Cohomology of some Artin groups and monoids

Graham Ellis

Emil Skoldberg∗

Mathematics Department, National University of Ireland, Galway

2nd March 2006

Abstract

We review Squier’s resolution for spherical Artin groups and present it in a gener-ality that applies to any group whose Cayley graph has certain lattice-like properties.Then, using elementary topological techniques, we show that the resolution holds fora non-spherical Artin group A if it holds for each Artin group defined by a full sub-graph of the Coxeter graph of A with no ∞-edges. This provides an alternative proofand generalisation of Charney and Davis’s solution of the K(π, 1) problem for FCArtin groups. We also explain how a lemma of Appel and Schupp implies that theresolution holds for an Artin group A if any three standard generators of the associ-ated Coxeter group generate an infinite group. We use the resolution to calculate theintegral cohomology of some non-spherical Artin groups and all 4-generator Artinmonoids whose corresponding Coxeter group is compact hyperbolic. We also give aformula for the second integral homology of a large class of Artin monoids. The finalsection of the paper includes details of computer techniques used in the cohomologycalculations.

1 Introduction

The K(π, 1) conjecture for an Artin group A asserts that a certain finite-dimensionalspace is an Eilenberg-Mac Lane space with fundamental group A. It is known tohold for various classes of Artin group [7, 8, 10, 9, 23, 30, 32]. In this paper we: (i)review Squier’s proof of the conjecture for spherical Artin groups [32] with a viewto clearly identifing why it does not apply to the non-spherical case, (ii) explainhow Squier’s method can be applied to certain groups, such as nilpotent groups withfree lower central quotients, which are not Artin groups, (iii) prove the conjecture insome new cases (Theorems 4), (iv) calculate the integral cohomology of some non-spherical Artin groups satisfying the conjecture, (v) calculate the integral cohomologyrings of some non-spherical Artin monoids. The interest in these monoids is that

∗This author was supported by Marie Curie fellowship HPMD-CT-2001-00079

1

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their cohomology coincides with that of the corresponding Artin group whenever theK(π, 1) conjecture holds. Both (i) and (ii) have already been treated in papers byDehornoy and LaFont [14] and Charney, Meier and Whittlesey [9]. However, we feelthat our approach to them is sufficiently different to merit interest.

The integral cohomology rings H∗(A, Z) are known for all the spherical Artingroups A. The calculation for spherical Artin groups of type An (the pure braidgroups) is due to F.V. Vainshtein [33]. The calculation for Artin groups of type Bn

and Dn is due to V.V. Gorjunov [20]. The seven remaining exceptional sphericalcases have been handled recently by C. Landi [26]. In the present paper we developtechniques and computer code aimed at computing the cohomology of non-sphericalArtin groups. An example application is the following.

Proposition 1 The Artin group A defined by the Coxeter graph

4

4∞

has integral cohomology groups

H0(A, Z) ∼= Z, H1(A, Z) ∼= Z4, H2(A, Z) ∼= Z

7,H3(A, Z) ∼= Z

7 ⊕ Z22, H4(A, Z) ∼= Z

5 ⊕ Z52, H5(A, Z) ∼= Z

2 ⊕ Z42

Hn(A, Z) = 0 (n ≥ 6).

These calculations are based on an explicit free resolution for certain Artin groups.The resolution can also be used to compute cohomology of finite index subgroups.As an example we consider the exceptional spherical Artin Group A with coxeterdiagram E8 and its even subgroup Ae (consisting of those elements of even length).The following kind of computation seems to be new.

Proposition 2 The even subgroup Ae of the Artin group A defined by the Coxetergraph

has integral cohomology groups

H0(Ae, Z) ∼= Z, H1(Ae, Z) ∼= Z2, H2(Ae, Z) ∼= Z,

H3(Ae, Z) ∼= Z2, H4(Ae, Z) ∼= Z22, H5(Ae, Z) ∼= Z2 ⊕ Z6

H6(Ae, Z) ∼= Z3 ⊕ Z6 H7(Ae, Z) ∼= Z6 H8(Ae, Z) ∼= Z2

Hn(Ae, Z) = 0 (n ≥ 9).

As further applications, we describe in Section 9 the intergal cohomology ringsof all those 4-generator Artin monoids corresponding to compact hyperbolic Coxetergroups. Also, Theorem 9 yields the following calculation.

Theorem 3 Let A be an Artin monoid defined by a Coxeter graph D. Then thesecond integral homology

H2(A, Z) ∼= (Z2)p ⊕ (Z)q

is a direct sum of p cyclic groups of order 2 and q infinite cyclic groups where p, qare defined below in Section 4 in terms of D.

3

An analogous formula for the second homology of Coxeter groups was obtainedpreviously by R. Howlett [24] using different methods.

Definition. Let us define an Artin group A to be flat if it satisfies the K(π, 1)conjecture. That is, if the CW-space XA introduced in Sections 3 and 4 below iscontractible.

Our technique for calculating the cohomology of an Artin group is to first provethat it is flat, and then to use computer code based on the free ZA-resolution arisingas the cellular chain complex of the space XA.

As mentioned above, many Artin groups are known to be flat. In particular,spherical Artin groups are flat [32]. In Section 8 we give a short proof of the following.

Theorem 4 An Artin group A with Coxeter graph D is flat if each Artin groupdefined by a full subgraph of D with no ∞-edges is flat.

This implies the result of Charney and Davis [7] that FC-Artin groups are flat. Infact, it generalizes their result since, for example, we shall use a lemma of Appel andSchupp [1] to prove the following in Section 7.

Theorem 5 [1] An Artin group is flat if any three standard generators in the cor-responding Coxeter group generate an infinite group.

Recall that an Artin group is of large type if its Coxeter graph is complete. Digress-ing from the main theme of the article we note that T.Brady and J.McCammond[2] have proved that all 3-generator Artin groups of large type admit a 2-dimensionalEilenberg-Mac Lane space whose universal cover is piecewise euclidean and non-positively curved. This yields an alternative explanation of why such groups areflat. The Eilenberg-Mac Lane space of Brady and McCammond involves l + m + n1-cells and the same number of 2-cells where the off-diagonal entries in the Coxetermatrix of the 3-generator Artin group are l,m, n ≥ 3. (In the sum l + m + n any ∞term has to be read as a 0.) As something of a curiosity we provide the followingsmaller Eilenberg-Mac Lane space.

Proposition 6 Let A be a 3-generator Artin group of large type. Then A admitsa 2-dimensional Eilenberg-MacLane space K(A, 1) involving at most six 1-cells andsix 2-cells and with a piecewise euclidean non-positively curved universal cover.

The free resolution from which we compute cohomology of Artin groups can infact be constructed for any group or monoid G whose Cayley graph (with respect toappropriate generators) has certain lattice-like properties similar to those of Artinmonoids. We thus introduce the notion of quasi-lattice order for a monoid in Sections3 and 4. This notion includes that of a quasi-lattice ordered group studied in [11].The notion also covers the locally left gaussian monoids considered P. Dehornoyand Y. Lafont [14] (see also [9]). By results of Garside and others [18, 4] all Artinmonoids, and all spherical Artin groups, are quasi-lattice ordered. So too are right-angled Artin groups [11]. Furthermore, we show in Proposition 9 that groups (suchas the Heisenberg group) arising as certain extensions of lattice ordered groups are

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themselves lattice ordered. We describe Squier’s free ZG-resolution of Z in the generalsetting of quasi-lattice ordered monoids G. A similar generalization has been givenrecently by P. Dehornoy and Y. Lafont in [14], and also by Charney, Meier andWhittlesey [9], but we feel that the differences in style between the approaches issufficient to justify any overlap with their papers. The benefit of working in themore general setting is that we can obtain, at little extra cost, results such as thefollowing.

Theorem 7 Let G be a nilpotent group for which the quotients Gk/Gk+1 of the termsGk in the lower central series are finitely generated free abelian groups. Let d denotethe rank of the abelian group ⊕k≥1Gk/Gk+1. Then there is an explicit d-dimensionalfree ZG-resolution of Z.

Such a resolution was obtained recently by K. Igusa and K.E. Orr [22] using differentmethods.

We should mention that an alternative approach to the cohomology of sphericalArtin groups has been developed by M. Salvetti [30]. He begins with a finite re-flection group W acting on R

n and considers the arrangement A in Cn consisting

of complexifications of all reflecting hyperplanes of W . The group W acts freelyon the space Y = C

n \ ∪H∈AH and a result of Deligne [12] says that the quotientY/W is a K(A, 1) space, where A is the Artin group corresponding to W . Salvetticonstructs a CW-space SW that is homotopy equivalent to Y/W . The cellular chaincomplex of the universal cover SW is thus a free ZA-resolution of Z. Loday [27] andDehornoy and Lafont [14] have asked for an algebraic proof of the asphericity of SW .Example 4 below clarifies that Squier’s algebraically derived resolution coincides withSalvetti’s resolution C∗(SW ). Thus Squier’s original proof [32], and also the variantof it described below, provide an algebraic verification of the asphericity of SW .

The space SW and homotopy equivalence SW ' Y/W exists even when the re-flection group W is infinite. Furthermore, the fundamental group of Y/W is alwaysthe Artin group A associated to W . It has been conjectured that the space Y/Wis always an Eilenberg-Mac Lane space. The conjecture has been proved in a manycases. In view of the homotopy equivalence SW ' Y/W the results below prove theconjecture in some new cases.

The article is organized into the following sections.

1. Introduction

2. Constructing classifying spaces

3. Lattice orders

4. Quasi-lattice orders

5. Aspherical Artin groups

6. Non-positive curvature

7. Aspherical Artin groups continued

8. Generalized FC Artin groups

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9. Cohomology rings

10. Computer techniques

2 Constructing classifying spaces

By a classifying space for a finitely generated group G we mean a connected CW-space B(G) with homotopy groups π1B(G) ∼= G and πiB(G) = 0 for i 6= 1. Theuniversal cover of B(G) is a contractible CW-space X endowed with a free actionof G. We recall a naive construction of the space X which, in many cases, leads toa practical method for calculating a small free ZG-resolution arising as the cellularchain complex C∗(X).

We construct X recursively, defining the (n + 1)-skeleton X(n + 1) in terms ofX(n). The 0-skeleton consists of precisely one 0-cell for each element in G. Leftmultiplication by G yields a fixed-point free action of G on X(0).

Suppose that we have constructed the n-dimensional space X(n) with a fixed-point free action of G where each g ∈ G acts in each dimension k by permuting thek-cells. Suppose also that πkX(n) = 0 for 0 ≤ k ≤ n − 1. We construct X(n + 1) intwo steps.

Step 1. The space X(n) is, by hypothesis, homotopy equivalent to a wedge ofn-spheres. By removing from X(n) one appropriate n-cell for each n-sphere in thewedge, we obtain a contractible CW-space Y (n) where X(n − 1) ⊂ Y (n) ⊂ X(n).Note that Y (n) is a maximal contractible subcomplex of X(n). (The space Y (0) isa point, and Y (1) is a tree.)

We define an n-cell en of X(n) to be complementary if it is not in Y (n). Theunion Y (n) ∪ en is homotopy equivalent to an n-sphere Sn for each complementaryn-cell en. Moreover, the sphere Sn is actually homeomorphic to a CW-subspace ofY (n)∪ en. This homoemorphism can be used to attach an (n + 1)-cell en+1 to X(n).Furthermore, for each g ∈ G we can attach an (n + 1)-cell g · en+1 by specifying thatits boundary is the image under g of the boundary of en+1.

Suppose Prop(X(n + 1)) is some readily checked boolean valued property which,when true, implies that πnX(n + 1) = 0.

Step 2. We initially set X(n + 1) = X(n) and, while Prop(X(n + 1)) is false,perform the following. We choose a complementary n-cell en and, for each g ∈ G,attach an (n + 1)-cell g · en+1 to X(n + 1) in the manner just explained.

By construction, πnX(n + 1) = 0 and the free G-action on X(n) extends toX(n + 1). The required space X is taken to be the union of the skeleta X(n).

The 1-skeleton X(1) can be viewed as the Cayley graph Γ(G,x) of the group Gwith respect to a suitable generating set x. Recall that this is a directed graph withone vertex g[] for each g ∈ G and with an edge from g[] to gx[] for each g ∈ G,x ∈ x.

The cellular chain complex C∗(X) is a free ZG-resolution of Z. For specific groupsG we would like to explicitly compute this resolution in such a way that the number of

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free generators in a given dimension is not too large. The crux to such a computationis to define an appropriate property Prop(X(n + 1)) in Step 2.

For finite groups G a definition of Prop(X(n + 1)) is given in [15] and the ZG-resolution C∗(X) is implemented in the software package HAP [16]. It turns outthat the resolution is a practical method for calculating the low dimensional integralcohomology of finite groups such as the Mathieu simple group M24.

An analysis of the above construction of X shows that it is in fact valid for certainmonoids G. In the following two sections we give a more explicit description of theconstruction for a class of infinite monoids G.

3 Lattice orders

Let G be a group or, more generally, a monoid. Let x be a finite subset of G, andlet G+ denote the submonoid of G generated by x. We refer to the elements in x asgenerators. The elements of G+ are precisely those that can be written as positivewords in the generators (i.e. words involving no inverses of generators). For g, h ∈ Gwe define

g ≤ h if and only if h = gk for some k ∈ G+.

Definition. We say that G is lattice ordered with respect to x if it is left cancellative(vu = vw ⇒ u = w) and both of the following hold:

1. The elements of G form a connected poset under the relation ≤, and this posetis in fact a lattice.

2. For all elements g ≤ h in G there is no infinite ascending chain from g to h.

Note that G+ can have no elements of finite order since it is a poset with respectto ≤. (In fact, G has no elements of finite order by Theorem 8.) Note that G isconnected as a poset if and only if the set

x± = x ∪ {x′ : x′ is a left inverse of x ∈ x}

generates G as a monoid. Note also that left cancellativity implies that, for allg, u, v ∈ G, the least upper bound gu ∨ gv is equal to g(u ∨ v).

A basic example of a lattice ordered monoid is the free abelian group Zn with x

equal to any set of n generators of the group. A much less obvious example is the3-strand braid group with presentation 〈x, y : xyx = yxy〉. Lattice ordered monoidsare slightly more general than the left Gaussian monoids considered by Dehornoy andLafont [14]. (The only invertible element in a left Gaussian monoid is the identity).Lattice ordered groups have been studied by J. Crisp amd M. Laca [11]. The moregeneral notion of quasi-lattice order will be described in Section 4.

For the remained of this section we assume that the monoid G is lattice orderedwith respect to the set x. Let ∆ denote the least upper bound of the finite set ofgenerators x, let 1 denote the identity element in G, and set

D∆ = {g ∈ G+ : 1 ≤ g ≤ ∆}.

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It is not difficult to see that D∆ must be a finite lattice. Our aim in this sectionis to explain how this finite lattice completely determines a reasonably small freeZG-resolution CG

∗ of Z. In Section 10 we describe computer methods which use onlya knowledge of D∆ to

(a) convert any positive word in the generators x to a canonical form.

(b) calculate the lowest common multiple, or join, of two elements in G+.

In the resolution CG∗ each CG

n is a free ZG-module freely generated by symbols[A] where A ranges over the subsets of x of size n. To describe the boundary mapsdn:CG

n → CGn−1 we introduce a contractible CW -space X admitting a free cellular

G-action, and define CG∗ = C∗(X) to be the cellular chain complex of X. The space

X is constructed as in Section 2 but with the details of Steps 1 and 2 more carefullyspecified.

Recall that the Cayley graph Γ(G,x) is a directed graph with one vertex g[] foreach g ∈ G and with an edge from g[] to gx[] for each g ∈ G,x ∈ x. We denote suchan edge by g[x]. Left cancellativity and the lack of torsion in G implies that an edgeis uniquely determined by its boundary vertices. The Cayley graph can be viewed asthe Hasse diagram of the lattice of G, with the lattice D∆ a subgraph. The Cayleygraph can also be regarded as a 1-dimensional CW-space with 0-cells the verticesand 1-cells the edges. As in Section 2 we take this space to be the 1-skeleton X(1) ofour desired space X. Note that X(1) is path connected because of the connectivityassumption on G. Furthermore, since G is left cancellative, left multiplication ofvertices by an element g ∈ G extends to a faithful graph morphism X(1) → X(1).This action of G on X(1) is free in the sense that it induces a free ZG-structure onthe cellular chain complex C∗X(2).

We order the generators x and for each pair of generators x1 < x2 we choosegenerators x1, x2 ∈ x and positive words w1, w2 such that the least upper boundx1 ∨ x2 is of the form x1 ∨ x2 = x1w1x1 = x2w2x2. This choice can, for instance, beperformed using a canonical form for words. The word x1w1x1(x2w2x2)

−1 determinesa path γ12 (i.e. a sequence of edges) in the Cayley graph which starts and ends at thevertex 1[]. We construct a maximal tree Y (1) in X(1) in such a way that preciselyone edge of γ12 is a complementary edge for each pair of generators x1 < x2. Thiscompletes Step 1 of the construction of X(2).

We construct X(2) by attaching a 2-cell g[x1, x2] to X(1) for each pair of genera-tors x1 < x2 and g ∈ G. The 2-cell g[x1, x2] is attached so that, starting at the 0-cellg[], its boundary is the path γ12.

x1 x2

x1 x2

w1 w2g[x1, x2]

8

It is convenient to fix some total order on G which is compatible with the lattice. Wecan then give each 2-cell g[x1, x2] an orientation by specifying that edge g[x1] occurspositively in the boundary if x1 < x2.

To complete Step 2 we must verify that π1X(2) = 0. Let γ be any path in X(1)starting and ending at a common vertex.

x1

x2

x3

xm

γ =

We say that a 2-dimensional CW-subspace in X(2) fills γ if its boundary is homotopicto γ, the homotopy taking place entirely in X(1). There is a canonical filling Fγ whichdepends on our total order on G. To define this, note that the Cayley graph Γ(G,x)is acyclic (meaning that it has no directed cycles). Therefore γ is acyclic and thuscontains at least one source (a vertex which is the target of no edge in γ). Let g[] bethe least source in γ, from which, say, the edges g[x] and g[y] start. If x 6= y then Fγ

is the union of the closure of the 2-cell g[x, y] and Fγ′ where the path γ′ is obtainedfrom γ by deleting the edges g[x], g[y] and inserting the remaining oriented edges ofthe boundary of g[x, y].

x1

x2

x3

xm

γ′

g[x,y]x

y

If x = y then the Fγ = Fγ′ where the circuit γ′ is obtained from γ by simply deletingthe edges g[x], g[y] and their initial vertex. If γ′ is empty then Fγ is deemed to beempty. To see that this recursive definition terminates let u ∈ G be the least upperbound of all the vertices in γ. Then u is also the least upper bound of the verticesin γ′. Furthermore, each vertex in γ′ is greater than or equal to some vertex in γ.The recursion terminates because condition (2) in the definition of a lattice orderingimplies that there are only finitely many vertices less than u and greater than somevertex in γ.

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The above discussion uses all of the defining properties of a lattice order to es-tablish that X(2) is a simply connected space with free G-action.

Suppose now that the n-skeleton X(n) has been constructed for some n ≥ 2 insuch a way that there is one k-cell g[x1, · · · , xk] for each g ∈ G and each subset ofk ≤ n generators in x. Suppose also that for 2 ≤ k ≤ n we have defined a canonicalk-dimensional CW-subspace Fγ of X(k) for any map γ:Sk−1 → X(k − 1), where theboundary of Fγ is homotopy equivalent to the image of γ, the homotopy taking placeinside X(n − 1).

For each set of n+1 generators x1, x2, . . . , xn+1 ∈ x we choose generators x1, . . . ,xn+1 ∈ x and positive words w1, . . . , wn+1 such that the least upper bound x1 ∨ . . .∨xn+1 is equal to xiwixi for each 1 ≤ i ≤ n+1. For each i the word xiwixi determinesa path pi in the Cayley graph Γ(G,x), starting at the vertex [] and ending at thevertex (x1 ∨ . . . ∨ xn+1)[].

p1 pn+1

Each pair of distinct paths pi, pj determines a closed circuit γij = pip−1j . Let Fij

denote the 2-dimesnional canonical filling of γij . For each triple of distinct pathspi, pj, pk the union Fij ∪ Fik ∪ Fjk is a 2-dimensional CW-subspace of X(2) equal tothe image of some map γijk:S

2 → X(2). Let Fijk denote the 3-dimensional fillingof γijk. Proceeding inductively, the n + 1 paths p1, . . . , pn+1 give rise to a mapγ1,...,n+1:S

n → X(n) whose image is the union ∪1≤i≤n+1F{1,...,n+1}\{i}. We constructthe maximal contractible subspace Y (n) in X(n) in such a way that precisely one(n−1)-cell in the image of γ1,...,n+1 is a complementary cell for each tuple of generatorsx1, . . . , xn+1. This completes Step 1 of the construction of X(n + 1).

We construct X(n+1) by attaching an (n+1)-cell g[x1, . . . , xn+1] to X(n) for each(n + 1)-tuple of generators x1, . . . , xn+1 and g ∈ G. The (n + 1)-cell g[x1, . . . , xn+1]is attached via the map γ1,...,n+1.

To complete this recursive definition of X(n + 1) we need to define a canonical(n + 1)-dimensional CW-subspace Fγ in X(n + 1) for any map γ:Sn → X(n). Theboundary of Fγ must be homotopy equivalent to the image of γ. Let U denote theimage γ, and let us write FU instead of Fγ .

The 1-skeleton U(1) of U is a finite directed subgraph of Γ(G,x) and as such isacyclic. It thus contains a source vertex g[]. If this source vertex is the start of atleast n + 1 distinct edges g[x1], · · · , g[xn+1] in U(1), then the canonical filling FU isthe union of the closure of the (n + 1)-cell g[x1, · · · , xn+1] and the CW-space FU ′

where U ′ is a certain n-dimensional CW-space of X(n). To define U ′ let B denotethe boundary of g[x1, · · · , xn+1]. Let Ai = {1, . . . , nn+1} \ {i} and set V equal to

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the interior of the union of the closures of those cells g[Ai] which lie in U . ThenU ′ = (U ∪ B) \ V .

If the source vertex g[] is not the start of at least n + 1 distinct edges, then let U ′

be the space obtained by removing from U the closures of those n-cells g[x1, · · · , xn]whose edges g[xi] all start at g[]. In this case we set FU = FU ′ .

As in the case n = 2, this canonical filling procedure terminates because of theexistence of least upper bounds in G and the finite ascending chain condition on G.

The boundary homomorphism dn:CGn → CG

n−1 is defined on generators by

dn([A]) =∑

g[B]⊂UA

sign (g,B)g[B]

where g[B] ranges over all (n − 1)-cells in the CW-space UA, and sign(g,B) is theorientation with which it occurs.

In summary, we have shown the following.

Theorem 8 For any lattice ordered left cancellative monoid G there is a contractibleCW-space X on which G acts freely and which has one cell g[A] of dimension |A| foreach g ∈ G and A ⊆ x. Consequently, CG

∗ = C∗(X) is a free ZG-resolution of Z.

Example 1. Let A3 be the group with presentation 〈x, y, z : xyx = yxy, yzy =zyz, xz = zx〉. This is the pure braid group on four strings and is known to be latticeordered with respect to x = {x, y, z} [4]. Let us define the canonical form of anyelement in the positive monoid A

+3 to be the representative word in the generators

with least lexicographical order. In our space X for this group the boundaries of the2-cells [x, y], [y, z] and [x, z] can be pictured as follows.

x

x

x

x

x

y

y

yy

y

y z

z

zz

z

The given canonical form for words yields three paths p1, p2, p3 from the identityelement to the least upper bound of the three generators.

x

x

x

x

x

x

y

y

y

y

y

y

z

zz

11

Canonically filling the three circuits γ12 = p1p−12 , γ13 = p1p

−13 , γ23 = p2p

−13 , we obtain

a CW-structure on the boundary of the 3-cell [x, y, z] in X.

x

xx

xx

x

x

x

yy

y

y

y

yy

y

z

z

z

z

z

z

z

z

Figure 1.

From the above pictures we see that the boundary maps in the ZA3-resolutionCA3

∗ = C∗(X) are given by

d1([x]) = [] − x[]d1([y]) = [] − y[]d1([z]) = [] − z[]

d2([x, y]) = [x] + x[y] + xy[x] − [y] − y[x] − yx[y]d2([y, z]) = [y] + y[z] + yz[y] − [z] − z[y] − zy[z]d2([x, z]) = [x] + x[z] − [z] − z[x]

d3([x, y, z]) = (1−z+yz−xyz)[x, y]+(1−x+yx−zyx)[y, z]+(y + xzy − 1 − zy − yxzy − xy)[x, z]

�

Many examples of lattice ordered groups can be found in [14]. The followingeasily proved result can also be used to generate further examples.

Proposition 9 Let 1 → N → G → Q → 1 be an extension of groups. If N isinfinite cyclic and Q is lattice ordered with respect to a set of n generators, then Gis lattice ordered with respect to a set of n + 1 generators.

Example 2. Consider the central extensions Z → G → Z × Z of the free abeliangroup on two generators by the infinite cyclic group. Since H2(Z × Z, Z) ∼= Z thegroup G occuring in these extensions is parametrized by non-negative integers nand can be presented as G =< x, y, z : [x, y]zn = 1, [x, z] = 1, [y, z] = 1 >. ByProposition 9 the group G is lattice ordered with respect to the generators x, y, z.By Theorem 8 it has a 3-dimensional classifying space B(G) involving three 1-cells,three 2-cells and one 3-cell. The 3-cell is attached according to the following picture.

12

x

xx

x

y

y

y

y

z

z

zz

z

zn−1

Note that when n = 1 the group G is the Heisenberg group. �

Example 3. Generalizing the preceding example, let G be a nilpotent group suchthat the quotients Gk/Gk+1 of the terms in the lower central series are finitely gen-erated free abelian groups of rank dk say. Here G1 = G and Gk+1 = [G,Gk]. NowG/G2 is free abelian and is thus a lattice ordered group with respect to a set of d1

generators. The group G/G3 is an extension of the lattice ordered group G/G2 bythe free abelian group G2/G3. By applying Proposition 9 precisely d2 times we seethat G/G3 is a lattice ordered group on a set of d1 + d2 generators. Similary, G isa lattice ordered group on a set of d = d1 + · · · + dc generators (with c such thatGc+1 = 1). Theorem 7 thus follows from Theorem 8. (We should remark that theexistence of the resolution in Theorem 7 also follows directly from a result of C.T.C.Wall [34] which expresses a free resolution for a group G as a twisted tensor productof free resolutions for a normal subgroup N and quotient group G/N . Wall’s resultdoes not provide an explicit formula for the boundary homomorphism in the twistedtensor product. The possible interest in a lattice order proof is that it provides onemethod of computing the boundary map.) �

Our interest in lattice orders stems primarily from Artin groups. We now recallthe basics on these.

A Coxeter matrix is a symmetric n × n matrix each of whose entries m(i, j) isa positive integer or ∞ with m(i, j) = 1 if and only if i = j. Such a matrix canbe represented by the Coxeter graph D with n vertices, and with an edge joiningvertices i and j if m(i, j) ≥ 3. We refer to m(i, j) as an edge label and tend to omitit from pictures of Coxeter graphs in the case m(i, j) = 3. An Artin group AD, Artinmonoid GD and Coxeter group WD is assigned to each Coxeter graph as follows. TheArtin group AD is generated by elements x = {x1, · · · xn} subject to the relations

(xixj)m(i,j) = (xjxi)m(i,j)

for all i 6= j, where (xy)m denotes the word xyxyx . . . of length m. The Artin monoidGD is the monoid presented by these generators and relators. The Coxeter group WD

is the group satisfying the additional relations x2 = 1 for x ∈ x.The Artin group AD and monoid GD are said to be spherical or of finite type if

WD is finite.

It is proved in [29] that GD is always a submonoid of AD. It is shown in [4] thatan Artin group (or monoid) is lattice ordered with respect to x if and only if it isspherical.

13

In subsequent sections we extend Theorem 8 to cover all Artin monoids andcertain non-spherical Artin groups.

4 Quasi-lattice orders

Definition. We say that a group or left cancellative monoid G is quasi-lattice orderedwith respect to a finite subset x ⊂ G if both of the following hold:

1′ The elements of G form a connected poset under the relation ≤, and there is aunique least upper bound for each finite subset of G which is bounded above.

2 For any elements g ≤ h in G there is no infinite ascending chain from g to h.

Left exactness implies that, for all g, u, v ∈ G, the elements u, v admit an upperbound if and only if the elements gu, gv admit one.

All Artin monoids are quasi-lattice ordered [4], and so too are all spherical Artingroups. It is shown in [11] that graph products of quasi-lattice ordered groups arequasi-lattice ordered.

We now explain how the construction of the space X and free ZG-chain complexCG∗ = C∗(X) of Section 3 extends to the more general case of quasi-lattice ordered

monoids G, and that the proof of contractibility/exactness also extends under re-strictive assumptions. The assumptions do not hold for non-spherical Artin groups,but they do hold for all Artin monoids.

The construction of the 1-skeleton X(1) as the Cayley graph Γ(G,x) does notinvolve upper bounds and cleary extends to quasi-lattice odered monoids.

The definition of the 2-cells g[x1, x2] extends subject to the existence of a leastupper bound x1 ∨ x2 for the generators x1, x2 ∈ x. So in the quasi-lattice orderedcase we attach one 2-cell g[x1, x2] for each g ∈ G and each pair of generators x1, x2

admitting an upper bound.The definition of a 3-cell g[x1, x2, x3] requires the existence of an upper bound

x1 ∨ x2 ∨ x3. It also requires the construction canonical fillings Fij of circuits γij =pip

−1j where pi are paths corresponding to positive word representations of x1∨x2∨x3.

Furthermore, the canonical filling Fγ of a path γ:S1 → X(1) requires the existence ofan upper bound for certain pairs of vertices in γ. However, in the particular paths γij

the vertices all admit a common upper bound, namely xi ∨xj . Moreover, all verticesin the construction of Fij are bounded above by xi ∨xj. It follows that the definitionof the 3-cell g[x1, x2, x3] is valid whenever the generators x1, x2, x3 admit a commonupper bound. So in the quasi-lattice ordered case we attach one 3-cell g[x1, x2, x3]for each g ∈ G and each triple of generators x1, x2, x3 admitting an upper bound.

Similar comments apply in higher dimensions. Thus, for a quasi-lattice orderedgroup or monoid G the CW-space X has has one cell g[A] of dimension n = |A| foreach g ∈ G and subset A ⊆ x admitting a common upper bound.

We are unable to prove that X is contractible in general. But Theorem 10 andProposition 11 are two easy results in this direction.

14

Theorem 10 Let G be a quasi-lattice ordered monoid G in which every pair of ele-ments admits some lower bound. Then the CW-space X is contractible. ConsequentlyCG∗ = C∗(X) is a free ZG-resolution of Z with one free generator in dimension

n = |A| for each subset A ⊆ x admitting a common upper bound.

Proof. Let U be an n-dimensional CW-subspace of X homotopy equivalent to ann-sphere. The canonical filling used in the proof of Theorem 8 needs to be modifiedbecause in the current setting there can be distinct 1-cells g[x1], · · · , g[xn] in U forwhich no n-cell g[x1, · · · , xn] exists in X. To produce a filling FU we choose someelement l ∈ G that is a lower bound for all the vertices of U . For each 0-cell g[]in U there is a (canonical) path pg in the Cayley graph from l to g[] representing apositive word. For each 1-cell g[x] in U there is a closed path γg,x consisting of thepath pg followed by the edge g[x] followed by the path p−1

gx . Since the vertices in thispath γg,x admit a common upper bound, the canonical filling procedure of Section3 can be used to construct a 2-dimensional CW-subspace Fg,x whose boundary isγg,x. For each 2-cell g[x, y] we let γg,x,y denote its union with the union of the fillingsFh,z corresponding to the edges h[z] in the boundary of the 2-cell. The CW-spaceγg,x,y is homotopic to a 2-sphere, and its vertices admit a common upper bound.The canonical filling procedure of Section 3 can be used to construct a 3-dimensionalCW-subspace Fg,x,y whose boundary is γg,x,y.

The above procedure extends to give an (n + 1)-dimensional CW-subspace Fe foreach n-cell e in U . Taking the union of these spaces Fe, where e ranges over all n-cellsin U , we obtain an (n + 1)-dimensional subspace FU whose boundary is U .

U

e

Fe

l

This shows that X is contractible. �

A resolution for (certain) quasi-lattice ordered monoids has been obtained pre-viously by Dehornoy and Lafont [14]. Their resolution has the same number ofgenerators as in Theorem 10.

Theorem 10 applies only to quasi-lattice ordered monoids in which every pair ofelements admits some lower bound. Suppose, however, that G is an arbitrary quasi-lattice ordered monoid (or group). We should then consider the simplicial complexKG whose vertex set is G and whose simplices are the non-empty finite subsets σ ofG that admit an upper bound.

15

Proposition 11 Let G be a quasi-lattice ordered monoid. The CW-space X is con-tractible if and only if H0(K

G) = Z and Hn(KG) = 0 for n ≥ 1.

Proof. Let G be quasi-lattice ordered with respect to the set x. Let G+ be thepositive submonoid (which is also quasi-lattice ordered), and let X+ be the CW-space corresponding to G+. We regard X+ as a CW-subspace of X. For each g ∈ Glet X+

g denote the image of X+ under the (left multiplication) action of g. Thus X+g

consists of the cells g.e for all cells e ∈ X+. The subspaces {X+g : g ∈ G} cover X.

If two elements g, h ∈ G admit a least upper bound g ∨ h then X+g ∩ X+

h = X+g∨h

is contractible. If g and h have no upper bound then X+g ∩ X+

h is empty. Theresult follows from the spectral sequence of a union of subspaces (see for instance [5],Chapter VII-5).

�

Example 4 Let G = GD be the Artin monoid associated to an arbitrary n × nCoxeter matrix (m(i, j)) represented by graph D. The monoid G is generated byx = {x1, · · · , xn}. For convenience we denote m(i, j) by m(xi, xj) and define xi < xj

if i < j. We give an explicit description of the chain complex CG∗ = C∗(X) in

dimensions ≤ 3, and a computational formula for the boundary in higher dimensions.There is precisely one generator [] in dimension 0, and precisely n generators

[x] (x ∈ x) in dimension 1. The ZG-module CG2 is freely generated by unordered

pairs [x, y] of distinct generators in x with m(x, y) 6= ∞. The ZG-module CG3 is

freely generated by unordered triples [x, y, z] of distinct generators in x for which theCoxeter matrix

1 m(x, y) m(x, z)m(y, x) 1 m(y, z)m(z, x) m(z, y) 1

is represented by one of the following five Coxeter graphs. These graphs correspond-ing to all possible 3-generator spherical Artin groups [25].

4

5

A3 :

B3 :

H3 :

I2(m) × A1 :

A1 × A1 × A1 :

m

In the following we assume x < y in a pair [x, y]. In a triple [x, y, z] we assumethat the generator x corresponds to the left-hand vertex in the Coxeter graph, thegenerator z corresponds to the right-hand vertex, and that either x < y < z ory < z < x or z < x < y.

The first two boundary maps in the chain complex CG∗ are given by:

d1([x]) = [] − x[]

d2([x, y]) = (1 + xy + · · · (xy)k)[x]+(x + xyx + · · · (xy)k−1x)[y]−(1 + yx + · · · (yx)k)[y]−(y + yxy + · · · (yx)k−1y)[x] if m(x, y) = 2k + 1

16

d2([x, y]) = (1 + xy + · · · (xy)k−1)[x]+(x + xyx + · · · (xy)k−1x)[y]−(1 + yx + · · · (yx)k−1)[y]−(y + yxy + · · · (yx)k−1y)[x] if m(x, y) = 2k

If D = A3 then, from Example 1, we have:

d3([x, y, z]) = (1 − z + yz − xyz)[x, y]+(1 − x + yx − zyx)[y, z]−(1 − y − xzy + zy + yxzy + xy)[x, z]

If D = B3 then from the following subgraph of the Cayley graph of GD (whichcan also be thought of as a maximal acyclic subgraph of the Cayley graph of theCoxeter group WD)

x

xx

xx

x

x

x

x

x

x

x

x

x

y

y

y

y

y

y

y

y

y

y

y

y

y

y

y

z

z

z

zz

zz

z

zz

zz

z

z

Figure 2.

we see that:

d3([x, y, z]) = (1 − z + yz − zyz − xyz + zxyz − yzxyz+zyzxyz)[x, y]

+(1 − x + yx − zyx + yzyx − xyzyx)[y, z]

−(1 − y + xy + zy − zxy − yzy + yzxy + xyzy−xyzxy − zyzxy + zxyzxy − yzxyzxy)[x, z]

If D = H3 we could read the formula for d3([x, y, z]) from the following subgraphof the Cayley graph of GD (after labelling its edges appropriately).

17

Figure 3

However, in this case, rather than visually deriving the boundary formula we usesome standard algebra properties of the interval 1 ≤ g ≤ ∆ in the lattice GD, where∆ is the least upper bound of the generators x. This interval can be viewed as amaximal acyclic subgraph of the Cayley graph of the Coxeter group WD, and such aCayley graph is known to form the 1-skeleton of a |x|-dimensional convex polytope(see for instance [17]).

Let A ⊆ x be a subset of generators of size m. Let WA be the subgroup of theCoxeter group WD generated by the images of the elements of A. The group WA isfinite if and only if the set A has a lowest common multiple in GD. Note that whenWA is finite, each element in WA can be expressed as a positive word in the imagesof the elements of A. For each a ∈ A let Sa be the subgroup of WA generated by theimages of A \ {a} and let Ta be a subset of positive words in GD representing a lefttransversal of Sa in WA. The above visual approach to the boundary map dn([A])routinely translates into the following general algebraic description:

dm([A]) =∑

a∈A

(∑

t∈Ta

(−1)1+|a|+|t|t)[A \ {a}] ,

where |a| denotes the position of a in the set A with respect to the ordering of x, and |t|denotes the length of any reduced word representing t ∈ GD. In the translation fromgeometry to algebra the sign (−1)1+|a|+|t| can be justified by thinking of generatorsas reflections.

Armed with this general formula for dm([A]) we can compute the boundary ho-momorphism d3, for D = H3, to be:

18

d3([x, y, z]) = (1 − z + yz − xyz − zyz + zxyz + yzyz − yzxyz−xyzyz + xyzxyz + zyzxyz − zxyzxyz−yzyzxyz + yzxyzxyz + xyzyzxyz−xyzxyzxyz − zyzxyzxyz + zxyzxyzxyz−yzxyzxyzxyz + zyzxyzxyzxyz)[x, y]

+(1 − x + yx − zyx + yzyx − xyzyx − zyzyx+zxyzyx − yzxyzyx + zyzxyzyx−yzyzxyzyx + xyzyzxyzyx)[y, z]

−(1 − y + xy + zy − zxy − yzy + yzxy + xyzy+zyzy − xyzxy − zyzxy − zxyzy + zxyzxy+yzyzxy + yzxyzy − yzxyzxy − xyzyzxy−zyzxyzy + xyzxyzxy + zyzxyzxy+yzyzxyzy − zxyzxyzxy − yzyzxyzxy−xyzyzxyzy + yzxyzxyzxy + xyzyzxyzxy−xyzxyzxyzxy − zyzxyzxyzxy + zxyzxyzxyzxy−yzxyzxyzxyzxy)[x, z]

If D = A1 × A1 × A1 then from the following subgraph of the Cayley graph ofGD

x

x

x

x

y

yy

y

z

zz

z

we see that:

d3([x, y, z]) = (1 − z)[x, y] + (1 − x)[y, z] + (y − 1)[x, z]

If D = A1 × I2(m) then from the following subgraph of the Cayley graph of GD

19

x

x x

x

xy y

y

y

z

z

z

z

we see that:

d3([x, y, z]) = (1 − z)[x, y]−(1 + xy + · · · (xy)k)[x, z]+(y + yxy + · · · (yx)k−1y)[x, z]−(x + xyx + · · · (xy)k−1x)[y, z]+(1 + yx + · · · (yx)k)[y, z] if m(x, y) = 2k + 1

d3([x, y, z]) = (1 − z)[x, y]−(1 + xy + · · · (xy)k−1)[x, z]+(y + yxy + · · · (yx)k−1y)[x, z]−(x + xyx + · · · (xy)k−1x)[y, z]+(1 + yx + · · · (yx)k−1)[y, z] if m(x, y) = 2k

This completes Example 4.

The chain complex CG∗ described in Example 4 was first considered by C. Squier

[32] and, independently, by M. Salvetti [30]. Squier describes the chain complex usingthe language of monoids and lattices. Salvetti uses the quite different language ofhyperplane arrangements and parabolic subgroups. Squier uses a spectral sequenceargument to prove that the complex is exact for Artin groups of finite type. Salvettiuses a result of Deligne [12] to show that it is exact for Artin groups of finite type.The papers [27] [14] ask for an algebraic proof of the exactness of Salvetti’s complex.It is now clear that such a proof is contained in Squier’swork. Our proof of Theorem1 yields an alternative algebraic proof.

As an exercise one can use the Squier-Salvetti complex to derive the surpris-ingly simple formula in Proposition 12 for the second integral homology of an ar-bitrary Artin monoid. The exercise requires the observation that, after tensoringwith the integers, the boundary homomorphisms are determined by the followingformulae on generators: d2([x, y]) = 0 if m(x, y) is even, d2([x, y]) = [y] − [x] if

20

m(x, y) is odd, d3([x, y, z]) = 2[x, z] for a corresponding Coxeter diagram D = A3,d3([x, y, z]) = [x, z] − [y, z] for a corresponding coxeter diagram I × A2(m) with modd, and d3([x, y, z] = 0 otherwise.

(An analogous result for arbitrary Coxeter groups was obtained by R. Howlett [24]using classical techniques. Our method is adapted in [21] to obtain a direct proof ofHowlett’s formula and a new formula for the third homology of Coxeter groups.)

Proposition 12 Let G be an arbitrary Artin monoid defined by a Coxeter graph D.The second integral homology

H2(G, Z) ∼= (Z2)p ⊕ (Z)q

is a direct sum of p cyclic groups of order 2 and q infinite cyclic groups where p, qare defined below in terms of D.

In order to define p let P denote the set of pairs {s, t} of non-adjacent verticesin D. For two such pairs define {s, t} ≡ {s′, t′} if t = t′ and s is connected to s′

by an odd labelled edge. This generates an equivalence relation on P . Say that anequivalence class is torsion if it is represented by a pair {s, t} for which there existsa vertex v connected to both s and t by edges labelled with a 3. Then p is equal tothe number of torsion equivalence classes in P . We set q1 equal to the number ofnon-torsion equivalence classes in P . We set q2 equal to the number of pairs of {s, t}of vertices connected by an even labelled edge.

Let Dodd be the graph obtained from D by removing all edges labelled by aneven integer or labelled by ∞. The graph Dodd has the same vertices as D. The firstintegral homology of this graph, H1(Dodd, Z), is a free abelian group of rank q3 say.We set q = q1 + q2 + q3.

To illustrate this formula we consider the affine braid group An defined by the(n + 1)-sided polygonal Coxeter graph with each edge labelled by 3.

An

Proposition 12 and [10] imply that this group has second homology

H2(A2, Z) = Z,

H2(A3, Z) = Z2 ⊕ Z2 ⊕ Z,

H2(An, Z) = Z2 ⊕ Z, n ≥ 4.

The remaining sections of the paper are concerned with proving that the Squier-Salvetti complex is contractible for certain non-spherical Artin groups, and using thecomplex to compute the cohomology rings of some non-spherical Artin groups andmonoids.

21

5 Aspherical Artin groups

Let PD = 〈x : r〉 be the presentation of the Artin group AD corresponding to someCoxeter graph D. Associated to this presentation is the 2-dimensional CW-spaceK(PD) with one 0-cell, with one 1-cell for each generator, and with one 2-cell foreach relator attached so that it spells the relator. Thus K(PD) is the orbit spaceX(2)/AD got by quotienting the 2-skeleton of the Squier-Salvetti complex X by thefree action of AD.

Definition. Recall that we say the Artin group AD is flat if its Squier-Salvetticomplex X is contractible.

Lemma 13 If π2K(PD) = 0 then the Artin group AD is flat.

Proof. If π2K(PD) = 0 then no three distinct standard generators in the Artinmonoid GD admit a common upper bound (because, if they did, they would deter-mine a CW-decomposition of S2 representing a non-trivial element in π2K(PD)).If no three generators admit an upper bound, then the Squier-Salvetti complex Xis precisely the universal cover of K(PD). Since X is 2-dimensional and π2X =π2K(PD) = 0 we have that X is contractible. �

Given a Coxeter graph D let us define the extended graph D to be the completegraph obtained from D by inserting an edge labelled by 2 between vertices i and jwhenever mij = 2 in the Coxeter matrix.

Definition Let us say that the Coxeter graph D is aspherical if its extended graphD contains no triangles with edge labels l,m, n satisfying 1

l+ 1

m+ 1

n> 1.

Lemma 14 If π2K(PD) = 0 then the graph D is aspherical.

Proof. Suppose π2(PD) = 0. Then no three distinct standard generators in theArtin monoid GD admit a common upper bound, and hence the associated Coxetergroup WD is infinite. The well-known classification of finite 3-generator Coxetergroups (see [25]) implies that the Coxeter graph D is aspherical. �

The converse of Lemma 14 is also true and can be deduced from a lemma ofK.Appel and P. Schupp [1]. They use ”small cancellation theory” to obtain variousresults on Artin groups of extra-large type (namely, those with off-diagonal entriesmij ≥ 4 in the Coxeter matrix). For intance, they prove [1, Theorem 2] that suchgroups are torsion free. It turns out that a key lemma in their proof can be used toprove the π2K(PD) = 0 for any aspherical Coxeter diagram D.

Before proving the converse of Lemma 14, and thus completing the proof of The-orem 5, we first digress to consider a result of T. Brady and J. McCammond [2].

6 Non-positive curvature

Recall that a 2-dimensional CW-space Y is said to be piecewise euclidean if it isequipped with a metric in such a way that each of its 2-cells is isometric to a convex

22

polygonal disc in the euclidean plane. The metric will associate numerical angles toeach ”corner” of a 2-cell in Y . Let v be any vertex (that is, zero cell) in Y and letNv be an arbitrary neighbourhood of v in Y which is homeomorphic to an open disc.The disc Nv contains a sequence of corners at v, and we say that Y is non-positivelycurved at v if the sum of the angles of these corners is at least 2π. The space Y issaid to be non-positively curved if it is non-positively curved at each vertex.

Brady and McCammond [2] have shown that any 3-generator Artin group AD

of large type admits a 2-dimensional Eilenberg-Mac Lane space with a piecewiseeuclidean, non-positively curved universal cover. Their Eilenberg-Mac Lane spaceinvolves l + m + n 1-cells and l + m + n 2-cells where l,m, n ≥ 3 are the edge labelsin D. (In the sum l + m + n any ∞ term has to be read as 0.) As something ofa curiosity we give a variant of their proof which leads to a non-positively curvedpiecewise euclidean structure on the cover of an Eilenberg-Mac Lane space with atmost six 1-cells and six 2-cells. Since spaces with non-positive curvature have trivialsecond homotopy group, this yields a converse to Lemma 14 for 3-generators Artingroups of large type.

Proof of Proposition 6.

We explain the proof with reference to a particular 3-vertex Coxeter graph D.The same arguments apply, essentially unchanged, to an arbitrary 3-vertex graph oflarge type. We consider the particular graph D

5

4

with edge labels l = 5, m = 4, n = 3. The Artin group AD has standard presentation

PD = 〈x, y, z | xyxyx = yxyxy, yzyz = zyzy, zxz = xzx〉.

The group AD also admits the presentation

P ′D = 〈x, y, z, a, b, c : xy = a, yz = b, zx = c, a2x = ya2, b2 = zby, cx = yc〉.

Let K be the 2-dimensional CW-space associated to the presentation P ′D. In the

universal cover K the 1-cells are oriented and labelled by the generators x, y, z, a, b, c.

23

The 2-cells of K are of the following six types.

a

a

a

aa

b

b

bb

c cc

x

x

xx

y

yy

y

z

z

z

z

We can give K a metric in which each of the cells is euclidean and where: theedges labelled by x, y, z each have length 1; consecutive edges with the same labelare parallel; the edges labelled by a, b, c are given lengths such that the angles are asshown in the following star graph of P ′

D. This graph has one vertex for each generatorand one vertex for the inverse of each generator in the presentation. There is a singleundirected edge between vertices u and v−1 if the word uv corresponds to preciselyone corner of a relator. There are two undirected edges between vertices u and v−1

if the word uv corresponds to more than one corner of a relator. In the followingrepresentation of the star graph the edges are represented by solid lines and labelledwith the angle of the corner. Dotted lines simply mean that their end vertices shouldbe identified.

a b c xx yy z z

a−1 b−1 c−1 x−1x−1 y−1y−1 z−1 z−1

π

ππ

38π3

8π

38π3

8π38π

38π 3

8π38π3

8π

38π3

8π38π

14π1

4π14π

58π5

8π58π

58π

58π5

8π

The left-hand section of the graph is the star graph of the presentation 〈x, y, a | xy =a, a2x = ya2〉. The middle section corresponds to the presentation 〈y, z, b | yz =b, b2 = zby〉. The right-hand section corresponds to the presentation 〈z, x, c | zx =c, cz = xc〉. Proposition 6 is proved for the particular D by observing that: (i) the

24

angles shown in the star graph are consistent with the 2-cells of K being euclidean;(ii) the labels on the edges in any loop in the star graph sum to at least 2π.

The same arguments hold for an arbitrary 3-generator Coxeter graph D of largetype. The star graph will involve at most three sections, and each section will haveone of just three possible forms (each of which is illustrated above.) �

7 Aspherical Artin groups continued

Let us now return to Theorem 5. For its proof we need [1, Lemma 8]. We restate thelemma in a form convenient for our present purposes, and we give an alternative proof.Given a reduced word w in the free group on two elements x, y, say w = xε1

1 xε22 · · · xεn

n

with xi ∈ {x, y}, xi 6= xi+1, εi 6= 0, one defines the syllable length of w to be ||w|| = n.

Lemma 15 [1] Let K be the 2-dimensional CW-space associated to the Artin pre-sentation 〈x, y : (xy)k = (yx)k〉, k ≥ 2. Let M be a CW-subspace of the universalcover K and suppose that M is a compact, connected, simply connected surface. Letw be a reduced word spelling the boundary of M . Then ||w|| ≥ k.

Proof. If k = 2 then K is the usual CW-decomposition of the torus and the lemmais obvious. For k ≥ 3 we use a Tietze-equivalent presentation as in the proof ofProposition 6. Let us consider the particular (but representative) case where k =5. The Tiezte-equivalence corresponds to a subdivision of M into triangular andquadrilateral cells.

x

x

y

y

a a

a

a

a

π/4

3π/8

3π/8

5π/8

5π/8

3π/8

3π/8

The surface M can be regarded as a subcomplex M ′ of the universal cover of the 2-dimensional CW-space K ′ associated to the presentation 〈x, y, a : a = xy, a2x = ya2〉.Angles have been chosen so as to ensure that all edges in M ′ labelled by a point inthe same direction which we deem to be north. All edges labelled by x or y lie on

25

one of two roughly diagonal axes. Furthermore, no two triangles can have a commonedge. A typical example of M ′ is the following.

x

x

y

y

a

a

a

a

a

a

a

a

M ′ M ′′

By removing any leftmost and rightmost triangles from M ′ we can obtain a surfaceM ′′ whose boundary word has the same number of syllables as that of M ′ (on substi-tuting a = xy). Clearly the left-hand vertical side of M ′′ involves at least two edgeslabelled by a, as does the right-hand vertical side. Each side contributes at least foursyllables to the boundary word, and the top and bottom ”horizontal” portions of theboundary each contribute at least one syllable each. Hence the boundary word w ofM has at least ten syllables, proving that ||w|| ≥ 2k. The same proof holds for allk ≥ 3. �

The following result, in conjunction with Lemma 13, completes the proof of The-orem 5.

Theorem 16 If the Coxeter graph D is aspherical then π2K(PD) = 0.

Proof. Let K = K(PD) be the 2-dimensional CW-space associated to an Artinpresentation. If π2K = π2(K) is non-trivial then there must exist a CW-subspace Min K which is homeomorphic to a 2-sphere. Suppose that D is aspherical. We mustprove that no such M exists.

Suppose such an M existed. Its 2-cells are bounded by polygons correspondingto the relators of PD. By removing 0-cells and 1-cells if necessary (and invoking Lyn-don’s result that a one-relator group is aspherical when the relator is not a properpower), we can give M a coarser CW-structure in which: (i) the boundary of any 2-cell corresponds to some word w = w(x, y) involving just two of the Artin generatorsx, y; (ii) there exists no coarsening with fewer 2-cells that satisfies (i). If the genera-tors of the boundary word w(x, y) of a 2-cell satisfy the Artin relation (xy)k = (yx)k

26

then, by Lemma 15, ||w(x, y)|| ≥ 2k. In the coarser CW-structure there may be0-cells which lie in the boundary of precisely two 1-cells. Such 0-cells can be removedto yield a CW-structure on M in which each 0-cell lies in the boundary of at leastthree 1-cells. A 1-cell can now be labelled by a power xi of an Artin generator. ByLemma 15 a 2-cell with boundary word w(x, y) has to have at least 2k 1-cells in itsboundary (where (xy)k = (yx)k).

We thus have a cellular decomposition of the sphere M in which each vertex hasdegree at least 3, and each 2-cell is bounded by at least 2k edges where k dependson the two generators in its boundary word. Let us give each 2-cell the metric of aregular euclidean polygon with each edge of length 1. Each corner of a 2-cell willhave interior angle at least 2π(k − 1)/k. The asphericity of the Coxeter graph Densures that at each vertex in M the angles sum to at least 2π. We have thus giventhe sphere M a piecewise euclidean and non-positively curved cell structure. Butthat contradicts the piecewise euclidean version of the Gauss-Bonnet theorem for a2-sphere [19]. Hence π2K must be trivial. �

8 Generalized FC Artin groups

We now turn to the proof of Theorem 4. Recall that an Artin group AD is is said tobe a flag complex Artin group or FC Artin group if any full subgraph of its Coxetergraph D with no infinite edges defines a spherical Artin group. (A subgraph of Dis full if an edge of D belongs to the subgraph whenever its two boundary verticesbelong to the subgraph.) The main result of [7] establishes that every FC Artingroup admits a finite Eilenberg-Mac Lane space. We shall generalise this.

Let us say that an Artin group AD is a generalised flag complex Artin group orGFC Artin group if any full and connected subgraph D′ of D, with D′ involving noinfinite edges, defines a flat Artin group. For example, the Artin group in Proposition1 is GFC by virtue of Theorem 16 and the solution to the K(π, 1) conjecture for affinebraid groups [10]. We restate Theorem 4 as follows.

Theorem 17 Every GFC Artin group is flat.

Proof. Let XD denote the Squier-Salvetti CW-complex associated to a Coxeter graphD. Let BD = XD/AD denote the orbit space obtained by killing the action of theArtin group AD.

Suppose that AD is a GFC Artin group with standard generating set x. We shalluse induction on the number of infinity edges in D and the number of connectedcomponents in D to show that AD is flat (i.e. that XD is contractible).

If there are no infinity edges and the graph D is connected then AD is flat bydefinition.

If D is not connected then AD is a direct product AD = AD′ × AD′′ of twonon-trivial Artin groups AD′ and AD′′ where the graph D is the disjoint union of D′

and D′′. Observe that the space XD is precisely the direct product XD′ × XD′′ ofthe Squier-Salvetti CW-complexes XD′ , XD′′ . Thus XD is contractible if and only ifboth XD′ and XD′′ are contractible. Hence, by induction on the number of connected

27

components in D, it suffices to prove the theorem for a GFC Artin group AD wherethe graph D is connected.

Suppose that the Coxeter graph D is connected. Suppose that there is an infinityedge in D whose endpoints correspond to the generators a, b ∈ x. Let Aa be thesubgroup of AD generated by x \ {a}, and A

a,bthe subgroup generated by x \ {a, b}.

Let D \ {a} denote the graph obtained from D by removing vertex a and all edgesincident with a. Let D\{a, b} be the subgraph obtained by removing vertices a, b andall edges incident with them. There are clearly surjective homomorphisms AD\{a} →Aa and AD\{a,b} → A

a,b. The main result in [28] shows that these surjections are in

fact isomorphisms. Note that each of the groups AD\{a}, AD\{b}, AD\{a,b} is a GFCArtin group whose Coxeter graph has fewer infinitiy edges than there are in D.

Suppose that D has n ≥ 1 infinity edges. As an inductive hypothesis assume thatthe theorem holds for all GFC Artin groups whose Coxeter graphs involve fewer thann infinity edges. Thus we can assume that BD\{a}, BD\{b}, BD\{a,b} are classifyingspaces for the subgroups Aa,Ab

,Aa,b

. Consider the homotopy pushout

BD\{a,b} //

��

BD\{a}

��BD\{b} // W

The space W = BD\{a}∪BD\{b} is precisely the orbit space W = BD. Now a theoremof J.H.C. Whitehead (see for example [5], Chapter II-7) the space W is a classifyingspace. Hence its universal cover XD is contractible. �

9 Cohomology rings

In this section we describe the integral cohomology ring for each 4-generator Artinmonoid whose associated Coxeter group is compact hyperbolic. The list of suchCoxeter groups is given in [25]. The computations were obtained using an implemen-tation of the Squier-Salvetti resolution [31] written in the functional programminglanguage Haskell. A GAP implementation of the resolution is avaiable in the HAP

cohomology package [16].

1. For the Artin monoid G defined by the graph • 4 • • 5 • the ad-

ditive structure of the integral cohomology is: H0(G, Z) = Z, H1(G, Z) =Z

2, H2(G, Z) = Z3, H3(G, Z) = Z

3. The cohomology ring is

H∗(G, Z) =Z[s1, s2, t1, t2, u]

(s1t1 − 2s1t2 − s2t1, s2t2 − 15u) + J≥4

where |s1| = |s2| = 1, |t1| = |t2| = 2, and |u| = 3. The ideal J≥4 is generatedby the monomials of degree 4.

2. For the monoid G defined by • • 5 • • the additive integral coho-

mology structure is: H0(G, Z) = Z, H1(G, Z) = Z, H2(G, Z) = Z, H3(G, Z) =

28

Z2. The cohomology ring is

H∗(G, Z) =Z[s, t, u1, u2]

(st + 15u1 + 15u2) + J≥4

where |s| = 1, |t| = 2, and |u1| = |u2| = 3.

3. For G defined by • 5 • • 5 • the additive cohomology structure

is: H0(G, Z) = Z, H1(G, Z) = Z, H2(G, Z) = Z, H3(G, Z) = Z2. The

cohomology ring is

H∗(G, Z) =Z[s, t, u1, u2]

(st + 15u1 + 15u2) + J≥4

where |s| = 1, |t| = 2, and |u1| = |u2| = 3.

4. For G defined by•

• 5 •

~~~~~~~

@@@@

@@@

•

the additive cohomology structure is: H0(G, Z) = Z, H1(G, Z) = Z, H2(G, Z) =Z

2, H3(G, Z) = Z3 ⊕ Z2. The cohomology ring is

H∗(G, Z) =Z[s, t1, t2, u1, u2, u3]

(st1 − st2 − 15u1 − 15u2, 2u3) + J≥4

where |s| = 1, |t1| = |t2| = 2, and |u1| = |u2| = |u3| = 3.

5. For G defined by• •

• •

the additive cohomology structure is: H0(G, Z) = Z, H1(G, Z) = Z, H2(G, Z) =Z, H3(G, Z) = Z

2 ⊕ Z22. The cohomology ring is

H∗(G, Z) =Z[s, t, u1, u2, u3, u4]

(st, 2u3, 2u4) + J≥4

where |s| = 1, |t| = 2, and |u1| = |u2| = |u3| = |u4| = 3.

6. For G defined by• •

• 4 •

29

the additive cohomology structure is: H0(G, Z) = Z, H1(G, Z) = Z, H2(G, Z) =Z, H3(G, Z) = Z

2 ⊕ Z22. The cohomology ring is

H∗(G, Z) =Z[s, t, u1, u2, u3, u4]

(st + 3u1 + 3u2, 2u3, 2u4) + J≥4

where |s| = 1, |t| = 2, and |u1| = |u2| = |u3| = |u4| = 3.

7. For G defined by

• 4 •

• 4 •

the additive cohomology strudture is: H0(G, Z) = Z, H1(G, Z) = Z2, H2(G, Z) =

Z4, H3(G, Z) = Z

4. The cohomology ring is

H∗(G, Z) =Z[s1,s2,t1,t2,t3,u]

(2t2s1−t2s2−4t3s1+2t3s2,2t1s1−4t1s2−3t2s1−8t3s1+4t3s2,3u−3t1s1−3t2s1−8t3s1+4t3s2)+J≥4

where |s1| = |s2| = 1, |t1| = |t2| = |t3| = 2, and |u| = 3.

8. For G defined by

• 5 •

• 4 •

the additive cohomology structure is: H0(G, Z) = Z, H1(G, Z) = Z, H2(G, Z) =Z

3, H3(G, Z) = Z4. The cohomology ring is

H∗(G, Z) =Z[s, t1, t2, t3, u1, u2, u3, u4]

(6u2 − 15u4 − t1s, 15u1 − 6u3 + t2s, 3u2 + 3u3 + t3s) + J≥4

where |s| = 1, |t1| = |t2| = |t3| = 2, and |u1| = |u2| = |u3| = |u4| = 3.

9. For G defined by• •

• 5 •

the additive cohomology structure is: H0(G, Z) = Z, H1(G, Z) = Z, H2(G, Z) =Z, H3(G, Z) = Z

2 ⊕ Z22. The cohomology ring is

H∗(G, Z) =Z[s, t, u1, u2, u3, u4]

(st, 2u3, 2u4) + J≥4

where |s| = 1, |t| = 2, and |u1| = |u2| = |u3| = |u4| = 3.

30

10. For G defined by

• 5 •

• 5 •

the additive cohomology structure is given by H0(G, Z) = Z, H1(G, Z) =Z, H2(G, Z) = Z

3, H3(G, Z) = Z4. The cohomology ring is

H∗(G, Z) =Z[s, t1, t2, t3, u1, u2, u3, u4]

(15u1 − t1s, t2s, 15u2 − t3s) + J≥4

where |s| = 1, |t1| = |t2| = |t3| = 2, and |u1| = |u2| = |u3| = |u4| = 3.

10 Computer techniques

Let G be a quasi-lattice ordered monoid with respect to a subsset x. Our descriptionof the resolution CG

∗ = C∗(X) in Theorem 10 is algorithmic, but for the fact that itassumes the existence of methods for determining: (i) least upper bounds of elementsin G+, (ii) canonical forms for elements of G+. We now provide these methods.

From the proof of Theorem 8, the generators x satisfy a set of relations r = {rx,y :x 6= y ∈ x with join x ∨ y ∈ G}. The relation rx,y is of the form xu = yv for somepositive words u, v ∈ G+. The sets x and r are the input data for computations.Each relation can be represented as a word xuy−1v−1 in the free group F (x) on theset x. Note that any path in the Cayley graph Γ(G,x) represents a unique word inF (x). However, if G is not a group then certain words in F (x) will not represent anypath in Γ(G,x). Also, a word in F (x) can represent many isomorphic paths withdistinct starting vertices.

All computations are based on a function ReversedWord(). This inputs a word win F (x) and, if it terminates, outputs a word wR in F (x). We regard w as potentialpath in the Cayley graph Γ(G,x), starting at some vertex a and ending at somevertex b.

x1

x2xn

b

a

The function terminates if w represents some path in the Cayley graph all of whosevertices admit a common upper bound. The output word wR is obtained using theword reversing procedure due to Dehornoy [13]. This is a slight generalization ofthe 1-dimensional ”canonical filling” used in the proof of Theorem 10 and has thefollowing geometric description.

31

As before, we say that a vertex in the potential path represented by w is a sourceif it is the start of two edges. A source is removable if either: i) both edges of thesource are labelled by the same generator, or ii) the edges are labelled by distinctgenerators x, y whose join x ∨ y exists in G. The word wR is computed recursively.

If w represents a potential path with no removable source then wR = w.If the path contains a source then wR = w′R where w′ is the word constructed as

follows. Find the first removable source (starting from the left say). If both edgestouching this source are labelled by the same generator x ∈ x then w′ is the wordgot by deleting these edges (or cancelling x with x−1).

xx

w w′

If the edges touching the source are labelled by distinct generators x, y ∈ x then w′

is the word got by applying the relation rx,y.

x

x y

y

w w′

rx,y

Note that the path of w′ lies “above” the path of w.Suppose that the word w represents a path in the Cayley graph all of whose

vertices admit an upper bound. Let u ∈ G denote the least upper bound of thevertices. Then u is also the least upper bound of the vertices in the path of w′. Thefunction ReversedWord(w) terminates in this case because there are only finitelymany elements in the lattice lying above the vertices of w and below u.

Let u, v be two positive words in the generators x whose images in G admit acommon upper bound. The function ReversedWord() can be used to determinewhether u and v represent the same element in G. We simply apply the function tothe word w = v−1u. They represent the same element if and only if wR is the emptyword. If wR is not the empty word then it is of the form wR = v′u′−1 where u′, v′

are positive words. The words uu′ and vv′ represent the least upper bound in G of uand v. Furthermore, u ≤ v if and only if v′ is the empty word. If u ≤ v then v = uu′.

The function ReversedWord() can be used to convert a positive word w in thegenerators x into a positive word wC with a canonical form. One way to do this isto define wC to be the lexicographically least word representing the image of w in G.The word wC can be computed using the recursion

wC = x(x′C)

where x is the lexicographically least generator such that x ≤ w, and x′ is a positiveword such that w = xx′.

32

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