ACTION-INDUCED NESTED CLASSES OF GROUPS AND JUMP (CO)HOMOLOGY

NANSEN PETROSYAN

Abstract. Using fixed-point-free group actions, we set up a scheme to define nested classes of groups indexed over ordinals. Restricting to cellular actions on CW-complexes, we find new classes of groups as well as new characterizations for some well-known classes. We extend some of the properties of the cohomological dimension of a group to groups with jump (co)homology and study their implications to cellular actions on finite dimensional CW-complexes that satisfy a quite general homological condition.

1. Introduction

When studying properties of a class of groups, constructing actions on topological spaces can give us useful and often essential information. Many types of groups are defined in this way. One such example is Kropholler’s class of hierarchically defined H𝔉-groups. This class is defined inductively by cellular actions on finite dimensional contractible CW-complexes (see [16]). It turns out to be a very large class of groups, containing all countable elementary amenable groups, all countable linear groups, and all groups with finite virtual cohomological dimension. There has been extensive research on the class of H𝔉-groups (e.g. [8], [17], [19], [20], [28]). We mention, for instance, that Kropholler and Mislin showed that any H𝔉-group of type FP_{∞} has a finite dimensional model for the classifying space for proper actions \underline{E}G (see [17]).

Motivated by the construction of H𝔉-groups, we introduce a method of defining such action-induced, nested classes of groups as follows.

Definition 1.1. Let 𝒳 be a class of groups. Suppose 𝒫 is a collection of topological spaces. Let 𝒜 be a restriction on the action of a group G that acts on a space in 𝒫 such that the induced action of each subgroup of G on this space also has the same restriction. H(𝒫, 𝒜 , 𝒳 ) is defined to be the smallest class of groups containing 𝒳 with the property that if a group G acts by 𝒜 on a space in 𝒫 such that all its isotropy groups are in H(𝒫, 𝒜 , 𝒳 ), then G is also in H(𝒫, 𝒜 , 𝒳 ).

Date: June 22, 2012.

Key words and phrases. cellular action, jump cohomology, Hirsch length.

2010 Mathematics Subject Classification. 20J05.

The author was supported by the Research Fund KU Leuven and FWO-Flanders Research Fellowship.

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There is also a hierarchical definition of H(𝒫, 𝒜 , 𝒳 )-groups by using ordinal numbers. We have that a group is in H(𝒫, 𝒜 , 𝒳 ) if and only if it belongs to H_{α}(𝒫, 𝒜 , 𝒳 ) for some ordinal α.

It is worth pointing out that although we restrict ourselves to cellular actions on finite dimensional CW-complexes, the definition allows one to consider many other types of actions, such as for example, algebraic actions on varieties, isometric actions on metric spaces, actions on manifolds and Lie groups, just to name a few.

When 𝒳 contains only the trivial group and 𝒜 defines the action to be cellular, we denote H(𝒫, 𝒜 , 𝒳 ) by H^{cell}(𝒫).

Identifying the subclass H_{1}^{cell}(𝒫) can be seen as the first step in understanding properties of groups in H^{cell}(𝒫). In general, H_{1}^{cell}(𝒫) consists of groups that can act freely and cellularly on a finite dimensional CW-complex of 𝒫. If the collection 𝒫 contains all 2-dimensional CW-complexes, then H_{1}^{cell}(𝒫) = H^{cell}(𝒫) is the class of all groups. This follows from the fact that any group can be realized as the fundamental group of a 2-dimensional CW-complex.

In Section 4, we consider the following collection of CW-complexes.

Definition 1.2. Suppose R is an integral domain of characteristic zero. The collection 𝒫_{R} consists of all finite dimensional CW-complexes X such that there is an integer k ≥ 0 (depending on X) such that for all i > k, H_{i}(X, ℤ) ⊗ R = 0 and H_{k}(X, ℤ) is finitely generated, infinite and R-torsion-free.

The homological conditions defining the collection 𝒫_{R} are quite natural for a CWcomplex. For instance, when R = ℚ, CW-complexes that have finitely generated integral homology groups, such as finitely dominated ones are in 𝒫_{R}.

The class H^{cell}(𝒫_{R}), characterized by the collection 𝒫_{R}, is the largest among the classes of groups we consider. It contains all H𝔉-groups and it is closed under taking subgroups, extensions, countable directed unions, HNN-extensions, and amalgamated products.

It follows from a theorem of Adem and Smith (see [2]) that all groups with periodic cohomology are in the subclass H_{1}^{cell}(𝒫_{R}). We show that H_{1}^{cell}(𝒫_{R}) also contains all groups with finite virtual cohomological dimension and every group in H_{1}^{cell}(𝒫_{R}) has jump cohomology over R (see Proposition 4.6).

Definition 1.3. Let R be a commutative ring with a unit. A discrete group G has jump (co)homology over R if there exists an integer k ≥ 0, such that for each subgroup H of G we have hd_{R}(H) = ∞ (cd_{R}(H) = ∞) or hd_{R}(H) ≤ k (cd_{R}(H) ≤ k). The smallest of all such k is called the jump height and denoted by hjh_{R}(G) (cjh_{R}(G)).

Jump cohomology is closely related to several well-known yet stronger finiteness conditions, such as spli ℤG < ∞, where spli ℤG is the supremum of the projective lengths of

ACTION-INDUCED NESTED CLASSES OF GROUPS AND JUMP (CO)HOMOLOGY 3

the injective ℤG-modules; the finitistic dimension of ℤG, fin.dim ℤG < ∞; and the property of G admitting a finite dimensional model for \underline{E}G which is known to be equivalent to jump cohomology for torsion-free LH𝔉-groups (see [24], [28]).

It turns out that for solvable groups homological jump height over ℚ coincides with the Hirsch length (see Proposition 3.7). Using this fact, we obtain the following result regarding solvable group actions on complexes in the collection 𝒫_{ℚ}.

Theorem 1.4 (see Corollary 4.3). Let G be a solvable group and let X be a finite dimensional G-CW-complex. Suppose there exists an integer k such that ⊕_{i≥k}H_{i}(X, ℤ) is finitely generated and infinite. Then G has finite Hirsch length if and only if there is an upper bound on the Hirsch lengths of all the stabilizer subgroups.

The paper is structured as follows. In Section 2, we introduce H(𝒫, 𝒜 , 𝒳 )-groups and derive some of their basic properties. Then, specializing to cellular actions, we construct the classes H^{cell}(𝒫_{i}), 1 ≤ i ≤ 5 that give new characterizations to some well-known classes of groups. We also construct the class of H^{cell}(𝒫_{6})-groups which contains all countable elementary amenable groups of finite Hirsch length and show that it satisfies a Tits alternative-like property. The focus of Section 3 is entirely on jump cohomology. In Section 4, we define the class H^{cell}(𝒫_{R}) and analyze its closure operations. We then investigate the subclass H_{1}^{cell}(𝒫_{R}) and its relation to jump cohomology. We end with Section 5 where we state some open questions.

2. The class of H(𝒫, 𝒜 , 𝒳 )-groups

In [16], Kropholler considered a fascinating class of hierarchically defined groups which he denoted by H𝔛. Let 𝔛 be a class of groups. H𝔛 can be defined as the smallest class of groups containing 𝔛 with the property that if the group G acts cellularly on a finite dimensional contractible CW-complex with all stabilizer subgroups in H𝔛, then G is in H𝔛.

As with H𝔛-groups, there exists an inductive definition of H(𝒫, 𝒜 , 𝒳 )-groups using ordinal numbers as follows:

(a) Let H_{0}(𝒫, 𝒜 , 𝒳 ) = 𝒳 .

(b) For an ordinal β > 0, define H_{β}(𝒫, 𝒜 , 𝒳 ) to be the class of groups that can act by 𝒜 on a space X ∈ 𝒫 such that each isotropy group is in H_{α}(𝒫, 𝒜 , 𝒳 ) for some α < β (α can depend on the isotropy group).

Clearly, a group is in H(𝒫, 𝒜 , 𝒳 ) if and only if it is in H_{α}(𝒫, 𝒜 , 𝒳 ) for some α.

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Lemma 2.1. Let 𝒳 be a subgroup closed class of groups. Then H(𝒫, 𝒜 , 𝒳 ) is subgroup closed. In addition, if H(𝒫, 𝒜 , 𝒳 )𝒳 = H(𝒫, 𝒜 , 𝒳 ), then H(𝒫, 𝒜 , 𝒳 ) is also extension closed.

Proof. We will only prove extension closure. The proof for subgroup closure is similar and straightforward.

Suppose we have a short exact sequence of groups 1 → K → G → Q → 1, such that K ∈ H_{α}(𝒫, 𝒜 , 𝒳 ) and Q ∈ H_{β}(𝒫, 𝒜 , 𝒳 ). We will use transfinite induction on the ordinal β to show that G ∈ H_{α+β}(𝒫, 𝒜 , 𝒳 ).

We start with β = 0 as the trivial case. Suppose β > 0 and assume that for each γ < β, if Q ∈ H_{γ}(𝒫, 𝒜 , 𝒳 ), then G ∈ H_{α+γ}(𝒫, 𝒜 , 𝒳 ). By definition, Q acts on a space X ∈ 𝒫 such that each isotropy subgroup is in H_{θ}(𝒫, 𝒜 , 𝒳 ) for some θ < β. Using the given epimorphism of G onto Q, we can construct an induced action of G on X. The isotropy subgroups of this action are extensions of K by the isotropy subgroups of Q. By induction, each of these groups is in H_{α+θ}(𝒫, 𝒜 , 𝒳 ) for some θ < β. This shows that G ∈ H_{α+β}(𝒫, 𝒜 , 𝒳 ), as claimed. □

The hypothesis of the lemma are clearly satisfied if we take 𝒳 as the class containing only the trivial group. Another class of groups for which these hypothesis hold is the class of finite groups 𝔉. Hence, the class of H𝔉-groups is subgroup and extension closed (see 2.2 of [16]).

There exists a natural partial ordering on the set of triples (𝒫, 𝒜 , 𝒳 ) defining the H(𝒫, 𝒜 , 𝒳 )-classes as follows: let (𝒫, 𝒜 , 𝒳 ) and (𝒫′, 𝒜′, 𝒳 ′) be the triples defining the groups H(𝒫, 𝒜 , 𝒳 ) and H(𝒫′, 𝒜′, 𝒳 ′), respectively. Then (𝒫, 𝒜 , 𝒳 ) ≤ (𝒫′, 𝒜′, 𝒳 ′), if 𝒫 ⊆ 𝒫′, every action with restriction 𝒜 also has restriction 𝒜′, and 𝒳 ⊆ 𝒳 ′.

Clearly, when (𝒫, 𝒜 , 𝒳 ) ≤ (𝒫′, 𝒜′, 𝒳 ′), then H_{α}(𝒫, 𝒜 , 𝒳 ) ⊆ H_{α}(𝒫′, 𝒜′, 𝒳 ′) for each ordinal α and H(𝒫, 𝒜 , 𝒳 ) ⊆ H(𝒫′, 𝒜′, 𝒳 ′).

Next, we will consider classes defined using cellular actions on finite dimensional CWcomplexes. When 𝒫 ⊆ {X | X is a finite dimensional CW-complex}, 𝒜 defines the action to be cellular, and the class 𝒳 consists only of the trivial group, we denote H(𝒫, 𝒜 , 𝒳 ) by H^{cell}(𝒫). When defining H^{cell}(𝒫)-classes, we will follow the convention that whenever a manifold is in 𝒫, then this means that the collection 𝒫 contains all possible CW-complexes associated to this manifold. We will also assume throughout that whenever a group acts cellularly on a CW-complex, then the action of the stabilizer group of any cell fixes that cell pointwise.

Proposition 2.2. Let 𝒫_{1} = {𝕊^{1}}. Then, H^{cell}(𝒫_{1}) is the class of finite solvable groups.

Proof. First, we will show that the class of finite solvable groups is in H^{cell}(𝒫_{1}).

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Suppose G is a finite solvable group. We proceed by induction on the length n of the given decomposition of G into cyclic factors.

Let H be a normal subgroup of length n − 1. Since G/H acts freely and cellularly on 𝕊^{1}, from extension closure, we have that G ∈ H^{cell}(𝒫_{1}).

Conversely, suppose G ∈ H^{cell}(𝒫_{1}). By transfinite induction, we can assume that G acts cellularly on a circle such that each stabilizer subgroup is finite solvable. The following easy lemma finishes the proof. □

Recall that a group G is said to act effectively (or faithfully) on a space X if for every e ≠ g ∈ G there exists x ∈ X such that gx ≠ x.

Lemma 2.3. A group G acts effectively and cellularly on 𝕊^{1} endowed with a CW-structure if and only if it is a subgroup of a finite dihedral group.

Proof. Viewing 𝕊^{1} as a regular polygon with vertices as the 0-cell, it is evident that the lemma is just a reformulation of the definition of a dihedral group. □

Proposition 2.4. Let 𝒫_{2} = {𝕋^{m} | m ∈ ℕ}. Then, H^{cell}(𝒫_{2}) = H_{1}^{cell}(𝒫_{2}) is the class of finite groups.

Proof. We will show that the class of finite groups is inside H_{1}^{cell}(𝒫_{2}), i.e. any finite group acts freely on a torus.

Suppose G is a finite group. Let g ∈ G and let H = ⟨g⟩. Construct a free and cellular action of H on 𝕊^{1}. Let X_{g} = Map_{H}(G, 𝕊^{1}) be the set of H-equivariant set maps from G to 𝕊^{1} where H acts on G by left translation. Let g_{1}, g_{2}, …, g_{n} be coset representatives of G/H. There exists a bijection

ϕ : X_{g} → ∏_{i=1}^{n} 𝕊^{1},

given by evaluating a map in X_{g} at g_{i} for each 1 ≤ i ≤ n. We now give X_{g} the topology and CW-structure coming from the product on the right hand side via the bijection ϕ. This structure is independent of the coset representatives. We define the action of G on X_{g} by (x_{0}f)(x) = f(xx_{0}) for f ∈ X_{g}, x_{0}, x ∈ G. Since H acts freely on 𝕊^{1}, the intersection of each stabilizer group of the action of G on X_{g} with H is trivial.

Let X = ∏_{g∈G} X_{g}. Then G act freely on X, because any stabilizer subgroup is the intersection of the stabilizer subgroups of the actions of G on X_{g} for all g ∈ G.

It is left to show that any group in H^{cell}(𝒫_{2}) is finite. But if G ∈ H^{cell}(𝒫_{2}), again using transfinite induction, it follows that G acts cellularly on a torus with all stabilizer subgroups finite. Hence, it is finite. □

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We note that the collection 𝒫_{2} in the theorem can be enlarged by considering instead all finite CW-complexes homotopy equivalent to a torus. Then the same conclusion will still hold.

The next result shows that the class of finite groups can also be obtained by replacing tori with spheres.

Proposition 2.5. Let 𝒫_{3} = {𝕊^{m} | m ∈ ℕ}. Then, H^{cell}(𝒫_{3}) is the class of finite groups.

Proof. By transfinite induction, it follows that any group in H^{cell}(𝒫_{3}) is finite.

In the following lemma, we will show that a group G of order n acts on 𝕊^{n−2} without fixed points. Illman’s theorem will then show that 𝕊^{n−2} has a G-invariant triangulation, and therefore a G-CW-complex structure (see [14]). By induction on the order of the group, we can immediately conclude that G is in H^{cell}(𝒫_{3}). □

Lemma 2.6. Let G be a finite group. Then G acts isometrically on 𝕊^{|G|−2} without fixed points.

Proof. Let n = |G| and consider the group ring ℝG. Define

V = {∑_{i=1}^{n} x_{i}g_{i} | ∑_{i=1}^{n} x_{i} = 0}.

Then V is a hyperplane of dimension n − 1. The space of vectors in V of unit length is then a sphere 𝕊^{n−2}. Note that G acts isometrically on V and hence on 𝕊^{n−2}. The resulting action is fixed-point-free. □

Proposition 2.7. Let 𝒫_{4} = {𝕊^{1}, ℝ}. Then, H^{cell}(𝒫_{4}) is the class of polycyclic groups.

Proof. By extension closure, it follows that H^{cell}(𝒫_{4}) contains all polycyclic groups.

For the converse, suppose G ∈ H^{cell}(𝒫_{4}). Then G acts cellularly on 𝕊^{1} or ℝ such that each stabilizer subgroup is polycyclic. If G acts on 𝕊^{1}, then by Lemma 2.3, it follows that G is polycyclic. If G acts on ℝ, the next lemma finishes the proof. □

Lemma 2.8. A group G acts effectively and cellularly on ℝ endowed with a CW-structure if and only if G is a subgroup of the infinite dihedral group.

Proof. We can assume that the cellular structure on ℝ is just given by the integers ℤ as the 0-cells and unit intervals as the 1-cells. Viewing ℤ as a metric space with the standard distance function, it immediately follows that the lemma is a reformulation of the definition of the infinite dihedral group as the group of isometries of ℤ. □

Theorem 2.9. Let 𝒫_{5} = {𝕊^{m}, ℝ | m ∈ ℕ}. Then, H^{cell}(𝒫_{5}) is the class of virtually polycyclic groups.

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Proof. Since {𝒫_{3}, 𝒫_{4}} ≤ 𝒫_{5}, we have {H^{cell}(𝒫_{3}), H^{cell}(𝒫_{4})} ⊆ H^{cell}(𝒫_{5}). Then, H^{cell}(𝒫_{5}) contains all finite, all polycyclic, and hence all virtually polycyclic groups.

Conversely, suppose G ∈ H_{α}^{cell}(𝒫_{5}) for some ordinal α > 0. Proceeding by transfinite induction, we can assume that G acts cellularly on 𝕊^{n} for some n ≥ 1 or on ℝ with all stabilizer subgroups virtually polycyclic. Without loss of generality, we can assume that the action is effective. Lemma 2.8 finishes the proof. □

In 1979, Olshanskii disproved the Baer Conjecture by constructing a non-virtually polycyclic, namely simple and torsion-free, Noetherian group (see [21]). In the next theorem, we prove that there is a large, subgroup and extension closed class of groups that satisfies Baer criterion.

Theorem 2.10. Let K denote the class of all countable elementary amenable groups of finite Hirsch length and set 𝒫_{6} = {X | X = 𝕊^{m}, m ∈ ℕ, or X is a locally finite tree}. Then K ⊊ H^{cell}(𝒫_{6}) and every group in H^{cell}(𝒫_{6}) either contains a free subgroup on two generators or it is in K. In particular, every Noetherian group in H^{cell}(𝒫_{6}) is virtually polycyclic.

Proof. The class H^{cell}(𝒫_{6}) contains all finite groups and all finitely generated abelian groups. Given a group G, which is a countable directed union of either finite groups or finitely generated abelian groups G_{i}, i ∈ ℕ of the same Hirsch length, one can construct a locally finite tree X with G acting on X so that each isotropy subgroup is a conjugate of some G_{i}. This shows that the class H^{cell}(𝒫_{6}) contains all countable locally finite and all countable abelian groups of finite Hirsch length. Therefore, by extension closure, it contains all countable, locally finite by virtually solvable groups of finite Hirsch length. Now, by a result of Hillman and Linnel (see [12]), H^{cell}(𝒫_{6}) contains K.

Next, we suppose G ∈ H_{α}^{cell}(𝒫_{6}), for some ordinal α > 0, does not contain a free subgroup on two generators. We will show that G is in K.

By transfinite induction, we can assume that G acts cellularly on a CW-complex X which is a sphere or a locally finite tree where all stabilizer subgroups are in K.

Suppose X is a sphere. Let v be a 0-cell of X. Clearly, G/G_{v} is finite and hence, G is countable elementary amenable of finite Hirsch length.

Let X be a locally finite tree. We can assume that the action of G on X is effective. Since G does not contain a free group on two generators, by a theorem of Pays and Valette (see [23]), G either fixes a vertex, an edge, an end, or a pair of ends. If G fixes a vertex or an edge, then we are done by induction. If G fixes a pair of ends then there exists a subgroup of index at most two fixing one of the two ends. Therefore, it is left to check the case where G fixes an end.

Suppose ω is an end fixed by G. Then for any g ∈ G and a half ray r ∈ ω, gr ∈ ω.

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We recall a homomorphism f : G → ℤ defined by f(g) = d(gx, x_{0}) − d(x, x_{0}), where x_{0} is a set vertex on a half ray r of ω and x is a point on r that lies between x_{0} and infinity such that gx is also on r. One can easily check that the definition is independent of the choice of x_{0} and x.

Let K be the kernel of f. It is left to show that K is in K. Let r ∈ ω and let x_{i}, i ∈ ℕ, be vertices of r such that x_{i+1} lies between x_{i} and infinity for each i. Let

K_{i} = {g ∈ K | g(x_{j}) = x_{j}, for each j ≥ i}.

It follows that K = ∪_{i∈ℕ}K_{i}. As K_{i} is a subgroup of G_{x_{i}}, it is in K for each i ∈ ℕ. Thus, K is elementary amenable and h(K) = sup{h(K_{i}) | i ∈ ℕ} = h(K_{1}) < ∞ since K_{i} is a finite index subgroup of K_{i+1} for each i ∈ ℕ. □

Similarly to the definition of H^{cell}(𝒫), if 𝒫 ⊆ {X | X is a metric space}, 𝒳 is the class containing only the trivial group and 𝒜 defines the action to be isometric, we denote H(𝒫, 𝒜 , 𝒳 ) by H^{iso}(𝒫). We assume that ℝ is the Euclidean line, 𝕊^{m} comes with the canonical subspace metric and 𝕋^{m} is the flat torus. We also assume that any locally finite tree comes with the usual metric where each edge has length equal to one.

Theorem 2.11. H^{cell}(𝒫_{i}) = H^{iso}(𝒫_{i}) for each 1 ≤ i ≤ 6.

Proof. We point out that all the previous results on H^{cell}(𝒫_{i}) hold when we replace cellular actions by isometric actions. The only nontrivial cases to prove are that H_{1}^{iso}(𝒫_{2}) is the class of finite groups and H^{cell}(𝒫_{6}) = H^{iso}(𝒫_{6}).

The first assertion says that every finite group acts freely and isometrically on a flat torus. But this follows from an analogous construction as in the proof of Proposition 2.4 of the torus X_{g}. By constructing an isometric action of H on 𝕊^{1}, it easily follows that the induced action of G on the flat torus X_{g} is also isometric. Hence, the action of G on X = ∏_{g∈G} X_{g} is again isometric.

For the second assertion, we use transfinite induction to show that H^{cell}(𝒫_{6}) ⊆ H^{iso}(𝒫_{6}). Let G ∈ H^{cell}(𝒫_{6}). Then, by induction, G acts cellularly on a sphere or a locally finite tree with all cell stabilizers in H^{iso}(𝒫_{6}). We can assume that the action is effective. Suppose G acts on a sphere. Then G is finite and hence in H^{iso}(𝒫_{3}) ⊂ H^{iso}(𝒫_{6}). Suppose G acts on a locally finite tree. The action is clearly isometric. Therefore, G ∈ H^{iso}(𝒫_{6}). The proof of the opposite inclusion is similar. □

The next result addresses the question of proper nesting of subclasses in H^{cell}(𝒫_{i}) corresponding to finite ordinals.

Theorem 2.12. Let ω be the least infinite ordinal. Then we have

(a) H_{ω}^{cell}(𝒫_{i}) = H^{cell}(𝒫_{i}) for i = 1, 3, 4, 5.

(b) H_{k}^{cell}(𝒫_{i}) ⊊ H_{k+1}^{cell}(𝒫_{i}) for i = 1, 3, 4, 5, 6 and each k ∈ ℕ.

Proof. Let i = 1 or i = 4. Given a (finite when i = 1) polycyclic group G, by induction on l, the minimal length of a subnormal series with cyclic factors of G, it follows that G ∈ H_{l}^{cell}(𝒫_{i}). This proves the assertion (a) for H^{cell}(𝒫_{1}) and H^{cell}(𝒫_{4}).

Let i = 3 or i = 5. Given a virtually polycyclic group G, it is poly-ℤ-by-finite. We let N be the poly-ℤ normal subgroup of G and denote the quotient by H. Induction on the order of H together with Lemma 2.6 shows that H ∈ H_{β}^{cell}(𝒫_{i}) for |β| < ∞. By induction on the length l(N) of N, it follows that N ∈ H_{α}^{cell}(𝒫_{i}) for |α| < ∞. Therefore, by Lemma 2.1, G ∈ H_{α+β}^{cell}(𝒫_{i}). This finishes the proof of (a).

To prove (b) in the cases i = 1, 3, observe that for a given nonnegative integer k and an odd prime number p, the elementary abelian p-group C_{p}^{k+1} is in H_{k+1}^{cell}(𝒫_{i}). We will show that it is not in H_{k}^{cell}(𝒫_{i}).

First, we recall the following now classical result on finite group actions. It states that if an elementary abelian p-group of rank r acts on a sphere, then s ≥ r − 1 where s is the maximal rank of the stabilizer subgroup of any cell (see [1]).

Now, by induction, we can assume C_{p}^{k} is not in H_{k−1}^{cell}(𝒫_{i}). Suppose, by a way of contradiction, that C_{p}^{k+1} is in H_{k}^{cell}(𝒫_{i}). Then this group acts cellularly on a sphere with all stabilizer subgroups in H_{k−1}^{cell}(𝒫_{i}). But then, there exists a stabilizer subgroup of rank at least k. Thus, we have arrived to a contradiction.

To prove assertion (b) in the remaining cases i = 4, 5, 6, we note that for a given nonnegative integer k, the free abelian group ℤ^{k+1} is in H_{k+1}^{cell}(𝒫_{i}). We will prove that it is not inside H_{k}^{cell}(𝒫_{i}).

First, let us state a corollary of Theorem 4.2 which will be used in the proof. It says that if Γ is a group with finite homological dimension hd(Γ) acting cellularly on an n-dimensional CW-complex homotopy equivalent to a closed connected manifold M of dimension q, then b ≥ hd(Γ) − n + q where b is the maximal homological dimension of the stabilizer subgroup of any cell.

Returning to our problem, by induction we can assume ℤ^{k} is not in H_{k−1}^{cell}(𝒫_{i}). Suppose ℤ^{k+1} is in H_{k}^{cell}(𝒫_{i}). Then it acts cellularly on a sphere or on a contractible 1-dimensional CW-complex with all stabilizer subgroups in H_{k−1}^{cell}(𝒫_{i}). So, by the corollary, we know that there exists a stabilizer subgroup of rank at least k. This is a contradiction. □

3. Jump (Co)homology

In [24], we considered a new (co)homological condition for groups called jump (co)homology. In this section, we take a closer look at groups with this property.

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Definition 3.1. Let R be a commutative ring with a unit. A discrete group G has jump (co)homology over R if there exists an integer k ≥ 0, such that for each subgroup H of G we have hd_{R}(H) = ∞ (cd_{R}(H) = ∞) or hd_{R}(H) ≤ k (cd_{R}(H) ≤ k). The smallest of all such k will be called the jump height and denoted by hjh_{R}(G) (cjh_{R}(G)).

When R = ℤ, we will simply say that G has jump (co)homology with jump height hjh(G) (cjh(G)).

It follows directly that if a group G has finite virtual (co)homological dimension over R, then it must have jump (co)homology of height vhd_{R}(G) (vcd_{R}(G)) over R.

The converse is clearly not true, as it is evident for instance from the example of the group ℚ/ℤ. This is an infinite torsion group that has jump (co)homology of height zero, but it does not have finite virtual (co)homological dimension.

It is natural to ask how jump cohomology is related to other homological invariants. We recall the following finiteness conditions on a group.

(1) G admits a finite dimensional model of the classifying space for proper actions, \underline{E}G ([17], [18]).

(2) G is of type Φ. A group G is of type Φ if for every ℤG-module M, pd_{ℤG}M < ∞ if and only if pd_{ℤH}M < ∞ for every finite subgroup H of G ([27]).

(3) the supremum of the projective lengths of the injective ℤG-modules, spli ℤG < ∞.

(4) the supremum of the injective lengths of the projective ℤG-modules, silp ℤG < ∞.

(5) G has finite generalized cohomological dimension defined as

sup{n ∈ ℕ | ∃M ℤ-free, ∃F ℤG-free : Ext_{ℤG}(M, F) ≠ 0}.

This is an invariant defined by Ikenaga in relation to his study of Farrell-Tate cohomology in ([13]).

(6) G has finite Gorenstein cohomological dimension Gcd(G). The Gorenstein cohomological dimension of a group G, denoted Gcd(G), is the Gorenstein projective dimension of R as a trivial RG-module ([6]).

(7) G admits a complete resolution in the strong sense ([8], [26]).

(8) G admits a complete resolution.

(9) G has finite finitistic dimension, denoted fin.dim ℤG, which is is the supremum of the projective dimensions of the ℤG-modules of finite projective dimension.

(10) G has jump cohomology. It is known that

(1) ⇒ (2) ⇒ (3) ⇔ (4) ⇔ (5) ⇔ (6) ⇔ (7) ⇒ (8) ⇒ (9) ⇒ (10).

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The implications (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇔ (5) ⇔ (6) ⇔ (7) ⇒ (8) ⇒ (9) are well-known (see [26], [27], [28], [8], [3]). The equivalence of spli ℤG and silp ℤG has recently been proven by Emmanouil in [10]. (9) ⇒ (10) is a direct consequence of the fact that finite finitistic dimension is a subgroup closed property (see [27, 3.3]).

In general, it has been conjectured that the first nine conditions on a group are equivalent (see [26], [28], [8]) and (2) ⇔ (9) has been verified for the class of LH𝔉-groups and (1) ⇔ (9) for the class of all LH𝔉-groups that have a bound on the order of their finite subgroups (see [28]). The equivalence (1) ⇔ (10) is known to hold for torsion-free LH𝔉groups (see [24], [28]), and in [7], it was shown that any locally finite group (which is a group of jump height 0) of cardinality ℵ_{n} has an (n + 1)-dimensional model for the classifying space for proper actions.

Not every group G with jump cohomology has a finite dimensional model for \underline{E}G. From a theorem of Chen (see [7, 6.12] or [22, 7.6]), it follows that the elementary abelian group ⊕_{ℵ_{ω}}C_{2}, which has jump cohomology, does not have jump cohomology over ℚ. In particular, it has infinite rational cohomological dimension and hence cannot have a finite dimensional model for the classifying space for proper actions.

Recently, in [11, 4.4, 4.8], Gandini showed that the first Grigorchuk group has infinite rational cohomological dimension and it is not in LH𝔉. Yet this group has jump cohomology of height zero. It also has jump cohomology of height one over ℚ (see the proof of [11, 4.7]).

Now, let us turn our attention to properties of jump cohomology. First, we show that groups with jump (co)homology satisfy relations similar to those of groups with finite (co)homological dimension. Whenever the proofs of homological and cohomological parts are analogous, we only prove one and omit the proof of the other.

Let R be a commutative ring with a unit. Recall that a discrete group G is said to be R-torsion-free if every torsion element of G has order invertible in R. A group G is said to be R-torsion if it is torsion and every element of G has order not invertible in R.

Lemma 3.2. Let G be a group and let R be a commutative ring with a unit.

(a) If G has jump (co)homology over R, then every subgroup S < G has jump (co)homology over R with hjh_{R}(S) ≤ hjh_{R}(G) (cjh_{R}(S) ≤ cjh_{R}(G)).

(b) If G has jump (co)homology of height zero over R, then G is a torsion group. Conversely, if G is an R-torsion group, then G has jump (co)homology of height zero over R.

(c) If G has jump cohomology over R, then G has jump homology over R and hjh_{R}(G) ≤ cjh_{R}(G). If G is countable, then G has jump cohomology over R if and only if G has jump homology over R and hjh_{R}(G) ≤ cjh_{R}(G) ≤ hjh_{R}(G) + 1.

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Proof. Part (a) follows immediately because, by definition, jump cohomology is a subgroup closed property.

The first part of the claim in part (b) follows from the fact that an infinite cyclic group has (co)homological dimension one over R.

Now, let G be an R-torsion group and let H be a nontrivial subgroup generated by a single element of G. By assumption, |H| · 1_{R} is not invertible in R. Therefore, cd_{R}(H) = hd_{R}(H) = ∞. This shows that any nontrivial subgroup of G has infinite (co)homological dimension over R.

To prove part (c), let G be countable and let H < G. Then, by Theorem 4.6 of [4], we have hd_{R}(H) ≤ cd_{R}(H) ≤ hd_{R}(H) + 1. This proves hjh_{R}(G) ≤ cjh_{R}(G) ≤ hjh_{R}(G) + 1.

In general, suppose G has jump cohomology over R. Let H < G so that hd_{R}(H) < ∞. Since homology of groups commutes with direct limits, we can find a finitely generated subgroup H′ of H such that hd_{R}(H′) = hd_{R}(H). Then, hd_{R}(H′) ≤ cd_{R}(H′) ≤ hd_{R}(H′)+1 implies hd_{R}(H) ≤ cjh_{R}(G). □

Remark 3.3. Note that over the integers, part (b) of the lemma states that a group is torsion if and only if it has jump (co)homology of height zero.

More generally, a finite group G has (co)homological dimension zero only if it is Rtorsion-free. If G contains R-torsion, then cd_{R}(G) = hd_{R}(G) = ∞. This implies that every finite group has jump (co)homology of height zero over R. It follows that every locally finite group has jump homology of height zero over R.

Hence, we see that relaxing the definition of (co)homological dimension of a group in this way allows one to consider groups with large torsion subgroups.

In the next lemma, we derive relations between jump heights of groups forming a short exact sequence.

Lemma 3.4. Let G be a group and let R be a commutative ring with a unit.

(a) Suppose H is a finite index subgroup of G. Then G has jump (co)homology over R if and only if H has jump (co)homology over R. Moreover, hjh_{R}(G) = hjh_{R}(H) (cjh_{R}(G) = cjh_{R}(H)).

(b) Suppose 1 → T ⟶^{ι}G⟶^{π}Q → 1 is a short exact sequence of groups such that Q has jump (co)homology and T is R-torsion. Then G has jump (co)homology of height hjh_{R}(G) ≤ hjh_{R}(Q) (cjh_{R}(G) ≤ cjh_{R}(Q)).

(c) Suppose 1 → K⟶^{ι}G⟶^{π}Q → 1 is a short exact sequence of groups such that Q has finite virtual (co)homological dimension over R and K has jump (co)homology over R. Then G has jump (co)homology of height hjh_{R}(G) ≤ hjh_{R}(K) + vhd_{R}(Q) (cjh_{R}(G) ≤ cjh_{R}(K) + vcd_{R}(Q)) over R.

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Proof. To prove part (a), suppose that H has jump cohomology over R. Let T < G with cd_{R}(T) < ∞. Since T ∩ H is a finite index subgroup of T, cd_{R}(T ∩ H) = cd_{R}(T). Therefore, cd_{R}(T) ≤ cjh_{R}(H) and cjh_{R}(G) = cjh_{R}(H). For part (b), let H < G with cd_{R}(H) < ∞. Since π|_{H} : H → Q is injective, it follows that cjh_{R}(G) ≤ cjh_{R}(Q). To prove part (c), by (a), we can assume cd_{R}(Q) < ∞. Let H < G with cd_{R}(H) < ∞. Since cd_{R}(H) ≤ cd_{R}(ι^{−1}(H)) + cd_{R}(π(H)), we have cd_{R}(H) ≤ cjh_{R}(K) + cd_{R}(Q). □

Remark 3.5. The cohomological part of the inequality in (c) cannot be generalized for short exact sequences where the group Q has jump cohomology, but not necessarily finite virtual cohomological dimension. For example, consider the short exact sequence 0 → ℤ → ℚ → ℚ/ℤ → 0. Note that cjh(ℤ) = cd(ℤ) = 1, cjh(ℚ/ℤ) = 0, and yet, cjh(ℚ) = cd(ℚ) = 2.

We end this section by relating jump (co)homology to the Hirsch length of solvable groups. In what follows, let R be an integral domain of characteristic zero. First, we recall a well-known result.

Theorem 3.6 (Stammbach, [25]). Let G be an R-torsion-free solvable group with Hirsch length h < ∞. Then, hd_{R}(G) = h and there exist an 𝔽G-module A additively isomorphic to the fractional field 𝔽 of R, such that H_{h}^{𝔽}(G, A) ≅ 𝔽.

Proposition 3.7. Let G be a solvable group.

(a) If h(G) < ∞, then G has jump homology over R with hjh_{R}(G) ≤ h(G).

(b) If G is R-torsion-free with jump homology over R, then h(G) = hd_{R}(G) = hjh_{R}(G).

Proof. Suppose h(G) < ∞. Let H < G with hd_{R}(H) < ∞. By Stammbach’s theorem, hd_{R}(H) = h(H). This shows that hjh_{R}(G) ≤ h(G).

To prove (b), suppose h(G) = ∞. Note that for each positive integer n, there is a subgroup H of G such that n ≤ h(H) < ∞. But then n ≤ hd_{R}(H) < ∞. This shows that G does not have jump homology over R which is a contradiction. Therefore, h(G) < ∞ and Stammbach’s theorem give us the desired equalities. □

4. The class H^{cell}(𝒫_{R})

An interesting type of groups that have jump cohomology are groups with periodic cohomology (see [24]). For torsion-free groups it has been conjectured by Talelli (see [26]) that the notions of periodic cohomology and finite cohomological dimension are equivalent. A deep theorem of Adem and Smith shows the relevance of Talelli’s conjecture to the study of free group actions on homology spheres.

ACTION-INDUCED NESTED CLASSES OF GROUPS AND JUMP (CO)HOMOLOGY 14

Theorem 4.1 (Adem-Smith, [2]). A group G has periodic cohomology induced by a cup product map if and only if G acts freely and cellularly on a finite dimensional complex homotopy equivalent to a sphere. If, in addition, G is countable, then it acts freely and properly discontinuously on some 𝕊^{n} × ℝ^{k}.

With this formulation, Talelli’s conjecture asserts that if a torsion-free group acts freely and properly discontinuously on some 𝕊^{n} × ℝ^{k}, then it in fact can act freely and properly discontinuously on some Euclidean space.

In [15], Jo and Nucinkis studied an analogous question for groups that can have torsion. They showed that an elementary amenable group G acting freely and properly discontinuously on 𝕊^{n} × ℝ^{k} admits a finite dimensional model for \underline{E}G.

It is not difficult to show that any group that acts cellularly on a finite dimensional contractible CW-complex such that all the stabilizer subgroups are finite, i.e. it is in the class H_{1}𝔉, has jump cohomology over any commutative ring with a unit. In the next theorem, we will prove a more general result about group actions on CW-complexes comprising the collection 𝒫_{R} of which 𝕊^{n} × ℝ^{k} is an example.

Theorem 4.2. Suppose R is an integral domain of characteristic zero. Let G be a group and let X be an n-dimensional G-CW-complex. Suppose there exist an integer k such that for all i > k, H_{i}(X, ℤ) ⊗ R = 0 and H_{k}(X, ℤ) ≅ ℤ^{m} ⊕ F where m > 0 and F is an R-torsion-free finite group. If all the stabilizer subgroups of the action of G on X have jump (co)homology over R uniformly bounded by an integer b, then G has jump (co)homology over R of height at most b + n − k + \frac{1}{2}m(m − 1).

Proof. We prove the cohomological statement and note that the homological part is analogous.

The action of G on the complex X induces an action on H_{k}(X, ℤ), which may be nontrivial. Since H_{k}(X, ℤ) ≅ ℤ^{m} ⊕F, the matrix group GL_{m}(ℤ) is a finite index subgroup of Aut(H_{k}(X, ℤ)). Thus, we can find a finite index subgroup of G that acts separately on the two summands ℤ^{m} and F, and acts trivially on F. In view of (a) of Lemma 3.4, we can assume that G acts this way. Then, the action is induced by a representation of ρ : G → GL_{m}(ℤ).

Suppose, by a way contradiction, that G has a subgroup of finite cohomological dimension over R larger than our estimate. Without loss of generality, we can assume this for G itself. Let H = ker(ρ). Since

vcd_{R}(ρ(G)) ≤ vcd(GL_{m}(ℤ)) = \frac{1}{2}m(m − 1),

we have cd_{R}(H) > b + n − k.

Let h = cd_{R}(H). Then, H_{R}^{i}(H, M) = 0 for all i > h and all RH-modules M, and there exist a RH-module A such that H_{R}^{h}(H, A) ≠ 0.

Consider the double complex Hom_{ℤ}(P_{∗}, C^{∗}(X, A)), where P_{∗} is a projective resolution of R over RH and C^{∗}(X, A) is the cellular co-chain complex of X with RH-module coefficients A. The natural bi-grading of the complex gives us the spectral sequence,

E_{2}^{p,q}(A) = H_{R}^{p}(H, H^{q}(X, A)) ⟹ H_{R}^{p+q}(H, C^{∗}(X, A)).

Observe that E_{2}^{p,q}(A) = 0 when p > h or q > k. The corner argument then entails,

H_{R}^{h+k}(H, C^{∗}(X, A)) ≅ H_{R}^{h}(H, H^{k}(X, A)).

By the Universal Coefficient Theorem, we also have

0 → Ext_{ℤ}^{1}(H_{k−1}(X), A) → H^{k}(X, A) → Hom_{ℤ}(H_{k}(X), A) → 0.

The associated long exact sequence in cohomology implies that H_{R}^{h}(H, H^{k}(X, A)) surjects onto H_{R}^{h}(H, Hom_{ℤ}(H_{k}(X), A)). On the other hand,

H_{R}^{h}(H, Hom_{ℤ}(H_{k}(X), A)) ≅ H_{R}^{h}(H, A^{m})

≅ \underbrace{H_{R}^{h}(H, A) ⊕ ⋯ ⊕ H_{R}^{h}(H, A)}_{m} ≠ 0.

Therefore, H_{R}^{h+k}(H, C^{∗}(X, A)) ≠ 0, showing that dim(H_{R}^{∗}(H, C^{∗}(X, A))) = h + k > b + n. The double complex also gives the E_{1}-term spectral sequence,

E_{1}^{p,q}(A) = ⨁_{σ∈Σ_{p}}H_{R}^{q}(H_{σ}, A) ⟹ H_{R}^{p+q}(H, C^{∗}(X, A)),

where X_{p} is the collection of all the p-cells and Σ_{p} denotes a set of representatives of all the H-orbits in X_{p}.

According to our hypotheses, E_{1}^{p,q} = 0 when p > n or q > b. This implies that dim(H_{R}^{∗}(H, C^{∗}(X, A))) ≤ b + n, contradicting our earlier estimate. □

We have already seen in Proposition 3.7 that the Hirsch length of a solvable group equals its homological jump height over a field of characteristic zero. Using this fact, we can give an application of Theorem 4.2 to solvable groups.

Corollary 4.3. Let G be a solvable group and let X be a finite dimensional G-CW-complex. Suppose there exists an integer k such that ⊕_{i≥k}H_{i}(X, ℤ) is finitely generated and infinite. Then G has finite Hirsch length if and only if there is an upper bound on the Hirsch lengths of all the stabilizer subgroups.

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Proof. Let b be a bound on the Hirsch lengths of all the stabilizer subgroups. By Proposition 3.7, every stabilizer subgroup has jump homology over ℚ with jump height equal to its Hirsch length. It can now be easily seen that the group G and the complex X satisfy the hypotheses of Theorem 4.2 for R = ℚ. Therefore, h(G) = hjh_{ℚ}(G) < ∞. □

Let us now make several observations about the class H^{cell}(𝒫_{R}) characterized by the collection 𝒫_{R} which is the largest among all the classes with have considered thus far.

Remark 4.4.

(a) Since 𝒫_{1} ≤ {𝒫_{2}, 𝒫_{3}, 𝒫_{4}} and {𝒫_{3}, 𝒫_{4}} ≤ 𝒫_{5} ≤ 𝒫_{6} ≤ 𝒫_{R}, for each ordinal α we

have

H_{α}^{cell}(𝒫_{1}) ⊆ {H_{α}^{cell}(𝒫_{2}), H_{α}^{cell}(𝒫_{3}), H_{α}^{cell}(𝒫_{4})}

and

{H_{α}^{cell}(𝒫_{3}), H_{α}^{cell}(𝒫_{4})} ⊆ H_{α}^{cell}(𝒫_{5}) ⊆ H_{α}^{cell}(𝒫_{6}) ⊆ H_{α}^{cell}(𝒫_{R}).

(b) Since H^{cell}(𝒫_{R}) contains all finite groups and any finite dimensional contractible complex is in 𝒫_{R}, the class H^{cell}(𝒫_{R}) contains the class of H𝔉-groups. Moreover, a simple transfinite induction shows that H_{α}𝔉 ⊆ H_{α+1}^{cell}(𝒫_{R}) for all ordinals α.

Lemma 4.5. H^{cell}(𝒫_{R}) is closed under countable directed unions, amalgamated products, and HNN-extensions.

Proof. Any group that is either a countable directed union, an amalgamated product, or HNN-extension of groups in H^{cell}(𝒫_{R}) is a fundamental group of a graph of groups with vertex groups in H^{cell}(𝒫_{R}). Let G be such a group for a graph of groups Y. Let X be the universal covering tree of Y. Then X is a 1-dimensional contractible G-CW-complex such that each stabilizer subgroup is in H^{cell}(𝒫_{R}). It follows that G ∈ H^{cell}(𝒫_{R}). □

Proposition 4.6. Let J_{R} be the class of groups with jump cohomology over R and V be the class of groups of finite virtual cohomological dimension. Then,V ⊆ H_{1}^{cell}(𝒫_{R}) ⊆ J_{R}.

Proof. If G ∈ H_{1}^{cell}(𝒫_{R}), then there is a finite dimensional free-G-CW-complex X satisfying the hypotheses of Theorem 4.2. So, G has jump cohomology.

Suppose G ∈ V. Let N ◁ G such that [G : N] < ∞ and cd(N) < ∞. Let X be a finite dimensional contractible free-N-CW-complex. Proceeding as in the proof of Proposition 2.4, we can construct a finite dimensional, contractible CW-complex Y = Map_{N}(G, X). Since N acts freely on Y, the intersection of each isotropy group of the action of G on Y with N is trivial.

Let F = G/N. Then F acts freely and cellularly on a torus T. Let G act on T through the epimorphism of G onto F and let G act on the product Y × T through the diagonal action. We claim that this action is free. Indeed, the isotropy groups of the action of G on T are N and the isotropy groups of the action of G on Y intersect trivially with N. □

ACTION-INDUCED NESTED CLASSES OF GROUPS AND JUMP (CO)HOMOLOGY 17

Remark 4.7.

(a) In Section 3, we pointed out that the group ⊕_{ℵ_{ω}}C_{2} has jump cohomology of height zero over ℤ but it does not have jump cohomology over ℚ. This in fact shows that ⊕_{ℵ_{ω}}C_{2} cannot be in H_{1}^{cell}(𝒫_{ℤ}). Indeed, if to the contrary, ⊕_{ℵ_{ω}}C_{2} was in this subclass, then it would also be in H_{1}^{cell}(𝒫_{ℚ}), since 𝒫_{ℤ} ≤ 𝒫_{ℚ}. But then, by applying Theorem 4.2 we could deduce that ⊕_{ℵ_{ω}}C_{2} has jump cohomology over ℚ which would be a contradiction. We can therefore conclude that ⊕_{ℵ_{ω}}C_{2} ∈ J_{ℤ} ∖ H_{1}^{cell}(𝒫_{ℤ}).

(b) Let 𝔏 be the class containing only the trivial group. In [24], we have shown that H𝔏 ∩ J_{ℤ} = H_{1}𝔏. Combining this together with Proposition 4.6, implies

H𝔏 ∩ H_{1}^{cell}(𝒫_{ℤ}) = H_{1}𝔏.

(c) It is known that Thompson’s group F is of type FP_{∞} but not in H𝔉 (see [5], [16]). This group can be defined by the presentation,

⟨x_{0}, x_{1}, x_{2}, ...|x_{i}^{−1}x_{n}x_{i} = x_{n+1} for all i < n and n ∈ ℕ⟩.

We point out that F does not have jump (co)homology over R, because it has an infinite rank abelian subgroup 𝔸_{ℵ_{0}} = ⟨x_{0}x_{1}^{−1}, x_{2}x_{3}^{−1}, x_{4}x_{5}^{−1}, ...⟩. This shows that H_{1}^{cell}(𝒫_{R}) does not contain F.

(d) In fact, the group 𝔸_{ℵ_{0}}, being a countable directed union of ℤ^{i}, i ∈ ℕ, acts cellularly on a 1-dimensional contractible CW-complex with all stabilizer subgroups in H_{1}^{cell}(𝒫_{R}). Therefore, 𝔸_{ℵ_{0}} ∈ H_{2}^{cell}(𝒫_{R}). Since 𝔸_{ℵ_{0}} does not have jump cohomology, this establishes H_{2}^{cell}(𝒫_{R}) ≠ H_{1}^{cell}(𝒫_{R}). Dicks, Kropholler, Leary, and Thomas (see [7, 7.12]) have used free abelian groups of large rank to show that A_{ℵ_{ω}} ∈ H_{3}𝔉 ∖ H_{2}𝔉. In fact, since A_{ℵ_{ω}} is torsion-free, this entails A_{ℵ_{ω}} ∈ H_{3}𝔏 ∖ H_{2}𝔏. Also, we know that H_{3}𝔏 ⊆ H_{3}^{cell}(𝒫_{R}). To summarize, we have just established 𝔸_{ℵ_{0}} ∈ H_{2}^{cell}(𝒫_{R}) ∖ H_{1}^{cell}(𝒫_{R}) and 𝔸_{ℵ_{ω}} ∈ H_{3}^{cell}(𝒫_{R}).

5. Questions

We end with some questions that we hope will provide further motivation to study the classes of groups we have discussed in this paper.

Question 1. Does there exist a group with jump cohomology over 𝔽 that is not in H_{1}^{cell}(𝒫_{𝔽}) for some field 𝔽 of characteristic zero?

It seems that the first Grigorchuk groups is a good candidate to consider since it has jump cohomology of height one over any field of characteristic zero.

Question 2. Is there an H^{cell}(𝒫_{R})-group that is not in H𝔉 for some integral domain R of characteristic zero?

For an example that addresses this question one has to look outside of the class of countable elementary amenable groups since this class is in H𝔉.

Question 3. Does there exist an H_{1}^{cell}(𝒫_{𝔽})-group, where 𝔽 is a field of characteristic zero, that does not satisfy one of the finiteness conditions (1)-(9) stated in Section 3?

We note that from Corollary 4.3 it follows that all countable solvable H_{1}^{cell}(𝒫_{𝔽})-groups have finite Hirsch length and hence satisfy conditions (1)-(9) (see [9]).

Question 4. Is there a field 𝔽 of characteristic zero such that the classes H_{1}^{cell}(𝒫_{𝔽}) and H_{1}𝔉 are distinct?

Let us point out that a well-known conjecture of Kropholler and Mislin asserts that the class H_{1}𝔉 coincides with the class of all groups G such that G admits a finite dimensional model for \underline{E}G. Also, the example of the first Grigorchuk group shows that either Question 1 or Question 4 must have a positive answer.

Acknowledgments

I owe many thanks to Alain Valette for fruitful conversations about group actions on trees and for recalling the result of Lemma 2.6. I am also thankful to Jean-François Lafont and Olympia Talelli for valuable interactions.

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Department of Mathematics, KU Leuven, Kortrijk, Belgium

E-mail address: Nansen.Petrosyan@kuleuven-kortrijk.be