Determining the action dimension of an Artin group by using its complex of abelian subgroups
DDetermining the action dimension of an Artingroup by using its complex of abeliansubgroups
Michael W. Davis ∗ Jingyin HuangAugust 16, 2016
Abstract
Suppose that (
W, S ) is a Coxeter system with associated Artingroup A and with a simplicial complex L as its nerve. We define thenotion of a “standard abelian subgroup” in A . The poset of suchsubgroups in A is parameterized by the poset of simplices in a certainsubdivision L (cid:11) of L . This complex of standard abelian subgroupsis used to generalize an earlier result from the case of right-angledArtin groups to case of general Artin groups, by calculating, in manyinstances, the smallest dimension of a manifold model for BA . (Thisis the “action dimension” of A denoted actdim A .) If H d ( L ; Z / (cid:54) = 0,where d = dim L , then actdim A ≥ d + 2. Moreover, when the K ( π, A , the inequality is an equality. AMS classification numbers . Primary: 20F36, 20F55, 20F65,57S30, 57Q35, Secondary: 20J06, 32S22
Keywords : action dimension, Artin group, Coxeter group
Introduction
Given a discrete, torsion-free group π , its geometric dimension , gd π , is thesmallest dimension of a model for its classifying space Bπ by a CW complex;its action dimension , actdim π , is the smallest dimension of a manifold model ∗ Partially supported by an NSA grant. a r X i v : . [ m a t h . G T ] A ug or Bπ . From general principles, actdim π ≤ π . Taking the universalcover of such a manifold model, we see that actdim π can alternately bedefined as the smallest dimension of a contractible manifold which admits aproper π -action. The basic method for calculating actdim π comes from workof Bestvina, Kapovich and Kleiner [3]. They show that if there is a finitecomplex K such that the 0-skeleton of the cone (of infinite radius) on K coarsely embeds in π and if the van Kampen obstruction for embedding K in S m is nonzero, then actdim π ≥ m +2 (such a complex K is an m -obstructor ).A Coxeter matrix ( m st ) on a set S is an ( S × S )-symmetric matrix with1 s on the diagonal and with each off-diagonal entry either an integer ≥ ∞ . Associated to a Coxeter matrix there is a Coxeter group W defined by the presentation with set of generators S and with relations:( st ) m st = 1. The pair ( W, S ) is a
Coxeter system . For a subset T ≤ S , thesubgroup generated by T is denoted W T and called a special subgroup . Thepair ( W T , T ) also is a Coxeter system. The subset T is spherical if W T isfinite. Let S ( W, S ) denote the poset of spherical subsets of S (including theempty set). There is a simplicial complex, denoted by L ( W, S ) (or simply by L ) called the nerve of ( W, S ). Its simplices are the elements of S ( W, S ) > ∅ .In other words, the vertex set of L is S and its simplices are precisely thenonempty spherical subsets of S . Note that a subset { s, t } < S of cardinality2 is an edge of L if and only if m st (cid:54) = ∞ .Given a Coxeter matrix ( m st ), there is another group A called the asso-ciated Artin group . It has a generator a s for each vertex s ∈ S and for eachedge { s, t } of L , an Artin relation : a s a t . . . (cid:124) (cid:123)(cid:122) (cid:125) m st terms = a t a s . . . (cid:124) (cid:123)(cid:122) (cid:125) m st terms . (0.1)The special subgroup A T is the subgroup generated by { a s | s ∈ T } ; it canbe identified with the Artin group associated to ( W T , T ). The subgroup A T is spherical if T is a spherical.A spherical Coxeter group W T acts as a group generated by linear re-flections on R n , where n = Card T . After tensoring with C , it becomes alinear reflection group on C n . Deligne [14] proved that the complement ofthe arrangement of reflecting hyperplanes in C n is aspherical. Since W T actsfreely on this complement, it follows that its quotient by W T is a model for BA T . Salvetti [22] described a specific n -dimensional CW complex, Sal T ,called the Salvetti complex , which is a model for this quotient of the hyper-plane arrangement complement. Thus, gd A T ≤ dim(Sal T ) = Card T . (In2act, since the cohomological dimension, cd A T , is ≥ Card T the previous in-equality is an equality.) More generally, one can glue together the complexesSal T , with T spherical, to get a CW complex Sal S with fundamental group A (= A S ). The dimension of Sal S is dim L + 1. The K ( π, -Conjecture for A is the conjecture that Sal S is a model for BA . This conjecture is true inmany cases, for example, whenever L is a flag complex, cf. [7]. Thus, if the K ( π, A , then gd A = dim L + 1.A Coxeter matrix ( m st ) is right-angled if all its nondiagonal entries areeither 2 or ∞ . The associated Artin group A is a right-angled Artin group (abbreviated RAAG). The next theorem is one of our main results. ForRAAGs, it was proved in [1]. Main Theorem. (Theorem 5.2 in section 6).
Suppose that L is the nerveof a Coxeter system ( W, S ) and that A is the associated Artin group. Let d = dim L . If H d ( L ; Z / (cid:54) = 0) , then actdim A ≥ d + 2 . So, if the K ( π, -Conjecture holds for A , then actdim A = 2 d + 2 . The justification for the final sentence of this theorem is that if K ( π, A , then gd A = d + 1; so, the largest possible value forthe action dimension is 2 d + 2.For any simplicial complex K , there is another simplicial complex OK of the same dimension, defined by “doubling the vertices” of K (cf. sub-section 2.3). The complex OK is called the “octahedralization” of K . Inthe right-angled case, the Main Theorem was proved by using OL for anobstructor, where L = L ( W, S ).The principal innovation of this paper concerns a certain subdivsion L (cid:11) of L , called the “complex of standard abelian subgroups” of A . The complex L (cid:11) plays the same role for a general Artin group as L does in the right-angledcase; its simplices parameterize the “standard” free abelian subgroups in A .We think of OL (cid:11) as the boundary of the union of the standard flat subspacesin the universal cover of BA . The Main Theorem is proved by showing that(1) Cone OL (cid:11) coarsely embeds in A , and (2) when H d ( L ; Z / (cid:54) = 0, the vanKampen obstruction for OL (cid:11) in degree 2 d is not zero.When A is a RAAG, a converse to the Main Theorem also was proved in[1]: if H d ( L ; Z /
2) = 0 (and d (cid:54) = 2), then actdim A ≤ d + 1. For a generalArtin group A for which the K ( π, H d ( L ; Z ) = 0 (and d (cid:54) = 2), then actdim A ≤ d + 1.3 Preliminaries
Let (
W, S ) be a Coxeter system.A reflection in W is the conjugate of an element of S . Let R denote theset of all reflections in W . For any subset T < S , R T denotes the set ofreflections in W T , i.e., R T = R ∩ W T .The Coxeter diagram D of ( W, S ) is a labeled graph which records thesame information as does ( m st ). The vertex set of D is S and there is an edgebetween s and t whenever m st >
2. If m st = 3, the edge is left unlabeled,otherwise it is labeled m st . (The notation D ( W, S ) or D ( S ) also will be usedfor D .)The Coxeter system is irreducible if D is connected. A component T of( W, S ) is the vertex set of a connected component of D . Thus, if T , . . . , T k are the components of ( W, S ), then D is the disjoint union of the inducedgraphs on T i and W = W T × · · · × W T k . So, the diagram shows us how todecompose W as a direct product. In this paper we will only be concernedwith the diagrams of spherical Coxeter groups. Definition 1.1.
Suppose T is a spherical subset and that T , . . . T k are thevertex sets of the components of D ( T ). Then { T , . . . T k } is the decomposition of T (into irreducibles). Lemma 1.2. (cf. [4, Ch. IV, Exerc. 22, p. 40]).
Suppose ( W T , T ) is a spher-ical Coxeter system. Then there is a unique element w T of longest length in W T . It has the following properties: • w T is an involution, • w T conjugates T to itself. • (cid:96) ( w T ) = Card R T . Remark 1.3.
Let w T be the element of longest length in a spherical specialsubgroup W T . Let ι T : W T → W T denote inner automorphism by w T . ByLemma 1.2, ι T restricts to a permutation of T and this permutation inducesan automorphism of the Coxeter diagram D ( T ). It follows that w T belongsto the center of W T if and only if the restriction of ι T to T is the trivialpermutation. Suppose ( W T , T ) is irreducible. It turns out that the center4f W T is trivial if and only if the permutation ι T | T is nontrivial (this isbecause this condition is equivalent to the condition that w T does not actas the antipodal map, −
1, on the canonical representation). If this is thecase, we say W T is centerless . In particular, when W T is centerless, itsCoxeter diagram admits a nontrivial involution. Using this, the question ofthe centerlessness of W T easily can be decided, as follows. When Card T = 2, W T centerless if and only if it is a dihedral group I ( p ), with p odd. WhenCard T >
2, the groups of type A n , D n with n odd, and E are centerless,while those of type B n , D n with n even, H , H , F , E and E are not (cf.[4, Appendices I-IX, pp. 250-275] or [10, Remark 3.1.2, p. 125]). Remark 1.4.
It is proved in [21] that if W T is irreducible and not spherical,then its center is trivial.For later use we record the following technical lemma. Lemma 1.5. (The Highest Root Lemma).
Suppose ( W, S ) is spherical. Thenthere is a reflection r ∈ R such that r / ∈ W T for any proper subset T < S .Proof.
It suffices to prove this when (
W, S ) is irreducible. So, suppose this.If the diagram (
W, S ) is not type H , H or I ( p ) with p = 5 or p >
6, then(
W, S ) is associated with a root system in R n . Each root φ can be expressedas an integral linear combination of the simple roots { φ s } s ∈ S : φ = (cid:88) s ∈ S n s φ s , where the coefficients n s are either all ≥ ≤
0; the root φ is said tobe positive or negative , respectively. Moreover, there is a positive root φ r foreach reflection r ∈ R . It is proved in [4, Ch. VI § φ r which dominates all other positive roots φ ,in the sense that the coefficients of φ r − φ are all ≥
0. In particular, since φ r dominates the simple roots, the coefficients n s of φ r are (cid:54) = 0. On the otherhand, if r ∈ R T , then n s = 0 for all s ∈ S − T . Hence, if φ r is the highestroot, then r / ∈ R T for any proper subset T < S .When the diagram is type H , H or I ( p ), one still has a “root system”with the properties in the previous paragraph, except that the coefficientsneed not be integers. One then can prove the lemma directly in each of thesethree cases by simply writing down a reflection r which does not lie in anyof the R T . 5et S (cid:11) denote the set of irreducible nonempty spherical subsets of S . Inother words, S (cid:11) := { T ∈ S ( W, S ) > ∅ | D ( T ) is connected } . (1.1)By Lemma 1.5, one can choose a function r : S (cid:11) → R , denoted T (cid:55)→ r ( T ),such that r ( T ) ∈ R T − (cid:91) T (cid:48) The function r : S (cid:11) → R is injective.Proof. According to [4, Ch. IV, § w ∈ W , there is a well-defined subset S ( w ) ≤ S , such thatthe set of letters used in any reduced expression for w is precisely S ( w ). Byconstruction, S ( r ( T )) = T . Hence, if r ( T ) = r ( T ), then T = T , i.e., r isinjective. L (cid:11) of L . We want to define the subdivision L (cid:11) of L (= L ( W, S )). First, the vertexset of L (cid:11) is the set S (cid:11) defined in (1.1), i.e., S (cid:11) is the set of (vertex sets of)irreducible spherical subdiagrams of D ( W, S ). Note that S (cid:11) is a subposet of S ( W, S ). We regard T ∈ S (cid:11) as being the barycenter of the correspondingsimplex of L . Given a subset α of S (cid:11) , define its support : sp ( α ) = (cid:91) T ∈ α T. Next we give an inductive definition of what it means for a subset of S (cid:11) tobe “nested”. Definition 1.7. Let α be a subset of S (cid:11) such that sp ( α ) is spherical. Bydefinition, the ∅ is nested . (This is the base case of the inductive definition.)Let { T , . . . T k } be the set of maximal elements in α . Then α is nested if • sp ( α ) is spherical. 6 { T , . . . , T k } is the decomposition of sp ( α ) into irreducibles. • α Here is a more geometric description of L (cid:11) . Suppose T isa spherical subset and that σ is the corresponding geometric simplex in L .Since L (cid:11) is defined by applying the subdivision procedure to each simplex of L , it suffices to describe σ (cid:11) . Suppose { T , . . . , T k } is the decomposition of T into irreducibles and that σ i is the geometric simplex in L corresponding to T i . This gives a join decomposition: σ = σ ∗ · · · ∗ σ k . Suppose by inductionon dim σ that the subdivision has been defined for each proper subcomplexof σ . The subdivision σ (cid:11) is then defined by using one of the following tworules. • If k = 1, then T is irreducible and the barycenter b T of σ will be avertex of the subdivision. Define σ (cid:11) := ( ∂σ ) (cid:11) ∗ b T . (In other words, σ (cid:11) is formed by coning off ( ∂σ ) (cid:11) to b T .) • If k > 1, then σ (cid:11) := ( σ ) (cid:11) ∗ · · · ∗ ( σ k ) (cid:11) . { ps,eps } not found (or no BBox) Figure 1: Subdivision for D = { ps,eps } not found (or no BBox) The following lemma will be used in subsection 3.2. Its proof is straight-forward. 7 emma 1.9. (cf. [10, Cor. 3.5.4]). The subdivision L (cid:11) is a flag complex. Remark 1.10. Assuming this lemma, one can define L (cid:11) without using in-duction. First, there is a direct description of the 1-skeleton of L (cid:11) : the edgesare the 2-element subsets { T , T } of S (cid:11) such that T and T are either or-thogonal or comparable. Second, L (cid:11) is the flag complex determined by this1-skeleton. Suppose A is the Artin group associated to a Coxeter system ( W, S ) and that { a s } s ∈ S is its standard set of generators. There is a canonical epimorphism p : A → W defined by a s (cid:55)→ s . The pure Artin group P A is the kernel of p .There is a map i : w (cid:55)→ a w from W to A , which is a set theoretic sectionfor p . It is defined as follows. If s · · · s l is a reduced expression for w , then a w = a s · · · a s k (1.3)The element a w does not depend on the choice of reduced expression byTits’ solution to word problem for Coxeter groups. This asserts that anytwo reduced expressions for an element of W differ by “braid moves,” corre-sponding to Artin relations of type (0.1), e.g., see [12, § i : W → A is not a homomorphism.) Remark 1.11. For subsets T , T (cid:48) subsets of S , we have W T ∩ W T (cid:48) = W T ∩ T (cid:48) , A T ∩ A T (cid:48) = A T ∩ T (cid:48) , P A T ∩ P A T (cid:48) = P A T ∩ T (cid:48) . The first formula is proved in [4, Ch. IV.8, Thm. 2, p. 12], the second in [23].The third follows from the first two. When T is a spherical subset of S , W T is a finite reflection group on R n (where n is the cardinality of T ) and so, by complexification, also a reflectiongroup on C n . Let M T denote the complement of the union of reflectinghyperplanes in C n . Deligne [14] proved that π ( M T ) is the pure Artin group P A T and that M T is a model for BP A T . Even when W T is infinite it still8as a geometric representation as a reflection group on R n so that W T actsproperly on the interior of a certain convex cone (the “Tits cone”). Hence, W T acts properly on some convex open subset Ω T < C n . We also denote thecomplement of the union of reflecting hyperplanes in Ω T by M T . It is provedin [23] that π ( M T ) ∼ = P A T . The original version of the K ( π, M T ∼ BP A T . (For more details, see [7] or [23].)When T (cid:48) < T , there is an open convex subset in Ω T which only intersectsthe reflecting hyperplanes of W T (cid:48) and which, therefore, can be identified withthe product of Ω T (cid:48) and a disk. Taking complements of hyperplanes we getan inclusion M T (cid:48) (cid:44) → M T . On the level of fundamental groups, this inducesthe standard inclusion P A T (cid:48) (cid:44) → P A T .Next, we consider the abelianization P A ab of P A as well as the abelian-ization of the special subgroups P A T . Since P A is the fundamental groupof the hyperplane arrangement complement M S , we have P A ab = H ( M S ).There is one reflecting hyperplane for each reflection r ∈ R . Since each suchhyperplane is the intersection of a linear hyperplane with an open convexsubset of C n , H ( M S ) is generated by loops around these hyperplanes; infact, H ( M S ) is the free abelian group on R , denoted Z R (e.g., see [20]). So,we have proved the following. Lemma 1.12. P A ab is the free abelian group Z R . Similarly, for any T < S , ( P A T ) ab = Z R T . Let { e r } r ∈ R be the standard basis for Z R . For each fundamental reflection s ∈ S , the element ( a s ) ∈ P A is a loop around the reflecting hyperplanecorresponding to S , i.e., e s is represented by ( a s ) , where a s is a standardArtin generator. Any reflection r ∈ R can be written as r = wsw − for some s ∈ S and w ∈ W . Put ε r := ( a w )( a s ) ( a w ) − , (1.4)where a w is the element of A defined by (1.3) (any other element of A lyingabove w would serve as well). Note that ε r ∈ P A and that it projects to e r ∈ P A ab . Thus, { e r } r ∈ R is the standard basis for Z R = P A ab . (Usually wewill write the group operation in P A ab additively.)9 Configurations of standard abelian subgroups Suppose A T is a spherical Artin group. Define ∆ T ∈ A by∆ T = a w T , (2.1)where w T ∈ W T is the element of longest length defined in Lemma 1.2 and a w T is defined by (1.3). For any t ∈ T , we have w T t = ι ( t ) w T , where ι : T → T isthe permutation induced by w T . Moreover, there is a reduced expression for w T ending with the letter t as well as another reduced expression beginningwith ι ( t ). So, a w T ( a t ) − = ( a ι ( t ) ) − a w T . Hence,∆ T a t = a ι ( t ) ∆ T . It follows that (∆ T ) lies in the center, Z ( A T ), of A T and that ∆ T ∈ Z ( A T )if and only if ι : T → T is the identity permutation. If T is spherical andirreducible, put δ T := (cid:40) ∆ T , if ι = id;(∆ T ) , otherwise . Note that (∆ T ) ∈ P A T . The next lemma follows from [2, Thm. 4.7, p. 294]. Lemma 2.1. Suppose ( W T , T ) is spherical and irreducible. Then Z ( A T ) isinfinite cyclic with generator δ T . Similarly, Z ( P A T ) is infinite cyclic withgenerator (∆ T ) . If ( W T , T ) has more than one component, define δ T to be the centralelement given by product of the appropriate choice of ∆ T (cid:48) or (∆ T (cid:48) ) for eachcomponent T (cid:48) .We also can define corresponding central elements in P A T and ( P A T ) ab by ε T := (∆ T ) ∈ P A T , and e T := image of (∆ T ) in ( P A T ) ab . Lemma 2.2. Suppose T is spherical. Then e T ∈ ( P A T ) ab is the sum of thestandard basis elements: e T = (cid:88) r ∈ R T e r . roof. Consider the hyperplane arrangement complement, M T . If ∆ T isdefined by (2.1), then the central element (∆ T ) ∈ π ( M T ) = P A T is repre-sented by the “Hopf fiber”. (More precisely, the Hopf fiber is C ∗ and (∆ T ) is represented by S < C ∗ .) The Hopf fiber links each reflecting hyperplaneonce. (To see this, consider the complement in C n of a single linear hyper-plane, in which case it is obvious.) Hence, the class of the Hopf fiber in H ( M T ) is (cid:80) r ∈ R T e r .Here is an alternative algebraic proof of the above lemma. Alternate proof of Lemma 2.2. Let F ( T ) + be the free monoid on T . For anyword f = t · · · t l which is a reduced expression for the corresponding element w f of W T , let a f be the corresponding element of A T , i.e., a f = a w f = a · · · a l ,where a i is the Artin generator corresponding to t i . Pick a reduced expression f ∈ F ( T ) + for the element w T of longest length in W T . Let f denote thereverse of f , i.e., if f = t t · · · t k , then f = t k · · · t t . Note that • Since f · f = 1 in W T and since w f = w T is an involution (by Lemma 1.2), w f = w f . Since both f and f are reduced expressions for w T , the words f and f represent the same element in A T , namely, ∆ T . • The subset of W T represented by { ( t · · · t i − ) · t i · ( t · · · t i − ) − } ≤ i ≤ k is exactly the set of reflections in W T (cf. [4, Lemma 2, p. 6]).A simple calculation shows that ε T = (∆ T ) = a f · a f = x k x k − · · · x where x i = ( a · · · a i − ) · ( a i ) · ( a · · · a i − ) − for 1 ≤ i ≤ k (cf. (1.4)). By thediscussion in subsection 1.4, the images of the x i ’s in ( P A T ) ab are exactlythe e r ’s with r ∈ R T . The lemma follows.Let j : Z S (cid:11) → Z R = P A ab be the homomorphism defined by T (cid:55)→ e T andextending linearly. By combining Lemmas 1.6 and 2.2, we get the following. Lemma 2.3. The homomorphism j : Z S (cid:11) → Z R is injective.Proof. The standard inner product, ( x, y ) (cid:55)→ x · y on Z R is defined on standardbasis elements by ( e r , e r (cid:48) ) (cid:55)→ δ r,r (cid:48) . For each T ∈ S (cid:11) let r ( T ) be as in (1.2)and Lemma 1.6. Note that for any T ∈ S (cid:11) and r ∈ R , we have e T · e r = 1 if r ∈ R T and is 0 otherwise. It follows that for T, T (cid:48) ∈ S (cid:11) , we have: e T (cid:48) · e r ( T ) = (cid:40) , if T ≤ T (cid:48) ,0 , otherwise.11sing this one sees that the e T are linearly independent in Z R . Indeed,if (cid:80) x T e T = 0, after taking the inner product with e r ( T ) , we get x T = 0whenever T is a maximal element of S (cid:11) . Let S (cid:48)(cid:11) be S (cid:11) with its maximalelements removed. Repeat the argument for maximal elements of S (cid:48)(cid:11) . Afterfinitely many iterations, we get that x T = 0 for all T ∈ S (cid:11) . Suppose α is a d -simplex in L (cid:11) . We are going to define a subgroup H α < A which is free abelian of rank d + 1. Suppose T, T (cid:48) ∈ Vert α (where Vert α means the vertex set of α ). By Remark 1.10, T and T (cid:48) are either orthogonalor comparable. In either case δ T and δ T (cid:48) commute. (This is obvious if T and T (cid:48) are orthogonal. If T and T (cid:48) are comparable, then without loss ofgenerality T < T (cid:48) , in which case, δ T (cid:48) is centralizes A T < A T (cid:48) .)Define H α to be the subgroup of A sp ( α ) generated by { δ T } T ∈ Vert α . By theprevious paragraph, H α is abelian. It is called the standard abelian subgroup associated to α . Similarly, define P H α to be the subgroup of finite index in H α generated by { (∆ T ) } T ∈ Vert α . Also, denote by J α the image of P H α in( P A sp ( α ) ) ab . Lemma 2.4. The natural map Z Vert α → H α is an isomorphism; so, H α isfree abelian of rank ( d + 1) .Proof. It suffices to show that the homomorphism Z Vert α → ( P H α ) ab inducedby T (cid:55)→ e T is an isomorphism. This is immediate from Lemma 2.3. Corollary 2.5. If dim L = d , then cd A ≥ d + 1 . If the K ( π, -Conjectureholds for A , then gd A = d + 1 .Proof. If α is a d -simplex in L (cid:11) , then H α ∼ = Z d +1 , which has cohomologicaldimension d + 1. Since cd A ≥ cd H α , this proves the first sentence. If the K ( π, A , then its Salvetti complex is a model for BA and consequently, d + 1 ≥ gd A ≥ cd A .Let j : Z S (cid:11) → Z R = P A ab be the homomorphism defined in the sen-tence preceding Lemma 2.3. Let J := Im j < P A ab denote its image. ByLemma 2.3, j : Z S (cid:11) → J is an isomorphism. So, for any simplex α of L (cid:11) , J α = j ( Z Vert α ). The next result is a key lemma. Lemma 2.6. If α and β are simplices of L (cid:11) , then H α ∩ H β = H α ∩ β . roof. Since P H α has finite index in H α , the lemma will follow once weshow that P H α ∩ P H β = P H α ∩ β . Let h : P A → P A ab be the abelianizationhomomorphism. Recall that h ( P H α ) = J α and h | P H α is injective (Lemma2.4). It follows that if P H α ∩ P H β (cid:41) P H α ∩ β , then J α ∩ J β (cid:41) J α ∩ β . So itsuffices to show J α ∩ J β = J α ∩ β . But this is immediate from the equality: Z Vert α ∩ Z Vert β = Z Vert( α ∩ β ) , after using the isomorphism j to transport this equality to an equality in-volving subgroups of J .We record the following lemma concerning the geometry of standardabelian subgroups, eventhough we do not need it in the sequel. Lemma 2.7. For any simplex α ⊂ L (cid:11) , H α is quasi-isometrically embeddedin A .Proof. First, consider the case when A is spherical. Charney [6] proved thatspherical Artin groups are biautomatic and Gersten-Short [16] proved thatbiautomatic groups are semihyperbolic. The Algebraic Flat Torus Theo-rem from [5, p. 475] states that every monomorphism of a finitely generatedabelian group to a semihyperbolic group is a quasi-isometric embedding. Thegeneral case follows from the spherical case, since any standard abelian sub-group is contained in a spherical special subgroup and special subgroups areconvex in A [8]. Octahedralization For a finite set V , let O ( V ) denote the boundary com-plex of the octahedron (or cross polytope) on V . In other words, O ( V )is the simplicial complex with vertex set V × {± } such that a subset { ( v , ε ) , . . . , ( v k , ε k ) } of V × {± } spans a k -simplex if and only if its first co-ordinates v , . . . v k are distinct. Projection to the first factor V × {± } → V gives a simplicial projection p : O ( V ) → σ ( V ), where σ ( V ) means the fullsimplex on V . Any finite simplicial complex K with vertex set V is a sub-complex of σ ( V ). Its octahedralization OK is defined to be the inverse imageof K in O ( V ): OK := p − ( K ) ≤ O ( V ) . OK is of the form ( σ, ε ), where σ is a simplex of K and ε : Vert σ → {± } is a function. (This terminology comes from [1] or[13, § OK is the result of doubling the vertices of K .Given a finite simplicial complex or metric space K , putCone( K ) := K × [0 , ∞ ) / K × { } . (2.2)If K is a metric space, then there is an induced metric on Cone( K ) whichgives it the structure of a euclidean cone (cf. [5, p. 60]). The idea is to use theformula for the euclidean metric on R n in polar coordinates - the coordinatein K is the “angle” (and the metric on K gives the angular distance) and[0 , ∞ ) gives the radial coordinate. For example, if K is the round ( n − K ) is isometric to E n ; if K is a spherical ( n − K ) is isometric to the corresponding sector in E n .A spherical ( n − σ is all right if its edges all have length π/ σ is isometric to the intersection of the unit sphere in R n withthe positive orthant [0 , ∞ ) n . Its octahedralization Oσ is a triangulation of S n − and with the induced all right piecewise spherical metric, it is isometricto the round sphere. Hence, Cone( Oσ ) is isometric to euclidean space.Suppose K is given its all right piecewise spherical structure in which eachedge has length π/ 2. Then Cone( K ) is isometric to a configuration of thepositive orthants Cone( σ ) and Cone( OK ) is isometric to a configuration ofeuclidean spaces all passing through the cone point. (There is one euclideanspace for each σ ∈ K .) The link of the cone point in Cone( OK ) is OK . ByGromov’s Lemma (cf. [12, Appendix I.6, p. 516]), Cone( OK ) is CAT(0) ifand only if OK is a flag complex. Coordinate subspace arrangements Given a finite simplicial complex K , one can define an arrangement of coordinate subspaces in the euclideanspace R Vert K . Let S ( K ) denote the poset of simplices in K . For each σ ∈ S ( K ), there is the subspace R Vert σ ≤ R Vert K . The coordinate subspacearrangement corresponding to K is the collection of subspaces, A ( K ) := { R Vert σ } σ ∈S ( K ) . The intersection of these subspaces with the integer lattice Z Vert K gives the coordinate arrangement of free abelian subgroups associatedto K , A Z ( K ) := { Z Vert σ } σ ∈S ( K ) . (2.3)14he union of the elements of A ( K ) or A Z ( K ) is denoted E OK or Z OK , re-spectively: E OK := (cid:91) σ ∈S ( K ) R Vert σ , Z OK := (cid:91) σ ∈S ( K ) Z Vert σ . (2.4)Let { b v } v ∈ Vert K be the standard basis for R Vert K so that { b v } v ∈ Vert σ is thestandard basis for R Vert σ . The positive cone spanned by standard basis { b v } v ∈ Vert σ is the positive orthant , [0 , ∞ ) Vert σ < R Vert σ . Similarly, if ( σ, ε ) isa simplex of OK , where ε is a choice of signs v (cid:55)→ ε v ∈ {± } , we get theorthant spanned by { ε v b v } v ∈ Vert σ .The proof of the next lemma is immediate from the definitions. Lemma 2.8. Let { h v } v ∈ Vert K be a collection of elements in some group π indexed by vertices of a finite simplicial complex K . Suppose h v and h v (cid:48) commute whenever v and v (cid:48) are joined by an edge. For a simplex σ in K ,let H σ ≤ π denote the image of the homomorphism Z Vert σ → π induced by v (cid:55)→ h v . Suppose that • the homomorphism Z Vert σ → π is injective, • the conclusion of Lemma 2.6 holds, i.e., H σ ∩ H τ = H σ ∩ τ for all σ , τ in K .Then the collection of free abelian subgroups A = { H σ } σ ∈S ( K ) is isomorphicto the coordinate arrangement of abelian groups in Z OK in the obvious sense. By Lemmas 2.4 and 2.6, the lemma above applies to the simplicial com-plexes K = L (cid:11) and OK = OL (cid:11) . For each α ∈ L (cid:11) , let B α = { δ T } T ∈ Vert α be the standard basis for H α and let R Vert α be the euclidean space with basis B α , i.e., R Vert α = H α ⊗ R . Remark 2.9. These notions also make sense when K is infinite and A isan infinite arrangement of free abelian subgroups in π . If A is invariantunder conjugation, then the conjugation action induces an action π on OK .In several cases, such action plays an important role of understanding thegroup π . For example, if π is the mapping class group of closed hyperbolicsurface and A is the collection of abelian subgroups generated by Dehn twists,15hen K is the curve complex of the underlying surface. If π is a right-angledArtin group and A is the collection standard abelian subgroups and theirconjugations, then K is the flag complex of the “extension graph,” whichwas introduced by Kim and Koberda [17]. In this section we review some definitions from [3]. K ⊂ ∂π ” For a finite simplicial complex K , define Cone( K ) as in (2.2). We giveCone( K ) the euclidean cone metric after choosing a piecewise euclidean met-ric on K . Next we triangulate Cone( K ) so that • for each simplex σ of K , Cone( σ ) is a subcomplex, • the edge-length metric on the 0-skeleton, Cone( K ) , is bi-Lipschitzequivalent to the metric on Cone( K ) induced from the euclidean conemetric. Definition 3.1. Suppose π is a discrete, finitely generated group equippedwith a word metric. As in [3, p. 228], write “ K ⊂ ∂π ” to mean that there isa proper, expanding, Lipschitz map, Cone( K ) → π .The terms “proper” and “Lipschitz” have their usual meanings. The term“expanding” needs further explanation, which is given below.Two functions h : A → B and h : A → B to a metric space B diverge from one another if for every D > C i ≤ A i , i = 1 , 2, such that h ( A − C ) and h ( A − C ) are of distance > D apart.Following [3], call a proper map h : Cone( K ) → B is expanding if for anytwo disjoint simplices σ and τ in K , the maps h | Cone( σ ) and h | Cone( τ ) diverge.Similarly, if Cone( K ) is equipped with an edge-length metric as above, then h : Cone( K ) → B is expanding if for disjoint simplices σ , τ in K , h | Cone( σ ) and h | Cone( τ ) diverge.Note that the meaning of “ K ⊂ ∂π ” in Definition 3.1 does not dependon choices we made in the beginning of this subsection.It is shown in [3] that if the universal cover of Bπ has a Z -structure (i.e.,a Z -set compactification) with boundary denoted ∂π and if K ⊂ ∂π , then16here is a proper, expanding, Lipschitz map Cone( K ) → π . In other words, K ⊂ ∂π = ⇒ “ K ⊂ ∂π ”. Let K be a finite simplicial complex. The deleted simplicial product , ( K × K ) − ∆, is the union of all cells in K × K of the form σ × τ , where σ and τ are disjoint closed simplices in K . The configuration space , C ( K ) is thequotient of ( K × K ) − ∆ by the free involution which switches the factors.Let c : C ( K ) → R P ∞ classify the double cover. If w ∈ H ( R P ∞ ; Z / 2) isthe generator, then the van Kampen obstruction for K in degree m is thecohomology class vk m Z / ( K ) ∈ H m ( C ( K ); Z / 2) defined byvk m Z / ( K ) := c ∗ ( w m ) . It is an obstruction to embedding K in S m . Moreover, in the case m =2 dim K with dim K (cid:54) = 2, it (or actually an integral version of it) is thecomplete obstruction to embedding K in S m . If vk m Z / ( K ) (cid:54) = 0, then K is an m -obstructor .The main results of [1] concern the van Kampen obstruction vk m Z / ( OK )of an octahedralization. From a nonzero Z / M ∈ Z k ( K ; Z / 2) and a k -simplex ∆ in the support of M , one produces a 2 k -chain Ω ∈ C k ( C ( OK ); Z / M, ∆)satisfies a certain technical condition, called the “ ∗ -condition” in [1, p. 122],then Ω is a cycle and vk k Z / ( OK ) evaluates nontrivially on it. This gives thefollowing. Theorem 3.2. ([1, Thm. 5.2, p. 121]). Given a simplicial complex K , sup-pose there is a k -cycle M ∈ Z ( K ; Z / and a k -simplex ∆ in M such that ( M, ∆) satisfies the ∗ -condition. Then vk k Z / ( OK ) (cid:54) = 0 , i.e., OK is a k -obstructor. It is then remarked in [1] that when k = dim K and K is a flag complex,the ∗ -condition is automatically satisfied. It also is proved in [1, Thm. 5.1]that when there is no nontrivial cycle on K in the top degree, vk k Z / ( OK ) = 0.So, the following is a corollary of Theorem 3.2. Theorem 3.3. ([1, Thm. 5.4, p. 121]). Suppose K is a d -dimensional flagcomplex. Then vk d Z / ( OK ) (cid:54) = 0 if and only if H d ( K ; Z / (cid:54) = 0 . In otherwords, OK is a d -obstructor if and only if H d ( K ; Z / (cid:54) = 0 . 17n immediate corollary is the following. Theorem 3.4. Suppose a d -dimensional complex L is the nerve of ( W, S ) .Then OL (cid:11) is a d -obstructor if and only if H d ( L ; Z / (cid:54) = 0 .Proof. Theorem 3.3 applies, since by Lemma 1.9, L (cid:11) is a flag complex. Remark 3.5. Kevin Schreve has pointed out to us that below the top di-mension, it is easier for the ∗ -condition to be satisfied for L (cid:11) than for L .Indeed, suppose M is a k -cycle on L and ∆ is a k -simplex in M . The subdi-vision M (cid:11) is homologous to M . Moreover, if the barycenter of ∆ occurs in∆ (cid:11) , then the ∗ -condition holds for ( M (cid:11) , ∆ (cid:48) ) where ∆ (cid:48) is any k -simplex in ∆containing the barycenter. Definition 3.6. ([3, p. 225]). The obstructor dimension of π , denoted byobdim π , is ≥ m + 2 if there is an m -obstructor K such that “ K ⊂ ∂π ” , i.e.,obdim π := sup { m + 2 | ∃ an m -obstructor K with “ K ⊂ ∂π ” } If there is a proper, expanding, Lipschitz map from Cone( K ) into a con-tractible ( m + 1)-manifold, then vk m Z / ( K ) = 0. So, if “ K ⊂ ∂π ” and π actsproperly on a contractible ( m + 1)-manifold, then the van Kampen obstruc-tion of K vanishes in degree m (cf. [3, Thm. 15, p. 226]); hence,obdim π ≤ actdim π. (3.1) Example 3.7. (RAAGs, cf. [1]). Suppose A is a RAAG with d -dimensionalnerve L . The standard model for BA is a union of tori; moreover, it is alocally CAT(0) cubical complex of dimension d + 1. Its universal cover EA is CAT(0), hence, contractible. Therefore, gd A = d + 1. Consequently,actdim A ≤ d + 1). The lifts of the tori in BA to EA , which contain agiven base point, give an isometrically embedded copy of E OL in EA . Thus,OL ⊂ ∂A . First suppose that H d ( L ; Z / (cid:54) = 0. Since L is a flag complex,Theorem 3.2 implies that OL is a 2 d -obstructor; so, obdim A ≥ d + 2. Thisgives actdim A ≤ d +2 ≤ obdim A . By (3.1), both inequalities are equalities:actdim A = 2 d + 2 = obdim A. H d ( L ; Z / 2) = 0, then OL is not a 2 d -obstructor. So, when d (cid:54) = 2, OL embeds in S d . An argument of [1, Prop. 2.2] then shows that theright-angled Coxeter group with nerve OL , denoted W OL , acts properly on acontractible (2 d + 1)-manifold. Since A < W OL , so does A . So, in this case,obdim A ≤ actdim A ≤ d + 1. We will need a lemma from [19]. Before stating it as Lemma 4.1 below, weexplain some terminology.Subspaces A and B of a metric space X are coarsely equivalent , denoted A c = B , if their Hausdorff distance is finite. The subspace A is coarselycontained in B , denoted A c ⊂ B , if there is a positive real number R suchthat A is contained in the R -neighborhood N R ( B ) of B . In this case, wealso write A ⊂ R B . We use A ∩ R B to mean N R ( A ) ∩ N R ( B ). A subspace C is the coarse intersection of A and B , denoted by C = A c ∩ B , if theHausdorff distance between C and A ∩ R B is finite for all sufficiently large R . In general, the coarse intersection of A and B may not exist; however, ifit exists, then it is unique up to coarse equivalence.The word metric (with respect to some finite generating set) on a finitelygenerated group π gives it the structure of a metric space. Different choicesof a generating set give rise to bi-Lipschitz equivalent word metrics. So, thevarious notions of “coarseness” explained in the previous paragraph makesense for subsets of π . Lemma 4.1. ([19, Lemma 2.2]). Let H , H be subgroups of a finitely gen-erated group π . The coarse intersection of H and H exists and is coarselyequivalent to H ∩ H . Suppose H is a finitely generated subgroup of a finitely generated group π .With respect to the word metrics, the inclusion H (cid:44) → π obviously is Lipschitz.The next lemma, a standard result, implies that H (cid:44) → π is a proper map.For a proof, see [15]. 19 emma 4.2. ([15, Lemma 3.6]). Suppose H is a finitely generated subgroupof a finitely generated group π . Then there exists a function ρ : [0 , ∞ ) → [0 , ∞ ) . with lim t →∞ ρ ( t ) = + ∞ such that for any a, b ∈ H , if d H ( a, b ) ≥ t ,then d π ( i ( a ) , i ( b )) ≥ ρ ( t ) . (Here d π and d H denote the word metrics on thecorresponding groups.) We return to the situation of subsection 2.3: A = { H σ } σ ∈S ( K ) is an ar-rangement of free abelian subgroups in a finitely generated group π , indexedby the poset of simplices of a finite simplicial complex K . Put H = (cid:83) H σ .Further suppose A is isomorphic to A Z ( K ), the coordinate arrangement offree abelian groups in Z OK , defined by (2.3) and (2.4). The word metricson the H σ induce a metric on H . Since H σ is bi-Lipschitz to Z Vert σ , theisomorphism Cone( OK ) → H also is bi-Lipschitz. Theorem 4.3. Suppose A = { H σ } σ ∈S ( K ) is an arrangement of free abeliansubgroups isomorphic to the coordinate arrangement in Z OK . Then “ OK ⊂ ∂π ”. By Lemma 2.8, the arrangement of standard abelian subgroups in A , { H α } α ∈S ( L (cid:11) ) , is isomorphic to the coordinate arrangement A Z ( L (cid:11) ) in Z OL (cid:11) .So, the following theorem is a corollary of Theorem 4.3. It is one of our mainresults. Theorem 4.4. “ OL (cid:11) ⊂ ∂A ”.Proof of Theorem 4.3. Let i : Cone( OK ) ∼ = H (cid:44) → π be the inclusion. As weexplained above, i is Lipschitz and proper. The remaining issue is to showthat it is expanding. So, suppose ∆ = ( σ, ε ) and ∆ (cid:48) = ( σ (cid:48) , ε (cid:48) ) are disjointsimplices of OK . Let S and S (cid:48) denote the octahedral spheres Oσ and Oσ inside OK . It suffices to show i (Cone(∆) (0) ) ∩ r i (Cone(∆ (cid:48) ) (0) ) is finite for any r > . (4.1) Case 1: S = S (cid:48) . In this case σ = σ (cid:48) (and since ( σ, ε ) and ( σ (cid:48) , ε (cid:48) ) aredisjoint, ε (cid:48) = − ε ). Condition (4.1) follows from Lemma 4.2 applied to themonomorphism i : Z Vert σ → π . Given any two subcomplexes F and F of S = Oσ , We also deduce from Lemma 4.2 that for any r > 0, there exists r (cid:48) > i (Cone( F ) (0) ) ∩ r i (Cone( F ) (0) ) ⊂ r (cid:48) i (Cone( F ∩ F ) (0) ) . (4.2)20hen F ∩ F = ∅ , Cone( F ∩ F ) (0) is just the cone point. Case 2: S (cid:54) = S (cid:48) . By Lemma 4.1, i (Cone(∆) (0) ) ∩ r i (Cone(∆ (cid:48) ) (0) ) ⊂ i (Cone( S ) (0) ) ∩ r i (Cone( S (cid:48) ) (0) ) ⊂ r (cid:48) i (Cone( S ∩ S (cid:48) ) (0) ) (4.3)for some r (cid:48) > 0. Applying (4.2) to subcomplexes of S , S (cid:48) and S ∩ S (cid:48) respec-tively, we deduce that i (Cone(∆) (0) ) ∩ r i (Cone( S ∩ S (cid:48) ) (0) ) ⊂ r (cid:48) i (Cone(∆ ∩ S ∩ S (cid:48) ) (0) ) (4.4) i (Cone(∆ (cid:48) ) (0) ) ∩ r i (Cone( S ∩ S (cid:48) ) (0) ) ⊂ r (cid:48) i (Cone(∆ (cid:48) ∩ S ∩ S (cid:48) ) (0) ) (4.5) i (Cone(∆ ∩ S ∩ S (cid:48) ) (0) ) ∩ r i (Cone(∆ (cid:48) ∩ S ∩ S (cid:48) ) (0) ) is finite . (4.6)Condition (4.1) follows from (4.3), (4.4), (4.5) and (4.6). Remark 4.5. Theorem 4.3 and its proof can be extended to the case of anarrangement of free abelian subgroups A isomorphic to the intersection ofthe integer lattice with some real subspace arrangement (not necessarily acoordinate subspace arrangement). Remark 4.6. The proof of Theorem 4.3 can be simplified simpler if we knowthe abelian subgroups are quasi-isometrically embedded. This is indeed thecase for standard abelian subgroups in an Artin group (cf. Lemma 2.7). A First, consider the case when A is spherical. The action dimension of anybraid group is computed in [3]. More generally, by using Theorem 4.4, Le[18] computed the action dimension of any spherical Artin group. Here is theresult. Theorem 5.1. (cf. [3, p. 234], [18, Thm. 4.10]). Suppose A is a sphericalArtin group associated to ( W, S ) . Let T , . . . , T k be the irreducible componentsof D ( W, S ) and put d i = Card( T i ) − . Then actdim A = k + (cid:80) d i . In otherwords, actdim A = 2 gd A − k .Proof. The nerve of ( W, S ) is a simplex σ of dimension d = Card( S ) − k = 1, i.e., the case where ( W, S ) is irreducible. The21arycenter v of σ is then a vertex of σ (cid:11) ; the link of v in σ (cid:11) is ( ∂σ ) (cid:11) ; and σ (cid:11) is the cone, ( ∂σ ) (cid:11) ∗ v . Since ( ∂σ ) (cid:11) is a triangulation of S d − , it follows fromTheorem 4.4 that O ( ∂σ (cid:11) ) is a (2 d − Oσ (cid:11) = O ( ∂σ (cid:11) ) ∗ S is a (2 d − A ≤ d + 1. Since thehyperplane complement in S d +1 is a (2 d + 1)-manifold, actdim A ≥ d + 1.By (3.1), both inequalities are equalities; so, actdim A = 2 d + 1 = obdim A .Consider the general case, A = A T × · · · A T k . By [3, Lemma 6], for anytwo groups π , π , we have actdim( π × π ) ≤ actdim π + actdim π andobdim( π × π ) ≥ obdim π + obdim π . Therefore,actdim A ≤ k (cid:88) i =1 (2 d i + 1) ≤ obdim A, which proves the lemma.Using Theorems 3.4 and 4.4 we can extend the results about RAAGs inExample 3.7 to general Artin groups. This gives the following restatementof the Main Theorem in the Introduction. Theorem 5.2. Suppose A is an Artin group associated to a Coxeter systemwith nerve L of dimension d . If H d ( L ; Z / (cid:54) = 0) , then obdim A ≥ d + 2 .So, if the K ( π, -Conjecture holds for A , then actdim A = 2 d + 2 .Proof. By Theorems 3.4 and 4.4, obdim A ≥ d +2. If the K ( π, A , then, by Corollary 2.5, gd A = d + 1 and consequently, themaximum possible value for its action dimension is 2 d + 2.The other statement in Example 3.7 is that in the case of a RAAG, if H d ( L ; Z / 2) = 0, then actdim A ≤ d + 1. In her PhD thesis Giang Le [18]proved that the same conclusion holds for general Artin groups under theslightly weaker hypothesis that H d ( L ; Z ) = 0. Theorem 5.3. (Le [18]). Suppose A is an Artin group associated to a Cox-eter system with d -dimensional nerve L . Further suppose that the K ( π, -Conjecture holds for A . If H d ( L ; Z ) = 0 (and d (cid:54) = 2 ), then actdim A ≤ d +1 . Comments on Theorem 5.3. As explained in Example 3.7, when A is aRAAG, if H d ( L ; Z / 2) = 0, then OL is not a 2 d -obstructor and (provided d (cid:54) = 2), OL embeds as a full subcomplex of some flag triangulation J of S d (cf. the proof of Proposition 2.2 in [1]). Since the Davis complex for the22ight-angled Coxeter group W J is a contractible (2 d + 1)-manifold and since A < W OL < W J , we see that actdim A ≤ d +1. Similarly, for a general Artingroup A , the vanishing of H d ( L ; Z / 2) implies that OL (cid:11) is a full subcomplexof some flag triangulation S d . However, we do not know how to deducefrom this that A acts properly on a contractible (2 d + 1)-manifold. Le’sargument in [18] is different. The condition H d ( L ; Z ) = 0 implies H d ( L ; Z ) =0 and that H d − ( L ; Z ) is torsion-free. Standard arguments show that onecan attach cells to L to embed it into a contractible complex L (cid:48) of the samedimension d (provided d (cid:54) = 2). We can assume L (cid:48) is a simplicial complex. Foreach simplex σ of L , there is a model for BA σ by a (2 d + 1)-manifold withboundary. Le proves Theorem 5.3 by showing that one can glue togethercopies of these manifolds along their boundaries, in a fashion dictated by L (cid:48) ,to obtain another manifold with boundary M with fundamental group A sothat when the K ( π, A , M is a model for BA . Remark 5.4. The Action Dimension Conjecture is the conjecture that if agroup π acts properly on a contractible manifold M , then the (cid:96) -Betti num-bers of π vanish above the middle dimension of M . In [1, § 7] it is explainedhow the calculations for RAAGs provide evidence for this conjecture. Thesame discussion applies to the results in this paper for general Artin groups.To wit, when the K ( π, A , it is provedin [11] that the (cid:96) -Betti number of A in degree ( k + 1) is equal to the ordinary(reduced) Betti number of L (or L (cid:11) ) in degree k ; in particular, the (cid:96) -Bettinumbers of A vanish in degrees > d + 1, when d = dim L . References [1] G. Avramidi, M. W. Davis, B. Okun, and K. 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