PROJECTIVE RESOLUTIONS FOR GRAPH PRODUCTS

Daniel E. Cohen

Queen Mary and Westfield College, London University

Abstract.

Let Γ be a finite graph together with a group G_{v} at each vertex v. The graph product G(Γ) is obtained from the free product of all G_{v} by factoring out by the normal subgroup generated by {g^{−1}h^{−1}gh; g ∈ G_{v}, h ∈ G_{w}} for all adjacent v, w.

In this note we construct a projective resolution for G(Γ) given projective resolutions for each G_{v}, and obtain some applications.

Let Γ be a finite graph together with a group G_{v} at each vertex v. The graph product G(Γ) is obtained from the free product of all G_{v} by factoring out by the normal subgroup generated by {g^{−1}h^{−1}gh; g ∈ G_{v}, h ∈ G_{w}} for all adjacent v, w.

In this note we construct a projective resolution for G(Γ) given projective resolutions for each G_{v}, and obtain some applications. This is quite easy to do, since G(Γ) is built up from the vertex groups by direct products and amalgamated free products.

Let A be an arbitrary group, and let R be a commutative ring. A projective resolution for A is an exact sequence

P : … → P_{n} →^{d_{n}} P_{n−1} → … P_{1} → P_{0} → R → 0

of projective (right) RA-modules. We shall always assume that P_{0} = RA.

Let G = A ∗_{C} B and let Q and N be projective resolutions of B and C. Suppose that N ⊗_{RC} RA and N ⊗_{RC} RB are summands of P and Q respectively. Let S_{k} be the quotient of (P_{k} ⊗_{RA} RG) ⊕ (Q_{k} ⊗_{RB} RG) by the submodule generated by all (n ⊗_{RC} 1, −n ⊗_{RC} 1), with the obvious map from S_{k} to S_{k−1}.

Lemma 1. In the above situation,

… S_{k} → S_{k−1} → … → S_{1} → RG → R → 0

is a projective resolution for G.

Proof. It is easy to check that each S_{k} is projective.

Let I_{A} be the augmentation ideal of A, with similar notation for B, C, G. We can regard the resolution P as finishing with P_{2} → P_{1} → I_{A} → 0, and this sequence

Typeset by AMS-TEX

Key words and phrases. graph products, projective resoltuions, Euler characteristics.

is exact. It is then immediate, by elementary properties of exact sequences, that we have an exact sequence

… S_{n} → S_{n−1} → … → S_{2} → S_{1} → M → 0,

where M is defined to be the quotient of (I_{A} ⊗_{RA} RG) ⊕ (I_{B} ⊗_{RB} RG) by the submodule generated by all ((c − 1) ⊗_{RC} 1, −(c − 1) ⊗_{RC} 1).

However, according to Lemma 4.3 and Theorem 4.7 of [6], M is exactly I_{G}, as required. □

Now let H = A × B. We find a projective resolution for H as the tensor product of P and Q. Precisely, we define (P⊗^{+}Q)_{n} to be ∑_{i+j=n} P_{i} ⊗_{R} Q_{j}, where i and j are non-zero. The superscript + is used because the tensor product is defined to allow i and j to be zero. Our modification will, in fact, lead to a messy formula for the resolution of H. It is used because at a later point it ensures that we do not have summands repeated unnecessarily.

Note that (P⊗^{+}Q)_{n} is an H -module, as are P_{n}⊗_{RA}RH and Q_{n}⊗_{RB} RH. These are plainly projective modules. Further, these last two modules are isomorphic to P_{n} ⊗_{R} RB and RA ⊗_{R} Q_{n}, and we identify the isomorphic modules.

Let T be (P⊗^{+}Q) ⊕ (P ⊗_{RA} RH) ⊕ (Q ⊗_{RB} RH) in dimensions greater than 0, while T_{0} = RH. We define a boundary operator on T in the obvious way on the last two summands. On P_{i} ⊗_{R} Q_{j} we define it to map p⊗q to (pd_{i})⊗q +(−1)^{i}p⊗(qd_{j}). When i or j is 1, one (or both) of these terms lies in the third or second of the summands.

As is well-known (for instance, V.1.1 of [3]) we have the following lemma.

Lemma 2. T is a projective resolution for H.

Now let Γ be a finite graph with vertex group G_{v} at the vertex v. Let Δ be any full subgraph of Γ, with the corresponding graph product G(Δ). There is an obvious homomorphism from G(Δ) to G(Γ) induced by inclusion of the vertex groups, and another homomorphism from G(Γ) to G(Δ) induced by the inclusion on those vertex groups with vertices in Δ and trivial on the other vertex groups. It follows that we have a monomorphism, which we regard as inclusion, from G(Δ) to G(Γ).

Let v be any vertex, let Δ be the full subgraph on all vertices except v, and let Ω be the full subgraph on v and all vertices adjacent to to v. Then G(Γ) = (G_{v} × G(Ω ∩ Δ) ∗_{G(Ω∩Δ)} G(Δ). This was proved by Green [7], and it is easy to obtain homomorphisms between these two groups which are inverses of each other. From this, and our lemmas, it is easy to find a projective resolution for G(Γ), but we need some more notation. We let P^{v} be a projective resolution of G_{v}, with dimension 0 being omitted for later notational convenience.

It is convenient to regard the vertices as having a specified total order. Let K be a complete subgraph of Γ whose vertex set (in the given order) is {v_{1}, …, v_{k}}. Let P^{K} = (P^{v_{1}} ⊗_{R}^{+} … ⊗_{R}^{+} P^{v_{k}}) ⊗_{RG(K)} G(Γ). As in the case of a direct product (which corresponds to the complete graph on two vertices), a boundary operator is defined on ∑ P^{L}, where the sum is taken over all complete subgraphs L of K.

We note that there is a bijection between the complete subgraphs of Ω ∩ Δ and the complete subgraphs of Ω which contain v and at least one other vertex. We will choose v to be the first vertex of the ordering.

Using the two lemmas, the theorem below now follows easily by induction on the number of vertices of Γ.

Theorem. ∑ P^{K}, taken over all complete subgraphs K of Γ, forms the positivedimensional part of a projective resolution for G(Γ).

Recall that a group is F P_{n} (F P_{∞}) if it has a projective resolution finitely generated in dimensions ≤ n (in all dimensions), it is of finite cohomological dimension if it has a projective resolution zero in all but finitely many dimensions, and it is F P if it has a projective resolution finitely generated in all dimensions and zero in all but finitely many. The following corollary is immediate from the theorem.

Corollary 1. Let the vertex groups of a graph product over a finite graph all be F P_{n} (for any n ≤ ∞), of finite cohomological dimension, or F P. Then the graph product has the same property.

Chiswell [4] defines an Euler characteristic for any F P group, and he obtains in [5] a formula for the Euler characteristic of a graph product, using the inductive decomposition of a graph product in terms of amalgamated free products and direct products. Our theorem (which relies on this decompostion) makes it clearer where Chiswell’s formula comes from. However, his proof also holds for Brown’s definition [3] of the Euler characteristic, which ours does not appear to.

Let G be a group, and let P be a finitely generated projective RG-module. Then Hattori [8] and Stallings [9] define a trace function t_{P}, which is an element of R(G/G′). Let τ_{P} be the coefficient of 1 in t_{P}. When G is F P take a projective resolution P which is finitely generated in all dimensions and zero in all but finitely many. Chiswell defines the Euler characteristic of G to be ∑_{0}^{∞} τ_{P_{i}}.

Let G be a subgroup of H, and let P be a finitely generated projective RGmodule. Then P ⊗_{RG} RH is a finitely generated projective RH-module, and t_{P ⊗_{RG}RH} is the image in R(H/H′) of t_{P}; hence τ_{P ⊗_{RG}RH} = τ_{P}. Also, if Q is a finitely generated projective RF -module, for some group F, then P ⊗_{R}Q is a finitely generated projective (G×F)-module, and t_{P ×_{R}Q} = t_{P} ⊗t_{Q} in R(G×F)/(G×F)′ = R(G/G′) ⊗_{R} R(F/F′); hence τ_{P ⊗_{R}Q} = τ_{P} τ_{Q}. These facts can be found in [2], and are easy to prove directly.

Let Γ be a finite graph, with an F P group at each vertex. For each complete subgraph K of Γ, with vertices v_{1}, …, v_{k}, let χ_{K} = (χ_{v_{1}} − 1) … (χ_{v_{k}} − 1), where χ_{v_{i}} is the Euler characteristic of the group at the vertex v_{i}. From our main theorem and the remarks in the previous paragraph, we immediately deduce the following formula due to Chiswell [5] (our formula differs slightly from his, because he includes the empty subgraph and we do not.

Corollary 2. With the above notation, χ(GΓ) = ∑ χ_{K}, the sum being taken over all complete subgraphs of Γ.

Let ⟨Y; S⟩ be a presentation of a group. This group then has a free resoltuion whose basis in dimension 1 is (a set bijective with) Y, and whose basis in dimension 2 is (a set bijective with) S. The boundary operator in dimension 2 can be described by means of the Fox derivatives.

Take such a presentation ⟨Y_{v}; S_{v}⟩ for each vertex group of a graph product, and take the corresponding free resolutions. Then the graph product has a presentation ⟨⋃ Y_{v}; ⋃ S_{v} ∪ C, where C is the set of all [a, b] for a ∈ Y_{v}, b ∈ Y_{w} and all adjacent vertices v and w such that v < w in the chosen ordering of the graph.

Consider the resolution of the graph product obtained by our theorem from the free resolutions of the vertex groups. It will be a free resolution whose basis in dimension 1 will be ⋃ Y_{v}. In dimension 2 it will have two kinds of basis elements. The members of ⋃ S_{v} form the first kind, while the second kind consists of all a⊗b, where a ∈ Y_{v}, b ∈ Y_{w}, for all adjacent vertices v and w such that v < w. Thus this basis is bijective with the basis given by the presentation of the graph product. Further, the boundary operator given by the theorem is exactly that given by the presentation.

In dimension 3 there are four kinds of basis elements. The first kind consists of all basis elements in dimension 3 of the resolutions of the vertex groups. The second kind consists of all a ⊗ s where a ∈ Y_{v}, s ∈ S_{w}, for all adjacent vertices v and w such that v < w. The third kind consists of all s ⊗ b where s ∈ S_{v}, b ∈ Y_{w}, for all adjacent vertices v and w such that v < w. The fourth kind consists of all a ⊗ b ⊗ c where a ∈ Y_{u}, b ∈ Y_{v}, c ∈ Y_{w} and {u, v, w} is a complete subgraph with u < v < w.

The boundary operator can be explicitly described for the basis elements of the second, third, and fourth kinds. When this is done, we recover a result of Baik, Howie, and Pride [1].

References

1. Y-G. Baik, J.Howie and S.J. Pride, The identity problem for graph products, J. Algebra (to appear).

2. H. Bass, Euler characteristics and characters of discrete groups, Invent. Math. 35 (1976), 155–196.

K.S. Brown, Cohomology of groups, Graduate texts in mathematics 87, Springer-Verlag, 1982.

4. I.M.Chiswell, Euler characteristics of groups, Math. Zeit. 147 (1976), 1-11.

5. I.M. Chiswell, The Euler characteristic of a graph product.

6. D.E. Cohen, Groups of cohomolgical dimension one, Springer lecture notes in mathematics, Springer-Verlag, 1972.

7. E.R. Green, Graph products of groups, Ph.D. Thesis, University of Leeds, 1990.

8. A. Hattori, Rank element of a projective module, Nagoya Math. J. 25 (1965), 113–120.

9. J.R. Stallings, Centerless groups — an algebraic formulation of Gottlieb’s theorem, Topology 4 (1965), 129–134.

School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS, England

E-mail address: D.E.Cohen@uk.ac.qmw.maths