some t ∈ T , α (cid:23) R t . Recall that T α denotes the subset of generators s such that α (cid:23) R s , and by Lemma 2.13, T α ∈ S f . The observation above can now be stated as follows: [ α ] T lies in αF ∩ D + k − if and only if T ∩ T α = ∅ . In particular αF ∩ D + k − is non-empty, since α = e and so for at least one s ∈ S , s ∈ T α and it follows that [ α ] s ∈ D + k − .Let Y be the subcomplex of αF spanned by the cosets [ α ] T such that T ∩ T α = ∅ . Thenstatement (a) is equivalent to showing that Y is contractible. Let Y be the subcomplex of Y spanned by [ α ] T such that T ⊆ T α . Then Y is contractible since it contains a maximal element[ α ] T α . Define a projection map p : Y → Y , [ α ] R [ α ] R ∩ T α . We claim that p is a deformation retraction and hence Y is also contractible. To see this, let R denotethe poset of subsets R ⊆ S with R ∩ T α = ∅ . Then cubes in Y are in one-to-one correspondencewith intervals I = [ R , R ] in R . This cube, together with its projection p ( I ) = [ R ∩ T α , R ∩ T α ]spans a larger cube C I = [ R ∩ T α , R ], since R ⊂ R implies R ∩ T α ⊂ R . Then the restrictionof p to the cube C I is a deformation retraction of C I onto C I ∩ Y (see Figure 2). Moreover, if I ′ = [ R , R ] ⊂ I is a subinterval, and C I ′ = [ R ∩ T α , R ] is the corresponding face of C I , then p | C I restricts to the corresponding deformation retraction of C I ′ onto C I ′ ∩ Y , i.e. ( p | C I ) | C I ′ = p | C I ′ . Itfollows that the restrictions of p to all such cubes C I for I ∈ T glue together along common facesto give the desired deformation retraction of Y onto Y . (cid:3) DELIGNE COMPLEX FOR ARTIN MONOIDS 14 Convexity and CAT(0) in FC-type case
In this section we will look at the geometric relationship between the monoid Deligne complex, D +Γ and the Deligne complex, D Γ . By Lemma 3.7, we may view D +Γ as a subcomplex of D Γ .Our main goal in this section is to prove the following theorem. Theorem 5.1.
The inclusion map ι : D +Γ → D Γ is a locally isometric embedding. Before proving the theorem, we discuss some consequences. While the theorem holds for allArtin groups, it has many additional implications for FC-type Artin groups since in that case, D Γ is CAT(0). Corollary 5.2. If A Γ is an FC-type Artin group, then the inclusion map ι : D +Γ → D Γ is anisometric embedding, hence D +Γ is CAT(0) and its image is convex in D Γ . Here is a proof of the corollary assuming Theorem 5.1
Proof. If A Γ is FC-type, then D Γ is CAT(0). In a CAT(0) space local geodesics are globallygeodesic, thus a local isometry ι takes a geodesic between x and y to a geodesic between ι ( x ) and ι ( y ). Since distance is measured by the length of geodesics, it follows that ι is a (globally) isometricembedding. Moreover, since geodesics in a CAT(0) space are unique, this implies that the geodesicin D Γ between two points in the image of ι also lies in the image of ι . That is, ι ( D +Γ ) is convexin D Γ . The CAT(0) condition is inherited by any convex subspace, so we conclude that D +Γ isCAT(0). (cid:3) Another important consequence of the above corollary is that if A Γ is FC-type, then for any twovertices x, y in D +Γ , their cubical convex hull lies entirely in D +Γ , that is, any minimal length edgepath in D Γ between x and y remains inside D +Γ .Next consider the action of A +Γ on D +Γ . For any cube complex X , a continuous map g : X → X that takes cubes isometrically to cubes, is distance non-increasing. In the case of a group action,this is sufficient to show that G acts by isometries since the inverse map g − is also distance non-increasing. In the case of a monoid action, this need not be true. The map g need not be surjectiveand it may decrease distances. Indeed, the action of a non-trivial element α ∈ A +Γ on D +Γ is neversurjective (in particular, the image does not contain [ e ] ∅ ). Another consequence of Corollary 5.2,however, is that the action is distance preserving. Corollary 5.3.
Suppose that A Γ is an FC-type Artin group. Then action of α ∈ A Γ+ induces anisometric embedding D +Γ → D +Γ .Proof. The action of α on D Γ induces an isometry, so this fact, combined with Corollary 5.2, showsthat translation by α preserve distances between points. (cid:3) Proof of 5.1.
To prove Theorem 5.1, we will apply the following lemma. A proof of thelemma can be found in [HW08], but we include an outline of the proof below for the completeness.
Definition 5.4.
A subcomplex K of a simplicial complex L is said to be a full subcomplex if anycollection of vertices in K that spans a simplex in L , also spans a simplex in K . Lemma 5.5.
Let X be a cube complex and Y ⊆ X a subcomplex. Suppose that for every vertex v ∈ Y , the link of v in Y is full subcomplex of the link of v in X . Then the inclusion map Y → X is a locally isometric embedding, where the metrics are given by minimal path lengths in Y ,respectively X . DELIGNE COMPLEX FOR ARTIN MONOIDS 15
Proof.
Here is sketch of the proof. To be a local geodesic in a cube complex, a path p must firstbe piecewise linear, that is p = p p . . . p k where each p i is a straight line lying in a single cube. Inaddition, at the point x i where p i meets p i +1 , the tangent vectors to these two segments must havedistance at least π in the link of x i . Thus to show that any path that is locally geodesic in Y isalso locally geodesic in X , we must show that for any point y ∈ Y , two points in lk Y ( y ) of distance ≥ π , are also of distance ≥ π in lk X ( y ).By standard arguments, one can reduce to checking the case where y = v is a vertex in Y . Let ℓ , ℓ be two points in lk Y ( v ) ⊆ lk X ( v ). The distance between them measured in lk X ( v ), is theminimal length of a path γ connecting these two points. If such a path γ lies entirely in lk Y ( v ),then the distance from ℓ to ℓ in lk Y ( v ) is equal to their distance in lk X ( v ). If γ exits lk Y ( v ),then the fact that lk Y ( v ) is a full subcomplex of lk X ( v ), means that it enters a simplex containinga vertex w in lk X ( v ) r lk Y ( v ). Let γ ′ be a maximal segment of γ whose interior lies in the openstar of w . The subspace of the star of w spanned by w and γ ′ can be identified with a subspace ofthe 2-sphere, with w as the north pole and γ ′ a geodesic in the upper hemisphere with endpointson the equator. Any such geodesic has length π , thus the distance between ℓ and ℓ in lk X ( v )is ≥ π . (cid:3) In light of this lemma, to prove Theorem 5.1, it remains to show that for any vertex v in D +Γ ,the link of v in D +Γ (denoted lk D + ( v )) is a full subcomplex of the link of v in D Γ ( denoted lk D ( v )).We approach this problem by splitting the link of a vertex into two pieces, the upward link and thedownward link, such that the link of v is the join of the upward and downward links. Definition 5.6. If v is a vertex corresponding to the monoid coset [ α ] T , then any vertex in lk D + ( v )corresponds to a coset which is either included in or contains [ α ] T . The vertices in lk D + ( v ) can bepartitioned into two sets according to the direction of this inclusion and we call the subcomplexesspanned by these sets the upward and downward links of v in D +Γ . We define upward and downwardlinks of vertices αA T in D Γ similarly.By Lemma 3.7, the upward link in lk D + ( v ) is a subcomplex of the upward link in lk D ( v ) andthe downward link in lk D + ( v ) is a subcomplex of the downward link in lk D ( v ).We now focus on the upward and downward links in turn. Lemma 5.7.
Let A Γ be an Artin group with monoid A Γ+ . For any vertex v = [ α ] T ∈ D +Γ , the map ι sends the upward link of [ α ] T in D +Γ to a full subcomplex of the upward link of αA T in D Γ .Proof. By Lemma 3.3 the vertices in the upward link can each be written as [ α ] R where R ∈ S f and R = T ∪ r for some r ∈ S . Now suppose that [ α ] T . . . [ α ] T n is a collection of vertices in theupward link, where each T i = T ∪ s i for some s i in the generating set S . Suppose further that theimages of these vertices under ι , αA T i , span a simplex in D Γ . This implies that T ′ = ∪ i ( T i ) is in S f . Thus [ α ] T ′ is a vertex in the complex D +Γ , and the vertices [ α ] T i span a simplex in the link of[ α ] T in D +Γ . (cid:3) Now we address the more difficult case of the downward link. The proof that downward links in D +Γ are mapped to full subcomplexes of downward links in D Γ will involve several steps, startingwith the following lemma. Lemma 5.8.
Let A Γ be an Artin group with monoid A Γ+ . For any vertex v = [ α ] T ∈ D +Γ , the map ι sends the one-skeleton of the downward link of [ α ] T to a full subgraph of the one-skeleton of thedownward link of αA T . DELIGNE COMPLEX FOR ARTIN MONOIDS 16
Proof.
Suppose [ a ] T and [ a ] T are vertices in the downward link of [ α ] T .Assume α is the minimal representative in [ α ] T . Then left multiplication by α preserves theinclusion relation on cosets and maps the downward link of [ e ] T (where e =identity) isomorphicallyto the downward link of [ α ] T . Thus we may assume without loss of generality that α = e , and a , a ∈ A + T .Let T = T ∩ T . By assumption, a A T , a A T , A T lie in a cube in D Γ spanned by cA T and A T for some c ∈ a A T ∩ a A T . By Lemma 2.11, any element of A T can be written in theform b ∆ − k where b ∈ A + T , k ≥ is the Garside element for A T . Thus we can write c = a b ∆ − k and likewise c = a b ∆ − j . Say k ≥ j . Then replacing b by b ∆ k − j , we may assumethat k = j , that is, c = a b ∆ − k = a b ∆ − k . If k = 0, then c lies in the monoid A + T . Hence the interval [[ c ] T , [ e ] T ] spans a cube in D +Γ .So suppose k >
0. Let d = gcd L (∆ k , ∆ k ) be the maximal left divisor, and write ∆ ki = dz i . Wewill show that cd ∈ a A + T ∩ a A + T . First, note that cd = a i b i z − i ∈ a i A T i for i = 1 ,
2, so itremains only to check that cd lies in the monoid. For this, note that ( a b ) − ( a b ) = z − z . Sincegcd L ( z , z ) = e , z − z is the unique minimal representative for this element. So we must have a i b i (cid:23) R z i . It follows that cd = a i b i z − i is in a i A + T i . (cid:3) Now we turn to the case of higher dimensional simplices in the downward link.
Lemma 5.9.
Let A Γ be an Artin group with monoid A Γ+ . For any vertex v ∈ D +Γ , the downwardlink of this vertex is flag.Proof. As before, left multiplication by α preserves the partially ordered set on the cosets. Thedownward link of v = [ α ] T is given by a copy of the Deligne monoid complex for A + T , left multipliedby α T . So it is sufficient to show this result in that case of a vertex [ e ] T , where e is the identityelement.Vertices in the downward link of [ e ] T are of the form [ a i ] T i , where a i ∈ A + T and T i = T \{ t i } , forsome t i ∈ T . Suppose we have a set of vertices in this form for 1 ≤ i ≤ n and each pair of verticesin this set spans an edge in the downward link. In other words, suppose that [ a i ] T i ∩ [ a j ] T j = ∅ forall pairs { i, j } . We would like to show that these vertices span an n -simplex in the downward linkby showing that ∩ i [ a i ] T i = ∅ .First we will find an expression for [ a i ] T i ∩ [ a j ] T j as a single coset. Let T ij = T i ∩ T j . The factthat [ a i ] T i ∩ [ a j ] T j = ∅ and [ a i ] T i = [ a j ] T j implies that T i = T j and T ij is a strict subset of thesesets. This means that if β ∈ A + T is in the intersection [ a i ] T i ∩ [ a j ] T j , then this intersection can bewritten as [ β ] T ij . However we would like a more precise expression for β . Claim : [ a i ] T i ∩ [ a j ] T j = ∅ ⇐⇒ lcm L ( a i , a j ) exists and [ a i ] T i ∩ [ a j ] T j = [lcm L ( a i , a j )] T ij . Proof of claim: ( ⇐ ) is immediate.( ⇒ ) Suppose [ a i ] T i ∩ [ a j ] T j = ∅ . Then there exists a common multiple x such that x = a i m i and x = a j m j for m i ∈ A + T i and m j ∈ A + T j . So lcm L ( a i , a j ) exists and x = lcm L ( a i , a j ) m for some m . Write lcm L ( a i , a j ) = a i b i = a j b j and compare x = lcm L ( a i , a j ) m = a i b i m = a j b j m to x = a i m i and x = a j m j . By cancellation of a i and a j on the left it follows that b i m ∈ A + T i and b j m ∈ A + T j .Therefore m ∈ A + T ij and x ∈ [lcm L ( a i , a j )] T ij . This shows [ a i ] T i ∩ [ a j ] T j ⊆ [lcm L ( a i , a j )] T ij . To show ⊇ , note that from above b i ∈ A + T i and b j ∈ A + T j . Therefore if y = lcm L ( a i , a j ) m for m ∈ A + T ij then it follows that y = a i b i m = a j b j m and since m ∈ A + T ij ⊂ A + T i and A + T j ,then y ∈ [ a i ] T i ∩ [ a j ] T j . DELIGNE COMPLEX FOR ARTIN MONOIDS 17
Now we turn our attention to the intersection ∩ i [ a i ] T i . Claim : if [ a i ] T i ∩ [ a j ] T j = [lcm L ( a i , a j )] T ij is non-empty for all pairs { i, j } , then ∩ i [ a i ] T i is alsonon-empty.Proof of claim: Set i = 1. We have that lcm L ( a , a j ) is in [ a ] T ∩ [ a j ] T j for all j and so we canwrite lcm L ( a , a j ) = a m j = a j n j for m j ∈ A + T and n j ∈ A + T j .Since, for every 2 ≤ j ≤ n, , m j is in A + T , and A + T is finite type (by definition), it followsthat lcm L ( { m j } ) exists and is in A + T . So a lcm L ( { m j } ) ∈ [ a ] T satisfies that a i is a left di-visor of this element for all i . Since they have a common multiple, they have a least commonmultiple lcm L ( { a i } ) for 1 ≤ i ≤ n .We now show that lcm L ( { a i } ) is in [ a ] T . Suppose it is not, then lcm L ( { a i } ) = a x for x ∈ A + T but x / ∈ A + T . Since a lcm L ( { m j } ) is a common multiple, the least common multiple is a left divisorof it, so a lcm L ( { m j } ) = a xy for some y ∈ A + . Then cancellation of a gives xy = lcm L ( { m j } ) ∈ A + T . This contradicts x / ∈ A + T .A similar argument shows that lcm L ( { a i } ) is in [ a k ] T k for any 1 ≤ k ≤ n . This shows thatlcm L ( { a i } ) is in the intersection ∩ i [ a i ] T i completing the proof. (cid:3) Since flag complexes are completely determined by their one-skeleton, combining Lemmas 5.8and 5.9, gives the desired result on downward links.
Lemma 5.10.
Let A Γ be an Artin group with monoid A Γ+ . For any vertex v = [ α ] T ∈ D +Γ , themap ι sends the downward link of [ α ] T in D +Γ to a full subcomplex of the downward link of αA T in D Γ . Finally we show that the link of a vertex in D +Γ is the join of the upward and downward links. Lemma 5.11.
Suppose that A Γ is an arbitrary Artin group and v = [ α ] T is a vertex in D +Γ . Thenthe map ι takes lk D + ( v ) to a full subcomplex of lk D ( ι ( v )) .Proof. Suppose that X is a set of vertices in lk D + ( v ), and suppose that these vertices span a simplexin lk D ( v ). The set X can be partitioned into two sets X u and X d in the upward and downwardlinks respectively. By Lemmas 5.7 and 5.10 the sets X u and X d must span simplices in lk D + ( v ). Ifeither X u or X d is empty then we are done.Assume X u and X d are non-empty. Simplices in lk D + ( v ) correspond to cubes in D +Γ containing v , or equivalently, intervals containing v . The simplex spanned by X u corresponds to an interval[ v, w ] for some w ∈ X u while the simplex spanned by X d corresponds to an interval [ z, v ] for some z ∈ X d . It follows that [ z, w ] is an interval containing v and the corresponding simplex in lk D + ( v )is precisely the span of X . (cid:3) Combining Lemma 5.5 with Lemma 5.11, this completes the proof of Theorem 5.1.6.
Properties of the monoid embedding
Our focus so far has been on the monoid Deligne complex and its relation to the full Delignecomplex. In this section we consider the relation between monoid Cayley graphs and the groupCayley graphs. Let S be the standard generating set for an Artin group A Γ and let G ( A Γ , S )denote the corresponding Cayley graph. As noted above, by a result of Paris [Par02] the Artinmonoid A Γ+ injects into the Artin group so we can identify elements of A Γ+ with vertices in thisCayley graph. DELIGNE COMPLEX FOR ARTIN MONOIDS 18
Definition 6.1.
The
Artin monoid Cayley graph , G + ( A Γ , S ), is the full subgraph of G ( A Γ , S )spanned by the vertices v ∈ A Γ+ .Note that when considering a path in the Artin monoid Cayley graph, one can traverse eitherforwards or backwards along edges, i.e. between monoid elements v and v · s for s ∈ § .Consider the induced metric on G + ( A Γ , S ). That is, the distance between two vertices a, b ∈ A Γ+ is the length of the shortest path in G + ( A Γ , S ) connecting them. It is interesting to askwhether G + ( A Γ , S ) embeds isometrically (respectively convexly) in G ( A Γ , S ), that is, whether some(respectively any) minimal length path from a to b in G + ( A Γ , S ) is also minimal length in G ( A Γ , S ).Since translation by a − does not preserve G + ( A Γ , S ), the problem does not simply reduce to thecase where a = e , and it seems quite subtle in general.We believe that a more promising approach is to use a slightly larger generating set, namely theset of minimal elements as defined below. Definition 6.2.
Let M be the set of all minimal elements in A Γ . That is M = { m ∈ A Γ+ | m (cid:22) L ∆ T , T ∈ S f } . The minimal elements in a finite type subgroup A T are in one-to-one correspondence with thenon-trivial elements in the Coxeter group W T . For the remainder of this section we denote thecorresponding Cayley graphs by G = G ( A Γ , M ) and G + = G + ( A Γ , M ).In the case of a finite type Artin group A Γ , one can algorithmically find a normal form forelements g ∈ A Γ which is geodesic with respect to this generating set [Cha95]. This normal formis obtained by first factoring g into a product g = a − b , where a, b ∈ A Γ+ and gcd L ( a, b ) = e , andthen factoring each of a and b into a product of minimals in a canonical way (called the right greedynormal form of a and b .) Proposition 6.3.
Suppose A Γ is a finite type Artin group, then G + ( A Γ , M ) embeds isometricallyin G ( A Γ , M ) .Proof. Let a, b be elements of A Γ+ . Then a minimal length edge path in Γ from a to b correspondsto a minimal length word in the generating set M representing the group element a − b . We claimthat at least one such path lies entirely in A Γ+ .To see this, let c = gcd L ( a, b ) and write a = ca ′ , b = cb ′ . Let a ′ = µ µ . . . µ k and b ′ = η η . . . η j be the right greedy normal forms. Then by [Cha95], the word w = µ − k . . . µ − η . . . η j is a minimallength representative for a − b = ( a ′ ) − ( b ′ ).Now consider the paths γ from e to a ′ and γ from e to b ′ given by the words µ . . . µ k and η . . . η j respectively. Both of these paths lie entirely in G + and the path that traverses γ in reversefollowed by γ is a minimal length edge path from a ′ to b ′ in both G + and G . Translating this pathby c gives a minimal length edge path from a to b which again lies entirely in G + . (cid:3) Recall that a subspace Y of a geodesic metric space X is said to be convex if every geodesic in X with endpoints in Y lies entirely in Y . It is said to be quasi-convex if there exists an N > X with endpoints in Y lies in the N -neighbourhood of Y . In light of theProposition 6.3, it is reasonable to ask whether G + is convex or at least quasi-convex, in G . Thisturns out to be false in general as the following example demonstrates. Example 6.4.
Let A Γ be an Artin group whose defining graph contains a subgraph Γ ′ of thefollowing form (here m can be any label). For example, this holds for any braid group with at leastfour generators. DELIGNE COMPLEX FOR ARTIN MONOIDS 19 Γ ′ = tsu m Consider the Cayley graph G of A Γ with respect to the set of minimal elements and the corre-sponding monoid Cayley graph G + . We claim that G + is not quasi-convex in G . To see this, wewill show that for every k ∈ N , there exist elements a, b in A Γ+ with a geodesic γ between themlying entirely in G + and a geodesic γ ′ in G which travels outside the k neighbourhood of γ .Fix k ∈ N and let n = k + 1. Consider the elements a = s n and b = t n . Then the geodesic γ pictured below lies entirely in G + . ts ts ts ts ts s n t n γ = e However, since ∆ s,u = su and ∆ t,u = tu , su, tu ∈ M . It follows that the path γ ′ below is also oflength 2 n , and hence a geodesic in G . Since n = k +1, this geodesic does not lie in a k -neighbourhoodof n . thus, we have constructed the required example. ts ts ts ts ts s n t n e tusu tusu tusu tusu tusu γ ′ = u − n length= n Many interesting questions remain regarding the relationship between both Deligne complexesand Cayley graphs of Artin monoids and their associated Artin groups. Some of these questionsare discussed in the introduction to this paper.
References [Alt98] Joseph A. Altobelli. The word problem for Artin groups of FC type.
Journal of Pure and Applied Algebra ,129(1):1–22, 1998.[BH11] Martin Bridson and Andre Haeflinger.
Metric Spaces Of Non-Positive Curvature . Springer-Verlag, 2011.[Boyd20] Rachael Boyd. Homological stability for Artin monoids.
Proceedings of the London Mathematical Society ,121(3):537–583, 2020.[Bri71] E. Brieskorn. Die Fundamentalgruppe des Raumes der regul¨aren Orbits einer endlichen komplexenSpiegelungsgruppe.
Inventiones Mathematicae , 12:57, 1971.[BS72] E. Brieskorn and K. Saito. Artin-Gruppen und Coxeter-Gruppen.
Inventiones Mathematicae , 17:245–271,1972.[CCC +
97] C. Coleman, R. Corran, J. Crisp, D. Easdown, R. Howlett, D. Jackson, and A. Ram. Artin groups andcoxeter groups. Translation of Brieskorn, E. and Saito, K.
Artin-Gruppen und Coxeter-Gruppen , 1997.[CD95a] Ruth Charney and Michael Davis. The K( π ,1)-problem for hyperplane complements associated to infinitereflection groups. Journal of the American Mathematical Society , 8(3):597–627, 1995.
DELIGNE COMPLEX FOR ARTIN MONOIDS 20 [CD95b] Ruth Charney and Michael Davis. Finite K( π ,1)’s for Artin Groups. Prospects in Topology, ed. by F.Quinn, Annals of Math Study 138 , 1995.[Cha95] Ruth Charney. Geodesic automation and growth functions for Artin groups of finite type.
MathematischeAnnalen , 301(1):307–324, 1995.[Cha07] R. Charney. An introduction to right-angled Artin groups.
Geometriae Dedicata , 125:141–158, 2007.[CMW19] Ruth Charney and Rose Morris-Wright. Artin groups of infinite type: trivial centers and acylindicalhyperbolicity.
Proceedings of the American Mathematical Society , 2019.[Cox33] H. S. M. Coxeter. The complete enumeration of finite groups of the form R i = ( R i R j ) k ij = 1. Journalof the London Mathematical Society , s1-10(1):21–25, 1933.[Dav08] M. W. Davis.
The geometry and topology of Coxeter groups , volume 32 of
London Mathematical SocietyMonographs Series . Princeton University Press, Princeton, NJ, 2008.[Del72] P. Deligne. Les immeubles des groupes de tresses g´en´eralis´es.
Inventiones Mathematicae , 17:273–302,1972.[Dob06] N. `E. Dobrinskaya. Configuration spaces of labeled particles and finite Eilenberg-Maclane complexes.
Proceedings of the Steklov Institute of Mathematics , 252(1):30–46, Jan 2006.[DP99] Patrick Dehornoy and Luis Paris. Gaussian Groups and Garside Groups, Two Generalisations of ArtinGroups.
Proceedings of the London Mathematical Society , 79(3):569–604, 1999.[Gar69] F. A. Garside. The braid group and other groups.
The Quarterly Journal of Mathematics , 20(1):235–254,01 1969.[God07] Eddy Godelle. Artin-Tits groups with CAT (0) Deligne complex.
Journal of Pure and Applied Algebra ,208(1):39–52, 2007.[GP12] Eddy Godelle and Luis Paris. Basic questions on Artin-Tits groups.
Configuration Spaces , pages 299–311,2012.[Hen85] Harrie Hendriks. Hyperplane complements of large type.
Invent. Math. , 79(2):375–381, 1985.[HW08] Fr´ed´eric Haglund and Daniel T. Wise. Special cube complexes. 17(5):1551–1620, 2008.[McD79] Dusa McDuff. On the classifying spaces of discrete monoids.
Topology , 18(4):313–320, 1979.[Mic99] J. Michel. A note on words in braid monoids.
Journal of Algebra , 215(1):366–377, 1999.[Ozo17] V. Ozornova. Discrete morse theory and a reformulation of the k( π , 1)-conjecture. Communications inAlgebra , 45(4):1760–1784, 2017.[Pao17] Giovanni Paolini. On the classifying space of Artin monoids.
Comm. Algebra , 45(11):4740–4757, 2017.[Par02] Luis Paris. Artin monoids inject in their groups.
Commentarii Mathematici Helvetici , 77(3):609–637,2002.[Par14] Luis Paris. K( π ,1) conjecture for Artin groups. Annales de la facult´e des sciences de ToulouseMath´ematiques , 23(2):361–415, 2014.[PS19] Giovanni Paolini and Mario Salvetti. Proof of the K ( π,
1) conjecture for affine Artin groups. arXiv:1907.11795 , 2019.[Sag12] Michah Sageev. CAT(0) cube complexes and groups.
Geometric group theory , (0), 2012.[Sal94] Mario Salvetti. The homotopy type of artin groups.
Mathematical Research Letters , 1:565–577, 01 1994.[vdL83] Harm van der Lek. The homotopy type of complex hyperplane complements.
PhD thesis, Nijmegen , 1983.[Vin71] `E. B. Vinberg. Discrete linear groups that are generated by reflections.