ALGEBRAIC INVARIANTS FOR RIGHT-ANGLED ARTIN GROUPS

STEFAN PAPADIMA AND ALEXANDER I. SUCIU^{1}

Abstract. A finite simplicial graph Γ determines a right-angled Artin group G_{Γ}, with generators corresponding to the vertices of Γ, and with a relation vw = wv for each pair of adjacent vertices. We compute the lower central series quotients, the Chen quotients, and the (first) resonance variety of G_{Γ}, directly from the graph Γ.

1. Introduction

1.1. Right-angled Artin groups. Let Γ = (V, E) be a finite simplicial graph. To such a graph, there is associated a right-angled Artin group, G_{Γ}, with a generator v for each vertex v ∈ V, and with a commutator relation vw = wv for each edge {v, w} ∈ E.

From the graph Γ, one builds a simplicial complex, Δ_{Γ}, called the flag complex, by filling in the complete subgraphs on k vertices by (k − 1)-simplices. In turn, the flag complex determines a CW-complex, K_{Γ}, called the cubical complex, by joining tori in the manner prescribed by Δ_{Γ}: the 0-cell corresponds to the empty simplex, the 1-cells correspond to vertices v ∈ V, the 2-cells correspond to edges e ∈ E, etc.

Much is known about the topology of the cubical complex K_{Γ}, and its fundamental group, G_{Γ}. First of all, K_{Γ} is an Eilenberg-MacLane K(G_{Γ}, 1) space, see Charney and Davis [5] and Meier and VanWyk [21]. The finiteness properties of the kernels of “diagonal” characters G_{Γ} → ℤ have been related in a very precise way to the homotopical and homological properties of Δ_{Γ} by Bestvina and Brady [3]. Finally, the geometric invariants of the type introduced by Bieri, Neumann, and Strebel [4] (invariants that hold subtle information about the finiteness properties of kernels of rational characters of G_{Γ}) have been determined by Meier, Meinert, and VanWyk [20].

1.2. Cohomology ring and formality. In this paper, we focus on a different set of invariants, mostly of an algebraic nature, for the right-angled Artin groups. The asphericity of K_{Γ} leads to an identification of the cohomology ring H^{∗}(G_{Γ}, 𝕜) with the exterior Stanley-Reisner ring E/J_{Γ}, where E is the exterior algebra over 𝕜, on generators {v^{∗}}_{v∈V}, and J_{Γ} is the ideal generated by the monomials corresponding to non-edges.

2000 Mathematics Subject Classification. Primary 20F36. Secondary 13F55, 20F14, 55P62, 57M07.

Key words and phrases. Graph, flag complex, cubical complex, right-angled Artin group, lower central series, holonomy Lie algebra, Chen Lie algebra, resonance variety.

1Partially supported by NSF grant DMS-0311142.

A crucial observation, due to Fröberg [10], is that E/J_{Γ} is a Koszul algebra. When combined with the fact that the group G_{Γ} is 1-formal (cf. Kapovich and Millson [17]), this implies that the space K_{Γ} is formal, see Proposition 3.3. In other words, all the rational homotopy theory of K_{Γ} (embodied essentially in the Malcev completion of G_{Γ}) can be extracted, at least in principle, from E/J_{Γ}.

We give here a complete description of several invariants of a right-angled Artin group G_{Γ}—the associated graded Lie algebra gr(G_{Γ}), the Chen groups gr_{k}(G_{Γ}/G′′_{Γ}), and the first resonance variety R^{1}(G_{Γ}; 𝕜)—solely in terms of the graph Γ. Our approach will enable us to derive in [25] similar results for the Bestvina-Brady groups associated to Γ.

1.3. Lower central series and clique polynomial. We start by analyzing the graded Lie algebra gr(G) associated to a right-angled Artin group G = G_{Γ}. In Theorem 3.4, we show that gr(G) is isomorphic, as a graded Lie algebra, with ℌ(G), the holonomy Lie algebra defined from the reduced diagonal map H_{2}(G) → H_{1}(G) ∧ H_{1}(G). Furthermore, the lower central series quotients are torsion-free, with ranks ϕ_{k} given by

(1.1)  ∏_{k=1}^{∞}(1 − t^{k})^{ϕ_{k}} = P_{Γ}(−t).

Here P_{Γ}(t) = ∑_{k≥0} f_{k}(Γ)t^{k} is the clique polynomial of Γ, with coefficient f_{k}(Γ) equal to the number of complete k-subgraphs of Γ (in other words, P_{Γ}(t) is the f-polynomial of the flag complex Δ_{Γ}). This LCS formula is a manifestation of the Koszul duality between the universal enveloping algebra of ℌ(G) and the cohomology ring H^{∗}(G, 𝕜) = E/J (this duality was first noted by Shelton and Yuzvinsky [31]).

1.4. Chen groups and cut polynomial. Next, we determine the Chen Lie algebra, gr(G/G′′). In Theorem 4.1 we show that the Chen groups of G = G_{Γ} are also torsionfree, with ranks θ_{k} given by

(1.2)  ∑_{k=2}^{∞} θ_{k}t^{k} = Q_{Γ} (\frac{t}{1 − t}) .

Here Q_{Γ}(t) = ∑_{j≥2} c_{j}(Γ)t^{j} is the “cut polynomial” of Γ, with coefficient c_{j}(Γ) equal to ∑_{W⊂V : |W|=j} b̃_{0}(Γ_{W}), where b̃_{0}(Γ_{W}) is one less than the number of components of the full subgraph on W.

The proof of (1.2) requires several steps. The first step, based on [23], is to identify, at least rationally, gr(G/G′′) with ℌ(G)/ℌ(G)′′. The next step, based on [12] (see also [30]), is to replace the computation of the Chen ranks by that of the Betti numbers in the linear strand of the free resolution of E/J over E. A formula from [1] and [2] now expresses these Betti numbers in terms of the corresponding Betti numbers of the polynomial Stanley-Reisner ring, S/I, which in turn can be computed from the cut numbers of Γ via a well-known formula of Hochster [16].

It is interesting to note that the Chen ranks θ_{k} often carry more information than the LCS ranks ϕ_{k}. To illustrate this phenomenon, we give in §6.3 examples of graphs with the same clique polynomial, but different cut polynomials.

1.5. Resonance varieties and cut sets. The numerical invariants ϕ_{k}(G) and θ_{k}(G) only reflect the additive structure of the respective graded Lie algebras, gr(G) and gr(G/G′′). To capture more refined information, related to the Lie bracket, we turn to the resonance variety R^{1}(G; 𝕜). First defined by Falk [9] in the context of hyperplane arrangements (over the field 𝕜 = ℂ), this subvariety of the affine space H^{1}(G, 𝕜) depends only on the holonomy Lie algebra ℌ(G).

It turns out that the resonance variety of G_{Γ} is determined by the cut sets of Γ. More precisely, we show in Theorem 5.5 that

(1.3)  R^{1}(G_{Γ}, 𝕜) = ⋃_{Γ_{W} disconnected^{W⊂V}} H_{W},

where H_{W} is the coordinate subspace spanned by W inside H^{1}(G_{Γ}, 𝕜) = 𝕜^{|V|}. From this description, it becomes apparent that R^{1}(G_{Γ}, ℝ) is the complement of the BNS invariant Σ^{1}(G_{Γ}).

To illustrate the strength of resonance, we give in §6.4 examples of graphs sharing the same clique polynomial, and the same cut polynomial, but having different resonance varieties. The difference is rather subtle: both varieties have the same number of components, of the same dimension, but the respective intersection lattices are inequivalent.

1.6. Higher-dimensional cubical complexes. In [18], Kim and Roush defined simply-connected analogues of the cubical complex K_{Γ}. The CW-complexes K_{Γ}^{q} (q ≥ 1) are obtained by joining products of (2q + 1)-dimensional spheres in the manner prescribed by Δ_{Γ}. For q sufficiently large, these spaces are both formal and coformal. In §7, we compute the rational homology and homotopy groups of the loop spaces ΩK_{Γ}^{q}, purely in terms of the clique polynomial of Γ.

We conclude with Example 7.3, which shows once again the usefulness of resonance varieties. Let Γ and Γ′ be two trees on the same number of vertices, but with different number of extremal vertices. Then K_{Γ}^{q} and K_{Γ′}^{q} have the same rational homotopy groups, yet non-isomorphic homotopy Lie algebras. The difference in the respective Whitehead product structures can be traced back to the different number of components in R^{1}(G_{Γ}, ℚ) and R^{1}(G_{Γ′}, ℚ).

2. Lie algebras and formality

2.1. Holonomy Lie algebra. Fix a ground ring R, equal to either ℤ or a field 𝕜. Let A be a connected, graded algebra over R, with graded pieces A^{i}, i ≥ 0. Assume A^{1} is a finitely generated R-module (torsion-free if R = ℤ), and A^{1} · A^{1} = 0, i.e., a^{2} = 0, for all a ∈ A^{1}. Then, the multiplication map in degree 1 descends to an R-linear map μ: A^{1} ∧ A^{1} → A^{2}.

Let Lie(A_{1}) be the graded free Lie algebra on A_{1}, where A_{i} = (A^{i})^{#} denotes the dual R-module. The holonomy Lie algebra of A, denoted ℌ(A), is the quotient of Lie(A_{1}) by the ideal generated by the image of the comultiplication map

(2.1)  \xymatrix{∇: A_{2} = (A^{2})^{#} \ar[r]^{μ^{#}} & (A^{1} ∧ A^{1})^{#} ≅ A_{1} ∧ A_{1} = Lie_{2}(A_{1})} .

Note that ℌ(A) inherits a natural grading from Lie(A_{1}); denote by ℌ(A)_{k} the k-th graded piece.

Now let G be a finitely presented group. Then A = H^{∗}(G; R) is a connected, graded-commutative R-algebra, with A^{1} finitely generated (and torsion-free if R = ℤ). If R = ℤ, or R = 𝕜 and char 𝕜 ≠ 2, then automatically A^{1} · A^{1} = 0. Otherwise, we need to assume that the abelianization H_{1}(G) is torsion-free, in which case it can be showed again that A^{1} · A^{1} = 0. In either case, we can define the holonomy Lie algebra of G, with coefficients in R, to be

(2.2)  ℌ_{R}(G) = ℌ(H^{∗}(G; R)).

In the integral case, simply write ℌ(G) = ℌ_{ℤ}(G), and note that ℌ(G)⊗ℚ = ℌ_{ℚ}(G). Moreover, note that, in the case when H_{1}(G) is torsion-free, the definition of ℌ(G) coincides with the one given in [23].

2.2. Formality properties. In Appendix A of [27], Quillen associates, in a functorial way, a Malcev Lie algebra, M_{G}, to any group G (see [23] for further details). A finitely presented group G is said to be 1-formal if M_{G} is isomorphic to the rational holonomy Lie algebra, ℌ_{ℚ}(G), completed with respect to the bracket length filtration. Equivalently, M_{G} is a quadratic Malcev Lie algebra.

Let X be a finite-type, connected CW-complex. Then X is formal, in the sense of Sullivan [33], if the rational homotopy type of X is determined by the rational cohomology ring H^{∗}(X; ℚ). As shown by Sullivan, the fundamental group of a formal space is 1-formal. The 1-formality of π_{1}(X) is not enough to insure the formality of X; indeed, simply-connected, non-formal spaces are easily constructed. Nevertheless, we have the following positive result in this direction.

Recall that a connected, graded algebra A over a field 𝕜 is called a Koszul algebra if Tor_{i}^{A}(𝕜, 𝕜)_{j} = 0, for all i ≠ j. A necessary condition is that A be the quotient of a free algebra on generators in degree 1 by an ideal I generated in degree 2. A sufficient condition is that I admit a quadratic Gröbner basis.

Proposition 2.1. Let X be a finite-type, connected CW-complex. Suppose H^{∗}(X; ℚ) is a Koszul algebra, and G = π_{1}(X) is 1-formal. Then X is a formal space.

Proof. By assumption, M_{G} ≅ { \atop ℌ_{ℚ}(G)}. This condition can be reinterpreted in terms of Sullivan’s minimal models, as follows:

𝔐_{1}(Ω_{dR}^{∗}X, d) ≅ 𝔐_{1}(H^{∗}(X; ℚ), d = 0),

where 𝔐(A, d) denotes the minimal model of a differential graded algebra (A, d), and 𝔐_{1} denotes the subalgebra generated by the degree 1 part of 𝔐, see [33]. On the other hand, formality of X means that

𝔐(Ω_{dR}^{∗}X, d) ≅ 𝔐(H^{∗}(X; ℚ), d = 0).

The Koszulness assumption on H^{∗}(X; ℚ) implies that 𝔐(Ω_{dR}^{∗}X, d) = 𝔐_{1}(Ω_{dR}^{∗}X, d) and 𝔐(H^{∗}(X; ℚ), d = 0) = 𝔐_{1}(H^{∗}(X; ℚ), d = 0). This observation (which follows readily from [26, Proposition 5.2]), finishes the proof. □

2.3. Lower central series. Let G be a group. The lower central series of G is the sequence of normal subgroups {γ_{k}G}_{k≥1}, defined inductively by γ_{1}G = G, γ_{2}G = G′, and γ_{k+1}G = (γ_{k}G, G), where (x, y) = xyx^{−1}y^{−1}. Observe that the successive quotients γ_{k}G/γ_{k+1}G are abelian groups. The direct sum of these quotients,

(2.3)  gr(G) = ⨁_{k≥1} γ_{k}G/γ_{k+1}G,

is the associated graded Lie algebra of G. The Lie bracket [x, y] is induced from the group commutator, while the grading is given by bracket length.

Assume G is finitely generated. Since gr(G) is generated as a Lie algebra by the degree 1 piece, gr_{1}(G) = G/G′, all the graded pieces gr_{k}(G) = γ_{k}G/γ_{k+1}G are finitely generated abelian groups. If, moreover, G is 1-formal, then

(2.4)  gr(G) ⊗ ℚ ≅ ℌ_{ℚ}(G)

as graded Lie algebras, as follows from Appendix A in [27].

Assume now G is finitely presented, and H_{1}(G) is torsion-free. Then, the canonical projection Lie(H_{1}(G)) ↠ gr(G) factors through an epimorphism Ψ_{G}: ℌ(G) ↠ gr(G). If, moreover, G is 1-formal and ℌ(G) is torsion-free, then the map Ψ_{G} yields an isomorphism of (integral) graded Lie algebras,

(2.5)  \xymatrix{Ψ_{G}: ℌ(G) \ar[r]^{≅} & gr(G)} .

The Chen Lie algebra of a group G is the associated graded Lie algebra of its maximal metabelian quotient, G/G′′. Assume G is 1-formal. Then, as shown in [23],

(2.6)  gr(G/G′′) ⊗ ℚ ≅ ℌ_{ℚ}(G)/ℌ_{ℚ}(G)′′.

Assume now G is finitely presented, and H_{1}(G) is torsion-free. Then, the map Ψ_{G} descends to an epimorphism of graded Lie algebras, Ψ_{G}^{(2)} : ℌ(G)/ℌ(G)′′ ↠ gr(G/G′′). If, moreover, G is 1-formal and ℌ(G)/ℌ(G)′′ is torsion-free, then, by [23, Theorem 1.2], the map Ψ_{G}^{(2)} yields an isomorphism

(2.7)  \xymatrix{Ψ_{G}^{(2)} : ℌ(G)/ℌ(G)′′ \ar[r]^{≅} & gr(G/G′′)} .

Of particular interest is the determination of the LCS ranks, ϕ_{k}(G) = rank gr_{k}(G) and Chen ranks, θ_{k}(G) = rank gr_{k}(G/G′′). Note that ϕ_{1}(G) = θ_{1}(G) = rank H_{1}(G); moreover, ϕ_{2}(G) = θ_{2}(G), ϕ_{3}(G) = θ_{3}(G), and ϕ_{k}(G) ≥ θ_{k}(G), for k ≥ 4.

Example 2.2. Let F_{n} be the free group of rank n. The associated graded Lie algebra gr(F_{n}) is isomorphic to the free Lie algebra L_{n} = Lie(ℤ^{n}), while gr(F_{n}/F′′_{n}) ≅ L_{n}/L′′_{n}. The LCS ranks are given by Witt’s formula: ϕ_{k}(F_{n}) = \frac{1}{k} ∑_{d|k} μ(d)n^{k/d}, or, equivalently,

(2.8)  ∏_{k=1}^{∞}(1 − t^{k})^{ϕ_{k}(F_{n})} = 1 − nt,

while the Chen ranks are given by K.-T. Chen’s formula: θ_{k}(F_{n}) = (k − 1)\binom{n+k−2}{k}, for k ≥ 2, or, equivalently,

(2.9)  ∑_{k=2}^{∞} θ_{k}(F_{n})t^{k} = 1 − \frac{1 − nt}{(1 − t)^{n}} ,

2.4. Homological algebra interpretation. Let X be a finite-type, connected CW complex, with fundamental group G = π_{1}(X) and rational cohomology ring A = H^{∗}(X; ℚ). Assume G is a 1-formal group. Using the isomorphism (2.4), Corollary 7.16 from Halperin and Stasheff [15] yields:

(2.10)  ϕ_{k}(G) = dim_{ℚ} Prim Ext_{A}^{k}(ℚ, ℚ)_{k}.

Here, Prim Ext_{A}(ℚ, ℚ) denotes the primitives in the bigraded Hopf Ext-algebra of A, where the upper degree is the resolution degree, and the lower degree is the internal degree.

We conclude this section with an analogue of formula (2.10) for the Chen ranks. The key tool is the following result of Fröberg and Löfwall [12].

Proposition 2.3. Let A be a graded algebra over a field 𝕜, with A^{1} finite-dimensional and A^{1} · A^{1} = 0. Assume A is generated in degree 1, i.e, the canonical map from the exterior algebra E = ⋀ A^{1} to A is surjective. Then:

(2.11)  (ℌ(A)′/ℌ(A)′′)_{k}^{#} = Tor_{k−1}^{E}(A, 𝕜)_{k}, for k ≥ 2.

Proof. In the case when char 𝕜 ≠ 2, the conclusion follows directly from Theorem 4.1(ii) in [12]. When char 𝕜 = 2, the minimal model proof of that Theorem still works, given our assumption that A^{1} · A^{1} = 0. □

Corollary 2.4. Let G be a finitely presented, 1-formal group. Let A be a gradedcommutative ℚ-algebra, such that dim A^{1} < ∞ and the map E = ⋀ A^{1} → A is surjective. Assume ℌ(A) = ℌ_{ℚ}(G). Then:

(2.12)  θ_{k}(G) = dim_{ℚ} Tor_{k−1}^{E}(A, ℚ)_{k}, for k ≥ 2.

Proof. Follows from (2.6) and Proposition 2.3. □

Given a group G as above, an algebra A with the required properties can be constructed as follows. Let K = ker (μ: H^{1}(G; ℚ) ∧ H^{1}(G; ℚ) → H^{2}(G, ℚ)), and E = ⋀ H^{1}(G; ℚ). Set A = E/(K + E^{≥3}). It is then readily checked that ℌ(A) = ℌ_{ℚ}(G).

In the case when G is the fundamental group of the complement of a complex hyperplane arrangement, and A is the Orlik-Solomon algebra of the arrangement, a different proof of formula (2.12) was given in [30].

3. Homology and lower central series

3.1. The cubical complex. By a graph Γ = (V_{Γ}, E_{Γ}) we mean a loopless, finite graph without multiple edges (i.e., a one-dimensional, finite simplicial complex), with vertex set V_{Γ} and edge set E_{Γ} ⊂ \binom{V_{Γ}}{2}. To such a graph, there corresponds a rightangled Artin group (or, graph group), denoted G_{Γ}, with presentation

(3.1)  G_{Γ} = ⟨v ∈ V_{Γ} | uv = vu if {u, v} ∈ E_{Γ}⟩.

The flag complex of Γ, denoted Δ_{Γ}, is the maximal simplicial complex with 1skeleton equal to Γ. The k-simplices of Δ_{Γ} are the (k + 1)-cliques of Γ, that is, the complete subgraphs on (k + 1) vertices.

Let K_{Γ} be the associated cubical complex, obtained by joining tori in the manner prescribed by Δ_{Γ}. More precisely, for each simplex σ of Δ_{Γ}, let T_{σ} be the torus formed by identifying parallel faces of a (|σ| + 1)-cube (T_{∅} is a point); then:

(3.2)  K_{Γ} = ⋃_{σ∈Δ_{Γ}} T_{σ}/(T_{σ} ∩ T_{σ′} = T_{τ} if σ ∩ σ′ = τ).

In other words, if (S^{1})^{×n} is the torus of dimension n = |V_{Γ}|, with the usual CWdecomposition, then K_{Γ} is the subcomplex obtained by deleting the cells corresponding to the non-faces of Δ_{Γ}.

Note that H_{k}(K_{Γ}) is free abelian, of rank equal to the number, f_{k}(Γ), of k-cliques in Γ (where f_{0}(Γ) = 1). In particular, the Poincaré polynomial of the cubical complex, Poin(K_{Γ}, t) = ∑_{k≥0} b_{k}(K_{Γ})t^{k}, equals the clique polynomial of the graph, P_{Γ}(t) = ∑_{k≥0} f_{k}(Γ)t^{k}.

It is readily seen that the fundamental group of K_{Γ} is the Artin group G_{Γ}. In fact, K_{Γ} is an Eilenberg-MacLane space of type K(G_{Γ}, 1), see [5], [21].

Example 3.1. Let Γ_{1} and Γ_{2} be two graphs (on distinct vertex sets). The join of the two graphs, Γ = Γ_{1} ∗ Γ_{2}, is the graph with vertex set V = V_{1} ∪ V_{2} and edge set E = E_{1} ∪ E_{2} ∪ {{v_{1}, v_{2}} | v_{1} ∈ V_{1} and v_{2} ∈ V_{2}}. Clearly, G_{Γ} = G_{Γ_{1}} × G_{Γ_{2}} and K_{Γ} = K_{Γ_{1}} × K_{Γ_{2}}.

For instance, let Γ = K_{n} be the complete graph on n vertices (the iterated join of n graphs on a single vertex). In this case, Δ_{Γ} is the (n − 1)-simplex, while K_{Γ} is the n-torus, with fundamental group G_{Γ} = ℤ^{n}.

Example 3.2. Suppose Γ = Γ_{1} ∐ Γ_{2} is the disjoint union of two subgraphs. Then G_{Γ} = G_{Γ_{1}} ∗ G_{Γ_{2}} and K_{Γ} = K_{Γ_{1}} ∨ K_{Γ_{2}}.

For instance, let Γ = \overline{K}_{n} be the complement of K_{n} (note that \overline{K}_{n} is the graph on n vertices, with empty edge set). In this case, Δ_{Γ} = \overline{K}_{n}, while K_{Γ} is the wedge of n circles, with fundamental group G_{Γ} = F_{n}.

For each subset W ⊂ V, let Γ_{W} be the full subgraph of Γ on vertex set W. Let G_{W} = G_{Γ_{W}} be the corresponding right-angled Artin group, and let K_{W} = K_{Γ_{W}} be the corresponding cubical complex. The inclusion W ⊂ V gives rise to a cellular inclusion map j_{W} : K_{W} → K_{Γ}. The induced homomorphism, (j_{W})_{#} : G_{W} → G_{Γ}, is a split injection, with retract G_{Γ} → G_{W} given on generators by v ↦ v if v ∈ W, and v ↦ 1 otherwise.

3.2. Cohomology ring and formality. Fix a labeling of the vertices of the graph Γ, and write V_{Γ} = {v_{1}, …, v_{n}}. As first noted by Kim and Roush [18], the cohomology ring of K_{Γ} is the quotient of E, the exterior algebra on generators v_{1}^{∗}, …, v_{n}^{∗} in degree 1, modulo the ideal J_{Γ} generated by the monomials v_{i}^{∗}v_{j}^{∗} for which {v_{i}, v_{j}} is not an edge of Γ. In other words,

(3.3)  H^{∗}(K_{Γ}) = E/J_{Γ}

is the exterior Stanley-Reisner ring of Γ (or, of the flag complex Δ_{Γ}). For example, if Γ = K_{n}, then H^{∗}(K_{Γ}) = E, while if Γ = \overline{K}_{n}, then H^{∗}(K_{Γ}) = E/𝔪^{2}, where 𝔪 is the ideal generated by v_{1}^{∗}, …, v_{n}^{∗}.

As noted by Fröberg [10] (and reproved by Shelton and Yuzvinsky [31]), the set {v_{i}^{∗}v_{j}^{∗} | {v_{i}, v_{j}} ∉ E_{Γ}} is a quadratic Gröbner basis for J_{Γ}; thus, H^{∗}(K_{Γ}, 𝕜) is a Koszul algebra, for any coefficient field 𝕜. Consequently, K_{Γ} is not only a K(π, 1)-space, but also a rational K(π, 1)-space, i.e., its Bousfield-Kan rationalization is aspherical, cf. [26].

As shown by Kapovich and Millson in [17, Theorem 16.10], all Artin groups are 1-formal; in particular, the right-angled Artin groups G_{Γ} are 1-formal. Using Proposition 2.1, we obtain:

Proposition 3.3. Let K_{Γ} be the cubical complex associated to a graph Γ. Then K_{Γ} is a formal space.

3.3. Associated graded Lie algebra. We are now ready to compute the graded Lie algebra associated to the lower central series filtration of a right-angled Artin group.

Theorem 3.4. Let Γ = (V, E) be a finite graph. Let G_{Γ} be the corresponding rightangled Artin group, with associated graded Lie algebra gr(G_{Γ}). Then:

(1) The holonomy Lie algebra of G_{Γ} has presentation

(3.4)  ℌ(G_{Γ}) = Lie(V)/([v, w] = 0 if {v, w} ∈ E).

(2) The graded abelian group ℌ(G_{Γ}) is torsion-free.

(3) The canonical projection Ψ: ℌ(G_{Γ}) → gr(G_{Γ}) is an isomorphism of graded Lie algebras over ℤ.

(4) The ranks of the graded pieces of gr(G_{Γ}) are given by

(3.5)  ∏_{k=1}^{∞}(1 − t^{k})^{ϕ_{k}(G_{Γ})} = P_{Γ}(−t),

where P_{Γ}(t) = ∑_{k≥0} f_{k}(Γ)t^{k} is the clique polynomial of the graph Γ.

Proof. Part (1) follows from the definition of the holonomy Lie algebra of G = G_{Γ}, and the identification of H^{∗}(G) with the exterior Stanley-Reisner ring of Γ.

To prove (2), it is enough to show that the Hilbert series of ℌ(G) ⊗ 𝕜 does not depend on the field 𝕜. By the Poincaré-Birkhoff-Witt theorem, it is enough to show that the Hilbert series of the universal enveloping algebra Uℌ_{𝕜}(G) does not depend on 𝕜. Now, as noted by Shelton and Yuzvinsky in [31], Uℌ_{𝕜}(G) = A_{𝕜}^{!}, where A_{𝕜} = H^{∗}(G; 𝕜), and A^{!} is the Koszul dual of a quadratic algebra A. But A_{𝕜} is a Koszul algebra, and so Hilb(A_{𝕜}^{!}, t) = Hilb(A_{𝕜}, −t)^{−1} by Koszul duality. Finally, note that Hilb(A_{𝕜}, t) = P_{Γ}(t) is independent of 𝕜.

Part (3) follows from Part (2), together with the 1-formality of G and formula (2.5).

Part (4) follows from Part (3), together with the proof of Part (2). □

4. Stanley-Reisner rings and Chen groups

4.1. Stanley-Reisner rings. Let Γ be a finite graph. Put a total order on the vertex set, and write V_{Γ} = {v_{1}, …, v_{n}}. As mentioned §3.2, the cohomology ring H^{∗}(G_{Γ}; 𝕜) is isomorphic to the exterior Stanley-Reisner ring, E/J_{Γ}, where E = ⋀_{𝕜}(v_{1}^{∗}, …, v_{n}^{∗}) and J_{Γ} = ideal ⟨v_{i}^{∗}v_{j}^{∗} | {v_{i}, v_{j}} ∉ E_{Γ}⟩. The Hilbert series of this graded ring equals the clique polynomial: Hilb(E/J_{Γ}, t) = P_{Γ}(t),

Associated to Γ there is also the (polynomial) Stanley-Reisner ring, S/I_{Γ}, where S = 𝕜[x_{1}, …, x_{n}] and I_{Γ} = ideal ⟨x_{i}x_{j} | {v_{i}, v_{j}} ∉ E_{Γ}⟩. As shown in [32], the Hilbert series of this graded ring is also determined by the clique polynomial:

(4.1)  Hilb(S/I_{Γ}, t) = P_{Γ}(\frac{t}{1 − t}).

A finitely generated, ℕ^{n}-graded module M over A = S or E has a minimal resolution by free A-modules and multi-homogeneous A-linear maps; such a resolution, F_{∗} → M, is unique up to isomorphism. Denote by β_{i,a}^{A}(M) the number of basis elements in F_{i} that are homogeneous of multi-degree a = (a_{1}, …, a_{n}), with a_{j} ≥ 0. By minimality of the resolution, β_{i,a}^{A}(M) = dim_{𝕜} Tor_{i}^{A}(M, 𝕜)_{a}.

The multi-graded Poincaré series of the two Stanley-Reisner rings are related by the following remarkable formula (see [2, (6.4)], [1, Proposition 2.1], or [8, Corollary 5.7]):

(4.2)  ∑_{i≥0} ∑_{a∈ℕ^{n}} β_{i,a}^{E}(E/J_{Γ})t^{i}u^{a} = ∑_{i≥0} ∑_{a∈ℕ^{n}} β_{i,a}^{S}(S/I_{Γ})\frac{t^{i}u^{a}}{∏_{j∈supp(a)}(1 − tu_{j})}.

Moreover, since I_{Γ} is a square-free monomial ideal, all the nonzero, multi-graded Betti numbers β_{i,a}^{S}(S/I_{Γ}) are also square-free, in the sense that a_{j} = 0 or 1; see Hochster [16].

4.2. Chen Lie algebra of G_{Γ}. Before stating our next result, we need to establish some terminology. For a graph Γ on vertex set V, let κ(Γ) be the connectivity of Γ, that is, the maximum integer r so that, for any subset W ⊂ V with |W| < r, the full subgraph of Γ on V \ W is connected. Also, let b̃_{0}(Γ) = rank H̃_{0}(Γ) be the number of components of Γ minus 1.

For each j ≥ 1, define the j-th cut number of Γ to be:

(4.3)  c_{j}(Γ) = ∑_{W⊂V : |W|=j} b̃_{0}(Γ_{W}).

Note that c_{1}(Γ) = 0, and also c_{j}(Γ) = 0, if j > |V| − κ(Γ). We are thus led to define the cut polynomial of Γ as

(4.4)  Q_{Γ}(t) = ∑_{j=2}^{|V|−κ(Γ)} c_{j}(Γ)t^{j}.

Theorem 4.1. Let Γ be a finite graph. Let G_{Γ} be the corresponding right-angled Artin group, with Chen Lie algebra gr(G_{Γ}/G′′_{Γ}). Then:

(1) The graded abelian group ℌ(G_{Γ})/ℌ(G_{Γ})′′ is torsion-free.

(2) The natural map Ψ^{(2)} : ℌ(G_{Γ})/ℌ(G_{Γ})′′ → gr(G_{Γ}/G′′_{Γ}) is an isomorphism of graded Lie algebras over ℤ.

(3) The ranks of the graded pieces of gr(G_{Γ}/G′′_{Γ}) are given by

(4.5)  ∑_{k=2}^{∞} θ_{k}(G_{Γ})t^{k} = Q_{Γ}(\frac{t}{1 − t}),

where Q_{Γ}(t) = ∑_{j≥2} c_{j}(Γ)t^{j} is the cut polynomial of Γ.

Proof. Let 𝕜 be a field. Applying Proposition 2.3 to the exterior Stanley-Reisner ring over 𝕜, we obtain:

(4.6)  dim_{𝕜} (ℌ_{𝕜}(G_{Γ})/ℌ_{𝕜}(G_{Γ})′′)_{k} = β_{k−1,k}^{E}(E/J_{Γ}), for k ≥ 2.

Using formula (4.2), we find:

(4.7)  ∑_{k≥2} β_{k−1,k}^{E}(E/J_{Γ})t^{k} = ∑_{i≥1} β_{i,i+1}^{S}(S/I_{Γ})(\frac{t}{1 − t})^{i+1}.

A well-known formula of Hochster [16, Theorem 5.1] gives:

(4.8)  β_{i,i+1}^{S}(S/I_{Γ}) = ∑_{W⊂V : |W|=i+1} dim_{𝕜} H̃_{0}(Γ_{W}, 𝕜) = c_{i+1}(Γ),

compare with [29, Proposition 2.1]. Combining (4.6), (4.7), and (4.8), we see that dim_{𝕜} (ℌ(G_{Γ})/ℌ(G_{Γ})′′)_{k} ⊗ 𝕜 is independent of 𝕜. This finishes the proof of Part (1).

Part (2) follows from Part (1), together with the 1-formality of G_{Γ} and formula (2.7).

Part (3), follows from Part (2), together with formulas (4.6)–(4.8). □

5. Cut sets and resonance varieties

5.1. Resonance variety of an algebra. Let A be a connected, graded algebra over a field 𝕜. Assume 0 < dim A^{1} < ∞, and A^{1} · A^{1} = 0. Then, for each a ∈ A^{1}, right-multiplication by a defines a cochain complex

(5.1)  (A, a): \xymatrix{A^{0} \ar[r]^{a} & A^{1} \ar[r]^{a} & A^{2}} .

The (first) resonance variety of A is the locus of points a in the affine space A^{1} where this complex fails to be exact in the middle:

(5.2)  R^{1}(A) = {a ∈ A^{1} | H^{1}(A, a) ≠ 0}.

Clearly, R^{1}(A) is a homogeneous algebraic variety; in particular, it contains 0 ∈ A^{1}.

Note that a nonzero element a ∈ A^{1} belongs to R^{1}(A) if and only if there is an element a′ ∈ A^{1} such that

(5.3)  a ∧ a′ ≠ 0 in A^{1} ∧ A^{1} and μ(a ∧ a′) = 0 in A^{2},

where μ: A^{1} ∧A^{1} → A^{2} is the multiplication map. In particular, R^{1}(A) depends only on the corestriction of μ to its image, and hence, only on the holonomy Lie algebra ℌ(A).

Lemma 5.1. Let φ: A → B be a morphism of graded algebras. If φ_{1} : A^{1} → B^{1} is injective, then φ_{1}(R^{1}(A)) ⊂ R^{1}(B).

Proof. We have the following commuting diagram:

\xymatrix{A^{1} ∧ A^{1} \ar[r]^{μ_{A}} \ar@{^{(}->}[d]^{φ_{1}∧φ_{1}} & A^{2} \ar[d]^{φ_{2}} \\ B^{1} ∧ B^{1} \ar[r]^{μ_{B}} & B^{2}}

Let a be a nonzero element in R^{1}(A) ⊂ A^{1}. Pick a′ ∈ A^{1} such that (5.3) holds. Then φ_{1}(a)∧φ_{1}(a′) ≠ 0, by injectivity of φ_{1}, and μ_{B}(φ_{1}(a)∧φ_{1}(a′)) = 0, by commutativity of the diagram. Hence, φ_{1}(a) belongs to R^{1}(B). □

In particular, if φ: A → B is an isomorphism of graded algebras, then the linear isomorphism φ_{1} : A^{1} → B^{1} restricts to an isomorphism R^{1}(A) → R^{1}(B) between the corresponding resonance varieties.

Lemma 5.2. Suppose the vector space A^{1} decomposes as a nontrivial direct sum U_{1} ⊕ U_{2}, with μ_{A}(u_{1} ∧ u_{2}) = 0 for all u_{1} ∈ U_{1} and u_{2} ∈ U_{2}. Then R^{1}(A) = A^{1}.

Proof. Let Z = {a ∈ A^{1} | a = u_{1} + u_{2}, with u_{i} ∈ U_{i} and u_{i} ≠ 0}. Clearly, Z is a nonempty, Zariski open subset of A^{1}, so it suffices to show that Z ⊂ R^{1}(A).

Let a = u_{1} + u_{2} ∈ Z. Then a ∧ u_{2} = u_{1} ∧ u_{2} ≠ 0 and μ_{A}(a ∧ u_{2}) = μ_{A}(u_{1} ∧ u_{2}) = 0. Thus, a belongs to R^{1}(A). □

5.2. Resonance variety of a group. Let G be a finitely presented group, with H_{1}(G) nonzero and torsion-free. Then, the cohomology ring A = H^{∗}(G; 𝕜) satisfies the condition a^{2} = 0 for all a ∈ A^{1}, and so the resonance variety of G (over the field 𝕜) can be defined as

(5.4)  R^{1}(G, 𝕜) = R^{1}(H^{∗}(G; 𝕜)).

Lemmas 5.1 and 5.2 have the following immediate corollaries.

Corollary 5.3. If α: G_{1} ↠ G_{2} is an epimorphism, then the induced monomorphism α^{∗}: H^{1}(G_{2}; 𝕜) → H^{1}(G_{1}; 𝕜) takes R^{1}(G_{2}, 𝕜) to R^{1}(G_{1}, 𝕜).

In particular, if G_{1} ≅ G_{2}, there is a linear isomorphism of ambient affine spaces, restricting to an isomorphism R^{1}(G_{1}, 𝕜) ≅ R^{1}(G_{2}, 𝕜). In other words, the ambient linear isomorphism type of the resonance variety is an invariant of the group.

Corollary 5.4. Suppose G = G_{1} ∗ G_{2} splits as a nontrivial free product of subgroups G_{1} and G_{2}, with H^{1}(G_{i}; 𝕜) ≠ 0. Then R^{1}(G, 𝕜) = H^{1}(G; 𝕜).

5.3. Resonance of right-angled Artin groups. Let Γ = (V, E) be a finite graph, and let G_{Γ} be the corresponding right-angled Artin group. Fix a field 𝕜. Write H_{V} = H^{1}(G_{Γ}; 𝕜) for the 𝕜-vector space with basis canonically indexed by V. If W is a subset of V, write H_{W} for the coordinate subspace spanned by W.

Theorem 5.5. Let Γ = (V, E) be a finite graph. Then:

(5.5)  R^{1}(G_{Γ}, 𝕜) = ⋃_{Γ_{W} disconnected^{W⊂V}} H_{W}.

Proof. Let U be the union of coordinate subspaces on the right side of (5.5). We need to show R^{1}(G_{Γ}, 𝕜) = U. For an element a = ∑_{v∈V} a_{v}v in H_{V}, we will write supp(a) = {v ∈ V | a_{v} ≠ 0} for its support.

Let a ∈ U. Then there is a subset W ⊂ V such that supp(a) ⊂ W and the graph Γ_{W} is disconnected. (Since 0 ∈ R^{1}(G_{Γ}, 𝕜), we may assume a ≠ 0, and so supp(a) ≠ ∅.) Since supp(a) ⊂ W, we have a ∈ H_{W}. Since Γ_{W} is disconnected, R^{1}(G_{W}, 𝕜) = H_{W}, by Corollary 5.4. Thus, a ∈ R^{1}(G_{W}, 𝕜). Now, Corollary 5.3, applied to the natural epimorphism G_{Γ} ↠ G_{W}, shows that R^{1}(G_{W}, 𝕜) ⊂ R^{1}(G_{Γ}, 𝕜). Hence, a ∈ R^{1}(G_{Γ}, 𝕜).

Let a ∉ U. Then, for every subset W ⊂ V containing supp(a), the graph Γ_{W} is connected. We need to show a ∉ R^{1}(G_{Γ}, 𝕜), i.e., if there is a′ = ∑_{v∈V} a′_{v}v ∈ H_{V} such that μ(a ∧ a′) = 0, then a ∧ a′ = 0.

Fix a linear order on V. Choose as basis for H^{2}(G_{Γ}, 𝕜) the set of oriented edges e = (v, w) with v < w. With these choices, the multiplication μ: H_{V} ∧ H_{V} → H^{2}(G_{Γ}, 𝕜) is given by

(5.6)  a ∧ a′ = ∑_{v<w} | \begin{matrix}a_{v} & a_{w} \\ a′_{v} & a′_{w} \end{matrix} | v ∧ w ↦ μ(a ∧ a′) = ∑_{e=(v,w)∈E} | \begin{matrix}a_{v} & a_{w} \\ a′_{v} & a′_{w} \end{matrix} | e.

The condition that μ(a ∧ a′) = 0 is thus equivalent to

(5.7)  |\begin{matrix}a_{v} & a_{w} \\ a′_{v} & a′_{w} \end{matrix}| = 0, for all \{v, w\} ∈ E.

To show that a ∧ a′ = 0, it is enough to prove the following.

Claim. Fix w ∈ supp(a). Then a′_{v} = \frac{a′_{w}}{a_{w}} a_{v}, for all v ∈ V.

Let v ∈ V. First suppose v ∈ supp(a). By assumption, the full subgraph on supp(a) is connected. So pick a path w = v_{0}, v_{1}, …, v_{k} = v in supp(a). Using condition (5.7)

for each edge of this path, we find a′_{v} = \frac{a′_{v_{k−1}}}{a_{v_{k−1}}} a_{v} = ⋯ = \frac{a′_{v_{0}}}{a_{v_{0}}} a_{v}, as claimed.

Now suppose v ∉ supp(a). Then, by assumption, the full subgraph on supp(a)∪{v} is connected. Hence, dist(v, supp(a)) = 1, that is, there is a vertex u ∈ supp(a) such that {u, v} ∈ E. Condition (5.7) on this edge implies a_{u}a′_{v} − a_{v}a′_{u} = 0. Since a_{v} = 0 and a_{u} ≠ 0, it follows that a′_{v} = 0, as claimed. □

Corollary 5.6. Let Γ = (V, E) be a finite graph, and let R^{1}(G_{Γ}, 𝕜) ⊂ H_{V} be the resonance variety of the group G_{Γ}. Then:

(1) The irreducible components of R^{1}(G_{Γ}, 𝕜) are the coordinate subspaces H_{W}, maximal among those for which Γ_{W} is disconnected.

(2) The codimension of R^{1}(G_{Γ}, 𝕜) equals the connectivity of Γ.

In particular, if Γ is disconnected, then R^{1}(G_{Γ}, 𝕜) = H_{V}.

Remark 5.7. Right-angled Artin groups share some common features with arrangement groups, that is, fundamental groups of complements of finite unions of hyperplanes in complex affine space. Indeed, if G is a group in either class, then G admits a finite presentation, with commutator relators only; it is 1-formal; and the resonance variety R^{1}(G, ℂ) is a union of linear subspaces. However, the intersection of these two classes of groups is fairly small. Indeed, a right-angled Artin group G_{Γ} is an arrangement group if and only if Γ is a complete multipartite graph. This assertion will be proved elsewhere, in a more general context. Let us only note here that a necessary condition for a group G to be an arrangement group is that any two distinct components of R^{1}(G, ℂ) intersect only at the origin, see [19]. This condition is violated by most graphs we will discuss in §6, for instance, circuits of length at least 5.

5.4. Bieri-Neumann-Strebel invariants. Let G be a finitely generated group. Pick a finite generating set for G, and let C(G) be the corresponding Cayley graph. Given an additive real character χ: G → ℝ, let C_{χ}(G) be the full subgraph on vertex set {g ∈ G | χ(g) ≥ 0}. In [4], Bieri, Neumann, and Strebel define the (first) BNS invariant of G to be:

(5.8)  Σ^{1}(G) = {χ ∈ Hom(G, ℝ) \ {0} | C_{χ}(G) is connected}.

Clearly, Σ^{1}(G) is a conical subset of the vector space Hom(G, ℝ) = H^{1}(G; ℝ). It turns out that Σ^{1}(G) does not depend on the choice of generating set for G, see [4].

The BNS invariants of right-angled Artin groups have been described in detail by Meier, Meinert, and VanWyk [20], [21]. Using these descriptions, together with our results above, we can identify the set Σ^{1}(G_{Γ}) in terms of the resonance variety R^{1}(G_{Γ}, ℝ).

Proposition 5.8. Let G_{Γ} be a right-angled Artin group.

(1) Σ^{1}(G_{Γ}) = H^{1}(G_{Γ}; ℝ) \ R^{1}(G_{Γ}, ℝ).

(2) Suppose χ: G_{Γ} → ℝ is a rational character, i.e., im(χ) ≅ ℤ. Then ker(χ) is finitely generated if and only if χ ∉ R^{1}(G_{Γ}, ℝ).

Proof. For a character χ: G_{Γ} → ℝ, let supp(χ) = {v ∈ V_{Γ} | χ(v) ≠ 0} be its support. Theorem 5.5 can be restated, as follows: χ is not in R^{1}(G_{Γ}, ℝ) if and only if, for any W ⊂ V_{Γ} with supp(χ) ⊂ W, the graph Γ_{W} is connected.

In [21], Meier and VanWyk show that a character χ: G_{Γ} → ℝ belongs to Σ^{1}(G_{Γ}) if and only if the full subgraph on supp(χ) is connected and dominant. (A subgraph Γ′ ⊂ Γ is dominant if any vertex v ∈ V_{Γ} \ V_{Γ′} has a neighbor in Γ′.)

Comparing these two characterizations finishes the proof of Part (1).

To prove Part (2), we need the following basic result from [4], concerning rational characters χ: G → ℝ: the group ker(χ) is finitely generated if and only if both χ and −χ belong to Σ^{1}(G). On the other hand, it was shown in [21] that Σ^{1}(G_{Γ}) = −Σ^{1}(G_{Γ}). Using these two facts, Part (2) follows from Part (1). □

A generalization to higher Σ-invariants and higher resonance varieties will be given in a forthcoming paper.

6. Examples

We now illustrate with a few examples how to compute the LCS ranks, the Chen ranks, and the resonance varieties of right-angled Artin groups, using the results from the three previous sections. We start by discussing methods for computing the two graph polynomials that appear in formulas (3.5) and (4.5).

6.1. Computing the clique and cut polynomials. When a graph Γ has no triangles (i.e., when Δ_{Γ} = Γ), the clique polynomial is easy to compute: P_{Γ}(t) = 1 + |V_{Γ}| t + |E_{Γ}| t^{2}. For an arbitrary graph, the computation of P_{Γ}(t) is more complicated, but a simple recursion formula is readily available.

Fix an edge e ∈ E_{Γ}. Let Γ \ e be the graph Γ with edge e removed, and let Γ \ N(e) be the graph obtained from Γ by removing the endpoints of e, all their neighbors, and all edges containing any one of these vertices. Using (4.1) and Theorem 5.1 from [28], we find:

(6.1)  P_{Γ}(t) = P_{Γ\e}(t) − t^{2}P_{Γ\N(e)}(t).

Next, we give some (partial) recursion formulas for the coefficients of the cut polynomial. An edge e ∈ E_{Γ} is called a bridge if it does not lie in any cycle; we say e is a near-bridge if the only cycles containing e are those incident on all vertices of Γ.

Lemma 6.1. Suppose the graph Γ has a near-bridge e. Then, for all j < |V_{Γ}|,

(6.2)  c_{j}(Γ) = c_{j}(Γ \ e) − \binom{|V_{Γ}| − 2}{j − 2}.

Proof. Put Γ′ = Γ \ e. Fix a proper subset W ⊂ V. If e ⊄ W, then clearly b̃_{0}(Γ_{W}) = b̃_{0}(Γ′_{W}). If e ⊂ W, then, by the assumption on e, we have b̃_{0}(Γ_{W}) = b̃_{0}(Γ′_{W}) − 1. Hence:

c_{j}(Γ) = ∑_{|W|=j, e⊄W} b̃_{0}(Γ_{W}) + ∑_{|W|=j, e⊂W} b̃_{0}(Γ_{W})

= ∑_{|W|=j, e⊄W} b̃_{0}(Γ′_{W}) + ∑_{|W|=j, e⊂W}(b̃_{0}(Γ′_{W}) − 1),

from which (6.2) follows at once. □

Lemma 6.2. Let Γ = Γ′ ∐ K_{1} be the disjoint union of a graph Γ′ with a singleton graph K_{1}. Then, for all j ≥ 2:

(6.3)  c_{j}(Γ) = c_{j}(Γ′) + c_{j−1}(Γ′) + \binom{|V_{Γ}| − 1}{j − 1}.

Proof. We have:

c_{j}(Γ) = ∑_{|W|=j, W⊂V_{Γ′}} b̃_{0}(Γ_{W}) + ∑_{|W|=j, W⊄V_{Γ′}} b̃_{0}(Γ_{W})

= ∑_{|W|=j, W⊂V_{Γ′}} b̃_{0}(Γ′_{W}) + ∑_{|W|=j−1, W⊂V_{Γ′}}(b̃_{0}(Γ′_{W}) + 1),

from which (6.3) follows at once. □

6.2. Trees. Let Γ be a tree on n vertices. In this case, Δ_{Γ} = Γ, and so the clique polynomial is P_{Γ}(t) = 1 + nt + (n − 1)t^{2}. Hence, by Theorem 3.4(4),

(6.4)  ∏_{k=1}^{∞}(1 − t^{k})^{ϕ_{k}(G_{Γ})} = (1 − t)(1 − (n − 1)t).

We claim the cut numbers of Γ are given by

(6.5)  c_{j}(Γ) = (j − 1)\binom{n − 1}{j}, for j = 2, …, n − 1.

For n = 1 or 2, there is nothing to prove. Let e be an extremal edge. Clearly, e is a bridge, and Γ \ e = Γ_{0} ∐ K_{1}, where Γ_{0} is a tree on n − 1 vertices. Formula (6.5) follows by induction on n (starting from n = 3, where it is obvious), using Lemmas 6.1 and 6.2.

Before proceeding, let us note that (6.5) is equivalent to β_{i,i+1}^{S}(S/I_{Γ}) = i\binom{n−1}{i+1}, for 1 ≤ i ≤ n − 2, a fact which can also be deduced from [11, Proposition 11].

Using Theorem 4.1(3) and formula (6.5), we find that the Chen ranks of G_{Γ} are given by θ_{k}(G_{Γ}) = (k − 1) · \binom{k+n−3}{k}, for all k ≥ 2, or, equivalently:

(6.6)  ∑_{k=2}^{∞} θ_{k}(G_{Γ})t^{k} = 1 − \frac{1 − (n − 1)t}{(1 − t)^{n−1}}.

Now consider the resonance variety R^{1}(G_{Γ}, 𝕜). The minimal cutsets of the tree Γ are the non-extremal vertices. Thus, R^{1}(G_{Γ}, 𝕜) consists of coordinate hyperplanes in 𝕜^{n}, one for each non-extremal vertex:

(6.7)  R^{1}(G_{Γ}, 𝕜) = ⋃_{v∈V : v non-extremal} H_{V\{v}}.

The above discussion shows that, for a tree Γ, the LCS ranks and the Chen ranks of G_{Γ} depend only on n = |V_{Γ}|, whereas the resonance variety R^{1}(G_{Γ}, 𝕜) also depends on the cut sets of the graph. We illustrate this phenomenon with an example.

Example 6.3. Let Γ and Γ′ be two trees on the same vertex set, but with different number of extremal vertices. (For example, take the Dynkin graphs Γ = A_{n} and Γ′ = D_{n}, with n ≥ 4; clearly, Γ has 2 extremal vertices, whereas Γ′ has 3 such vertices.) Then ϕ_{k}(G_{Γ}) = ϕ_{k}(G_{Γ′}) and θ_{k}(G_{Γ}) = θ_{k}(G_{Γ′}), for all k ≥ 1, yet R^{1}(G_{Γ}, 𝕜) ≅ R^{1}(G_{Γ′}, 𝕜). Hence, ℌ(G_{A_{n}}) and ℌ(G_{D_{n}}) are not isomorphic as graded Lie algebras, although they have the same graded ranks.

6.3. Circuits. Let Γ_{n} be a circuit on n vertices, n ≥ 4. Again, Δ_{Γ_{n}} = Γ_{n}, and so P_{Γ_{n}}(t) = 1 + nt + nt^{2}. Hence,

(6.8)  ∏_{k=1}^{∞}(1 − t^{k})^{ϕ_{k}(G_{Γn})} = 1 − nt + nt^{2}.

If n ≥ 5, the right side of (6.8) does not factor into linear factors in ℤ[t], and so the group G_{Γ_{n}} cannot decompose as an (iterated) semi-direct product of free groups, with trivial monodromy action in homology.

Deleting an edge from Γ_{n} yields the n-chain A_{n}. Clearly, any edge in Γ_{n} is a nearbridge; thus, by Lemma 6.1, c_{j}(Γ_{n}) = c_{j}(A_{n}) − \binom{n−2}{j−2}, for all j < n. On the other hand, c_{n}(Γ_{n}) = 0, since Γ_{n} is connected. Using formula (6.5), and plugging into (4.5) gives:

(6.9)  ∑_{k=2}^{∞} θ_{k}(G_{Γ_{n}})t^{k} = ∑_{j=2}^{n−2} ((j − 1)\binom{n − 1}{j} − \binom{n − 2}{j − 2}) (\frac{t}{1 − t})^{j}.

The resonance variety R^{1}(G_{Γ_{n}}, 𝕜) consists of n(n − 3)/2 codimension 2 subspaces in 𝕜^{n}, one for each pair of non-adjacent vertices:

(6.10)  R^{1}(G_{Γ_{n}}, 𝕜) = ⋃_{\{v,w\}∈E\overline{Γn}} H_{V\\{v,w\}}

Example 6.4. For each n ≥ 5, let Γ′_{n} be an (n−1)-circuit, with an extra vertex and an edge attaching it to the circuit. Clearly, P_{Γ′_{n}}(t) = P_{Γ_{n}}(t), and so ϕ_{k}(G_{Γ′_{n}}) = ϕ_{k}(G_{Γ_{n}}), for all k ≥ 1.

Although the LCS ranks of G_{Γ′_{n}} and G_{Γ_{n}} are the same, the Chen ranks are different. Indeed, an application of Lemmas 6.1 and 6.2 yields c_{j}(Γ′_{n}) = c_{j}(Γ_{n}) for j = 2, …, n− 2. On the other hand, c_{n−1}(Γ′_{n}) = 1, whereas c_{n−1}(Γ_{n}) = 0. Thus,

(6.11) ∑_{k=2}^{∞} θ_{k}(G_{Γ′_{n}})t^{k} = ∑_{j=2}^{n−2} ((j − 1)\binom{n − 1}{j} − \binom{n − 2}{j − 2}) (\frac{t}{1 − t})^{j} + (\frac{t}{1 − t})^{n−1}.

Consequently, θ_{k}(G_{Γ′_{n}}) = θ_{k}(G_{Γ_{n}}) for k ≤ n−2, but θ_{k}(G_{Γ′_{n}}) > θ_{k}(G_{Γ_{n}}), for k ≥ n−1.

6.4. A pair of graphs. We conclude this section with a pair of right-angled Artin groups that differ in a rather subtle way.

Example 6.5. Let Γ and Γ′ be the two graphs in Figure 1. Both graphs have clique polynomial P (t) = 1 + 6t + 9t^{2} + 4t^{3} and cut polynomial Q(t) = t^{2}(6 + 8t + 3t^{2}); thus, G_{Γ} and G_{Γ′} have the same LCS and Chen ranks. Their resonance varieties have the same number of irreducible components, each a codimension 2 linear subspace:

R^{1}(G_{Γ}, 𝕜) = H\overline{23} ∪ H\overline{25} ∪ H\overline{35} , R^{1}(G_{Γ′}, 𝕜) = H\overline{15} ∪ H\overline{25} ∪ H\overline{26} .

Yet the two varieties are not isomorphic, since dim(H\overline{23} ∩ H\overline{25} ∩ H\overline{35}) = 3, whereas dim(H\overline{15} ∩ H\overline{25} ∩ H\overline{26}) = 2.

[figure: 1; 2; 3; 4; 5; 6; 1; 2; 3; 4; 5; 6]

Figure 1. The graphs Γ and Γ′

7. Higher-dimensional cubical complexes

The cubical complex K_{Γ} associated to a graph Γ has natural, higher-dimensional analogues, introduced in [18]. For each integer q ≥ 0, let K_{Γ}^{q} be the CW-complex obtained from (S^{2q+1})^{×n}, n = |V_{Γ}|, by deleting the cells corresponding to the nonfaces of the flag complex Δ_{Γ}. Note that K_{Γ}^{0} = K_{Γ} = K(G_{Γ}, 1), while K_{Γ}^{q} is simplyconnected, for q ≥ 1.

Let ΩY denote the based loop space of a pointed CW-complex Y. In [18], Kim and Roush determined the Pontryagin ring structure of H_{∗}(ΩK_{Γ}^{q}). We now pursue this study, and determine the natural Hopf algebra structure on H_{∗}(ΩK_{Γ}^{q}; ℚ). As a byproduct, we compute explicitly the rational homology and homotopy groups of ΩK_{Γ}^{q}, in terms of the clique polynomial of Γ.

First, some background. The homotopy Lie algebra of a simply-connected, finitetype CW-complex Y is the graded ℚ-vector space

π_{∗}(ΩY) ⊗ ℚ = ⨁_{k≥1} π_{k}(ΩY) ⊗ ℚ,

with the graded Lie algebra structure coming from the Whitehead product. By the Milnor-Moore theorem [22], the universal enveloping algebra of π_{∗}(ΩY) ⊗ ℚ is isomorphic, as a Hopf algebra, to H_{∗}(ΩY, ℚ). Finally, we say Y is coformal if its rational homotopy type is determined by its homotopy Lie algebra.

Theorem 7.1. Let Γ = (V, E) be a finite graph, and q a positive integer. Then: (1) The homotopy Lie algebra of K_{Γ}^{q} has presentation

π_{∗}(ΩK_{Γ}^{q}) ⊗ ℚ = Lie(V[q])/([v, w] = 0 if {v, w} ∈ E),

where Lie(V[q]) denotes the free Lie algebra on generators from V, in degree 2q.

(2) The ranks of the graded pieces, Φ_{k}(K_{Γ}^{q}) = rank π_{k}(ΩK_{Γ}^{q}), vanish if 2q ∤ k, while the other ranks are given by the following homotopy LCS formula:

∏_{k=1}^{∞}(1 − t^{(2q+1)k})^{Φ_{2qk}(K_{Γ}^{q})} = P_{Γ}(−t^{2q+1}).

(3) The Hilbert series of H_{∗}(ΩK_{Γ}^{q}; ℚ) is given by

Poin(ΩK_{Γ}^{q}, t) = P_{Γ}(−t^{2q})^{−1}.

(4) If 2q + 1 > dim K_{Γ}, then K_{Γ}^{q} is both formal and coformal.

Proof. Let X = K_{Γ} and Y = K_{Γ}^{q}. Following [18], note that the ring H^{∗}(Y) is the q-rescaling of the ring H^{∗}(X), with rescaling factor of 2q + 1; in particular,

(7.1)  Poin(Y, t) = Poin(X, t^{2q+1}) = P_{Γ}(t^{2q+1}).

Now recall that H^{∗}(X; ℚ) is a Koszul algebra. Therefore, by [24, Theorem B], the graded Lie algebra π_{∗}(ΩY) ⊗ ℚ is the q-rescaling of gr(G_{Γ}) ⊗ ℚ, with rescaling factor of 2q. Part (1) follows from Theorem 3.4, Parts (1) and (3).

Parts (2) and (3) are direct consequences of Theorem C and formula (28) from [24].

As for Part (4), formality is guaranteed by [24, §4.1 and Proposition 4.4], while coformality follows from [24, Proposition 1.12]. □

Remark 7.2. The conclusion of Theorem 7.1 may not hold for an arbitrary subcomplex of (S^{2q+1})^{×n}. Indeed, let Y be the (2q + 1)(n − 1)-skeleton of (S^{2q+1})^{×n}. Then Y is formal. Nevertheless, the vanishing property from Part (2) and the coformality property from Part (4) fail for Y, as soon as 2q +1 > n−1 > 1; see [24, Example 8.5].

Example 7.3. Let Γ and Γ′ be two trees on the same vertex set, but with different number of extremal vertices, e.g., Γ = A_{n} and Γ′ = D_{n}, with n ≥ 4. Let Y = K_{Γ}^{q} and Y′ = K_{Γ′}^{q} be the corresponding higher cubical complexes. Then Y and Y′ are both formal and coformal spaces, by Theorem 7.1(4), and share the same Poincaré polynomial, by formula (7.1); moreover, the rational homology and homotopy groups of ΩY and ΩY′ have the same dimensions, by Theorem 7.1(3) and (2). Nevertheless, Y and Y′ are not (rationally) homotopy equivalent. Indeed, by the computation from Example 6.3, ℌ(G_{Γ}) ≅ ℌ̸(G_{Γ′}). Comparing the Lie algebra presentations from Theorems 3.4 and 7.1, we conclude that π_{∗}(ΩY) ⊗ ℚ ≅ π̸_{∗}(ΩY′) ⊗ ℚ.

Added in Proof. After completing this paper, we became aware of the work of Duchamp and Krob [6], [7]. Using Theorem 3.4(1), it is readily seen that Theorem 2.1 of [7] is equivalent to our Theorem 3.4(3), while Corollary II.16 and Theorem III.3 of [6] are equivalent to our Theorem 3.4(2) and (4), respectively. Thus, our methods give a completely different (and much shorter) proof of Duchamp and Krob’s results.

Acknowledgments. Computations were carried out with the help of GAP 4.4 [13] and Macaulay 2 [14].

This work was done while the authors were attending the program “Hyperplane Arrangements and Applications” at the Mathematical Sciences Research Institute in Berkeley, California, in Fall, 2004. We thank MSRI for its support and hospitality during this stay.

References

1. A. Aramova, L. Avramov, J. Herzog, Resolutions of monomial ideals and cohomology over exterior algebras, Trans. Amer. Math. Soc. 352 (1999), 579–594. MR 2000c:13021

2. A. Aramova, J. Herzog, T. Hibi, Gotzmann theorems for exterior algebras and combinatorics, Journal of Algebra 191 (1997), 174–211. MR 98c:13025

3. M. Bestvina, N. Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445–470. MR 98i:20039 4. R. Bieri, W. Neumann, R. Strebel, A geometric invariant of discrete groups, Invent. Math. 90

(1987), no. 3, 451–477. MR 89b:20108

5. R. Charney, M. Davis, Finite K(π, 1)s for Artin groups, Prospects in topology (Princeton, NJ, 1994), 110–124, Ann. of Math. Stud., Vol. 138, Princeton Univ. Press, Princeton, NJ, 1995. MR 97a:57001

6. G. Duchamp, D. Krob, The free partially commutative Lie algebra: bases and ranks, Adv. Math. 95 (1992), no. 1, 92–126. MR 93j:17009

7. , The lower central series of the free partially commutative group, Semigroup Forum 45 (1992), no. 3, 385–394. MR 93e:20047

8. D. Eisenbud, S. Popescu, S. Yuzvinsky, Hyperplane arrangement cohomology and monomials in the exterior algebra, Trans. Amer. Math. Soc. 355 (2003), no. 11, 4365–4383. MR 2004g:52036

9. M. Falk, Arrangements and cohomology, Ann. Combin. 1 (1997), no. 2, 135–157. MR 99g:52017 10. R. Fröberg, Determination of a class of Poincaré series, Math. Scand. 37 (1975), no. 1, 29–39.

MR 0404254

11. , Rings with monomial relations having linear resolutions, J. Pure Appl. Algebra 38 (1985), no. 2-3, 235–241. MR 87b:13020

12. R. Fröberg, C. Löfwall, Koszul homology and Lie algebras with application to generic forms and points, Homology Homotopy Appl. 4 (2002), no. 2, part 2, 227–258. MR 2003g:13018

13. The GAP Group, GAP–Groups, Algorithms, and Programming, Version 4.4 (2004); available

at http://www.gap-system.org.

14. D. Grayson, M. Stillman, Macaulay 2: a software system for research in algebraic geometry;

available at http://www.math.uiuc.edu/Macaulay2.

15. S. Halperin, J. Stasheff, Obstructions to homotopy equivalences, Adv. Math. 32 (1979), no. 3, 233–279. MR 80j:55016

16. M. Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes, in: Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975), pp. 171–223, Lecture Notes in Pure and Appl. Math., Vol. 26, Dekker, New York, 1977. MR 0441987

17. M. Kapovich, J. Millson, On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 88 (1998), no. 8, 5–95. MR 2001d:14024

18. K.H. Kim, F.W. Roush, Homology of certain algebras defined by graphs, J. Pure Appl. Algebra 17 (1980), no. 2, 179–186. MR 82e:05114b

19. A. Libgober, S. Yuzvinsky, Cohomology of the Orlik-Solomon algebras and local systems, Compositio Math. 21 (2000), no. 3, 337–361. MR 2001j:52032

20. J. Meier, H. Meinert, L. VanWyk, Higher generation subgroup sets and the Σ-invariants of graph groups, Comment. Math. Helv. 73 (1998), no. 1, 22–44. MR 99f:57002

21. J. Meier, L. VanWyk, The Bieri-Neumann-Strebel invariants for graph groups, Proc. London Math. Soc. (3) 71 (1995), no. 2, 263–280. MR 96h:20093

22. J.W. Milnor, J.C. Moore, On the structure of Hopf algebras, Ann. Math. 81 (1965), 211–264. MR 30 #4259

23. S. Papadima, A. Suciu, Chen Lie algebras, International Math. Research Notices 2004, no. 21, 1057–1086. MR 2004m:17043

24. , Homotopy Lie algebras, lower central series, and the Koszul property, Geometry & Topology 8 (2004), 1079–1125. MR 2005g:55022

25. , Algebraic invariants for Bestvina-Brady groups, preprint, 2005.

26. S. Papadima, S. Yuzvinsky, On rational K[π, 1] spaces and Koszul algebras, J. Pure Appl. Alg. 144 (1999), no. 2, 156–167. MR 2000k:55017

27. D. Quillen, Rational homotopy theory, Ann. of Math. 90 (1969), 205–295. MR 0258031

28. P. Renteln, The Hilbert series of the face ring of a flag complex, Graphs Combin. 18 (2002), no. 3, 605–619. MR 2003i:13025

29. M. Roth, A. Van Tuyl, On the linear strand of an edge ideal, preprint arXiv:math.AC/0411181.

30. H. Schenck, A. Suciu, Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence, to appear in Trans. Amer. Math. Soc., available at

   arXiv:math.AC/0502438.

31. B. Shelton, S. Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc. 56 (1997), no. 3, 477–490. MR 99c:16044

32. R. Stanley, Combinatorics and commutative algebra, Second edition, Progress in Mathematics, vol. 41, Birkhäuser, Boston, MA, 1996. MR 98h:05001

33. D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331. MR 0646078

Institute of Mathematics of the Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania

  E-mail address: Stefan.Papadima@imar.ro

Department of Mathematics, Northeastern University, Boston, MA 02115, USA

  E-mail address: a.suciu@neu.edu
  URL: http://www.math.neu.edu/~suciu