MORSE THEORY AND FINITENESS PROPERTIES OF GROUPS

Mladen Bestvina and Noel Brady

June 15 1996

Abstract. We examine the finiteness properties of certain subgroups of “right angled” Artin groups. In particular, we find an example of a group that is of type FP(ℤ) but is not finitely presented.

§1. Introduction

The two fundamental finiteness properties in group theory are the properties of being finitely generated and of being finitely presented. There are many examples of finitely generated groups that are not finitely presented. Indeed, there are uncountably many finitely generated groups, but only countably many finitely presented ones [Ne]. A remarkable example of one such group was constructed by Stallings in [St]. It can be described as the kernel of the homomorphism of the direct product F_{2} × F_{2} of free groups of rank 2 to the infinite cyclic group ℤ that sends both basis elements of both factors to 1 ∈ ℤ.

More general finiteness properties were introduced by C.T.C. Wall in [Wa 1]. A group H is said to be of type F_{n} if it has an Eilenberg-Mac Lane complex K(H, 1) with finite nskeleton. Equivalently, a group is of type F_{n} if it acts freely, faithfully, properly, cellularly, and cocompactly on an (n−1)-connected cell complex. Clearly, a group is finitely generated if and only if it is of type F_{1}, and is finitely presented if and only if it is of type F_{2}.

One can generalize this notion further by replacing the phrase “(n − 1)-connected” in the latter definition by the phrase “homologically (n − 1)-connected (with respect to a given coefficient ring R)”. This gives a weaker finiteness property, which we call FH_{n}(R). It has been an open question whether FH_{n}(ℤ) is equivalent to F_{n} for n > 1. It is known that FH_{1}(R) for any nontrivial R is equivalent to finite generation of the group, and hence to F_{1}.

The finiteness properties FP_{n}(R) used in the literature were introduced by Bieri [Bi 2] and are slightly weaker than the properties FH_{n}(R) above. Precise definitions

Typeset by AMS-TEX

The first author is supported by an NSF grant. He also thanks the Mathematics Department at Melbourne University for their support and hospitality while a part of this work was being carried out.

The second author received partial support from NSF Grant DMS9200433.

and the relationship between these properties are given in Section 3. The main point is that we use the topologically defined FH_{n}(R) to detect if our groups satisfy the finiteness conditions FP_{n}(R).

The first example of a group which is finitely presented but not of type FP_{3}(ℤ) was given by Stallings in [St]. It may be described as the kernel of the homomorphism F_{2}×F_{2}×F_{2} → ℤ that sends all basis elements to 1 ∈ ℤ. This was generalized in [Bi 3] to produce examples of groups which were of type FP_{n}(ℤ) but not of type FP_{n+1}(ℤ) for all n ≥ 1. Other examples of groups of type F_{n} but not of type FP_{n+1}(ℤ) for all n ≥ 1 were discovered later by Stuhler [Stu] and by Abels and Brown [AB].

In this paper, we give a general construction which provides examples of groups having one type of finiteness property but not another. See Section 6 for specific examples. In particular, we produce groups of type FP(ℤ) which are not finitely presented.

Our construction begins with the class of “right angled Artin groups”. These are also known as graph groups. We associate a right angled Artin group G_{L} to each finite flag complex L (a flag complex is a simplicial complex that is determined by its 1-skeleton). For example, if L is the n-sphere triangulated as the (n + 1)-fold join of 0-spheres, then G_{L} is the (n + 1)-fold direct product of free groups F_{2} considered in the Stallings-Bieri examples above.

Next we examine the finiteness properties of the kernel H_{L} of the homomorphism G_{L} → ℤ that “sends all basis elements to 1”. Now, each Artin group G_{L} acts in a nice fashion on a CAT(0) space X_{L} (analogous to the action of a free group on a tree). Our method consists of constructing a map f : X_{L} → ℝ which is equivariant with respect to the homomorphism G_{L} → ℤ and viewing it as a Morse function on X_{L}. This allows us to obtain information about the homotopy type of the point preimages or “level sets” of f. The key point is that H_{L} acts cocompactly on each level set, so this translates into information about finiteness properties of H_{L}.

This Morse theory point of view can be found in [Bi 1] and in [BNS]. The Bieri-Neumann-Strebel “geometric invariants” of [BNS] are related to homotopy types of halfspaces f^{−1}[t, ∞) (or more precisely to the homotopy type of the pro-system f^{−1}[t, ∞) as t → ∞).

The main theorem relates finiteness properties of H_{L} to the topology of L.

Main Theorem. Let L be a finite flag complex. Let G = G_{L} be the associated right angled Artin group, and H = H_{L} the subgroup of G_{L} described above. We compute homology with coefficients in a ring R with 0 ≠ 1.

(1) H ∈ FP_{n+1}(R) if and only if L is homologically n-connected.

(2) H ∈ FP(R) if and only if L is acyclic.

(3) H is finitely presented if and only if L is simply connected.

The subgroups H_{L} are interesting test cases for other questions in group theory. The equivalence of finite generation of H_{L} and connectivity of L was already proven in [MV]. H. Meinert [Me] has related connectivity properties of L to finite generation of H_{L} in the more general setting of graph products of groups. Recently, J. Meier, H. Meinert and L. vanWyk [MMV] have completely determined the Bieri-Neumann-Strebel Σ-invariants for the right angled Artin groups.

The outline of this paper is as follows. In Sections 2, 3 and 4 we give the definitions and properties of Morse functions in the category of affine cell complexes, and we discuss finiteness properties of groups. In section 5 we describe the Morse theory view of the homomorphism from the right angled Artin groups above to ℤ. Theorem 4.1 can now be interpreted as proving the finiteness properties of H_{L} in the Main Theorem from the topological properties of L.

We define the notion of sheets in Section 6. These are used in Sections 7 and 8 when we prove the reverse implications of the Main Theorem. Examples and applications of the Main Theorem are provided in Section 6.

It is worth noting that all implications in the Main Theorem follow directly from the homotopy description of the sublevel sets furnished in Theorem 8.6. However, we provide separate proofs of certain implications in Theorem 4.1, and Theorem 7.1. We feel that these demonstrate the power of the Morse theory and homology arguments alone. Another application of these ideas can be found in [B].

We thank Aldo Bernasconi and Steve Gersten for pointing out that the groups G_{L} we consider are indeed Artin groups. We also thank Robert Bieri and the referee for their comments.

§2. Morse theory on affine cell complexes

In this Section we develop Morse theory for affine cell complexes. This has some very interesting applications in geometric group theory. For example, it was the motivation for our construction in Section 5 below. In [B] it is used to construct an example of a hyperbolic group containing a finitely presented subgroup which is not hyperbolic.

Definition 2.1. Let X be a CW complex. We say that X is an affine cell-complex if it is equipped with the following structure. An integer m is given, and for each cell e of X we are given a convex polyhedral cell C_{e} ⊂ ℝ^{m} and a characteristic function χ_{e} : C_{e} → e such that the restriction of χ_{e} to any face of C_{e} is a characteristic function of another cell, possibly precomposed by a partial affine homeomorphism (=restriction of an affine homeomorphism) of ℝ^{m}. An admissible characteristic function for e is any function obtained from χ_{e} by precomposing with a partial affine homeomorphism of ℝ^{m}.

Thus the restriction of an admissible characteristic function to a face is an admissible characteristic function (of another cell).

Definition 2.2. A map f : X → ℝ defined on an affine cell-complex X is a Morse function if

• for every cell e of X fχ_{e} : C_{e} → ℝ extends to an affine map ℝ^{m} → ℝ, and fχ_{e} is constant only when dim e = 0, and

• the image of the 0-skeleton is discrete in ℝ.

The main concern of this paper is to determine the topology of the f preimages of closed intervals in ℝ. We use the following notation and terminology. For a closed subset J ⊂ ℝ denote by X_{J} the set f^{−1}(J), and also let X_{t} = X_{{t}}. We call the sets X_{t} level sets of the Morse function f. We refer to the set X_{(−∞,t]} as the sublevel set corresponding to X_{t}.

Lemma 2.3. If J ⊂ J′ ⊂ ℝ are non-empty, connected and X_{J′} \ X_{J} contains no vertices of X, then X_{J} ↪ X_{J′} is a homotopy equivalence.

Proof. For each cell e of X and each admissible characteristic function χ_{e} : C_{e} → X we construct a strong deformation retraction H_{t}^{χ_{e}} of C_{e} ∩ (fχ_{e})^{−1}(J′) to C_{e} ∩ (fχ_{e})^{−1}(J) so that

• if χ_{e} is precomposed by a partial affine homeomorphism h, then H_{t}^{χ_{e}} is conjugated by h, i.e. H_{t}^{χ_{e}h} = h^{−1}H_{t}^{χ_{e}}h and

• the restriction of H_{t}^{χ_{e}} to a face of C_{e} is the strong deformation retraction associated to that face.

The construction is by induction on dim e. The inductive step amounts to the following observation. If C is a convex cell in some Euclidean space and F and G are two disjoint faces with F top dimensional and G either top dimensional or just a vertex, then any strong deformation retraction from \overline{∂C \ F} to G extends to a strong deformation retraction from C to G.

The deformation retractions constructed above induce a strong deformation retraction from X_{J′} to X_{J}. □

Ascending and descending links.

Any affine cell complex can be equipped with a natural PL structure. This is done inductively over skeleta while at the same time proving that all admissible characteristic functions are PL. For the inductive step, first observe that every admissible attaching map ∂C → X^{(i)} is PL, since the restriction to each face of C is PL by the inductive hypothesis. Thus X^{(i+1)} can be given the PL structure as the adjunction space of X^{(i)} and the attaching maps defined on the boundary of convex (i+1)-cells in Euclidean space.

When α : A → B is a PL map between polyhedra and a ∈ A is an isolated point of α^{−1}α(a), then α induces a map Lk(a, A) → Lk(α(a), B) between links of a and α(a). We shall denote this map by α_{∗} (the point a is understood from the context).

In particular, when χ_{e} : C_{e} → X is an admissible characteristic function of a cell e of an affine cell complex X and w is a vertex of C_{e}, we have a well-defined map χ_{e∗} : Lk(w, C_{e}) → Lk(χ_{e}(w), X). We may then identify the link Lk(v, X) of a vertex v of X with

Lk(v, X) = ⋃{ χ_{e∗}(Lk(w, C_{e})) : χ_{e}(w) = v }.

Definition 2.4. The ascending link (the ↑-link) is

Lk_{↑}(v, X) = ⋃{ χ_{e∗}(Lk(w, C_{e})) : χ_{e}(w) = v and fχ_{e} has a minimum at w } ⊂ Lk(v, X)

and the descending link (the ↓-link) is

Lk_{↓}(v, X) = ⋃{ χ_{e∗}(Lk(w, C_{e})) : χ_{e}(w) = v and fχ_{e} has a maximum at w } ⊂ Lk(v, X).

Lemma 2.5. Let f : X → ℝ be a Morse function on an affine cell complex as above. Suppose J ⊂ J′ ⊂ ℝ are closed intervals, inf J = inf J′, and J′ \ J contains only one point r of f(0-cells). Then X_{J′} is homotopy equivalent to X_{J} with the copies of Lk_{↓}(v, X) (v a vertex with f(v) = r) coned off.

A similar statement holds when inf J = inf J′ is replaced by sup J = sup J′ and Lk_{↓}(v, X) by L_{↑}(v).

Proof. The argument is similar to that of Lemma 2.3. Since X_{J′∩(−∞,r]} ↪ X_{J′} is a homotopy equivalence, we may assume by Lemma 2.3 that sup J′ = r. Let r − ϵ = sup J. If a cell e of X has the property that min f|e > r then e is disjoint from X_{J′}. For any admissible characteristic function χ_{e} : C_{e} → X of any other cell e we construct, inductively on dim e, a strong deformation retraction of (fχ_{e})^{−1}(−∞, r] onto the subset

(fχ_{e})^{−1}(−∞, r − ϵ] ∪ ⋃{F : F is a face of C_{e} with fχ_{e}(F) ⊆ (−∞, r]}

satisfying both naturality properties from the proof of Lemma 2.3.

These strong deformation retractions induce a strong deformation retraction of X_{J′} onto X_{J} with the cones attached as stated in the Lemma. □

Unless otherwise indicated, all homology groups are taken with coefficients in a fixed commutative ring R with 0 ≠ 1.

Corollary 2.6. Let f : X → ℝ be a Morse function on an affine cell complex as above. Suppose that J ⊂ J′ ⊂ ℝ are connected.

(1) If each ↑-link and each ↓-link is homologically n-connected, then inclusion X_{J} ↪ X_{J′} induces isomorphism in H̃_{i} for i ≤ n and epimorphism in H̃_{n+1}.

(2) If each ↑-link and each ↓-link is simply connected, then inclusion X_{J} ↪ X_{J′} induces isomorphism in π_{1}.

(3) If each ↑-link and each ↓-link is connected, then the inclusion X_{J} ↪ X_{J′} induces an epimorphism in π_{1}.

Proof. Since f(X^{(0)}) is discrete, the proof follows by induction from Lemma 2.3, Lemma 2.5, the Mayer-Vietoris and the Seifert-Van Kampen Theorems. □

§3. Finiteness Properties of Groups

In this section H is a discrete group, and R is a commutative ring with 1 ≠ 0 which we consider as a trivial RH-module.

Definition 3.1. A group H is said to be of type FP_{n}(R) if there exists a resolution (exact sequence)

P_{n} → P_{n−1} → ⋯ → P_{0} → R

of the trivial RH-module R by finitely generated projective RH-modules P_{i}.

Definition 3.2. A group H is said to be of type FP(R) if there exists a finite resolution

0 → P_{n} → ⋯ → P_{0} → R

of R by finitely generated projectives over RH.

Definition 3.3. A group H is of type F_{n} if there exists a K(H, 1) with finite n-skeleton.

One often writes H ∈ FP_{n}(R) to mean H is of type FP_{n}(R).

An application of Schanuel’s Lemma from homological algebra gives us the following method of showing that a group H is of type FP_{n}(R) but not of type FP_{n+1}(R). A proof of this can be found in K. Brown’s book [Br 1] (pp. 192-193) for example.

Proposition 3.4. If there exists a resolution

0 → Z_{n} → P_{n} → ⋯ → P_{0} → R → 0

where the P_{i} are all finitely generated, projective over RH and Z_{n} is not finitely generated over RH. Then H is of type FP_{n}(R) but not of type FP_{n+1}(R). □

The most important way of producing projective resolutions (actually free resolutions) of R over RH comes from algebraic topology. The following definitions appeared in the introduction as natural generalizations of Wall’s finiteness conditions F_{n}.

Definition 3.5. A group H is said to be of type FH_{n}(R) if it acts freely, faithfully, properly discontinuously, cellularly, and cocompactly on a cell complex X such that H̃_{i}(X, R) = 0 for all i ≤ n − 1.

Definition 3.6. A group H is said to be of type FH(R) if it acts freely, faithfully, properly discontinuously, cellularly, and cocompactly on an R-acyclic cell complex X.

The following lemma relates the topological definitions FH to the classical finiteness conditions FP.

Lemma 3.7. The following implications hold for a group H, and ring R with 1 ≠ 0. (1) If H ∈ FH_{n}(R) then H ∈ FP_{n}(R).

(2) If H ∈ FH(R) then H ∈ FP(R).

(3) If H acts freely, faithfully, properly discontinuously, cellularly, and cocompactly on an (n − 1)-connected cell complex X, then H is of type F_{n}.

Proof. From the definitions 3.5 and 3.6 we see that in parts 1 and 2 the group H acts nicely on a cell complex X which satisfies certain homological properties. All one has to do is write out the reduced chain complex of X with coefficients in R. Note that the vanishing of reduced homology groups implies that this is a resolution of R. The RH-module structure comes from the action of H on X, so compactness of the quotient X/H implies finite generation of the corresponding chain groups as RH-modules.

For part 3 note that X/H can be made into a K(H, 1) by adding cells of dimension n + 1 and higher. □

In a similar fashion, Proposition 3.4 yields the following useful topological criterion which tells when a group is of type FP_{n}(R) but not of type FP_{n+1}(R).

Lemma 3.8. Suppose that a group H acts freely, faithfully, properly, cellularly, and cocompactly on a complex X which satisfies

• H̃_{i}(X, R) = 0 for 0 ≤ i ≤ n − 1

• H̃_{n}(X, R) is not finitely generated as an RH-module then H is of type FH_{n}(R) but not of type FP_{n+1}(R). □

There are other finiteness properties in the literature. Finiteness properties of groups defined in terms of free resolutions were introduced by Serre [Se]. One says that a group is of type FL_{n}(R) (respectively FL(R)) if it satisfies the conditions of Definition 3.1 (respectively Definition 3.2) with all occurences of the word “projective” replaced by the word “free”. A group is said to be of finite type, denoted F, if it has a finite Eilenberg-Mac Lane space.

The following set of implications shows the relationship between our properties FH and the classical finiteness conditions.

F_{n} → FH_{n} → FL_{n} → FP_{n}
F → FH → FL → FP

Example 6.3 (3) below shows that the implications F → FP and F_{n} → FH_{n} (n ≥ 2) cannot be reversed. Clearly, FH_{1} → F_{1}. In [Se] Serre asked if FP → FL holds. It is still not known if this holds or if the implication FL → FH holds. The conditions FP_{n} and FL_{n} are known to be equivalent. One simply takes direct sums of the projectives in the resolution with complements to obtain a resolution which is free up to dimension n. These ideas were first observed by Wall [Wa 1]. The condition FL_{n} is equivalent to FH_{n} for n = 1, 2. A group of type FL_{1} is, in particular, finitely generated, and is seen to be of type FH_{1} by considering its Cayley complex with respect to a given presentation (with finite generating set). If the group is of type FL_{2} then the first homology of the quotient of this Cayley complex is finitely generated. Thus we see that the group is of type FH_{2} by equivariantly adding 2-cells to the Cayley complex, and killing the first homology of the complex. Finally, it is not known if the implication FL_{n} → FH_{n} holds for n ≥ 3.

The next result says that if a finitely presented group H acts freely, faithfully, properly, cocompactly and cellularly on a cell complex Y then π_{1}(Y) is finitely generated modulo the H action. It is used in Section 8 to conclude that certain groups are not finitely presentable.

Proposition 3.9. Let H be a finitely presented group. Suppose that H acts freely, faithfully, properly discontinuously, cocompactly, and cellularly on a connected cell complex Y. Then it is possible to attach to Y finitely many H-orbits of 2-cells so that the resulting complex is simply connected.

Proof. Since H is finitely presented, H acts freely, faithfully, properly discontinuously, cocompactly, and cellularly on a simply connected cell complex Z. Construct H-equivariant cellular maps α : Y^{(2)} → Z and β : Z^{(1)} → Y^{(1)} ⊂ Y. By attaching finitely many orbits of 2-cells to Y (one for every orbit of 2-cells in Z) we may assume that β extends to an equivariant cellular map β̃ : Z^{(2)} → Y^{(2)}. The map Y^{(1)}×{0, 1} → Y defined by (y, 0) ↦ y, (y, 1) ↦ βα(y) extends to an equivariant cellular map F : Y^{(1)} × {0, 1} ∪ Y^{(0)} × [0, 1] → Y by connectivity of Y. After attaching finitely many orbits of 2-cells to Y (one for every orbit of 1-cells in Y) we may assume that F extends to an equivariant cellular map F̃ : Y^{(1)} × [0, 1] → Y.

We now claim that at this stage Y is simply connected. Indeed, let ℓ be a loop in Y^{(1)}. Then F̃ provides a homotopy between ℓ and βα(ℓ). The loop α(ℓ) is nullhomotopic in Z^{(2)}, and thus so is β̃α(ℓ) = βα(ℓ). □

§4. Kernels of Homomorphisms to ℤ.

In this section we assume that a group G acts freely, faithfully, properly, cocompactly, and cellularly on a contractible affine cell complex X and that the action preserves the affine structure. More precisely, if e is a cell of X with an admissible characteristic function χ_{e} and g ∈ G, then gχ_{e} is an admissible characteristic function for the cell g(e). We also assume that ϕ : G → ℤ is an epimorphism, and that f : X → ℝ is a ϕ-equivariant Morse function (where ℤ acts on ℝ by translations in the usual fashion). Let H denote the kernel of ϕ and observe that H acts freely and cocompactly on level sets X_{t}. We now investigate the relationship between finiteness properties of H and the homotopy types of ascending and descending links.

Theorem 4.1. Let f : X → ℝ be a ϕ-equivariant Morse function and let H = Ker(ϕ) be as above.

(1) Suppose that each ↑-link and each ↓-link is homologically n-connected. Then H ∈ FH_{n+1}(R).

(2) If all ↑-links and ↓-links are R-acyclic, then H ∈ FH(R).

(3) If all ↑-links and ↓-links are simply connected, then H is finitely presented.

Proof. (1) By Corollary 2.6, for t < s the inclusion X_{(−∞,t]} ↪ X_{(−∞,s]} induces isomorphisms in H̃_{i} for i ≤ n and an epimorphism in H̃_{n+1}. Since X = ∪_{r∈ℤ}X_{(−∞,r]} is acyclic, it follows that H̃_{i}(X_{(−∞,t]}) = 0 for i ≤ n and all t. Similarly, H̃_{i}(X_{[t,∞)}) = 0 for i ≤ n and all t. Since X = X_{(−∞,t]} ∪ X_{[t,∞)} is acyclic, it follows from Mayer-Vietoris that the intersection X_{t} is homologically n-connected. Thus H acts freely and cocompactly on a homologically n-connected complex and so is of type FH_{n+1}(R).

(2) It follows from the proof of (1) that X_{t} is acyclic.

(3) By (1) each X_{t} is connected. By Corollary 2.6 inclusion X_{t} ↪ X induces an isomorphism in fundamental groups. Hence X_{t} is simply connected and H is finitely presented. □

§5. Right-angled Artin groups

We define right angled Artin groups and their associated piecewise euclidean cubical complexes. The construction of the Eilenberg–Mac Lane spaces for the right angled Artin groups is not new. See for example [MV] and [CD].

Definition 5.1. A simplicial complex L is said to be a flag complex if every finite collection of vertices of L which are pairwise adjacent spans a simplex in L. We see that such complexes L are completely determined by their 1-skeleton.

It is worth noting that this definition does not impose any restrictions on the topology of L, since the first barycentric subdivision of a simplicial complex is a flag complex. For a proof see [Br 2], page 28.

Definition 5.2. The right angled Artin group G_{L} associated to the finite flag complex L has finite generating set { g_{1}, …, g_{N} } in one to one correspondence with the vertex set L^{(0)} = { v_{1}, …, v_{N} } and has finite presentation

G_{L} = ⟨ g_{1}, …, g_{N} : [g_{i}, g_{j}] = 1 for all edges {v_{i}, v_{j}} in L^{(1)} ⟩

Note that this is a subpresentation of the standard presentation of ℤ^{N}, so that there is a natural epimorphism G_{L} → ℤ^{N} taking the generators g_{i} to the standard basis elements of ℤ^{N}. Composing this with the epimorphism

ℤ^{N} → ℤ : (x_{1}, …, x_{N}) ↦ Σ_{i}x_{i}

gives an epimorphism ϕ : G_{L} → ℤ which sends all the generators g_{i} to 1 ∈ ℤ.

Thus there is a short exact sequence of groups

1 → H_{L} → G_{L} → ℤ → 1

and we are interested in finiteness properties of the kernel H_{L}.

We obtain information about H_{L} by constructing an Eilenberg-Mac Lane space Q_{L} for G_{L} which is a compact, piecewise euclidean cubical complex of nonpositive curvature, and a continuous map Q_{L} → S^{1} which induces the homomorphism G_{L} → ℤ. The universal cover X of Q_{L} is a CAT(0) metric space on which G_{L} acts cocompactly by deck transformations. The map Q_{L} → S^{1} lifts to a ϕ-equivariant Morse function

f : X → ℝ

with the ↑-links and ↓-links all isomorphic to L, so the results of Section 4 apply to give one set of implications in the Main Theorem.

We begin with some definitions. Denote by ⊔ ⊓^{n} the regular n-cube in ℝ^{n} with vertex at the origin, and edges defined by the unit basis vectors.

Definition 5.3. A piecewise euclidean cubical complex is a cell complex constructed from a finite, disjoint collection of regular cubes, by glueing their faces via isometries. We shall refer to these as PE cubical complexes.

Let us look at the local geometry of a PE cubical complex.

Definition 5.4. Let v be a vertex of a regular cube ⊔ ⊓^{n} in ℝ^{n}. The euclidean metric on ℝ^{n} gives an inner product on the tangent space T_{v}(ℝ^{n}). We define the link of v in ⊔ ⊓^{n} (denoted by Lk(v, ⊔ ⊓^{n}) ) to be the set of unit tangent vectors at v which point into ⊔ ⊓^{n}. It is a subset of the unit sphere S^{n−1} which is homeomorphic to an (n − 1)-simplex. It has a natural spherical metric in which all dihedral angles are right angles. Such spherical simplices are termed all right simplices.

Thus the link of a vertex in a PE cubical complex is a piecewise spherical cell complex, all of whose cells are all right simplices. The next definition gives a combinatorial version of nonpositive curvature for PE cubical complexes.

Definition 5.5. A PE cubical complex is said to be nonpositively curved if the link of each vertex is a flag complex.

The following lemma of Gromov [Gr] (pg. 122) (see also [Da] Lemma 9.3) tells when a piecewise spherical cell complex, whose cells are all right simplices, satisfies the CAT(1) inequality. (Gromov’s statement uses the “no triangle condition” instead of the “flag complex” condition below).

Lemma 5.6 (Gromov). Let L be an all right, piecewise spherical complex. Then L is a CAT(1) space if and only if it is a flag complex. □

A PE cubical complex whose links are all CAT(1) spaces satisfies a local CAT(0) inequality (see [Gr] pg. 120 or [GH] pg. 197), and so its universal cover is a CAT(0) metric space (and hence is contractible) — see [Gr] pg. 119 or [GH] pg. 193. There is a comprehensive survey of these results in section I of [Da].

The following definition will be useful in describing the link structure in our PE cubical complexes.

Definition 5.7. Let L be a simplicial complex, and let

{ S_{α}^{0} }_{α∈L^{(0)}}

be a collection of 0-spheres indexed by the set of vertices in L. The spherical complex associated to L is denoted by S(L) and is defined to be the union

S(L) = ⋃{ S_{α_{0}}^{0} ∗ ⋯ ∗ S_{α_{m}}^{0} : < α_{0}, …, α_{m} > an m-simplex of L }

Lemma 5.8. Let L be a flag complex. Then the associated spherical complex S(K) is also a flag complex.

Proof. Let π : S(L)^{(0)} → L^{(0)} be the 2-to-1 map which takes the two vertices of S_{α}^{0} to α for all α ∈ L^{(0)}.

It is clear from the definition of S(L) that the pair of vertices in each 0-sphere S_{α}^{0} are not adjacent. Therefore, given a collection {v_{0}, …, v_{k}} of pairwise adjacent vertices in S(L) the set {π(v_{0}), …, π(v_{k})} is a collection of distinct, pairwise adjacent vertices in L.

These vertices span a simplex in L (since L is a flag complex), and the original vertices span a simplex in the corresponding k-sphere in S(L). □

The following result about flag complexes will be used in local computations in Section 8. First we define the subcomplexes St(σ, L) and St′(σ, L) associated to a simplex σ in a simplicial complex L.

Definition 5.9. Let L be a simplicial complex, and σ a simplex of L. We denote by St(σ, L) the subcomplex of L consisting of all simplices which contain σ. This is called the closed star of σ in the literature.

We denote by St′(σ, L) the subcomplex of L consisting of all simplices which contain a face of σ.

Proposition 5.10. Let L be a flag complex equipped with the all right CAT(1) metric and let σ be a simplex of L.

(1) St′(σ, L) is contractible.

(2) Let τ be another simplex of L and assume that d(a, b) < π/2 for a point a ∈ τ and a point b of σ. Then σ ∩ τ ≠ ∅.

Proof. We see that St′(σ, L) is contractible as follows. For each simplex ρ in L, the closed star St(ρ, L) is contractible as it is a union of simplices, any subcollection of which intersects in a common face containing ρ.

Given a collection {v_{i}, …, v_{k}} of vertices of σ let ⟨v_{1}, …, v_{k}⟩ denote the face of σ which they span. Since L is a flag complex, we see that

St(v_{1}, L) ∩ ⋯ ∩ St(v_{k}, L) = St( ⟨v_{1}, …, v_{k}⟩ , L)

is contractible. Thus the union

St′(σ, L) = ⋃{ St(v, L) : v a vertex of σ }

is also contractible.

For (2) note that the simplicial map from St′(σ, L) to a 1-simplex (of length π/2), which maps σ to one vertex and the frontier F r(St′(σ, L)) to the other vertex, is distance nonincreasing in the spherical metric. We see this by noting that the map restricts to a distance nonincreasing simplicial map on each simplex of St′(σ, L).

Now suppose that a ∈ τ and b ∈ σ are such that d(a, b) < π/2, and that σ ∩ τ = ∅. Then a geodesic γ from a to b intersects F r(St′(σ, L)), and so γ ∩ St′(σ, L) maps onto the 1-simplex. But this implies that d(a, b) ≥ π/2, a contradiction. □

The following lemma also concerns the spherical geometry of all-right piecewise spherical simplicial complexes. It is due to Gromov (see [Gr] pg. 122 or [Da] Sublemma 3.13), and is the spherical geometry component of the proof of Lemma 5.6 above (Gromov’s Lemma). We shall need this result in the proof of Lemma 8.3.

Lemma 5.11. Let v be a vertex in an all-right, piecewise spherical simplicial complex, and let B be the ball (closed star) of radius π/2 about v. Let x, y ∈ ∂B (the sphere of radius π/2 about v), and let γ be a geodesic from x to y which intersects the interior of B. Then the length of γ is at least π. □

The cubical complex Q_{L}.

We define and describe properties of the PE cubical complex Q_{L} associated to a finite flag complex L. Specifically, we have.

Theorem 5.12. Let L be a finite flag complex, G_{L} the associated right angled Artin group, and ϕ : G_{L} → ℤ the epimorphism defined above. Then there exists a nonpositively curved PE cubical complex Q_{L} and a map l : Q_{L} → S^{1} satisfying the following conditions

(1) l induces the homomorphism ϕ : G_{L} → ℤ.

(2) Q_{L} has one vertex and the link of this vertex is isomorphic to S(L).

(3) The lift of l to the universal covers is a ϕ-equivariant Morse function f : X → ℝ.

(4) All ↑- and ↓-links of X (with respect to the Morse function f) are isomorphic to L.

Note that since the PE cubical complex Q_{L} is nonpositively curved, it is a K(G_{L}, 1).

Proof. First construct a geometric realization of L in ℝ^{N} (N equals the number of vertices of L) in the usual way; mapping each vertex v_{i} of L to the (endpoint of the) basis vector e_{i} and mapping an m-simplex

σ = { v_{i_{0}}, …, v_{i_{m}} }

to the convex hull in ℝ^{N} of the images of the v_{i_{j}}. For each such σ define ⊔ ⊓_{σ} to be the regular (m + 1)-cube based at the origin in ℝ^{N} with edges defined by the basis vectors { e_{i_{j}} : v_{i_{j}} ∈ σ }.

Now define Q_{L} to be the image of the union of cubes

⋃{ ⊔ ⊓_{σ} : σ a simplex of L }

under the projection ℝ^{N} → T^{N} = ℝ^{N}/ℤ^{N}. Here ℤ^{N} acts on ℝ^{N} by translations in the usual fashion, so T^{N} is just the standard N-torus.

Clearly, Q_{L} is a PE cubical complex. Since the vertices of each ⊔ ⊓_{σ} have integer coordinates, the complex Q_{L} has just one vertex. Each m-simplex σ in L corresponds to an (m + 1)-cube ⊔ ⊓_{σ} in ℝ^{N} which descends to a torus in the quotient Q_{L}. This torus contributes an m-sphere (triangulated as the (m + 1)-fold join of 0-spheres) to the link of the single vertex in Q_{L}. Thus the link of this vertex in Q_{L} is precisely the complex S(L). This proves property (2).

Since L is a flag complex, Lemma 5.6 ensures that the link S(L) in Q_{L} is also flag, and so Q_{L} is nonpositively curved.

The linear map

l : ℝ^{N} → ℝ : (x_{1}, …, x_{N}) ↦ x_{1} + ⋯ + x_{N}

descends to a continuous map (fibration) ℝ^{N}/ℤ^{N} → S^{1}. Restricting to the subcomplex Q_{L} of the N-torus, gives a continuous map Q_{L} → S^{1} which we also denote by l.

The fundamental group of Q_{L} is easily computed from the 2-skeleton and is the Artin group G_{L}. The map l sends each basis vector in ℝ^{N} to the unit interval in ℝ, and so l_{∗} maps generators of π_{1}(Q_{L}) to 1 ∈ ℤ. Thus l_{∗} is the homomorphism ϕ : G_{L} → ℤ, and (1) holds.

Lifting l to the universal covers gives a continuous map

f : X → ℝ

which is clearly ϕ-equivariant. In order to establish (3), it remains to prove that f is a Morse function on X.

From the definition of f we see that for each m-cell e of X

fχ_{e} = τl

where the attaching map χ_{e} may need to be precomposed with an isometry ⊔ ⊓^{m} → ⊔ ⊓^{m}, and where l : ⊔ ⊓^{m} → ℝ is the restriction of the linear map

ℝ^{m} → ℝ : (x_{1}, …, x_{N}) ↦ x_{1} + ⋯ + x_{N}

and τ : ℝ → ℝ is an integer translation. Note that fχ_{e} is constant only when e is a vertex of X. Also, the f image of the 0-skeleton X^{(0)} is just ℤ ⊂ ℝ. Thus f is a Morse function on X.

Finally, note that the local picture of X at a vertex can be embedded in ℝ^{N} by mapping the vertex to the origin. The local picture of f at this vertex is then given by the linear map l : ℝ^{N} → ℝ defined above. Thus one sees that the ↑- and ↓-links are isomorphic to L. □

§6. Sheets

In this section we define and record properties of sheets. These are special flats in the universal cover X of Q_{L}, which we view as building blocks. This point of view enables us to determine the homology and homotopy type of (sub)level sets in Sections 7 and 8. We also state the Main Theorem and give some examples.

Recall that Q_{L} is the union of tori, one for every simplex of L.

Definition 6.1. A sheet in X is a flat which is a subcomplex of the preimage in X of one of the tori in Q_{L}.

Note that in general there are flats in X that are not sheets. Indeed, when L consists of 2 points, X is a tree, and there are countably many sheets, but uncountably many biinfinite geodesics.

Alternatively, we can describe sheets as follows. Any simplicial map L′ → L induces a map Q_{L′} → Q_{L}. When L′ is a single simplex of L, this map can be regarded as inclusion to a subcomplex (with underlying space a torus). A sheet is a component of the preimage in X of such a subcomplex as L′ ranges over all simplices in L.

Given a vertex v ∈ X and A ⊂ X a union of sheets, we define St(v, A) to be the cone on Lk(v, A), and consider it as a small neighborhood of v in A. Similarly, St_{↓}(v, A) is just the cone on Lk_{↓}(v, A). We also use St to denote the closed star of a simplex in a simplicial complex as in Proposition 5.10. It should be clear from the context what the notation refers to.

There is a natural retraction r_{v} : Lk(v, X) → Lk_{↓}(v, X). This map takes vertices in Lk_{↑}(v, X) to corresponding vertices in Lk_{↓}(v, X) by a central symmetry in v. It fixes the vertices of Lk_{↓}(v, X), and extends simplicially to give a map : Lk(v, X) → Lk_{↓}(v, X). The map r_{v} can be coned off to give a retraction of neighborhoods St(v, X) → St_{↓}(v, X) also denoted by r_{v}.

We record the following properties of sheets.

Proposition 6.2.

(1) X is covered by the sheets.

(2) The intersection of any collection of sheets is either empty, a vertex, or a sheet.

(3) All (sub)level sets of f restricted to a sheet are contractible. Moreover, all ↑- and ↓-links of the restriction are single simplices.

(4) The retraction r_{v} preserves sheets through v. More precisely, if A ⊆ X is a union of sheets and v ∈ A, then the restriction of r_{v} induces retractions Lk(v, A) → Lk_{↓}(v, A) and St(v, A) → St_{↓}(v, A) and these are also denoted r_{v}. Moreover, r_{v}^{−1}(Lk_{↓}(v, A)) = Lk(v, A).

Proof. Property (1) follows immediately from the definition of sheets and the fact that Q_{L} is a union of tori.

We see that property (2) holds for the intersection of any two sheets, and proceed by induction.

The restriction of f to a k-dimensional sheet is given by the linear map

l : ℝ^{k} → ℝ : (x_{1}, …, x_{k}) ↦ x_{1} + ⋯ + x_{k}

where ℝ^{k} has the usual cubing. So property (3) clearly holds.

Property (4) is immediate from the definitions of sheet and of the map r_{v}. □

Here is the Main Theorem and some examples.

Main Theorem. Let L be a finite flag complex. Let G = G_{L} be the associated right angled Artin group, and ϕ : G → ℤ the homomorphism with kernel H = H_{L} as in Theorem 5.12. We compute homology with coefficients in a ring R with 0 ≠ 1.

(1) H ∈ FP_{n+1}(R) if and only if L is homologically n-connected.

(2) H ∈ FP(R) if and only if L is acyclic.

(3) H is finitely presented if and only if L is simply connected.

Note that all 3 implications ⇐ follow from Theorem 5.12 and Theorem 4.1. In the next two sections we shall develop an understanding of (sub)level sets which is sufficient to prove the three implications ⇒.

Examples 6.3.

(1) For L = S^{n} and any R, H_{L} is of type FP_{n} but not of type FP_{n+1}. Triangulating S^{n} as the (n + 1)-fold join of 0-spheres, the group G_{L} is F_{2}^{n+1}, the (n + 1)-fold product of free groups of rank 2, and the homomorphism to ℤ sends each basis element of each coordinate F_{2} to 1 ∈ ℤ. For n = 1, 2 this is the Stallings’ example [St], and for all n this is due to Bieri [Bi 3].

(2) Let p be a prime and let L be the (appropriately triangulated) Moore space obtained from the n-sphere (n ≥ 1) by attaching an (n + 1)-cell via a degree p attaching map. Then L is acyclic over any field of characteristic ≠ p, but over fields of characteristic p it is only (n − 1)-connected. Thus H_{L} is of type FP over fields of characteristic ≠ p. Over a field of characteristic p H_{L} is of type FP_{n} but not F P_{n+1}. For example, for p = 3 and n = 1 H_{L} is an example of a group that is not finitely presented (and not even of type FP_{2}(ℤ/3)) but is of type FP(R) whenever 3 ∈ R is invertible.

Bieri and Strebel [BS] have constructed a group that is of type FP_{2}(F) for any field F, but is not of type FP(ℤ).

(3) Let L be an acyclic nonsimply-connected finite flag complex, say of dimension 2. Then H_{L} is FP (over all rings) but it is not finitely presented. This example answers in the negative a long standing question concerning finiteness properties of groups which first appeared in [Bi 2] Chapter I (2).

Furthermore, the cohomological dimension of H_{L} is 2 since the level sets are 2dimensional and acyclic. It is natural to ask if such groups H_{L} have 2-dimensional Eilenberg-Mac Lane spaces. Thus one obtains a family of potential counterexamples to the Eilenberg-Ganea conjecture.

If L is a flag triangulation of a spine of the Poincare homology sphere, then in Theorem 8.7 we show that H_{L} is either a counterexample to the Eilenberg-Ganea conjecture or that there is a counterexample to the Whitehead conjecture.

Recall that the Eilenberg-Ganea conjecture states that if a group H has cohomological dimension 2, then it has a 2-dimensional Eilenberg-Mac Lane space K(H, 1). The Whitehead conjecture states that every connected subcomplex of an aspherical 2-complex is aspherical.

§7. Homology of (sub)level sets

We now fix a finite flag complex L and the associated right angled Artin group G = G_{L}. Let X = Q̃_{L}, ϕ : G → ℤ, H = H_{L}, and f : X → ℝ be as in Theorem 5.12. The goal of this section is to compute homology groups of sets X_{J} for a closed connected J ⊆ ℝ.

Theorem 7.1. Let A be a (possibly infinite) union of sheets and vertices in X, and denote by A_{J} the intersection of A and the set X_{J}. Then

H_{∗}(A, A_{J}) ≅ ⨁_{v∉A_{J}} H_{∗}(St_{↓}(v, A), Lk_{↓}(v, A))

where the sum ranges over the vertices in A that are not contained in A_{J}.

Since H_{∗}(∅, ∅) = 0 we could be summing over all vertices in X not contained in X_{J}.

Proof. We first construct a homomorphism

Ψ_{A} : H_{∗}(A, A_{J}) → ⨁_{v∉A_{J}} H_{∗}(St_{↓}(v, A), Lk_{↓}(v, A))

and then show it is an isomorphism.

The coordinate of Ψ_{A} corresponding to v ∈ A \ A_{J} is defined as the composition

H_{∗}(A, A_{J}) → H_{∗}(A, A\Int St(v, A)) ≅ H_{∗}(St(v, A), Lk(v, A)) →^{r_{A∗}} H_{∗}(St_{↓}(v, A), Lk_{↓}(v, A)).

An element x of H_{∗}(A, A_{J}) can be represented by a relative cycle supported in a finite subcomplex of A, and then the coordinates of Ψ_{A}(x) corresponding to the vertices outside the support are 0. Therefore, Ψ_{A} is a well-defined homomorphism into the direct sum.

The homomorphism Ψ_{A} is natural, i.e. if A ⊂ A′ and A′_{J} = A′ ∩ X_{J}, then the diagram

H_{∗}(A, A_{J}) ⟶^{Ψ_{A}} ⨁ H_{∗}(St_{↓}(v, A), Lk_{↓}(v, A))
↓  ↓
H_{∗}(A′, A′_{J}) ⟶^{Ψ_{A′}} ⨁ H_{∗}(St_{↓}(v, A′), Lk_{↓}(v, A′))

commutes (the vertical arrows are induced by inclusion).

We now prove that Ψ_{A} is an isomorphism. Since homology commutes with direct limits, it suffices to prove this when A is a finite union of sheets and vertices of X. For such A we proceed by induction on the number of sheets and vertices in the union. When A is empty, a single sheet, or a vertex in A_{J} then both groups are zero. When A is a vertex in A \ A_{J}, then the only nonvanishing groups are in dimension 0, they are both isomorphic to the coefficient group, and Ψ_{A} is an isomorphism.

Now suppose that Ψ_{A} is an isomorphism whenever A is a union of ≤ k sheets and vertices, and let A be a union of k + 1 sheets and vertices. Write A = A′ ∪ S where S is a sheet or a vertex, and A′ is a union of k sheets and vertices. Note that A′ ∩ S is a union of ≤ k sheets and vertices (by Proposition 6.2(2)) and so Ψ_{A′∩S}, Ψ_{A′}, and Ψ_{S} are isomorphisms. That Ψ_{A} is an isomorphism now follows from the Mayer-Vietoris sequence, the naturality of Ψ and the 5-lemma. □

As a corollary, we have the following description of the homology of level and sublevel sets.

Corollary 7.2. There are isomorphisms of RH-modules (1) H̃_{∗}(X_{(−∞,t]}) ≅ ⨁_{v∈X_{(t,∞)}} H̃_{∗}(L).

(2) H̃_{∗}(X_{t}) ≅ ⨁_{v∉X_{t}} H̃_{∗}(L). □

We can now prove two more implications in the Main Theorem.

Proof of (1) and (2) of Main Theorem. Let n be the smallest integer such that H̃_{n}(L) ≠ 0. Then H̃_{n}(X_{t}) is not a finitely generated RH-module, so the claim follows from Lemma 3.8. □

§8. Homotopy type of (sub)level sets

In this section we see how to construct X using sheets; starting with sheets through a given vertex, and then adding sheets which pass through neighboring vertices. This view of X enables us to determine the homotopy type of the (sub)level sets in Theorem 8.6 below, and hence to prove part (3) of the Main Theorem.

We begin with some lemmas describing the intersections of certain collections of sheets and (sub)level sets.

Lemma 8.1. Let w be a vertex of X and let K be the union of a collection of sheets containing w. Let J be a closed interval in ℝ. Then

(1) K is contractible.

(2) All ↑- and ↓-links of K are contractible except possibly at w, and at w the two are naturally isomorphic.

(3) K_{J} = K ∩ X_{J} is homotopy equivalent to Lk_{↓}(w, K) when w ∉ X_{J} and it is contractible if w ∈ X_{J}.

Proof. K is a cone with conepoint w, hence contractible. We prove (2) for ↓-links, the other case being similar. Let x be any vertex of K different from w. There is a sheet S ⊆ K that contains w and x and is minimal with this property. Thus Lk_{↓}(x, K) is the union of the simplex σ = Lk_{↓}(x, S) and the simplices of the form Lk_{↓}(x, S′) where S′ ranges over sheets in K that pass through x and w. All these simplices contain σ, so the union is contractible.

There is a natural central symmetry defined on each sheet S through w. This maps Lk_{↑}(w, S) bijectively to Lk_{↓}(w, S), and these combine to give the isomorphism between Lk_{↑}(w, K) and Lk_{↓}(w, K).

We now prove (3). First suppose that w ∈ X_{J}. Then it follows from Lemma 2.5 that the inclusion K_{J} ↪ K is a homotopy equivalence, and thus K_{J} is contractible. If w ∉ X_{J}, assume for concreteness that w is above X_{J}, i.e. that f(w) > t for all t ∈ J. Choose ϵ ∈ (0, 1) such that f(w) − t > ϵ for all t ∈ J. Then again from Lemma 2.3 it follows that inclusions

K_{J} ↪ K_{(−∞,f(w)−ϵ]} ↩ K_{f(w)−ϵ}

are homotopy equivalences. Finally, to show that K_{f(w)−ϵ} is homotopy equivalent to Lk_{↓}(w, K) observe that the correspondence

Lk_{↓}(w, S) ↔ S ∩ X_{f(w)−ϵ}

is a 1-1 order preserving correspondence between elements of closed covers of Lk_{↓}(w, K) and K_{f(w)−ϵ} both of which are closed under intersections and have contractible elements. Hence Lk_{↓}(w, K) and K_{f(w)−ϵ} are homotopy equivalent (see [BoS] Section 8.2). □

The next lemma shows that the minimum distance from a vertex in X to a cube in X is attained at a vertex of the cube. It is used in Lemmas 8.4 and 8.5 below.

Lemma 8.2. Let X be a CAT(0) PE cubical complex, Q ⊂ X a cube, and c ∈ X a vertex. Then the minimal distance from c to Q is attained at a unique point of Q. Moreover, this point is a vertex of Q.

Proof. Suppose there are two points x and y of Q which realize this minimum distance. Then the geodesic [x, y] lies in Q, since cubes are convex in X. Applying the CAT(0) inequality to the geodesic triangle xyc we see that all interior points of [x, y] are closer to c than either of the endpoints. This contradicts the choice of x and y.

Let γ be a distance minimizing geodesic from c to Q, and suppose that γ does not end at a vertex of Q. Then γ meets Q in the interior of some face. Let e denote an edge of this face.

For each point p ∈ γ choose the minimal cube Q_{p} in X which contains p. Consider the collection of all such cubes Q_{p} for p ∈ γ.

We claim that there is a cellular map from the union of such cubes to a 1-simplex which takes γ to an interior point of the simplex. Hence, c could not be a vertex, and the Lemma is proven.

Let Q_{0} be the cube which contains the edge e and the initial segment of the geodesic γ, and is minimal among cubes with this property. Here we parameterize γ so that it starts at Q and ends at c. This cube contains the face of Q at which γ originates, and so has a metric product structure, defined as the product of a codimension one face times the edge e. Since γ minimizes the distance from c to Q we see that Q_{0} ∩ γ is perpendicular to the e-coordinate.

The metric product structure induces a codimension-1 metric product structure on the face of intersection of Q_{0} and the next cube Q_{1} of the collection along the geodesic γ. This extends uniquely to a codimension-1 product structure of Q_{1} and so on.

Finally, we map the e-coordinate onto a 1-simplex, collapsing the other coordinates to a point. The geodesic γ remains perpendicular to the e-coordinate throughout, and so maps to an interior point of the 1-simplex. □

Now we begin to build up X as a union of sheets. Choose a base vertex v in X and an ordering v = v_{1}, v_{2}, v_{3}, ⋯ of all vertices in X such that d(v, v_{i}) ≤ d(v, v_{j}) whenever i ≤ j. Denote by K_{i} the union of all sheets through v_{i}. We will study the homotopy type of K_{1} ∪ K_{2} ∪ ⋯ ∪ K_{n} by induction on n. To that end, we fix n > 1, let w = v_{n}, and let K = K_{n} ∩ (K_{1} ∪ K_{2} ∪ ⋯ K_{n−1}).

In Lemma 8.4 below, we give a description of K as the union of a collection of sheets containing v_{n}. This together with Lemma 8.1 ensures that K is contractible. This will be important in proving Theorem 8.6. First, we need to prove that sets like the K_{i} are convex in X.

Lemma 8.3. Let x ∈ X be a vertex, and let K_{x} denote the union of all sheets through x. Then K_{x} is a convex subset of X.

Proof. We break the proof into a number of steps. The first step gives a local formulation of the problem in terms of convexity of subcomplexes of all-right piecewise spherical simplicial complexes. In the second step we show that the purely combinatorial condition of “fullness” ensures the convexity of a subcomplex of an all-right piecewise spherical simplicial complex. Steps 3 and 4 reduce the problem to that of checking that

Lk_{↓}(y, K_{x}) ⊂ Lk_{↓}(y, X) = L

is a full subcomplex, for all vertices y ∈ K_{x}. Finally, in step 5 we calculate these ↓-links, and complete the proof.

Step 1: (Local formulation of convexity) We show that K_{x} is convex in X by checking that each geodesic in K_{x} is also a geodesic in X.

Note that a segment of a geodesic which lies inside a cube in K_{x} will also be locally a geodesic in X. So we only need check the points where the geodesic in K_{x} passes through vertices of X, or passes from one cube to another through a common face.

Suppose a geodesic in K_{x} passes through a vertex y. In order for it to be a local geodesic in X we need that the distance in Lk(y, X) between the two directions defined by this path should be at least π. This will hold if

Lk(y, K_{x}) ⊂ Lk(y, X)  is convex  (A)

where we say that a subcomplex M of an all-right piecewise spherical simplicial complex N is convex if it satisfies the following: For all points a, b ∈ M such that d_{N}(a, b) < π, the geodesic [a, b] is contained in M.

Similarly, if a geodesic in K_{x} intersects a cube Q in a single interior point, then it will be a local geodesic in X provided that the link of Q in K_{x} is a convex subcomplex of the link of Q in X. Let y be any vertex of Q, and let σ_{Q} be the simplex of Lk(y, X) determined by Q. There is a natural identification of the link of Q in K_{x} with Lk(σ_{Q}, Lk(y, K_{x})). Likewise the link of Q in X can be identified with Lk(σ_{Q}, Lk(y, X)), and so the convexity condition becomes

Lk(σ_{Q}, Lk(y, K_{x})) ⊂ Lk(σ_{Q}, Lk(y, X))  is convex  (B)

for all cubes Q in K_{x} and vertices y ∈ Q.

So we only have to verify conditions (A) and (B) above. In the following steps we reduce this to verifying a simple combinatorial condition at ↓-links.

Step 2: (Combinatorial condition for convexity in all-right piecewise spherical complexes) In this step we prove the following. Let M ⊂ N be a full subcomplex of an all-right piecewise spherical simplicial complex N. Then M is convex in N.

Recall that a subcomplex M ⊂ N is said to be full if it satisfies the following: If a set of vertices of M spans a simplex τ ⊂ N, then τ ⊂ M.

Suppose that M ⊂ N is full, and that a, b ∈ M satisfy d_{N}(a, b) < π. We have to show that the geodesic [a, b] is contained in M. Let σ_{a} (respectively σ_{b}) denote the minimal simplex of N which contains a (respectively b). Note that σ_{a} and σ_{b} are contained in M.

Now either a = b and the result is trivial, or a ≠ b and so the collection of all simplices σ which intersect [a, b] nontrivially in their interiors is not empty. We claim that the vertices of such σ are contained in the union of the set of vertices of σ_{a} and the set of vertices of σ_{b}. Hence, σ ⊂ M by fullness. But [a, b] is contained in the union of such σ, and so [a, b] ⊂ M as required.

Suppose σ is a simplex of N whose interior intersects [a, b] nontrivially, and let v ∈ σ be a vertex. The path [a, b] intersects the open star about v (=open ball in N of radius π/2 about v). Now if both d_{N}(v, a) and d_{N}(v, b) are at least π/2 Lemma 5.11 implies that the length of [a, b] is at least π contradicting the assumption d_{N}(a, b) < π. Thus one of d_{N}(a, v) and d_{N}(b, v) is strictly less than π/2. Suppose d_{N}(a, v) < π/2. Then either a = v or a lies in the interior of a simplex with vertex v. In either case, v is a vertex of σ_{a}, and we are done.

Step 3: (Reduction to vertices of K_{x}) Step 2 implies that, in order to check that properties (A) and (B) hold, we need only verify that

Lk(y, K_{x}) is a full subcomplex of Lk(y, X) (A′)

for all vertices y ∈ K_{x} and

Lk(σ_{Q}, Lk(y, K_{x})) is a full subcomplex of Lk(σ_{Q}, Lk(y, X)) (B′)

for all cubes Q ⊂ K_{x} and vertices y ∈ Q.

But property (B′) follows from property (A′) by the following observation. Let M be a full subcomplex of the simplicial complex N, and let σ ⊂ M be a simplex. Then Lk(σ, M) is a full subcomplex of Lk(σ, N).

Step 4: (Reduction to ↓-links) Property (A′) reduces to verifying the following statement.

Lk_{↓}(y, K_{x}) is a full subcomplex of Lk_{↓}(y, X) = L (A′′)

for each vertex y ∈ K_{x}.

This reduction follows from the following simple observation. Let M be a full subcomplex of the simplicial complex N. Then the associated spherical complex S(M) is a full subcomplex of the associated spherical complex S(N).

Step 5: (↓-link computations) Finally, we verify property (A′′). Let y ∈ K_{x} be a vertex. If y = x then Lk_{↓}(y, K_{x}) = Lk_{↓}(y, X) is clearly convex. So we assume that y ≠ x. Then the geodesic [x, y] defines a unique point of Lk(y, X). Taking the r_{y} image gives a point p ∈ L = Lk_{↓}(y, X). Since K_{x} is a geodesic cone on x we see that p ∈ Lk_{↓}(y, K_{x}), and that in fact Lk_{↓}(y, K_{x}) is just the union U of all simplices σ ⊂ L which contain p. Note that simplicial structure on U can be described by: τ is a simplex of U if and only if there exists a simplex σ containing both τ and p.

This is seen to be full in L as follows. Let σ_{p} denote the minimal simplex of L which contains p. Then σ_{p} is a face of each simplex containing p. In particular, if v ∈ U is a vertex, then either v ∈ σ_{p} or {v} ∪ σ_{p} spans a simplex. Hence, if {v_{1}, …, v_{j}} ⊂ U is a collection of vertices which span a simplex σ ⊂ L, then {v_{1}, …, v_{j}} ∪ σ_{p} spans a simplex of U. Thus σ ⊂ U, and so U is a full subcomplex of L. □

Lemma 8.4. Let K = K_{n} ∩ (K_{1} ∪ ⋯ ∪ K_{n−1}) be as above. Then K is the union of those sheets that contain w and at least one v_{j} with j < n.

Proof. This union is contained in K, so we need only prove the reverse inclusion. That is, given x ∈ K, there exists a sheet S ⊂ X through v_{n} which contains some v_{j} (j < n) and contains x.

First we prove this for vertices v ∈ K. Now v ∈ K implies that v ∈ K_{n} ∩ K_{i} for some i < n. Thus v_{i}, v_{n} ∈ K_{v} where K_{v} denotes the union of all sheets through v.

By Lemma 8.3 K_{v} is convex, and so contains the geodesic [v_{i}, v_{n}]. Let Q denote the minimal cube containing v_{n} and the end segment of the geodesic [v_{i}, v_{n}]. By minimality, Q ⊂ K_{v} and so Q lies in a sheet S through v. We claim that one of the vertices of Q is in the set {v_{1}, …, v_{n−1}}. Thus the sheet S contains v, v_{n} and some v_{j} (j < n) as required.

To see this claim, note that the ordering on the v_{i} implies that the geodesic [v_{1}, v_{i}] is not longer than [v_{1}, v_{n}], and so the CAT(0) inequality implies that the distance from v_{1} to any interior point of [v_{i}, v_{n}] is strictly smaller than d(v_{1}, v_{n}). In particular this is true for the points of Q which lie in the interior of [v_{i}, v_{n}]. Thus v_{n} is not the closest point of Q to v_{1}, and so Lemma 8.2 ensures that one of the other vertices of Q is v_{j} for some j < n.

Finally, suppose that x ∈ K is not a vertex. Then x ∈ S′ ⊂ K_{n} ∩ K_{i} for some sheet S′ and some i < n. We know that each vertex of S′ is contained in a sheet through v_{n} and some v_{j} (j < n). Since there are only finitely many sheets through v_{n}, then one such sheet S must contain a maximal, general position subset of vertices of S′, and hence contains all of S′ as required. □

Next we investigate the link of w = v_{n} in K.

Lemma 8.5. Lk_{↓}(w, K) ≅ Lk_{↑}(w, K) is contractible.

Proof. Denote by a the point in Lk(w, X) determined by the geodesic [w, v] and let b = r(a) ∈ Lk_{↓}(w, X) be the image of a under the retraction r_{w} (see section 6). Let σ denote the smallest simplex of Lk_{↓}(w, X) that contains b.

Let T denote the collection of simplices τ in L = Lk_{↓}(w, X) such that the sheet through w corresponding to τ contains one of the v_{i} (i ≤ n − 1). By Lemma 8.4 Lk_{↓}(w, K) and Lk_{↑}(w, K) are isomorphic to the simplicial complex determined by T . Claim 2 below implies that this complex is actually just St′(σ, L) which is contractible by Proposition 5.10 (1).

Claim 1. Every face of σ is in the collection T .

Indeed, let S_{0} be the sheet through w such that Lk_{↓}(w, S_{0}) = σ. We know from Proposition 6.2 (4) that Lk(w, S_{0}) contains a. Let Q be the smallest cube that contains w and such that Lk(w, Q) contains a. Thus Q ⊂ S_{0} and σ = r_{w}(Lk(w, Q)). We shall show that each vertex of Q which is distance one from w is in the collection {v_{1}, …, v_{n−1}}. Hence, σ and every face of σ is in T .

Let v′ be a vertex of Q which is distance one from w. Then the angle at w defined by [w, v] and [w, v′] is less than π/2. Thus the distance from v to a point on the edge [w, v′] which is close to w is less than d(v, w). Lemma 8.2 implies that

d(v, v′) = d(v , [w, v′] ) < d(v, w)

and so v′ ∈ {v_{1}, …, v_{n−1}}.

Claim 2. T = { τ a simplex of L : τ ∩ σ ≠ ∅ }

For the inclusion ⊂, let τ be an element of T , and let S be the corresponding sheet in K. Then S contains v_{i} for some i < n. In the triangle Δ(w, v, v_{i}) we have d(v, w) ≥ d(v, v_{i}) and thus the angle at w is < π/2. It follows that Lk(w, S) contains a point within distance < π/2 from the point a defined above, and hence τ = Lk_{↓}(w, S) = r_{w}(Lk(w, S)) contains a point within distance < π/2 from b = r_{w}(a). Thus it must intersect σ by Proposition 5.10 (2).

Now for the reverse inclusion. It is clear that if S_{1} ⊂ S_{2} are sheets and Lk_{↓}(w, S_{1}) is in T , then Lk_{↓}(w, S_{2}) is also in T . In particular, if a vertex of a simplex τ of L is in T , then τ is in T . If τ ∩ σ ≠ ∅ then τ contains a vertex of σ. But we’ve seen that all vertices of σ are in T in Claim 1. Hence τ ∈ T . □

Now we are in a position to determine the homotopy type of sublevel sets X_{J}.

Theorem 8.6. X_{J} is homotopy equivalent to the wedge of L’s, one for every vertex of X not in X_{J}.

Note: When L is connected, the choice of basepoints is irrelevant. When L is disconnected, different copies of L in the above wedge might have basepoints in different components.

Proof. We write X_{J} as the increasing union of sets of the form

X(n) = X_{J} ∩ (K_{1} ∪ K_{2} ∪ ⋯ ∪ K_{n}) .

We may express X(n) as the union X(n) = X(n−1)∪(X_{J} ∩K_{n}). Note that the intersection

X(n − 1) ∩ (X_{J} ∩ K_{n}) = X_{J} ∩ (K_{n} ∩ (K_{1} ∪ K_{2} ∪ ⋯ ∪ K_{n−1}))

is contractible. This follows from Lemma 8.1 (3) (setting K_{w} = K_{n}∩(K_{1}∪K_{2}∪⋯∪K_{n−1})) and Lemma 8.5.

Now Lemma 8.1 (3) implies that X_{J} ∩ K_{n} is either contractible or homotopy equivalent to L depending on whether or not v_{n} ∈ X_{J}. Thus, it follows inductively that X(n) is homotopy equivalent to the wedge of L’s, one for every v_{i}, i ≤ n not contained in X_{J}. Further, inclusions X(n − 1) ↪ X(n) respect the wedge structure, so the theorem follows. □

We can now complete the proof of the Main Theorem.

Proof of (3) of Main Theorem. If L is not connected then part (1) of the Main Theorem implies that H will not be finitely generated, and hence not finitely presented.

If L is connected but not simply-connected, then π_{1}(X_{t}) is the free product of π_{1}(L)’s, one for every vertex not in X_{t}. If H were finitely presented, then Proposition 3.9 would imply that π_{1}(X_{t}) is generated by the H-translates of finitely many loops. Since X is contractible, each of these finitely many loops is null homotopic in X, and hence is null homotopic in some block X_{[t−T,t+T]}. Since H acts by “horizontal” translations, all the Horbits of these loops will be null homotopic in X_{[t−T,t+T]}. This implies that the inclusion X_{t} ↪ X_{[t−T,t+T]} induces the trivial map on π_{1}. But Corollary 2.6 (3) says that this inclusion induces an epimorphism in π_{1}, and thus X_{[t−T,t+T]} is simply connected, contradicting Theorem 8.6 above. □

We end this section with an examination of the level sets in the case when L is a flag triangulation of a spine of the Poincare homology sphere. We have the following result which shows that at least one of the Eilenberg-Ganea and Whitehead conjectures must be false.

Theorem 8.7. Let L be a flag triangulation of a spine of the Poincare homology sphere. Then either H_{L} is a counterexample to the Eilenberg-Ganea conjecture or there is a counterexample to the Whitehead conjecture.

We begin by observing that there are ‘arbitrarily large’ copies of L which are quasiisometrically embedded in the level set X_{t}. Given a vertex v ∈ X we identify the complex L with the descending link Lk_{↓}(v, X). Thus each point x ∈ L determines a unique geodesic g_{x} ⊂ K_{v} through v.

Definition 8.8. Let t ∈ ℝ and v ∈ X be a vertex such that f(v) > t, and M ⊂ L be a subcomplex of L = Lk_{↓}(v, X). Define the shadow of M on X_{t} to be the set

S_{v,M} = ⋃{ g_{x} : x ∈ M } ∩ X_{t}

Note that S_{v,L} is homeomorphic to L, by the map which sends each simplex σ ⊂ L to S_{v,σ} ⊂ S_{v,L}. Give L a metric by requiring that each 2-simplex is an equilateral triangle with sidelengths equal to |f(v) − t| and then taking the induced path metric. The homeomorphism above becomes a quasi-isometry, with constants which are independent of |f(v) − t|. To see this note that this is obviously true if S_{v,L} inherited its metric from K_{v} (the quasi-isometry constants would be no larger than the cardinality of the 1-skeleton of L). But K_{v} is a convex subset of X by Lemma 8.3. Hence we obtain the same quasiisometry inequality for S_{v,L} as a subset of X.

Proof of Theorem 8.7. If H_{L} does not have geometric dimension 2 we are done. Otherwise, let Y be a contractible 2-complex on which H_{L} acts freely, faithfully, properly and cellularly. Since Y is contractible we can define an H_{L}-equivariant map ϕ : X_{t} → Y.

Now X_{t} and ϕ(X_{t}) are quasi-isometric to each other. Thus point preimages of ϕ will have diameters bounded by the quasi-isometry constants. We denote the restriction ϕ|_{S_{v,L}} by ϕ_{v}.

By the observation following Definition 8.8 above, we may choose a vertex v ∈ X so that |f(v) − t| is large in comparison with the quasi-isometry constants. For each point x ∈ S_{v,L} the geodesic [x, v] determines a unique minimal simplex σ ⊂ L = Lk_{↓}(v, X). The preimage ϕ_{v}^{−1}(ϕ_{v}(x)) will be contained in the contractible (by Proposition 5.10 (1)) subcomplex S_{v,St′(σ,L)}. For vertices v with |f(v) − t| large enough one can define a left homotopy inverse to ϕ_{v} by taking each vertex of ϕ_{v}(S_{v,L}) to a point of its ϕ_{v}-preimage, and extending over skeleta.

Thus the shadow S_{v,L} is a homotopy retract of ϕ_{v}(S_{v,L}), and so we have

π_{2}(ϕ_{v}(S_{v,L})) ⊃ π_{2}(S_{v,L}) = π_{2}(L) = π_{2}(L̃) = H_{2}(S^{3} − {120 points}) ≠ 0

The last equality above follows from the fact that L is a spine of the Poincare homology 3-sphere. In conclusion ϕ_{v}(S_{v.L}) ⊂ Y is a connected subcomplex of the contractible 2complex Y which is not aspherical, and so gives a counterexample to the Whitehead conjecture. □

References

[AB] H. Abels and K.S. Brown, Finiteness properties of solvable S-arithmetic groups: An example, J. Pure Appl. Algebra 44 (1987), 77–83.

[Bi 1] R. Bieri, The geometric invariants of a group: A survey with emphasis on the homotopical approach,

Geometric group theory, LMS Lecture Notes (Niblo and Roller, eds.), vol. 181, 1993, pp. 24–36. [Bi 2] R. Bieri, Homological Dimension of Discrete Groups, Queen Mary College Mathematics Notes,

1976. [Bi 3] R. Bieri, Normal subgroups in duality groups and in groups of cohomological dimension 2, Jour.

Pure App. Algebra 7 (1976), 35–52.

[BS] R. Bieri and R. Strebel, Valuations and finitely presented metabilian groups, Proc. London Math. Soc. 41 (1980), 439–464.

[BNS] R. Bieri, W. D. Neumann and R. Strebel, A geometric invariant of discrete groups, Invent. Math. 90 (1987), 451–477.

[BoS] A. Borel and J-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1974), 244–297.

[B] N. Brady, Branched coverings of cubical complexes and subgroups of hyperbolic groups, Preprint, University of Utah (1995).

[Br 1] K. Brown, Cohomology of Groups, GTM 87, Springer-Verlag, New York, 1982. [Br 2] K. Brown, Buildings, Springer-Verlag, New York, 1989.

[CD] R. Charney and M. Davis, Finite K(π, 1)’s for Artin groups, Preprint. Ohio State University.

[Da] M. Davis, Nonpositive curvature and reflection groups, Proceedings of the Eleventh Annual Workshop in Geometric Topology, Park City, Utah (Hosted by BYU), 1994.

[GH] E. Ghys and P. de la Harpe (eds.), Sur les Groupes Hyperboliques d’apres Mikhael Gromov, Prog. Math. vol. 83, Birkhauser, Boston, 1990.

[Gr] M. Gromov, Hyperbolic groups, Essays in Group Theory, M.S.R.I. Publ. 8, Springer, New York 1897 (S.M. Gersten, ed.).

[Me] H. Meinert, The Bieri-Neumann-Strebel invariant for graph products of groups, preprint.

[MV] J. Meier and L. VanWyk, The Bieri-Neumann-Strebel invariants for graph groups, Proc. London Math Soc. (3) 71 (1995), 263–280.

[MMV]J. Meier, H. Meinert, L. vanWyk, Higher Generation subgroup sets and the Σ-invariants of graph

groups, Ask John.

[Ne] B.H. Neumann, Some remarks on infinite groups, Jour. London Math. Soc. 12 (1937), 120–127.

[Se] J.P. Serre, Cohomologie des groupes discretes, Ann. of Math. Studies 70 (1971), 77–169.

[St] J.R. Stallings, A finitely presented group whose 3-dimensional integral homology is not finitely generated, Amer. Jour. Math. 85 (1963), 541–543.

[Stu] Stuhler, Homological properties of certain arithmetic groups in the function field case, Inven. Math. 57 (1980), 263–282.

[Wa] C.T.C. Wall, Finiteness conditions for CW-complexes, Ann. of Math. 81 (1965), 56–69.

Department of Mathematics, University of Utah, Salt Lake City, UT 84112

   E-mail address: bestvina@math.utah.edu and brady@math.utah.edu