Algebra and topology of right-angled Artin groups

Transcription

1 Algebra and topology of right-angled Artin groups Alex Suciu Northeastern University Boston, Massachusetts (visiting the University of Warwick) Algebra Seminar University of Leeds October 19, 2009 Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

2 Outline 1 Artin groups The braid groups Serre s realization problem Artin groups associated to labeled graphs Serre s problem for Artin groups Right-angled Artin groups 2 Toric complexes and right-angled Artin groups Partial products of spaces Toric complexes Associated graded Lie algebra Holonomy Lie algebra Chen Lie algebra 3 Resonance varieties The resonance varieties of a space Resonance of toric complexes Resonance of smooth, quasi-projective varieties Serre s problem for right-angled Artin groups revisited Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

3 Artin groups The braid groups The braid group Definition (E. Artin 1926/1947) σ i σ i+1 σ i = σ i+1 σ i σ i+1 1 i n 2 B n = σ 1,..., σ n 1 σ i σ j = σ j σ i i j > 1 Fits into exact sequence 1 P n B n π S n 1, where S n is the symmetric group, π sends a braid to the corresponding permutation, and P n is the group of pure" braids. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

4 Artin groups Serre s realization problem Serre s realization problem Problem Which finitely presented groups G are (quasi-) projective, i.e., can be realized as G = π 1 (M), with M a connected, smooth, complex (quasi-) projective variety? Theorem (J.-P. Serre 1958) All finite groups are projective. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

5 Artin groups Serre s realization problem Theorem (Fox Neuwirth 1962) Both P n and B n are quasi-projective groups. P n = π 1 (F(C, n)), where F(C, n) = C n \ {z i = z j } is the configuration space of n ordered points in C, or, the complement of the braid arrangement. B n = π 1 (C(C, n)), where C(C, n) = F(C, n)/s n = C n \ { n = 0} is the configuration space of n unordered points in C, or, the complement of the discriminant hypersurface. Remark As we shall see later, neither P n nor B n is a projective group. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

6 Artin groups Artin groups associated to labeled graphs Artin groups Let Γ = (V, E) be a finite, simple graph, and l: E Z 2 an edge-labeling. The associated Artin group: G Γ,l = v V } vwv {{ } = wvw }{{ }, for e = {v, w} E. l(e) l(e) E.g.: (Γ, l) Dynkin diagram of type A n 1 G Γ,l = B n The corresponding Coxeter group, W Γ,l = G Γ,l / v 2 = 1 v V fits into exact sequence 1 P Γ,l G π Γ,l W Γ,l 1. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

7 Artin groups Serre s problem for Artin groups Serre s problem for Artin groups Theorem (Brieskorn 1971) If W Γ,l is finite, then G Γ,l is quasi-projective. Idea: let A Γ,l = reflection arrangement of type W Γ,l (over C) X Γ,l = C n \ H A Γ,l H, where n = A Γ,l P Γ,l = π 1 (X Γ,l ) then: G Γ,l = π 1 (X Γ,l /W Γ,l ) = π 1 (C n \ {δ Γ,l = 0}) Theorem (Kapovich Millson 1998) There exist infinitely many (Γ, l) such that G Γ,l is not quasi-projective. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

8 Artin groups Right-angled Artin groups Right-angled Artin groups Important particular case: l(e) = 2, for all e E. Simply write: G Γ = v V vw = wv if {v, w} E. Γ = K n (discrete graph) G Γ = F n Γ = K n (complete graph) G Γ = Z n Γ = Γ Γ G Γ = G Γ G Γ Γ = Γ Γ G Γ = G Γ G Γ Theorem (Kim Makar-Limanov Neggers Roush 80/Droms 87) Γ = Γ G Γ = GΓ Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

9 Artin groups Right-angled Artin groups Serre s problem for right-angled Artin groups Theorem (Dimca Papadima S. 2009) The following are equivalent: 1 G Γ is a quasi-projective group 2 Γ = K n1 K nr 3 G Γ = F n1 F nr 1 G Γ is a projective group 2 Γ = K 2r 3 G Γ = Z 2r Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

10 Toric complexes and right-angled Artin groups Partial products of spaces Partial product construction Input: L, a simplicial complex on [n] = {1,..., n}. (X, A), a pair of topological spaces, A. Output: (X, A) L = (X, A) σ X n σ L where (X, A) σ = {x X n x i A if i / σ}. Interpolates between A n and X n. Converts simplicial joins to direct products: (X, A) K L = (X, A) K (X, A) L. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

11 Toric complexes and right-angled Artin groups Toric complexes Toric complexes Definition The toric complex associated to L is the space T L = (S 1, ) L. In other words: Circle S 1 = e 0 e 1, with basepoint = e 0. Torus T n = (S 1 ) n, with product cell structure: (k 1)-simplex σ = {i 1,..., i k } k-cell e σ = ei 1 1 ei 1 k T L = σ L eσ. Examples: T = T n points = n S 1 T n 1 = (n 1)-skeleton of T n T n 1 = T n Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

12 Toric complexes and right-angled Artin groups Toric complexes k-cells in T L (k 1)-simplices in L. C CW (T L ) is a subcomplex of C CW (T n ); thus, all k = 0, and H k (T L, Z) = C simplicial k 1 (L, Z) = Z # (k 1)-simplices of L. H (T L, k) is the exterior Stanley-Reisner ring k L = E/J L, where E = exterior algebra (over k) on generators v1,..., v n in degree 1; J L = ideal generated by all monomials vi 1 vi k corresponding to simplices {i 1,..., i k } / L. Clearly, π 1 (T L ) = G Γ, where Γ = L (1). T L is formal, and so G Γ is 1-formal. (Notbohm Ray 2005) In fact, G Γ,l is 1-formal. (Kapovich Millson) Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

13 Toric complexes and right-angled Artin groups Toric complexes K (G Γ, 1) = T Γ, where Γ is the flag complex of Γ. (Davis Charney 1995, Meier VanWyk 1995) Hence, H (G Γ, k) = k (v 1,..., v n )/J Γ, where J Γ = ideal(vi vj {v i, v j } / E(Γ)) Since J Γ is a quadratic monomial ideal, A = E/J Γ is a Koszul algebra (Fröberg 1975), i.e., Tor A i (k, k) j = 0, for all i j. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

14 Toric complexes and right-angled Artin groups Associated graded Lie algebra Associated graded Lie algebra Let G be a finitely-generated group. Define: LCS series: G = G 1 G 2 G k, where G k+1 = [G k, G] LCS quotients: gr k G = G k /G k+1 (f.g. abelian groups) LCS ranks: φ k (G) = rank(gr k G) Associated graded Lie algebra: gr(g) = k 1 gr k (G), with Lie bracket [, ]: gr k gr l gr k+l induced by group commutator. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

15 Toric complexes and right-angled Artin groups Associated graded Lie algebra Example (Witt, Magnus) Let G = F n (free group of rank n). Then gr G = Lie n (free Lie algebra of rank n), with LCS ranks given by Explicitly: φ k (F n ) = 1 k (1 t k ) φ k = 1 nt. k=1 d k µ(d)nk/d, where µ is Möbius function. Example (Kohno 1985) Let G = P n. Then: (1 t k ) φ k = (1 t) (1 (n 1)t) k=1 In other words, φ k (P n ) = φ k (F 1 F n 1 ). Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

16 Toric complexes and right-angled Artin groups Holonomy Lie algebra Holonomy Lie algebra Definition (Chen 1977, Markl Papadima 1992) Let G be a finitely generated group, with H 1 = H 1 (G, Z) torsion-free. The holonomy Lie algebra of G is the quadratic, graded Lie algebra h G = Lie(H 1 )/ideal(im( )), where : H 2 (G, Z) H 1 H 1 = Lie 2 (H 1 ) is the comultiplication map. Let G = π 1 (X) and A = H (X, Q). (Löfwall 1986) U(h G Q) = k 1 Extk A (Q, Q) k. There is a canonical epimorphism h G gr(g). (Sullivan 1977) If G is 1-formal, then h G Q gr(g) Q. Example G = F n, then clearly h G = Lie n, and so h G = gr(g). Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

17 Toric complexes and right-angled Artin groups Holonomy Lie algebra Let Γ = (V, E) graph, and P Γ (t) = k 0 f k(γ)t k its clique polynomial. Theorem (Duchamp Krob 1992, Papadima S. 2006) For G = G Γ : 1 gr(g) = h G. 2 Graded pieces are torsion-free, with ranks given by Idea of proof: (1 t k ) φ k = P Γ ( t). k=1 1 A = k V /J Γ h G k = L Γ := Lie(V)/([v, w] = 0 if {v, w} E). 2 Shelton Yuzvinsky: U(L Γ ) = A! (Koszul dual). 3 Koszul duality: Hilb(A!, t) Hilb(A, t) = 1. 4 Hilb(h G k, t) independent of k h G torsion-free. 5 But h G gr(g) is iso over Q (by 1-formality) iso over Z. 6 LCS formula follows from (3) and PBW. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

18 Toric complexes and right-angled Artin groups Chen Lie algebra Chen Lie algebras Definition (Chen 1951) The Chen Lie algebra of a finitely generated group G is gr(g/g ), i.e., the assoc. graded Lie algebra of its maximal metabelian quotient. Write θ k (G) = rank gr k (G/G ) for the Chen ranks. Facts: gr(g) gr(g/g ), and so φ k (G) θ k (G), with equality for k 3. The map h G gr(g) induces epimorphism h G /h G gr(g/g ). (P. S. 2004) If G is 1-formal, then h G /h G Q gr(g/g ) Q. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

19 Toric complexes and right-angled Artin groups Chen Lie algebra Example (Chen 1951) gr(f n /F n ) is torsion-free, with ranks θ 1 = n, and ( ) n + k 2 θ k = (k 1), for k 2. k Example (Cohen S. 1995) gr(p n /P n ) is torsion-free, with ranks θ 1 = ( ) n 2, θ2 = ( n 3), and ( ) n + 1 θ k = (k 1), for k 3. 4 In particular, θ k (P n ) θ k (F 1 F n 1 ), for n 4 and k 4. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

20 Toric complexes and right-angled Artin groups Chen Lie algebra The Chen Lie algebra of a RAAG Theorem (P. S. 2006) For G = G Γ : 1 gr(g/g ) = h G /h G. 2 Graded pieces are torsion-free, with ranks given by ( t θ k t k = Q Γ 1 t k=2 ), where Q Γ (t) = j 2 c j(γ)t j is the cut polynomial" of Γ, with c j (Γ) = W V: W =j b 0 (Γ W ). Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

21 Idea of proof: Toric complexes and right-angled Artin groups Chen Lie algebra 1 Write A := H (G, k) = E/J Γ, where E = k (v 1,..., v n ). 2 Write h = h G k. 3 By Fröberg and Löfwall (2002) ( h /h ) k = Tor E k 1 (A, k) k, for k 2 4 By Aramova Herzog Hibi & Aramova Avramov Herzog (97-99): dim k Tor E k 1 (E/J Γ, k) k = ( ) t i+1 dim k Tor S i (S/I Γ, k) i+1, 1 t k 2 i 1 where S = k[x 1,..., x n ] and I Γ = ideal x i x j {v i, v j } / E. 5 By Hochster (1977): dim k Tor S i (S/I Γ, k) i+1 = dim k H0 (Γ W, k) = c i+1 (Γ). W V: W =i+1 6 The answer is independent of k h G /h G is torsion-free. 7 Using formality of G Γ, together with h G /h G Q gr(g/g ) Q ends the proof. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

22 Toric complexes and right-angled Artin groups Chen Lie algebra Example Let Γ be a pentagon, and Γ a square with an edge attached to a vertex. Then: P Γ = P Γ = 1 5t + 5t 2, and so φ k (G Γ ) = φ k (G Γ ), for all k 1. Q Γ = 5t 2 + 5t 3 but Q Γ = 5t 2 + 5t 3 + t 4, and so θ k (G Γ ) θ k (G Γ ), for k 4. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

23 Resonance varieties The resonance varieties of a space Resonance varieties Let X be a connected CW-complex with finite k-skeleton (k 1). Let k be a field; if char k = 2, assume H 1 (X, Z) has no 2-torsion. Let A = H (X, k). Then: a A 1 a 2 = 0. Thus, get cochain complex (A, a): A 0 a A 1 a A 2 Definition (Falk 1997, Matei S. 2000) The resonance varieties of X (over k) are the algebraic sets R i d (X, k) = {a A1 dim k H i (A, a) d}, defined for all integers 0 i k and d > 0. R i d are homogeneous subvarieties of A1 = H 1 (X, k) R i 1 Ri 2 Ri b i +1 =, where b i = b i (X, k). R 1 d (X, k) depends only on G = π 1(X), so denote it by R d (G, k). Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

24 Resonance varieties Resonance of toric complexes Resonance of toric complexes Recall A = H (T L, k) is the exterior Stanley-Reisner ring of L. Identify A 1 = k V the k-vector space with basis {v v V}. Theorem (Papadima S. 2009) R i d (T L, k) = W V P σ L dim k Hi 1 σ e (lk LW (σ),k) d V\W k W, where L W is the subcomplex induced by L on W, and lk K (σ) is the link of a simplex σ in a subcomplex K L. In particular: R 1 (G Γ, k) = W V Γ W disconnected k W. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

25 Resonance varieties Resonance of toric complexes Example Let Γ and Γ be the two graphs above. Both have P(t) = 1 + 6t + 9t 2 + 4t 3, and Q(t) = t 2 (6 + 8t + 3t 2 ). Thus, G Γ and G Γ have the same LCS and Chen ranks. Each resonance variety has 3 components, of codimension 2: R 1 (G Γ, k) = k 23 k 25 k 35, R 1 (G Γ, k) = k 15 k 25 k 26. Yet the two varieties are not isomorphic, since dim(k 23 k 25 k 35 ) = 3, but dim(k 15 k 25 k 26 ) = 2. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

26 Resonance varieties Resonance of smooth, quasi-projective varieties Projective varieties Let M be a connected, smooth, complex projective variety. Then: 1 H (M, Z) admits a Hodge structure 2 Hence, the odd Betti numbers of M are even 3 M is formal (Deligne Griffiths Morgan Sullivan 1975) This puts strong restrictions on G = π 1 (M): 1 b 1 (G) is even 2 G is 1-formal, i.e., its Malcev Lie algebra m(g) is quadratic 3 G cannot split non-trivially as a free product (Gromov 1989) In particular, B n is not a projective group, since b 1 (B n ) = 1. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

27 Resonance varieties Resonance of smooth, quasi-projective varieties Quasi-projective varieties Let X be a connected, smooth, quasi-projective variety. Then: H (X, Z) has a mixed Hodge structure (Deligne ) X = CP n \ {hyperplane arrangement} X is formal (Brieskorn 1973) W 1 (H 1 (X, C)) = 0 π 1 (X) is 1-formal X = CP n \ {hypersurface} π 1 (X) is 1-formal (Morgan 1978) (Kohno 1983) Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

28 Resonance varieties Resonance of smooth, quasi-projective varieties A structure theorem Theorem (D. P. S. 2009) Let X be a smooth, quasi-projective variety, and G = π 1 (X). Let {L α } α be the non-zero irred components of R 1 (G). If G is 1-formal, then 1 Each L α is a p-isotropic linear subspace of H 1 (G, C), with dim L α 2p + 2, for some p = p(α) {0, 1}. 2 If α β, then L α L β = {0}. 3 R d (G) = {0} α L α, where the union is over all α for which dim L α > d + p(α). Furthermore, 4 If X is compact, then G is 1-formal, and each L α is 1-isotropic. 5 If W 1 (H 1 (X, C)) = 0, then G is 1-formal, and each L α is 0-isotropic. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

29 Resonance varieties Resonance of smooth, quasi-projective varieties Proof uses two basic ingredients: A structure theorem for Vd i (X, C) the jump loci for cohomology with coefficients in rank 1 local systems on a quasi-projective variety X (Arapura 1997). A tangent cone theorem", equating TC 1 (Vd 1(G, C)) with R1 d (G, C), for G a 1-formal group. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

30 Resonance varieties Serre s problem for right-angled Artin groups revisited Serre s problem for RAAGs revisited Using this theorem, together with the computation of R 1 (G Γ, C), the characterization of those graphs Γ for which G Γ can be realized as a (quasi-) projective fundamental group follows. Example Let Γ be the graph above. The maximal disconnected subgraphs are Γ {134} and Γ {124}. Thus: R 1 (G Γ, C) = C {134} C {124}. But C {134} C {124} = C {14}, which is a non-zero subspace. But recall G Γ is 1-formal. Thus, G Γ is not a quasi-projective group. Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31

31 Resonance varieties Serre s problem for right-angled Artin groups revisited References A. Dimca, S. Papadima, A. Suciu, Topology and geometry of cohomology jump loci, Duke Math. Journal 148 (2009), no. 3, S. Papadima, A. Suciu, Algebraic invariants for right-angled Artin groups, Math. Annalen 334 (2006), no. 3, , Toric complexes and Artin kernels, Advances in Math. 220 (2009), no. 2, Alex Suciu (Northeastern University) Right-angled Artin groups University of Leeds, Oct / 31