Classifying spaces for the family of virtually abelian subgroups of orientable 3-manifold groups

Porfirio L. León Álvarez; Luis Jorge Sánchez Saldaña

doi:10.1515/forum-2022-0006

Article Open Access

Classifying spaces for the family of virtually abelian subgroups of orientable 3-manifold groups

Porfirio L. León Álvarez and Luis Jorge Sánchez Saldaña

Published/Copyright: June 29, 2022

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From the journal Forum Mathematicum Volume 34 Issue 5

Abstract

For a group G, let ℱ n be the family of all subgroups of G containing a subgroup of finite index isomorphic to ℤ r for some r = 0 , 1 , 2 , … , n . Joecken, Lafont and Sánchez Saldaña computed the ℱ 1 -geometric dimension of 3-manifold groups. As a natural extension of the aforementioned result, the goal of this article is to compute the ℱ n -geometric dimension of 3-manifold groups for all n ≥ 2 .

Keywords: Cohomological dimension; geometric dimension; virtually abelian groups; acylindrical splittings; classifying spaces; families of subgroups

MSC 2010: 55R35; 57S25; 20F65; 57K31

1 Introduction

Given a group Γ, we say a collection ℱ of subgroups of Γ is a family if it is non-empty, closed under conjugation and under taking subgroups. Fix a group Γ and a family ℱ of subgroups of Γ. We say that a Γ-CW-complex X is a model for the classifying space E ℱ ⁢ Γ if every isotropy group of X belongs to the family ℱ and the fixed point set X H is contractible whenever H belongs to ℱ . It can be shown that a model for the classifying space E ℱ ⁢ Γ always exists and it is unique up to Γ-homotopy equivalence. We define the ℱ -geometric dimension of Γ by

gd ℱ ⁡ ( Γ ) = min ⁡ { n ∈ ℕ ∣ there is a model for ⁢ E ℱ ⁢ Γ ⁢ of dimension ⁢ n } .

Examples of families of subgroups are: the family that only consists of the trivial subgroup { 1 } , and the family F IN of finite subgroups of Γ. The ℱ -geometric dimension has been widely studied in the last decades, and the present paper contributes to this topic.

Let n ≥ 0 be an integer. A group is said to be virtually ℤ n if it contains a subgroup of finite index isomorphic to ℤ n . Define the family

ℱ n = { H ≤ Γ ∣ H ⁢ is virtually ⁢ ℤ r ⁢ for some ⁢ 0 ≤ r ≤ n } .

The families { 1 } , ℱ 0 = F IN and ℱ 1 = V CYC are relevant due to its connection with the Farrell–Jones and Baum–Connes isomorphism conjectures. The families ℱ n have been recently studied by several authors; see for instance [7, 27, 10, 28].

In [12], Joecken, Lafont and Sánchez Saldaña computed the ℱ 1 -geometric dimension of 3-manifold groups, that is, fundamental groups of connected, closed, orientable 3-manifolds. The main goal of the present article is to set up a natural extension of this result: we explicitly compute the ℱ k -geometric dimension of 3-manifold groups for all k ≥ 2 . Actually, for a 3-manifold group we have ℱ 3 = ℱ k for all k ≥ 4 ; see Corollary 7.2. Thus, our computations only deal with the families ℱ 2 and ℱ 3 .

Throughout the present article, we always consider orientable manifolds. To state our first main theorem, we use the prime decomposition of a 3-manifold. Recall that a 3-manifold P is prime if P = P 1 ⁢ # ⁢ P 2 implies that P 1 or P 2 is the 3-sphere. It is well known that every connected, closed 3-manifold has a unique decomposition as connected sum of prime 3-manifolds; see Theorem 2.5.

Theorem 1.1.

Let M be a connected, closed, and oriented 3-manifold. Let P 1 , P 2 , … , P r be the pieces in the prime decomposition of M. Set Γ = π 1 ⁢ ( M ) and Γ i = π 1 ⁢ ( P i ) . Then, for all k ≥ 2 ,

gd ℱ k ⁡ ( Γ ) = { 0 if ⁢ M = ℝ ⁢ P 3 ⁢ # ⁢ ℝ ⁢ P 3 , 2 if ⁢ r ≥ 2 , Γ i ∈ ℱ k ⁢ for all ⁢ 1 ≤ i ≤ r , and ⁢ Γ ⁢ is not virtually cyclic, max ⁡ { gd ℱ k ⁡ ( Γ i ) ∣ 1 ≤ i ≤ r } otherwise.

Moreover, the following assertions hold:

Γ i ∈ ℱ 2 if and only if P i is modelled on 𝕊 3 or 𝕊 2 × 𝔼 or P i = ℝ ⁢ P 3 ⁢ # ⁢ ℝ ⁢ P 3 .
Γ i ∈ ℱ 3 if and only if Γ i ∈ ℱ 2 or Γ i is modelled on 𝔼 3 .

In the light of the previous theorem, the task of computing the ℱ k -geometric dimension of a 3-manifold group is reduced to computing the ℱ k -geometric dimension of fundamental groups of prime 3-manifolds. If M is a prime manifold, then we can cut-off along certain embedded tori in such a way that the resulting connected components are either hyperbolic or Seifert fiber 3-manifolds. This is the famous JSJ-decomposition after the work of Perelman; see Theorem 2.6.

Theorem 1.2.

Let M be a connected, closed, and oriented prime 3-manifold. Let N 1 , N 2 , … , N r be the pieces in the minimal JSJ-decomposition of M. Set Γ = π 1 ⁢ ( M ) and Γ i = π 1 ⁢ ( N i ) . If k ≥ 2 , then

gd ℱ k ⁡ ( Γ ) = { 2 if M is modelled on ⁢ Sol , max ⁡ { gd ℱ k ⁡ ( Γ i ) ∣ 1 ≤ i ≤ r } otherwise.

From the previous theorem, it is clear that our next task is to compute the ℱ k -geometric dimension of the fundamental group of all possible JSJ-pieces, that is, the fundamental groups of hyperbolic and Seifert fibered 3-manifolds. We accomplished this and we summarize our results in Table 1. In the gd ℱ 2 column, we reference the theorems where the computations were carried out. The last column is justified in Proposition 7.1.

Table 1

ℱ k -geometric dimension of the pieces JSJ.

Type of manifold	gd ℱ 2 ⁡ ( Γ ) , proved in	gd ℱ k ⁡ ( Γ ) , k ≥ 3
Hyperbolic with empty boundary	3, Theorem 3.1	3
Hyperbolic with non-empty boundary	3, Theorem 3.1	3
Seifert fiber with base orbifold B which is either bad or modelled on 𝕊 2	0, Theorem 4.1	0
Seifert fiber with base orbifold B modelled on ℍ 2 , and boundary empty or non-empty	2, Theorem 4.2	2
Seifert fiber modelled on 𝔼 3 with empty boundary and base orbifold B modelled on 𝔼 2	5, Theorem 4.3	0
Seifert fiber modelled on Nil with empty boundary and base orbifold B modelled on 𝔼 2	3, Theorem 4.3	3
Seifert fibered with non-empty and base orbifold B modelled on 𝔼 2	0, Theorem 4.4	0

Outline of the paper.

In Section 2, we set up the preliminaries such as the definition and some properties of the ℱ -geometric dimension, Bass–Serre theory, 3-manifold theory, and some push-out type constructions for classifying spaces for families. In Section 3, Section 4 and Section 5, we compute the ℱ 2 -geometric dimension of hyperbolic, Seifert fiber and Sol -manifolds, respectively. In Section 6, we recall the notion of acylindrical splitting for the fundamental group of a graph of groups. Then we prove two results that will be useful for using the computations of the previous sections to prove our main theorems. We can think of the results in Section 6 as the tools that help us glue together the classifying spaces of the building pieces of a 3-manifold (the prime and JSJ pieces). Finally, in Section 7 we prove Theorem 1.1 and Theorem 1.2.

2 Preliminaries

2.1 Geometric and cohomological dimension for families

Given a group Γ, we say a collection ℱ of subgroups of Γ is a family if it is non-empty, closed under conjugation and under taking subgroups. Fix a group Γ and a family ℱ of subgroups of Γ. We say that a Γ-CW-complex X is a model for the classifying space E ℱ ⁢ Γ if every isotropy group of X belongs to the family ℱ and the fixed point set X H is contractible whenever H belongs to ℱ . It can be shown that a model for the classifying space E ℱ ⁢ Γ always exists and is unique up to Γ-homotopy equivalence. We define the ℱ -geometric dimension of Γ by

gd ℱ ⁡ ( Γ ) = min ⁡ { n ∈ ℕ ∣ there is a model for ⁢ E ℱ ⁢ Γ ⁢ of dimension ⁢ n } .

The orbit category 𝒪 ℱ ⁢ Γ is the category whose objects are G-homogenous spaces Γ / H with H ∈ ℱ and morphisms are Γ-functions. The category of Bredon modules is the category whose objects are contravariant functors M : 𝒪 ℱ ⁢ Γ → Ab from the orbit category to the category of abelian groups and morphisms are natural transformations f : M → N . This is an abelian category with enough projectives. The constant Bredon module ℤ ¯ : 𝒪 ℱ ⁢ Γ → Ab is defined in objects by ℤ ¯ ⁢ ( Γ / H ) = ℤ and in morphisms by ℤ ¯ ⁢ ( φ ) = id ℤ . We define the ℱ -cohomological dimension of Γ by

cd ℱ ⁡ ( Γ ) = min ⁡ { n ∈ ℕ ∣ there is a projective resolution of ⁢ ℤ ¯ ⁢ of dimension ⁢ n } .

The proof of the following proposition is implicit in [19, Proof of Theorem 3.1] and [20, Proposition 5.1]; see also [21, Theorem 2.3].

Proposition 2.1.

Let G be a group. Let F and G be families of subgroups of G such that F ⊆ G . If X is a model for E G ⁢ G , then

gd ℱ ⁡ ( G ) ≤ max ⁡ { gd ℱ ∩ G σ ⁡ ( G σ ) + dim ⁡ ( σ ) ∣ σ ⁢ is a cell of ⁢ X } .

Lemma 2.2.

Let n ≥ 1 and let G be a finitely generated group. Assume that for all 1 ≤ r ≤ n , G is not virtually Z r . Then gd F n ⁡ ( G ) ≥ 2 .

Proof.

It is easy to see that gd ℱ n ⁡ ( G ) = 0 if and only if G ∈ ℱ n . Thus, by hypothesis, gd ℱ n ⁡ ( G ) ≠ 0 . Now assume gd ℱ n ⁡ ( G ) = 1 . Then G acts on a tree with stabilizers in ℱ n . In particular, every element g ∈ G has a fixed point. Since G is finitely generated, by [30, Corollary 3, p. 65], G has a global fixed point in X, but this contradicts our hypothesis that G ∉ ℱ n . ∎

2.2 Bass–Serre theory

In this section, we remember some notation and results of the Bass–Serre theory that we will use later.

A graph Y (in the sense of Serre) consists of a set of vertices V ⁢ ( Y ) , a set of edges E ⁢ ( Y ) and two functions E ⁢ ( Y ) → V ⁢ ( Y ) × V ⁢ ( Y ) , y ↦ ( o ⁢ ( Y ) , t ⁢ ( Y ) ) , and E ⁢ ( Y ) → E ⁢ ( Y ) , y ↦ y ¯ , satisfying y ¯ ¯ = y and o ⁢ ( y ) = t ⁢ ( y ¯ ) for all y ∈ E ⁢ ( Y ) . The geometric realization of Y is the quotient space V ( Y ) ⊔ ( E ( Y ) × I ) / ∼ , where V ⁢ ( Y ) and E ⁢ ( Y ) have the discrete topology, and the equivalence relation in V ⁢ ( Y ) ⊔ ( E ⁢ ( Y ) × I ) is given as follows: for every y ∈ E ⁢ ( Y ) and t ∈ I ,

( y , t ) ∼ ( y ¯ , 1 - t ) , ( y , 0 ) ∼ o ⁢ ( y ) , ( y , 1 ) ∼ t ⁢ ( y ) .

A graph of groups 𝐘 consists of a graph Y, a group Y P for each P ∈ V ⁢ ( Y ) , and a group Y y for each y ∈ E ⁢ ( Y ) , together with monomorphisms φ y : Y y → Y t ⁢ ( y ) . One requires in addition Y y ¯ = Y y . Suppose that the group G acts without inversions on the graph Y, i.e. for every g ∈ G and y ∈ E ⁢ ( Y ) we have g ⁢ y ≠ y ¯ . Then we have an induced graph of groups with underlying graph Y / G by associating to each vertex (resp. edge) the isotropy group of a preimage under the quotient map Y → Y / G .

Given a graph of groups 𝐘 , one of the classic theorems of Bass–Serre theory provides the existence of a group G = π 1 ⁢ ( 𝐘 ) , called the fundamental group of the graph of groups 𝐘 , and the tree T (a graph with no cycles), called the Bass–Serre tree of 𝐘 , such that G acts on T without inversions, and the induced graph of groups is isomorphic to 𝐘 . The identification G = π 1 ⁢ ( 𝐘 ) is called a splitting of G.

As a direct consequence of [9, Lemma 1.1] we get the following result, which will be useful later on.

Proposition 2.3.

Let H be a group virtually Z n acting on a tree T. Then exactly one of the following happens:

H fixes a vertex of T.
H acts co-compactly in a unique geodesic line γ of T.

2.3 3-manifolds and their decompositions

In this section, we revisit some results of 3-manifolds that we use later. For more details, see [29, 2].

A Seifert fibered space is a 3-manifold M with a decomposition of M into a disjoint union of circles, called fibres, such that each circle has a tubular neighborhood in M that is isomorphic to a fibered solid torus or Klein bottle. If we collapse each of these circles, we obtain a surface B that has a natural orbifold structure; we call B the base orbifold of M. Such an orbifold B has its orbifold fundamental group π 1 orb ⁢ ( B ) , which is not necessarily the fundamental group of the underlying topological space, but it is related to the fundamental group of M via the following lemma.

Lemma 2.4 ([29, lemma 3.2]).

Let M be a Seifert fiber space with base orbifold B. Let Γ be the fundamental group of M. Then there is an exact sequence

1 → K → Γ → π 1 orb ⁢ ( B ) → 1 ,

where K denotes the cyclic subgroup of Γ generated by a regular fiber. The group K is infinite except in cases where M is covered by S 3 .

A 2-orbifold B is of exactly one of the following types depending on the structure of its universal orbifold covering: bad, spheric, hyperbolic or flat. In this paper, we will divide the computation of the ℱ k -geometric dimension of a Seifert manifold M depending on the type of its base orbifold B.

The following is a well-known theorem of Kneser (existence) [15] and Milnor (uniqueness) [23]; see [25, Theorem 2.1.2].

Theorem 2.5 (Prime decomposition).

Let M be a closed, oriented 3-manifold. Then P 1 ⁢ # ⁢ ⋯ ⁢ # ⁢ P n , where each P i is prime. Furthermore, this decomposition is unique up to order and homeomorphism.

Another well-known result we will need is the Jaco–Shalen [11] and Johanson [13] decomposition, after Perelman’s work; see [2, Theorem 1.7.6].

Theorem 2.6 (JSJ-decomposition).

For a closed, prime, oriented 3-manifold M, there is a collection T ⊆ M of disjoint incompressible tori, i.e. two-sided property embedded and π 1 -injective, such that each component of M - T is either a hyperbolic or a Seifert fibered manifold. A minimal such collection T is unique up to isotopy.

It is a consequence of the uniformization theorem that compact surfaces (2-manifolds) admit Riemannian metrics with constant curvature, that is, compact surfaces admit geometric structures modelled on 𝕊 2 , 𝔼 2 or ℍ 2 ; see [25, Theorem 1.1.1]. In dimension three, we are not guaranteed constant curvature. Thurston demonstrated that there are eight 3-dimensional maximal geometries up to equivalence ([29, Theorem 5.1]): 𝕊 3 , 𝔼 3 , ℍ 3 , 𝕊 2 × 𝔼 , ℍ 2 × 𝔼 , PSL ~ 2 ⁢ ( ℝ ) , Nil , and Sol . A manifold M is called geometric if there is a geometry X and a discrete subgroup Γ ≤ isom ⁡ ( X ) with free Γ-action on X such that M is diffeomorphic to the quotient X / Γ ; we also say that M admits a geometric structure modelled on X. Similarly, a manifold with non-empty boundary is geometric if its interior is geometric. It is worth saying that a 3-manifold is Seifert fiber if and only if it admits a geometry modelled on 𝕊 3 , 𝔼 3 , 𝕊 2 × 𝔼 , ℍ 2 × 𝔼 , PSL ~ 2 ⁢ ( ℝ ) , or Nil ; see, for example, [2, Theorem 1.8.1] or [29, Theorem 5.3]. We will use this fact throughout the paper.

2.4 Push-out constructions for classifying spaces

In this section, we revisit some results that help us construct classifying spaces using homotopy push-outs of other classifying spaces.

Definition 2.7.

Let Γ be a finitely generated group, and let ℱ ⊂ ℱ ′ be a pair of families of subgroups of Γ. We say a collection 𝒜 = { A α } α ∈ I of subgroups of Γ is adapted to the pair ( ℱ , ℱ ′ ) if the following conditions hold:

For all A , B ∈ 𝒜 , either A = B or A ∩ B ∈ ℱ .
The collection 𝒜 is closed under conjugation.
Every A ∈ 𝒜 is self-normalizing, i.e. N Γ ⁢ ( A ) = A .
For all A ∈ ℱ ′ - ℱ , there is B ∈ 𝒜 such that A ≤ B .

Theorem 2.8 ([17, p. 302]).

Let F ⊂ F ′ be families of subgroups of Γ. Assume that the collection A = { H α } α ∈ I is adapted to the pair ( F , F ′ ) . Let H be a complete set of representatives of the conjugacy classes within A , and consider the cellular Γ-push-out

Then X is a model for E F ′ ⁢ Γ . In the above Γ-push-out, we require that either (i) f is the disjoint union of cellular H-maps, and g is an inclusion of Γ-CW-complexes, or that (ii) f is the disjoint union of inclusions of H-CW-complexes, and g is a cellular Γ-map.

Let φ : Γ → Γ 0 be a surjective homomorphism. If H is a subgroup of Γ 0 , we will denote by H ~ the subgroup φ - 1 ⁢ ( H ) of Γ. Given a family ℱ of subgroups of Γ 0 , we denote by ℱ ~ the smallest family of Γ that contains the set { H ~ : H ∈ ℱ } .

Theorem 2.9 ([12, Theorem 4.5.]).

Let F be a family of subgroups of the finitely generated discrete group Γ. Let φ : Γ → Γ 0 be a surjective homomorphism. Let F 0 ⊆ F 0 ′ be a nested pair of families of subgroups of Γ 0 satisfying F 0 ~ ⊆ F ⊆ F 0 ′ ~ , and let A = { A α } α ∈ I be a collection adapted to the pair F 0 ⊆ F 0 ′ . Let H be a complete set of representatives of the conjugacy classes in A ~ = { φ - 1 ⁢ ( A α ) } α ∈ I , and consider the following cellular Γ-push-out:

Then X is a model for E F ⁢ Γ . In the Γ-push-out above, we require that either (i) f is the disjoint union of cellular H ~ -maps, and g is an inclusion of Γ-CW-complexes, or that (ii) f is the disjoint union of inclusions of H ~ -CW-complexes, and g is the cellular Γ-map.

Remark 2.10.

Conditions (i) and (ii) at the end of the statements of Theorem 2.8 and Theorem 2.9 are required so that the Γ-push-outs in both statements are actually homotopy Γ-push-outs. It is worth saying that both conditions can always be achieved using a simple cylinder replacement trick. Let us explain how the map f is defined in the context of Theorem 2.9. For H ~ ∈ ℋ , we have a unique H-map E ℱ 0 ⁢ H → E ℱ 0 ⁢ Γ 0 , where the codomain is an H ~ -CW-complex by restriction. By a standard argument, we get an induced Γ-map Γ × H ~ E ℱ 0 ⁢ H → E ℱ 0 ⁢ Γ 0 , and f is the union of these maps running over all elements of ℋ . The construction of g is analogous. Both maps g and f can be chosen in such a way that they satisfy conditions (i) and (ii). By the equivariant cellular approximation theorem, f and g can be taken to be cellular Γ-maps. Also, the Γ-map g (resp. f) can be replaced, if necessary, by the inclusion i : ⊔ Γ × H ~ E ℱ 0 ⁢ H → Cyl ⁡ ( g ) , where Cyl ⁡ ( g ) is the mapping cylinder of g which equivariantly deformation retracts onto the original E ℱ 0 ⁢ Γ 0 , and therefore is still a model for the same classifying space. Also, note that the dimension of Cyl ⁡ ( g ) is max ⁡ { dim ⁡ ( E ℱ 0 ⁢ H ) + 1 , dim ⁡ ( E ℱ 0 ⁢ Γ 0 ) } ; for more details, see [20, Remark 2.5]. This cylinder construction will be used repeatedly (if necessary) throughout the paper.

3 The hyperbolic case

In this section with just one theorem, we compute the ℱ 2 -geometric dimension of hyperbolic manifolds with or without boundary.

Theorem 3.1.

Let M be a hyperbolic 3-manifold of finite volume (possibly with non-empty boundary) and let Γ = π 1 ⁢ ( M , x 0 ) . Then gd F 2 ⁡ ( Γ ) = 3 .

Proof.

We have two cases depending on whether the boundary of M is empty or not. First, suppose that M has empty boundary. From [29, Corollary 4.6.], Γ cannot contain a subgroup isomorphic to ℤ 2 . In consequence, ℱ 1 = ℱ k for all k ≥ 1 . Thus, by [12, Proposition 6.1.], gd ℱ 1 ⁡ ( Γ ) = 3 , and the statement follows.

Now suppose that M has non-empty boundary. First, let us point out two observations:

The first one is that Γ is torsion-free. In fact, since M is hyperbolic, and in particular aspherical, the universal covering ℍ 3 of M is a model for E ⁢ Γ , and therefore Γ has finite cohomological dimension. It follows from [5, Corollary 2.5] that Γ must be torsion-free.
The second observation is that every infinite virtually cyclic subgroup of Γ is isomorphic to ℤ . This follows from the first observation and the classification of virtually cyclic groups; see, for example, [14, Proposition 4].

Consider the collection ℬ of subgroups of Γ consisting of the following terms:

All conjugates of the fundamental groups of the cusps of M. All these groups are isomorphic to ℤ 2 since every cusp is homeomorphic to the product of a torus and an interval.
All maximal infinite virtually cyclic subgroups of Γ that are not subconjugate to the fundamental group of a cusp. All these groups are isomorphic to ℤ by observation (ii) above.

Note that Γ is hyperbolic relative to the collection given by the fundamental groups of the cusps of M; see, for example, [26, Example I, p. 4]. By [16, Theorem 2.6], the collection ℬ is adapted to ( ℱ 0 , ℱ 1 ) . We claim that ℬ is adapted to ( ℱ 0 , ℱ 2 ) . Let us verify each of the conditions in Definition 2.7 for ℬ and ℱ 0 ⊆ ℱ 2 . Conditions (ii) and (iii) follow directly from the proof of Lafont and Ortiz, since they do not depend on the families but only on the collection ℬ . Condition (i) also follows directly from the proof of Lafont and Ortiz, since such a condition only depends on the small family ℱ 0 . For condition (iv), we notice that Lafont and Ortiz verified that every infinite virtually cyclic subgroup of Γ is contained in an element of ℬ , and thus we only have to verify that every virtually ℤ 2 -subgroup of Γ is contained in an element of ℬ . In fact, if H is a virtually ℤ 2 -subgroup of Γ, then H does not contain a subgroup isomorphic to a free group in two generators. Therefore, by the Tits alternative for relatively hyperbolic groups, see for instance [3, Remark 3.5], H is subconjugate to the fundamental group of a cusp.

Now we are in a good shape to use Theorem 2.8 and Remark 2.10 to construct a model for E ℱ 2 ⁢ Γ . Let ℋ be a collection of representatives of conjugacy classes in ℬ . Then the space X defined by the Γ-push-out

(3.1)

given by Theorem 2.8, provides a model for E ℱ 2 ⁢ Γ . To construct this Γ-push-out, we use the cylinder construction described in Remark 2.10.

We claim that X can be chosen to be of dimension 3. Note that, by Remark 2.10, the dimension of X is

max ⁡ { dim ⁡ ( E ⁢ H ) + 1 , dim ⁡ ( E ⁢ Γ ) , dim ⁡ ( E ℱ 2 ⁢ H ) } .

We have seen that H is isomorphic to ℤ or ℤ 2 . Therefore, EH has a model of dimension 1 or 2, and in both cases there is a model for E ℱ 2 ⁢ H of dimension 0. The hyperbolic space ℍ 3 is a 3-dimensional model for E ⁢ Γ . This proves our claim. As a consequence, gd ℱ 2 ⁡ ( Γ ) ≤ 3 .

Now we are going to show that gd ℱ 2 ⁡ ( Γ ) ≥ 3 . It is well known that gd ℱ 2 ⁡ ( Γ ) ≥ cd ℱ 2 ⁡ ( Γ ) , and thus it is enough to show that cd ℱ 2 ⁡ ( Γ ) ≥ 3 . To show this last inequality, we will prove that

H ℱ 2 3 ⁢ ( Γ ; ℤ ¯ ) ≠ 0 .

The Mayer–Vietoris long exact sequence applied to (3.1) leads to

(3.2) ⋯ → H ℱ 2 3 ⁢ ( Γ ; ℤ ¯ ) → H 3 ⁢ ( Γ ; ℤ ¯ ) ⊕ ⊕ H ∈ ℋ H ℱ 2 3 ⁢ ( H ; ℤ ¯ ) → ⊕ H ∈ ℋ H 3 ⁢ ( H ; ℤ ¯ ) → ⋯ .

Since there is a 0-dimensional model for E ℱ 2 ⁢ H for all H ∈ ℋ , we have H ℱ 2 3 ⁢ ( H ; ℤ ¯ ) = 0 . Therefore,

⊕ H ∈ ℋ H ℱ 2 3 ⁢ ( H ; ℤ ¯ ) = 0 .

By [22, Theorem 1.1.7], there exists a 2-dimensional complex X ⊆ M such that X has the same homotopy type of M. Thus, the universal cover X ~ of X is a model for E ⁢ Γ . We conclude that

H 3 ⁢ ( Γ ; ℤ ¯ ) = H 3 ⁢ ( M ; ℤ ) = H 3 ⁢ ( X ; ℤ ) = 0 .

As a consequence of (3.2) and the observations in the previous paragraph, we get the following exact sequence:

⋯ → H 2 ⁢ ( Γ ; ℤ ¯ ) → 𝜑 ⊕ H ∈ ℋ H ≅ ℤ 2 H 2 ⁢ ( H ; ℤ ¯ ) → H ℱ 2 3 ⁢ ( Γ ; ℤ ¯ ) → 0 .

Let us describe the map φ more explicitly. Recall that this map is induced by the unique H equivariant map ⊔ H ∈ ℋ , H ≅ ℤ 2 E ⁢ H → E ⁢ Γ considering E ⁢ Γ as an H-CW-complex by restriction; see Remark 2.10. To give a concrete description of this inclusion map, we fix as a model for E ⁢ Γ the universal covering N ~ of the thick part N of M, that is, we chop off all cusps from M. Since every ℤ 2 -subgroup in ℋ is the fundamental group of a cusp, they correspond to the fundamental groups of the connected components of the boundary of N. Therefore, the map

⊔ H ∈ ℋ , H ≅ ℤ 2 Γ × H E ⁢ H → E ⁢ Γ

can be identified with the inclusion p - 1 ⁢ ( ∂ ⁡ N ) → N ~ , where p : N ~ → N is the universal covering map. On the other hand, we have the natural isomorphisms

H 2 ⁢ ( Γ ; ℤ ¯ ) ≅ H 2 ⁢ ( E ⁢ Γ / Γ ; ℤ ) = H 2 ⁢ ( N ; ℤ ) and H 2 ⁢ ( H ; ℤ ¯ ) ≅ H 2 ⁢ ( E ⁢ H / H ; ℤ ) ;

see, for example, [1, Example 4.3]. In conclusion, φ can be identified with the map induced in the Bredon cohomology by the inclusion ∂ ⁡ N ↪ N .

In order to show that H ℱ 2 3 ⁢ ( Γ ; ℤ ¯ ) ≠ 0 , it is enough to show that the map φ is not surjective. Using the long exact sequence of the pair ( N , ∂ ⁡ N ) , i.e.

⋯ → H 2 ⁢ ( N , ∂ ⁡ N ) → H 2 ⁢ ( N ) → 𝜑 H 2 ⁢ ( ∂ ⁡ N ) → H 3 ⁢ ( N , ∂ ⁡ N ) → 0 ,

we can see that φ is not surjective if and only if H 3 ⁢ ( N , ∂ ⁡ N ) ≠ 0 . By Poincaré duality, H 3 ⁢ ( N , ∂ ⁡ N ) ≅ H 0 ⁢ ( N ) = ℤ . Therefore, φ is not surjective. This finishes the proof. ∎

4 The Seifert fiber case

In this section, we compute the ℱ 2 -dimension of Seifert fiber manifolds with and without boundary. These computations are carried out in Theorem 4.1, Theorem 4.2, Theorem 4.3 and Theorem 4.4

4.1 Seifert fiber manifolds with bad or spherical base orbifold

Theorem 4.1 ([12, Proposition 5.1]).

Let M be a closed Seifert fiber 3-manifold with base orbifold B and fundamental group Γ. Assume that B is either a bad orbifold, or a good orbifold modelled on S 2 . Then Γ is virtually cyclic; in particular, gd F k ⁡ ( Γ ) = 0 for all k ≥ 1 .

4.2 Seifert fiber manifolds with hyperbolic base orbifold

Theorem 4.2.

Let M be a Seifert fiber 3-manifold (possibly with non-empty boundary) with base orbifold B and fundamental group Γ. Assume that B is modelled on H 2 . Then gd F 2 ⁡ ( Γ ) = 2 .

Proof.

First, we are going to show that gd ℱ 2 ⁡ ( Γ ) ≤ 2 . For this we will construct a model for E ℱ 2 ⁢ Γ of dimension 2 using Theorem 2.9. By Lemma 2.4, we obtain an exact sequence of groups

(4.1) 1 → K → Γ → 𝜑 Γ 0 → 1 ,

where Γ 0 = π 1 orb ⁢ ( B ) is the orbifold fundamental group of B and K is a cyclic infinite subgroup of Γ generated by a regular fiber. Let

ℱ n ′ = { H < Γ 0 : H ⁢ is virtually ⁢ ℤ r ⁢ for some ⁢ r = 0 , 1 , … , n } .

Note that, since Γ 0 is a Fuchsian group, it cannot have a subgroup isomorphic to ℤ 2 , and therefore ℱ 2 ′ = ℱ 1 ′ . By [16, Theorem 2.6], the collection

𝒜 = { H < Γ 0 : H ⁢ is virtually ⁢ ℤ ⁢ and maximal in ⁢ ℱ 1 ′ - ℱ 0 ′ }

is adapted to the pair ( ℱ 0 ′ , ℱ 1 ′ ) .

Consider the pulled-back families ℱ 0 ′ ~ and ℱ 1 ′ ~ which by definition are generated by { φ - 1 ⁢ ( L ) : L ∈ ℱ 0 ′ } and { φ - 1 ⁢ ( L ) : L ∈ ℱ 1 ′ } , respectively. We claim that

ℱ 0 ′ ~ ⊂ ℱ 2 ⊂ ℱ 1 ′ ~ .

The first inclusion follows from the fact that φ - 1 ⁢ ( L ) , with L ∈ ℱ 0 ′ , is a virtually cyclic subgroup of Γ. While the second follows from he following argument. For L ∈ ℱ 2 , we have three options: L is finite, L is virtually ℤ or L is virtually ℤ 2 . If L is virtually ℤ 2 , then there is a subgroup L ′ < L of finite index isomorphic to ℤ 2 . Note that φ ⁢ ( L ′ ) is a cyclic subgroup of Γ 0 that has finite index in φ ⁢ ( L ) , and thus φ ⁢ ( L ) ∈ ℱ 1 ′ . We conclude that L ⊆ φ - 1 ⁢ ( φ ⁢ ( L ) ) ∈ ℱ 1 ′ ~ . The other cases follow using a completely analogous argument. Let ℋ be a complete set of representatives of conjugacy classes in 𝒜 * = { φ - 1 ⁢ ( H ) : H ∈ 𝒜 } . Then, by Theorem 2.9, we have the following Γ-push-out that gives a model X for E ℱ 2 ⁢ Γ :

where H ~ stands for φ - 1 ⁢ ( H ) . In the above push-out, if needed, we can replace the upper horizontal arrow by its mapping cylinder to satisfy the conditions at the end of the statement of Theorem 2.9; see Remark 2.10.

We claim that X can be chosen to be of dimension 2. Note that the dimension of X is

max ⁡ { dim ⁡ ( E ℱ 0 ⁢ H ) + 1 , dim ⁡ ( E ℱ 0 ⁢ Γ 0 ) , dim ⁡ ( E ℱ 2 ⁢ H ~ ) } .

Since H ∈ 𝒜 , we have that H is virtually ℤ . Then a model for E ℱ 0 ⁢ H is ℝ for all H ~ ∈ ℋ . As a consequence of [4, Corollary 2.8], a model for E ℱ 0 ⁢ Γ 0 is ℍ 2 . Finally, we show that H ~ is virtually ℤ 2 for all H ~ ∈ ℋ , and in consequence the one point space is a model for E ℱ 2 ⁢ H ~ for all H ~ ∈ ℋ . Let H ~ ∈ ℋ . By the definition of ℋ , we have H ~ ∈ 𝒜 * . Then H ~ = φ - 1 ⁢ ( S ) for some S ∈ 𝒜 . The group S is virtually ℤ , i.e. there is a subgroup T < S such that T is a finite index subgroup of S isomorphic to ℤ . Thus, by (4.1), we have the following short exact sequence:

1 → K → φ - 1 ⁢ ( T ) → T → 1 .

Then φ - 1 ⁢ ( T ) is isomorphic to ℤ ⋊ ℤ , which is virtually ℤ 2 . It follows that H ~ is virtually ℤ 2 . Therefore, X can be constructed to be a 2-dimensional Γ-CW-complex.

The group Γ is finitely generated as it is the fundamental group of a compact manifold. On the other hand, Γ ∉ ℱ 2 since it has as a quotient the group Γ 0 (see (4.1)), which is either virtually non-cyclic free or virtually a surface group. Therefore, by Lemma 2.2, gd ℱ 2 ⁡ ( Γ ) ≥ 2 . ∎

4.3 Seifert fiber manifolds with flat base orbifold

Theorem 4.3.

Let M be a Seifert fiber closed 3-manifold (without boundary) with base orbifold B and fundamental group Γ. Suppose that B is modelled on E 2 . Then M is modelled on E 3 or is modelled on Nil . Moreover, the following assertions hold:

If M is modelled on 𝔼 3 , then gd ℱ 2 ⁡ ( Γ ) = 5 .
If M is modelled on Nil , then gd ℱ 2 ⁡ ( Γ ) = 3 .

Theorem 4.4.

Let M be a compact Seifert fiber 3-manifold, with base orbifold B and fundamental group Γ. Suppose that B is modelled on E 2 and M has non-empty boundary. Then M is diffeomorphic to T 2 × I or the twisted I-bundle over the Klein bottle. In particular, gd F k ⁡ ( Γ ) = 0 for all k ≥ 2 .

Before proving the above theorems, let us set some notation, for A ∈ GL 2 ⁡ ( ℤ ) :

A is elliptic if it has finite order.
A is parabolic if it is conjugate to a matrix of the form

(4.2) ( 1 s 0 1 )

for some s ≠ 0 .
A is hyperbolic if it is neither elliptic nor parabolic.

It is well known that every matrix in A ∈ GL 2 ⁡ ( ℤ ) is either elliptic, parabolic or hyperbolic. Moreover, A is elliptic (resp. parabolic, hyperbolic) if and only if A r is elliptic (resp. parabolic, hyperbolic) for all r ≥ 1 .

Theorem 4.5.

Suppose that M is a 3-manifold with fundamental group isomorphic to Γ = Z 2 ⋊ φ Z , where φ : Z → Aut ⁡ ( Z 2 ) = GL 2 ⁡ ( Z ) is a homomorphism. Then the following statements hold:

If φ ⁢ ( 1 ) is elliptic, then gd ℱ 2 ⁡ ( Γ ) = 5 .
If φ ⁢ ( 1 ) is parabolic, then gd ℱ 2 ⁡ ( Γ ) = 3 .
If φ ⁢ ( 1 ) is hyperbolic, then gd ℱ 2 ⁡ ( Γ ) = 2 .

For the proof of Theorem 4.5, we need the following three lemmas.

Lemma 4.6.

Let Γ be a virtually Z 3 group. Then gd F 2 ⁡ ( Γ ) = 5 .

Proof.

In [18, Proposition A], Lopes Onorio shows that gd ℱ 2 ⁡ ( ℤ 3 ) = 5 . In consequence, gd ℱ 2 ⁡ ( Γ ) ≥ 5 . On the other hand, in [27, Proposition 1.3.] and [24, Theorem 6.14] it is proven that gd ℱ 2 ⁡ ( Γ ) ≤ 5 . Therefore, gd ℱ 2 ⁡ ( Γ ) = 5 . ∎

Lemma 4.7.

Let Γ = Z 2 ⋊ φ Z with φ ⁢ ( 1 ) = A be a parabolic element in G ⁢ L 2 ⁢ ( Z ) . Then the following assertions hold:

The matrix A fixes an infinite cyclic subgroup of ℤ 2 . Moreover, the infinite cyclic maximal subgroup N fixed by A is normal in Γ and Γ / N is isomorphic to ℤ 2 or to ℤ ⋊ ℤ .
Consider the homomorphism π : Γ → Γ / N and let ℱ 1 ′ be the family of virtually cyclic subgroups of Γ / N . Then the pulled-back family ℱ 1 ′ ~ of ℱ 1 ′ is equal to the family ℱ 2 of Γ.

Proof.

First, we prove part (i). Since A is parabolic, without loss of generality we can assume that A is a matrix of the form (4.2). Therefore, A fixes a maximal infinite cyclic subgroup N of ℤ 2 . It is easy to see that N lies in the center of Γ, and it follows that N is a normal subgroup of Γ. Note that Γ / N = ( ℤ 2 / N ) ⋊ ℤ is isomorphic to ℤ 2 or to ℤ ⋊ ℤ .

Now, we proof part (ii). First, we show the inclusion ℱ 1 ′ ~ ⊆ ℱ 2 . Note that both Γ and Γ / N are torsion-free groups, and in consequence every virtually cyclic subgroup of them is either trivial or infinite cyclic. By the definition of ℱ 1 ′ ~ , it is enough to prove that π - 1 ⁢ ( L ) ∈ ℱ 2 for every infinite cyclic subgroup L of Γ / N . Let L be an infinite cyclic subgroup of Γ / N . Hence, π - 1 ⁢ ( L ) fits in the short exact sequence

1 → ℤ → π - 1 ⁢ ( L ) → ℤ → 1 ,

and we conclude that π - 1 ⁢ ( L ) is isomorphic to ℤ 2 or to ℤ ⋊ ℤ . In either case, π - 1 ⁢ ( L ) is virtually ℤ 2 .

Now, we show the inclusion ℱ 2 ⊆ ℱ 1 ′ ~ . Let L ∈ ℱ 2 . We have three cases: L is trivial, L is infinite cyclic or L is virtually ℤ 2 . If L trivial, there is nothing to prove. If L is isomorphic to ℤ , then π ⁢ ( L ) is trivial or infinite cyclic, and therefore π ⁢ ( L ) ∈ ℱ 1 ′ . Thus, L ⊆ π - 1 ⁢ ( π ⁢ ( L ) ) ∈ ℱ 1 ′ ~ . Finally, suppose that L is virtually ℤ 2 , i.e. there is a finite index subgroup T < L such that T is isomorphic ℤ 2 . We claim that T ∩ N is non-trivial. Suppose T ∩ N = 1 . Since N is a subgroup of the center of Γ, a generator h of N commutes with generators α and β of T. Thus, the subgroup generated by h , α , β is isomorphic to ℤ 3 , and by [20, Theorem 5.13 (iii)] this subgroup has ℱ 1 -dimension equal to 4. In consequence, gd ℱ 1 ⁡ ( Γ ) ≥ 4 , but this contradicts the fact that gd ℱ 1 ⁡ ( Γ ) = 3 (see [12, Proposition 5.4]). From our claim, since Γ / N is torsion-free, we can see that π ⁢ ( T ) is infinite cyclic. Since π ⁢ ( T ) is of finite index in π ⁢ ( L ) , it follows that π ⁢ ( L ) is virtually ℤ . Then π ⁢ ( L ) ∈ ℱ 1 ′ . Therefore, we obtain L < π - 1 ⁢ ( π ⁢ ( L ) ) ∈ ℱ 1 ′ ~ . ∎

Lemma 4.8.

Let Γ = Z 2 ⋊ φ Z with φ ⁢ ( 1 ) = A be a hyperbolic element in G ⁢ L 2 ⁢ ( Z ) . Let H be the subgroup Z 2 ⋊ { 0 } of Γ. Then the following assertions hold:

Every subgroup of Γ isomorphic to ℤ 2 is a subgroup of H , in particular the family ℱ 2 = ℱ 1 ∪ SUB ⁡ ( H ) where SUB ⁡ ( H ) is the family of all subgroups of H.
Let C be an infinite cyclic subgroup of Γ . Then

N Γ ⁢ C ≅ { ℤ if ⁢ C ≰ H , ℤ 2 if ⁢ C ≤ H .

Proof.

First, we proof the part (i) by contradiction. Suppose that there is a subgroup L of Γ isomorphic to ℤ 2 that is not a subgroup of H. From the short exact sequence

(4.3) 1 → H → Γ → 𝜓 ℤ → 1 ,

we obtain the following short exact sequence:

1 → L ∩ H → L → ψ ⁢ ( L ) → 1 .

Since L is not a subgroup of H = ker ⁡ ψ , we have that ψ ⁢ ( L ) is non-trivial. Then ψ ⁢ ( L ) = 〈 r ⁢ ℤ 〉 with r ≠ 0 . Since L is isomorphic to ℤ 2 and ψ ⁢ ( L ) = 〈 r ⁢ ℤ 〉 , it follows that L ∩ H is isomorphic to ℤ . In consequence, L = ( L ∩ H ) × 〈 r ⁢ ℤ 〉 . Thus, A r fixes an infinite cyclic subgroup of H ≅ ℤ 2 , and therefore A r is parabolic, which implies that A is parabolic. This is a contradiction since we were assuming that A is hyperbolic.

Now we are going to show that ℱ 2 = ℱ 1 ∪ SUB ⁡ ( H ) . It is clear that ℱ 1 ∪ SUB ⁡ ( H ) ⊆ ℱ 2 . It only remains to prove that ℱ 2 ⊆ ℱ 1 ∪ SUB ⁡ ( H ) . Let L ∈ ℱ 2 . Then L is either virtually cyclic or L is virtually ℤ 2 . If L is virtually cyclic, by definition, L ∈ ℱ 1 ⊆ ℱ 1 ∪ SUB ⁡ ( H ) . Now, suppose that L is virtually ℤ 2 . From (4.3), we get the following short exact sequence:

(4.4) 1 → L ∩ H → L → ψ ⁢ ( L ) → 1 .

Since L is virtually ℤ 2 and, by part (i), H contains all subgroups of Γ isomorphic to ℤ 2 , we have that L ∩ ℤ 2 is isomorphic to ℤ 2 . Then (4.4) is equivalent to

1 → ℤ 2 → L → ψ ⁢ ( L ) → 1 .

Once more, since L is virtually ℤ 2 , we conclude that ψ ⁢ ( L ) must be trivial. It follows that

L ∈ SUB ⁡ ( H ) ⊆ ℱ 1 ∪ SUB ⁡ ( H ) .

Now, we prove part (ii). First, suppose that C ≰ H . Then the elements of C are of the form ( ( 0 , 0 ) , l ) . Let ( ( x , y ) , w ) ∈ N Γ ⁢ C . Then

( ( x , y ) , w ) ⁢ ( ( 0 , 0 ) , l ) ⁢ ( ( x , y ) , w ) - 1 = ( ( x , y ) + A w ⁢ ( - x , - y ) , l ) .

It follows that A w ⁢ ( - x , - y ) = ( - x , - y ) . By the hypothesis, A is hyperbolic. Thus, A w is hyperbolic. Therefore, ( - x , - y ) = ( 0 , 0 ) . We conclude that N Γ ⁢ C = C ≅ ℤ .

Now, suppose that C ≤ H . From (4.3), we have the following short exact sequence:

1 → N Γ ⁢ C ∩ H → N Γ ⁢ C → ψ ⁢ ( N Γ ⁢ C ) → 1 .

By hypothesis, C ≤ H , so H < N Γ ⁢ C . Thus the sequence above is equivalent to

(4.5) 1 → H → N Γ ⁢ C → ψ ⁢ ( N Γ ⁢ C ) → 1 .

We are going to show that ψ ⁢ ( N Γ ⁢ C ) = 0 . For this it is enough to see that N Γ ⁢ C does not contain elements of the form ( ( a , b ) , l ) with l ≠ 0 . Suppose that C is generated by ( ( x , y ) , 0 ) . Then

( ( a , b ) , l ) ⁢ ( ( x , y ) , 0 ) ⁢ ( ( a , b ) , l ) - 1 = ( A l ⁢ ( x , y ) , 0 ) .

It follows that A l ⁢ ( x , y ) = ± ( x , y ) . But we are assuming that A is hyperbolic, which implies that A l is hyperbolic. Therefore, ψ ⁢ ( N Γ ⁢ C ) = 0 . We conclude from (4.5) that N Γ ⁢ C ≅ ℤ 2 . ∎

Proof of Theorem 4.5.

(i) We claim that Γ is virtually ℤ 3 . By hypothesis, A is elliptic, so A has finite order. Let n be the order of A. Then the subgroup { 0 } ⋊ 〈 n ⁢ ℤ 〉 acts trivially on the factor ℤ 2 . Therefore,

ℤ 2 ⋊ 〈 n ⁢ ℤ 〉 = ℤ 2 × 〈 n ⁢ ℤ 〉

is a finite index subgroup of ℤ 2 ⋊ φ ℤ isomorphic to ℤ 3 . Now, the claim follows from Lemma 4.6.

(ii) First, we construct a model for E ℱ 2 ⁢ Γ of dimension 3, and as a consequence we get gd ℱ 2 ⁡ ( Γ ) ≤ 3 . By Lemma 4.7 (i), the matrix A fixes a maximal infinite cyclic subgroup N, and Γ / N = ( ℤ 2 / N ) ⋊ ℤ is isomorphic to either ℤ 2 or ℤ ⋊ ℤ . In both cases, Γ / N is a 2-crystallographic group. Then, by [6], there is a 3-dimensional model Y for E ℱ 1 ′ ⁢ ( Γ / N ) where ℱ 1 ′ is the family of virtually cyclic subgroups of Γ / N . By Lemma 4.7 (ii), we have ℱ 1 ′ ~ = ℱ 2 . Then Y is also a model for E ℱ 2 ⁢ Γ with the Γ-action induced by π : Γ → Γ / N .

Now, we show gd ℱ 2 ⁡ ( Γ ) ≥ 3 . It is enough to prove that H ℱ 2 3 ⁢ ( Γ , ℤ ¯ ) ≠ 0 . Let Y be the model for E ℱ 2 ⁢ Γ that we mentioned in the previous paragraph. Then

H ℱ 2 3 ⁢ ( Γ ; ℤ ¯ ) = H 3 ⁢ ( Y / Γ ; ℤ ) = H 3 ⁢ ( Y / ( Γ / N ) ; ℤ ) .

The latter homology group was shown to be nonzero in [6, p. 8, proof of Theorem 1.1].

(iii) In order to prove gd ℱ 2 ⁡ ( Γ ) ≤ 2 , we construct a model for E ℱ 2 ⁢ Γ of dimension 2. Let H denote ℤ 2 ⋊ { 0 } . By Lemma 4.8 (i), the family ℱ 2 of Γ is the union ℱ 2 = ℱ 1 ∪ SUB ⁡ ( H ) , where SUB ⁡ ( H ) is the family of all subgroups of H. By [8, Lemma 4.4], the space X given by the following homotopy Γ-push-out is a model for E ℱ 2 ⁢ Γ :

(4.6)

where ℱ 1 ⁢ ( H ) = SUB ⁡ ( H ) ∩ ℱ 1 . Here the maps g and f are the unique (up to Γ-homotopy) Γ-maps given by the inclusions of families ℱ 1 ⁢ ( H ) ⊆ ℱ 1 and ℱ 1 ⁢ ( H ) ⊆ SUB ⁡ ( H ) . We claim that, with suitable choices, X is 2-dimensional. Let us explain first the idea of the construction. First, note that H is a normal subgroup of Γ, and E ⁢ ( Γ / N ) = E ⁢ ℤ = ℝ is a model for E SUB ⁡ ( H ) ⁢ Γ ; see, for example, the proof of [20, Corollary 2.10]. Next, we will show that there is a 3-dimensional model Y for E ℱ 1 ⁢ Γ such that the following assertion hold:

Y is the union of two Γ-subcomplexes Y 1 and Y 2 .
Every 3-cell in Y belongs to Y 2 .
Y 2 is a model for E ℱ 1 ⁢ ( H ) ⁢ Γ .
The map f is the inclusion Y 2 → Y .

Since f is an inclusion, the homotopy Γ-push-out (4.6) can be replaced by an honest Γ-pushout; see, for example, [32, Theorem 1.1]. That is, we take X = Y ∪ g ℝ = Y ⊔ ℝ / ∼ , where we identify x ∼ g ⁢ ( x ) for all x ∈ Y 2 . By (i)–(iv), we can see that every 3-cell of Y is being collapsed to a 1-dimensional space via g, and hence we get that X is a complex of dimension less than or equal to 2. From the explicit construction, we actually conclude that X is of dimension 2.

Let us construct Y, Y 1 and Y 2 . Since Γ is a torsion-free poly- ℤ group, by [20, Lemma 5.15 and Theorem 2.3], we get a model Y for E ℱ 1 ⁢ Γ via the following Γ-push-out:

(4.7)

where I is a set of representatives of commensuration classes in ℱ 1 - ℱ 0 . To construct the aforementioned Γ-pushout, we use a construction analogous to the one described in Remark 2.10, that is, we replace E ⁢ Γ with the mapping cylinder of the top horizontal arrow.

By Lemma 4.8 (ii), the set I is the disjoint union of

I 1 = { C ∈ I ∣ N Γ ⁢ C ≅ ℤ } = { C ∈ I ∣ C ≰ ℤ 2 ⋊ { 0 } } and I 2 = { C ∈ I ∣ N Γ ⁢ C ≅ ℤ 2 } = { C ∈ I ∣ C ≤ ℤ 2 ⋊ { 0 } } .

Therefore, the Γ-push-out (4.7) can be written as

Define Y 1 and Y 2 via the following Γ-push-outs:

We used the cylinder construction described in Remark 2.10 to construct all pushouts above. Clearly, Y = Y 1 ∪ Y 2 and Y 1 ∩ Y 2 = E ⁢ Γ . Since ℱ 1 ⁢ ( H ) is the family of all virtually cyclic subgroups of H and I 2 is the set of representatives of commensuration classes of infinite cyclic subgroups in H, we can use [20, Lemma 5.15 and Theorem 2.3] to prove that Y 2 is a model for E ℱ 1 ⁢ ( H ) ⁢ Γ .

We note the following assertions:

If C ∈ I 1 , then a model for E ⁢ N Γ ⁢ C ≅ E ⁢ ℤ is ℝ . Thus, the cylinder associated to E ⁢ N Γ ⁢ C ≅ E ⁢ ℤ contributes to E ℱ 1 ⁢ Γ with a subspace of dimension 2.
If C ∈ I 2 , then a model for E ⁢ N Γ ⁢ C ≅ E ⁢ ℤ 2 is ℝ 2 . Thus, the cylinder associated to E ⁢ N Γ ⁢ C contributes to E ℱ 1 ⁢ Γ with a subspace of dimension 3.
By [20, Theorem 5.13], E ⁢ Γ has a model of dimension 3.

As a consequence of the above observations, we conclude that every 3-cell of Y belongs to Y 2 . This concludes the construction of a 2-dimensional model for E ℱ 2 ⁢ Γ .

Clearly, Γ is finitely generated and Γ ∉ ℱ 2 . Therefore, by Lemma 2.2, gd ℱ 2 ⁡ ( Γ ) ≥ 2 . ∎

Proof of Theorem 4.3.

The first conclusion follows from [29, Theorem 5.3]. For part (i), we have that Γ is 3-crystallographic, and hence Γ is virtually ℤ 3 . Hence, part (i) follows from Lemma 4.6.

For part (ii), we have that Γ is torsion-free as it is the fundamental group of a spherical manifold. Then, by [31, Theorem N], we obtain a short exact sequence

1 → ℤ 2 → Γ → ℤ → 1 ,

where the morphism φ : ℤ → Aut ⁡ ( ℤ 2 ) is such that φ ⁢ ( 1 ) = A is parabolic. Hence, by Theorem 4.5 (ii), we have gd ℱ 2 ⁡ ( Γ ) = 3 . ∎

Proof of Theorem 4.4.

By [25, Theorem 1.2.2], M is modelled on 𝔼 3 , Nil , or M is diffeomorphic to T 2 × I or the twisted I-bundle over the Klein bottle. The 3-manifold M cannot be modelled on 𝔼 3 and Nil as any such manifold must have empty boundary; see [25, p. 60, paragraph 2 and last paragraph]. We conclude that M is diffeomorphic to T 2 × I or to the twisted I-bundle over the Klein bottle. The fundamental groups of T 2 × I and the twisted I-bundle over the Klein bottle are isomorphic to ℤ 2 and ℤ ⋊ ℤ , respectively. In either case, Γ is virtually ℤ 2 , and therefore gd ℱ 2 ⁡ ( Γ ) = 0 . ∎

5 The Sol case

In this section, we compute the ℱ 2 -dimension of manifolds modelled on Sol . This computation is relevant in the statement and proof of Theorem 1.2.

Proposition 5.1.

Let M be a connected closed 3-manifold modelled on Sol with fundamental group Γ. Then gd F 2 ⁡ ( Γ ) = 2 .

In order to prove Proposition 5.1, we need the following lemma. Let 𝒦 be the fundamental group of the Klein bottle.

Lemma 5.2.

Let K 1 and K 2 be copies of K , and let A be the index two Z 2 -subgroup of K . Assume that φ : A → A is a hyperbolic isomorphism. Consider Γ = K 1 * A K 2 , the amalgamated product associated to φ. Then the following assertions hold:

ℱ 2 = ℱ 1 ∪ ℱ ⁢ ( K 1 , K 2 ) , where ℱ ⁢ ( K 1 , K 2 ) is the smallest family of Γ containing K 1 and K 2 .
Let C be an infinite cyclic subgroup of Γ . Then

N Γ ⁢ C ⁢ is virtually ⁢ { ℤ 2 if ⁢ | C ∩ A | = ∞ , ℤ otherwise.

Proof.

(i) We show the inclusion ℱ 2 ⊆ ℱ 1 ∪ ℱ ⁢ ( K 1 , K 2 ) . Let S ∈ ℱ 2 . Then S is virtually cyclic or virtually ℤ 2 . If S is virtually cyclic, by definition, S ∈ ℱ 1 ⊆ ℱ 1 ∪ ℱ ⁢ ( K 1 , K 2 ) . Now, let S be virtually ℤ 2 . We have the following short exact sequence:

1 → A → Γ → 𝜓 D ∞ → 1 .

Let L be the subgroup isomorphic to ℤ of D ∞ of index 2. Then the subgroup Γ ′ = ψ - 1 ⁢ ( L ) ≅ A ⋊ φ L ≤ Γ is also of index 2. It follows that Γ ′ ∩ S is of finite index in S. Since S ≤ Γ is virtually ℤ 2 , Γ ′ ∩ S ≤ Γ ′ is also virtually ℤ 2 . By Lemma 4.8, Γ ′ ∩ S ≤ A . Therefore, in the short exact sequence

1 → A ∩ S → S → ψ ⁢ ( S ) → 1

we have that ψ ⁢ ( S ) is finite. If we think of D ∞ as the free product of two cyclic groups of order two, we can easily see that every finite subgroup of D ∞ is either trivial or a conjugate of one of the free factors. It follows that S is subconjugate to K 1 or K 2 . The other inclusion ℱ 1 ∪ ℱ ⁢ ( K 1 , K 2 ) ⊆ ℱ 2 is clear.

(ii) First, assume | C ∩ A | = ∞ . Recall that if C and C ′ are commensurable cyclic subgroups of Γ, then N Γ ⁢ C = N Γ ⁢ C ′ ; see [20, Lemma 5.15]. Thus, we can assume that C ≤ A . Then, by Lemma 4.8, N Γ ⁢ C ∩ Γ ′ is isomorphic to ℤ 2 . Now Γ ′ is of index 2 in Γ, and thus [ N Γ C : N Γ C ∩ Γ ′ ] ≤ 2 . It follows that N Γ ⁢ C is virtually to ℤ 2 . Now suppose that | C ∩ A | < ∞ , and thus C ∩ A is trivial because A is torsion-free. By Lemma 4.8, we have that N Γ ⁢ C ∩ Γ ′ is isomorphic to ℤ . Now Γ ′ is of index 2 in Γ, and thus [ N Γ C : N Γ C ∩ Γ ′ ] ≤ 2 . It follows that N Γ ⁢ C is virtually ℤ . ∎

Proof of Proposition 5.1.

In [2, Theorem 1.8.2, p. 17], Aschenbrenner, Friedl and Wilton show that if M is modelled on Sol , then either M is the mapping torus of ( T 2 , A ) with A Anosov or M is a double of K ⁢ × ~ ⁢ I , the twisted I-bundle over the Klein bottle. In the first case, we have that Γ = ℤ 2 ⋊ ψ ℤ with ψ ⁢ ( 1 ) = A hyperbolic. By Theorem 4.5 (iii), gd ℱ 2 ⁡ ( Γ ) = 2 .

Suppose now that M is a double of K ⁢ × ~ ⁢ I . In this case, by the Seifert–van Kampen theorem, Γ = K 1 ∗ ℤ 2 K 2 , where K 1 and K 2 are copies of the fundamental group of the Klein bottle 𝒦 , and ℤ 2 is embedded in K i as an index two subgroup. Let H be the index two ℤ 2 -subgroup of 𝒦 . In the proof of Lemma 5.2, we proved that Γ contains an index two subgroup isomorphic to ℤ 2 ⋊ φ ℤ , with φ : H → H being a hyperbolic isomorphism. It follows, from our previous case, that

gd ℱ 2 ⁡ ( Γ ) ≥ gd ℱ 2 ⁡ ( ℤ 2 ⋊ φ ℤ ) = 2 .

It remains to be proven that gd ℱ 2 ⁡ ( Γ ) ≤ 2 . For this we construct a model for E ℱ 2 ⁢ Γ of dimension 2. The construction of such a model follows the same strategy as in the proof of Theorem 4.5 (iii). We include the details for the sake of completeness.

By Lemma 5.2, ℱ 2 = ℱ 1 ∪ ℱ ⁢ ( K 1 , K 2 ) , where ℱ ⁢ ( K 1 , K 2 ) is the smallest family of Γ containing K 1 and K 2 . Then, by [8, Lemma 4.4], the following homotopy Γ-push-out gives a model for E ℱ 2 ⁢ Γ :

(5.1)

where ℱ 1 ⁢ ( K 1 , K 2 ) = ℱ ⁢ ( K 1 , K 2 ) ∩ ℱ 1 . We claim that, with suitable choices, X is 2-dimensional. Let us first explain the idea of the construction. First, note that H is a normal subgroup of Γ, and Γ / H ≅ ℤ 2 ∗ ℤ 2 . Let ℱ ⁢ ( ℤ 2 , ℤ 2 ) be the smallest family of ℤ 2 ∗ ℤ 2 containing both ℤ 2 factors, and notice that ℱ ⁢ ( K 1 , K 2 ) coincides with the smallest family of Γ containing all preimages of elements in ℱ ⁢ ( ℤ 2 , ℤ 2 ) under the projection Γ → ℤ 2 ∗ ℤ 2 . Recall that ℤ 2 ∗ ℤ 2 acts on ℝ by simplicial isometries. In fact, the first factor acts as a reflection through 0, and the second factor as a reflection through 1 2 , and with this action E ℱ ⁢ ( ℤ 2 , ℤ 2 ) ⁢ ( ℤ 2 * ℤ 2 ) = ℝ . Now it is easy to verify that

E ℱ ⁢ ( ℤ 2 , ℤ 2 ) ⁢ ( Γ / H ) = E ℱ ⁢ ( ℤ 2 , ℤ 2 ) ⁢ ( ℤ 2 * ℤ 2 ) = ℝ

is a model for E ℱ ⁢ ( K 1 , K 2 ) ⁢ Γ .

Next, we will show that there is a 3-dimensional model Y for E ℱ 1 ⁢ Γ such that the following assertions hold:

Y is the union of two Γ-subcomplexes Y 1 and Y 2 .
Every 3-cell in Y belongs to Y 2 .
Y 2 is model for E ℱ 1 ⁢ ( K 1 , K 2 ) ⁢ Γ .
The map f is the inclusion Y 2 → Y .

Since f is an inclusion, the homotopy Γ-push-out (5.1) can be replaced by an honest Γ-pushout; see, for example, [32, Theorem 1.1]. That is, we take X = Y ∪ g ℝ = Y ⊔ ℝ / ∼ , where we identify x ∼ g ⁢ ( x ) for all x ∈ Y 2 . By (i)–(iv), we can see that every 3-cell of Y is being collapsed to a 1-dimensional space via g, and hence we get that X is a complex of dimension less than or equal to 2. From the explicit construction, we actually conclude that X is of dimension 2. Let us construct Y, Y 1 and Y 2 . Since Γ is a torsion-free and virtually poly- ℤ group (it contains an index two subgroup isomorphic to ℤ 2 ⋊ ℤ ) , by [20, Lemma 5.15 and Theorem 2.3], we get a model Y for E ℱ 1 ⁢ Γ via the following Γ-push-out:

(5.2)

Since I is a set of representatives of commensuration classes, we can assume that every C ∈ I is a subgroup of the index two subgroup ℤ 2 ⋊ ℤ of Γ. Now by Lemma 5.2 (ii), we have that I is a disjoint union of

I 1 = { C ∈ I ∣ N Γ ⁢ C ⁢ is virtually ⁢ ℤ } = { C ∈ I ∣ C ≰ H } and I 2 = { C ∈ I ∣ N Γ ⁢ C ⁢ is virtually ⁢ ℤ 2 } = { C ∈ I ∣ C ≤ H } .

Then the Γ-push-out (5.2) can be written as

Define Y 1 and Y 2 via the following Γ-push-outs:

We used the cylinder construction described in Remark 2.10 to construct all pushouts above. Clearly, Y = Y 1 ∪ Y 2 and Y 1 ∩ Y 2 = E ⁢ Γ . Since ℱ 1 ⁢ ( K 1 , K 2 ) is the smallest family of Γ containing the virtually cyclic subgroups of K 1 and K 2 , a direct application of [20, Lemma 5.15 and Theorem 2.3] leads to the fact that Y 2 is a model for E ℱ 1 ⁢ ( K 1 , K 2 ) ⁢ Γ .

We note the following assertions:

If C ∈ I 1 , then N Γ ⁢ C is virtually ℤ . Thus, by [20, Theorem 5.13], E ⁢ N Γ ⁢ C has a 1-dimensional model. Thus, the cylinder associated to E ⁢ N Γ ⁢ C contributes to E ℱ 1 ⁢ Γ with a subspace of dimension 2.
If C ∈ I 2 , then N Γ ⁢ C is a torsion-free virtually ℤ 2 group. Thus, by [20, Theorem 5.13], E ⁢ N Γ ⁢ C has a 2-dimensional model. Thus, the cylinder associated to E ⁢ N Γ ⁢ C contributes to E ℱ 1 ⁢ Γ with a subspace of dimension 3.
By [20, Theorem 5.13], E ⁢ Γ has a model of dimension 3.

As a consequence of the above observations, we conclude that every 3-cell of Y belongs to Y 2 . This concludes the construction of a 2-dimensional model for E ℱ 2 ⁢ Γ . ∎

6 A small detour on acylindricity of groups acting on trees

In this section, we prove Lemma 6.2 and Theorem 6.3. These results will be relevant in the next section to compute the ℱ 2 -dimension of fundamental groups of prime manifolds from the ℱ 2 -dimension of the JSJ-pieces, and to compute the ℱ 2 -dimension of a 3-manifold group from the ℱ 2 -dimensions of the prime pieces.

In the following definition, we state the notion of acylindricity, which will be key to understand the abelian subgroups of a 3-manifold group.

Definition 6.1.

Let Y be a graph of groups with fundamental group G. The splitting G = π 1 ⁢ ( Y ) is acylindrical if there is an integer k such that, for every path γ of length k in the Bass–Serre tree T of Y, the stabilizer of γ is finite.

Recall that a geodesic line of a simplicial tree T is a simplicial embedding of ℝ in T, where ℝ has as vertex set ℤ and an edge joining any two consecutive integers.

Lemma 6.2.

Let Y be a graph of groups with fundamental group Γ and Bass–Serre tree T. Suppose that the splitting of Γ is acylindrical. Then the following assertions hold:

The setwise stabilizer of every geodesic line in T is virtually cyclic.
Every virtually ℤ n subgroup of Γ with n ≥ 2 fixes a vertex of T.

Proof.

(i) Let L be a geodesic line of T. Denote by Fix Γ ⁡ ( L ) (resp. Stab Γ ⁡ ( L ) ) the subgroup of all elements of Γ that fix L pointwise (resp. setwise). Consider L with simplicial structure induced by T. Note that the group of simplicial automorphisms of L, denoted by Aut ⁡ ( L ) , is isomorphic to the infinite dihedral group D ∞ . Since Stab Γ ⁡ ( L ) acts by simplicial automorphisms on L, we have a homomorphism Stab Γ ⁡ ( L ) → Aut ⁡ ( L ) = D ∞ with kernel Fix Γ ⁡ ( L ) and image being a certain subgroup D of D ∞ . In other words, we have a short exact sequence

1 → Fix Γ ⁡ ( L ) → Stab Γ ⁡ ( L ) → D → 1 .

Since L contains arbitrarily long paths of T, the acylindricity hypothesis implies that Fix Γ ⁡ ( L ) is finite. On the other hand, since D ∞ is a virtually cyclic group, D is also virtually cyclic. Therefore, Stab Γ ⁡ ( L ) is virtually cyclic.

(ii) Let H be a virtually ℤ n subgroup of Γ with n ≥ 2 . Then, by Proposition 2.3, exactly one of the following happens: either H fixes a vertex of T or H acts co-compactly in a unique geodesic line L of T. It remains to rule out the second possibility. Suppose that H acts co-compactly in a unique geodesic line L of T. Let S be a finite index subgroup of H isomorphic to ℤ n . Note that S acts by restriction on L, i.e. we have a homomorphism

S → 𝜑 Aut ⁡ ( L )

whose image contains an infinite subgroup of translations. Since Aut ⁡ ( L ) ≅ D ∞ , we have that φ ⁢ ( S ) is isomorphic either to ℤ or to D ∞ . But φ ⁢ ( S ) is not isomorphic to D ∞ ; otherwise, the abelian group S / ker ⁡ ( φ ) would be isomorphic to the non-abelian group D ∞ . The homomorphism

S → 𝜑 Aut ⁡ ( L )

induces the following short exact sequence

1 → ker ⁡ ( φ ) → S → φ ⁢ ( S ) ≅ ℤ → 1 .

Note that every element of ker ⁡ ( φ ) , by definition, acts trivially on L, and thus ker ⁡ ( φ ) ⊆ Fix Γ ⁡ ( L ) . On the other hand, since L is an (infinite) geodesic of T, it contains paths of arbitrary large length, and therefore Fix Γ ⁡ ( L ) fixes arbitrarily long paths. The acylindricity hypothesis implies that Fix Γ ⁡ ( L ) is finite. Hence, ker ⁡ ( φ ) is a finite subgroup of the torsion-free group S, and thus ker ⁡ ( φ ) is trivial. We conclude that S ≅ ℤ n embeds in ℤ , which is a contradiction. ∎

In the following theorem, we describe a 2-dimensional model T ~ for a classifying space with respect to a family that contains the family ℱ n . This T ~ will be used in the next section to compute the ℱ 2 dimension of a 3-manifold group using the prime splitting and the JSJ decomposition.

Theorem 6.3.

Let Y be a graph of groups with fundamental group Γ finitely generated and Bass–Serre tree T. Consider the collection A of all geodesics of T that admit a co-compact action of an infinite virtually cyclic subgroup of Γ. Then the space T ~ given by the homotopy Γ-push-out

is a model T ~ for E Iso Γ ⁡ ( T ~ ) ⁢ Γ where Iso Γ ⁡ ( T ~ ) is the family generated by the isotropy groups of T ~ , i.e. by coning-off on T the geodesics in A , we obtain a model for E Iso Γ ⁡ ( T ~ ) ⁢ Γ . Moreover, if the splitting Γ = π 1 ⁢ ( Y ) is acylindrical, then the family Iso Γ ⁡ ( T ~ ) contains the family F n of Γ for all n ≥ 0 .

Proof.

If H ≤ Γ acts co-compactly on the geodesic line γ of T and g ∈ Γ , then g ⁢ H ⁢ g - 1 acts co-compactly on g ⁢ γ . It follows that both ⊔ γ ∈ 𝒜 γ and ⊔ γ ∈ 𝒜 { * γ } are Γ-CW-complexes, and therefore the space T ~ is a Γ-CW-complex.

Clearly, T ~ K is non-empty if an only if K ∈ Iso Γ ⁡ ( T ~ ) . We will see that for K ∈ Iso Γ ⁡ ( T ~ ) the fixed point set T ~ K is contractible. We have two cases: T K ≠ ∅ or T K = ∅ . In the first case, we have that T K is a sub-tree of T. Thus, T ~ K is obtained from T K by coning-off some geodesic segments. Then it follows that the space T ~ K is contractible. In the second case, we have that T ~ K consists of a union of some cone points. Note that * γ ∈ T ~ K if and only if K ≤ Stab Γ ⁡ ( γ ) . By hypothesis, T K = ∅ . Then, from [30, Corollary 3], we have that there is a hyperbolic element h ∈ K that acts co-compactly on γ. Since h acts co-compactly on a unique geodesic of T, we conclude T ~ K = * γ , and therefore it is contractible.

We show that the family ℱ n of Γ is contained in Iso Γ ⁡ ( T ~ ) . Let K ∈ ℱ n . Then we have three cases: K is finite, K is virtually ℤ or K is virtually ℤ k with k ≥ 2 . If K is finite, then it is well known that K has a fixed point in T, and thus K ∈ Iso Γ ⁡ ( T ~ ) . If T is virtually ℤ , then, by Proposition 2.3, K fixes a vertex in T or it acts co-compactly on a unique geodesic γ H . In the first case, it is clear that K ∈ Iso Γ ⁡ ( T ~ ) , while for the second case K fixes * γ , and therefore K ∈ Iso Γ ⁡ ( T ~ ) . Finally, if K is virtually ℤ k with k ≥ 2 , we have by Lemma 6.2 that K fixes a point in T, and therefore K ∈ Iso Γ ⁡ ( T ~ ) . ∎

7 Proofs of Theorem 1.1 and Theorem 1.2

In this section, we prove the main theorems of the present paper, but beforehand we need some preliminary results.

Proposition 7.1.

Let Γ be the fundamental group of a 3-manifold M that is either hyperbolic, Seifert fiber possibly with non-empty boundary, or modelled on Sol . The following statements hold:

If M is not modelled on 𝔼 3 , then ℱ 2 = ℱ k for all k ≥ 3 . In particular, gd ℱ 2 ⁡ ( Γ ) = gd ℱ k ⁡ ( Γ ) for all k ≥ 3 .
If M is modelled on 𝔼 3 , then ℱ 3 = ℱ k for all k ≥ 4 . Moreover, gd ℱ k ⁡ ( Γ ) = 0 for all k ≥ 3 .

Proof.

By [18, Proposition A], gd ℱ 2 ⁡ ( ℤ 3 ) = 5 . As a consequence, if Γ contains a subgroup isomorphic to ℤ 3 , then gd ℱ 2 ⁡ ( Γ ) ≥ 5 . By the second column of Table 1 and Proposition 5.1, we conclude that Γ does not contain a ℤ 3 -subgroup unless M is modelled on 𝔼 3 . This proves the first item. The second item follows by noticing that if M is modelled on 𝔼 3 , then Γ is virtually ℤ 3 . ∎

Corollary 7.2.

Let Γ be the fundamental group of the 3-manifold M. Let H be a Z n -subgroup of Γ. Then n ≤ 3 . Moreover, Γ contains a Z 3 -subgroup if and only if one of the prime pieces of M is modelled on E 3 .

Proof.

Let n ≥ 2 . Let

Γ = Γ 1 ∗ ⋯ ∗ Γ r

be the splitting of Γ associated to the prime decomposition of M, and let H ≤ Γ be a ℤ n -subgroup of Γ. Then, by the Kurosh subgroup theorem, without loss of generality, H is a subgroup of Γ 1 . Next, we look at the graph of groups Y given by the JSJ decomposition of Γ 1 ; in particular, the vertex groups of Y are the fundamental groups of the JSJ pieces of Γ 1 , and the edge groups are isomorphic to the fundamental groups of the tori in the JSJ decomposition. Then, by Theorem 7.3, the splitting Γ 1 = π 1 ⁢ ( Y ) is acylindrical, and by Lemma 6.2, H fixes a vertex of the Bass–Serre tree of Y. Thus, H is conjugated to a subgroup of the fundamental group of a JSJ piece N of Γ 1 . By Proposition 7.1, we conclude n ≤ 3 . Moreover, if we assume n = 3 , by Proposition 7.1, such a piece must be modelled on 𝔼 3 . On the other hand, every manifold modelled on 𝔼 3 has empty boundary, and thus N is a prime piece of Γ. Finally, if M has a prime piece modelled on 𝔼 3 , it is clear that Γ contains a ℤ 3 -subgroup. ∎

Theorem 7.3 ([12, Proposition 8.2]).

Let M be a closed, oriented, connected, prime 3-manifold which is not geometric. Let Y be the graph of groups associated to its minimal JSJ decomposition. Then the splitting of G = π 1 ⁢ ( Y ) as the fundamental group of Y is acylindrical.

Proposition 7.4.

Suppose M is a closed, oriented, connected, prime 3-manifold which is not geometric. Let N 1 , N 2 , … , N r be the pieces of the minimal JSJ decomposition of M. Set Γ = π 1 ⁢ ( M ) and Γ i = π 1 ⁢ ( N i ) . If k ≥ 2 , then

max ⁡ { gd ℱ k ⁡ ( Γ i ) ∣ 1 ≤ i ≤ r } ≤ gd ℱ k ⁡ ( Γ ) ≤ max ⁡ { 2 , gd ℱ k ⁡ ( Γ i ) ∣ 1 ≤ i ≤ r } .

Proof.

For all 1 ≤ i ≤ r , the group Γ i is a subgroup of Γ. Then gd ℱ k ⁡ ( Γ i ) ≤ gd ℱ k ⁡ ( Γ ) , and the first inequality follows.

Now, we show the second inequality. Let Y the graph of groups associated to the JSJ decomposition of M with Bass–Serre tree T. By Theorem 7.3, the decomposition of π 1 ⁢ ( M ) = Γ is acylindrical. Therefore, we can use Theorem 6.3 to obtain a 2-dimensional space T ~ that is obtained from T by coning-off some geodesics of T. The space T ~ is a model for E Iso Γ ⁡ ( T ~ ) ⁢ Γ where Iso Γ ⁡ ( T ~ ) is the family generated by the isotropy groups of T ~ , and ℱ k ⊂ Iso Γ ⁡ ( T ~ ) . We have everything set up to apply Proposition 2.1, that is, we only have to compute

gd ℱ k ∩ Γ σ ⁡ ( Γ σ ) + dim ⁡ ( σ )

for each cell σ of T ~ . Once this is done, the proof will be complete.

If the 0-cell σ of T ~ belongs to T, then Γ σ = Γ i for some 1 ≤ i ≤ r . If the 0-cell σ belongs to T ~ - T , then Γ σ is the setwise stabilizer of a geodesic of T, and therefore is virtually cyclic by Lemma 6.2. Hence, in this case, gd ℱ k ∩ Γ σ ⁡ ( Γ σ ) + dim ⁡ ( σ ) = gd ℱ k ⁡ ( Γ i ) or 0.
If the 1-cell σ of T ~ belongs to T, then Γ σ is isomorphic to ℤ 2 . If the 1-cell σ has a vertex in T ~ - T , then Γ σ is virtually cyclic as in the previous item. Hence, in this case,

gd ℱ k ∩ Γ σ ⁡ ( Γ σ ) + dim ⁡ ( σ ) = 1 .
If σ is a 2-cell of T ~ , then it always contain a vertex of T ~ - T , and therefore Γ σ is virtually cyclic. Hence, in this case,

gd ℱ k ∩ Γ σ ⁡ ( Γ σ ) + dim ⁡ ( σ ) = 2 .

∎

Proof of Theorem 1.2.

If the minimal JSJ decomposition of M has only one piece, then the manifold cannot be modelled on Sol , since such manifolds are neither Seifert nor hyperbolic. Hence, if M has only one JSJ piece, the theorem follows. From now on, suppose that the minimal JSJ decomposition of M has more than one piece. We have two cases: M is geometric or not. If M is not geometric, we claim that

gd ℱ k ⁡ ( Γ ) = max ⁡ { gd ℱ k ⁡ ( Γ i ) ∣ 1 ≤ i ≤ r } .

By Proposition 7.4, it is enough to see that there is a piece N i in the minimal JSJ decomposition of M such that gd ℱ k ⁡ ( Γ i ) ≥ 2 . By definition, the pieces in the JSJ decomposition of M are either hyperbolic with boundary or Seifert fibers with boundary. If the JSJ decomposition of M has a hyperbolic piece or a Seifert fiber with base orbifold B modelled on ℍ 2 , then we are done, since by Table 1 the fundamental groups of these pieces have gd ℱ k equal to 3 and 2, respectively. It remains to see what happens if we only have Seifert fiber pieces with base orbifold B modelled on 𝔼 2 . By Theorem 4.4, these pieces are either diffeomorphic to T 2 × I or to the twisted I-bundle over the Klein bottle. If we have a piece of the form T 2 × I , then, by the minimality of the JSJ decomposition of M, we would have only one piece. Then M would be the mapping torus of a self-diffeomorphism of T 2 and, by [2, Theorem 1.10.1., p. 23], M would be geometric. Since we are in the non-geometric case, we discard this possibility. Lastly, we see what happens if we only have pieces homeomorphic to the twisted I-bundle over the Klein bottle. Note that these pieces only have only one boundary component, and therefore we only can have two such pieces glued by a diffeomorphism between their boundaries. In [2, p. 19, last paragraph], Aschenbrenner, Friedl and Wilton show that M is geometric, and once more we discard this possibility.

Suppose now that M is geometric with at least two JSJ pieces. Note that M is not hyperbolic; otherwise, the JSJ decomposition of M would have only one piece. If M is modelled on Sol , the theorem follows from Proposition 5.1. Finally, if M is neither hyperbolic nor modelled on Sol , then, by [2, Theorem 1.8.1, p. 17], M is a Seifert fiber. But this cannot happen since this would imply that we have only one JSJ piece. ∎

Theorem 7.5.

Let M be a closed, connected, oriented 3-manifold. Let P 1 , P 2 , … , P r be the pieces of the prime decomposition of M. Set Γ = π 1 ⁢ ( M ) and Γ i = π 1 ⁢ ( P i ) . If k ≥ 2 , then

max ⁡ { gd ℱ k ⁡ ( Γ i ) ∣ 1 ≤ i ≤ r } ≤ gd ℱ k ⁡ ( Γ ) ≤ max ⁡ { 2 , gd ℱ k ⁡ ( Γ i ) ∣ 1 ≤ i ≤ r } .

Proof.

Let T be the Bass–Serre tree of the splitting Γ = Γ 1 ∗ ⋯ ∗ Γ r . Since the edge stabilizers of T are trivial, the splitting of Γ is acylindrical. Now, the proof is completely analogous to the proof of Proposition 7.4 and the details are left to the reader. ∎

Lemma 7.6.

Let G = H 1 * … * H k with k ≥ 2 and H i ≠ 1 for all i. Then exactly one of the following assertions holds:

G is isomorphic to D ∞ with k = 2 and H 1 , H 2 isomorphic to ℤ 2 .
G contains a non-cyclic free subgroup.

Proof.

Consider G as the two-fold free product H 1 * ( H 2 * ⋯ * H k ) . By [2, Lemma 1.11.2, p. 24], this free product contains a non-cyclic free subgroup unless the two factors are isomorphic to ℤ 2 . By hypothesis, H i ≠ 1 for all i. Then H 2 * ⋯ * H k is isomorphic to ℤ 2 if and only if it has only one factor isomorphic to ℤ 2 . Therefore, the factors of G = H 1 * ( H 2 * ⋯ * H k ) are isomorphic to ℤ 2 if and only if k = 2 and H 1 , H 2 are isomorphic to ℤ 2 . ∎

Proof of Theorem 1.1.

If we only have one piece, the theorem follows as we are necessarily in the third case of our conclusion. From now on, suppose that we have at least two pieces in the prime decomposition. Then Γ = π 1 ⁢ ( P 1 ) ∗ π 1 ⁢ ( P 2 ) ∗ ⋯ ∗ π 1 ⁢ ( P r ) with r ≥ 2 . By Lemma 7.6, we have two cases: the group Γ is isomorphic to D ∞ with r = 2 and π 1 ⁢ ( P 1 ) , π 1 ⁢ ( P 2 ) are isomorphic to ℤ 2 , or Γ contains a non-cyclic free subgroup. In the first case, we are done, since π 1 ⁢ ( P 1 ) = ℤ 2 implies that P 1 is homeomorphic to ℝ ⁢ P 3 .

From now on, suppose also that Γ contains a non-cyclic free subgroup. Then Γ is not virtually abelian, and by Lemma 2.2 we get gd ℱ k ⁡ ( Γ ) ≥ 2 . Next, we consider two cases for a fixed k: gd ℱ k ⁡ ( Γ i ) = 0 for all i, or not. In the first case, by Theorem 7.5, gd ℱ k ⁡ ( Γ ) ≤ 2 . Therefore, gd ℱ k ⁡ ( Γ ) = 2 , and hence we are done in this case. In the second case, we have that there is a Γ s such that gd ℱ k ⁡ ( Γ s ) ≠ 0 , and by Lemma 2.2 we have gd ℱ k ⁡ ( Γ s ) ≥ 2 . Therefore, by Theorem 7.5,

gd ℱ k ⁡ ( Γ ) = max ⁡ { gd ℱ k ⁡ ( Γ i ) ∣ 1 ≤ i ≤ r } ,

and we are done in this final case.

To finish, we prove the moreover part of the statement. Assume that Γ = π 1 ⁢ ( M ) is virtually abelian. By [2, Theorem 1.11.1], M is spherical, ℝ ⁢ P 3 ⁢ # ⁢ ℝ ⁢ P 3 , 𝕊 1 × 𝕊 2 , or it is covered by a torus bundle. If Γ is covered by a torus bundle, then it has a finite index subgroup isomorphic to K = ℤ 2 ⋊ φ ℤ , and therefore K must also be virtually abelian. Since K is poly- ℤ of rank 3, K must be virtually ℤ 3 , and thus K and Γ are modelled on 𝔼 3 . Now, the moreover part follows easily. ∎

Communicated by Jan Bruinier

Funding source: Universidad Nacional Autónoma de México

Award Identifier / Grant number: PAPIIT-IA101221

Funding statement: Both authors were supported by grant PAPIIT-IA101221. The first author was supported by a doctoral scholarship of the Mexican Council of Science and Technology (CONACyT).

Acknowledgements

We thank Rita Jiménez Rolland for comments on a draft of the present article. We are grateful to the anonymous referee; his/her comments helped to improve the readability of the paper.

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Received: 2021-05-18

Published Online: 2022-06-29

Published in Print: 2022-09-01

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Keywords for this article

Cohomological dimension; geometric dimension; virtually abelian groups; acylindrical splittings; classifying spaces; families of subgroups

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