Injective generation of derived categories and other applications of cohomological invariants of infinite groups

Abstract In the study of the representation theory of infinite groups, cohomological invariants play a very useful role. In a recent paper, we proved a number of properties regarding how these invariants interact with each other, extending the scope of some results in the literature. In this short article, we look into several ways in which the behavior of these invariants can be applied in various areas.

In [7], we focus on the behavior of a range of cohomological invariants for infinite groups. In this paper, we look at various directions, often with not much connections between each other, in which our results from [7] can be applied.
The applications shown in this paper have been divided into two parts-Part 1 (Secs. 2 and 3) and Part 2 (Secs. [4][5][6][7][8]. The applications shown in Part 1 have to do with devising some useful generation properties of the derived unbounded category and the stable module category of large classes of infinite groups. In Part 2, we find applications of our invariants in some general questions related to the cohomology and representation theoretic properties of infinite groups. Although there are no apparent connections between the sections in Part 2, the applicability of these invariants and related properties is the unifying theme.
We start with some background on the cohomological invariants that we will be dealing with and recalling some of the results involving them that we will be applying throughout this paper.

Background on cohomological invariants
Most of the invariants that we deal with are either defined over group algebras or for groups over commutative rings: Definition 1.1. Let R be a ring. Denote by spliðRÞ and silpðRÞ the supremum over the projective dimension of injective R-modules and the supremum over the injective dimension of projective R-modules respectively (these invariants were first introduced in [18]).
The finitistic dimension of R, denoted fin: dimðRÞ, is defined to be the supremum over the projective dimension of all R-modules that have finite projective dimension.
When R ¼ AC, where C is a group and A is a commutative ring, kðACÞ is defined to the supremum over the projective dimension of those AC-modules that have finite projective dimension when restricted to finite subgroups of C. Definition 1.2. Let R be a ring. An R-module M is said to admit complete resolutions (over R) iff some high enough syzygy of it occurs as a kernel in a double infinite totally acyclic complex of R-projectives (similarly, M is said to admit weak complete resolutions over R iff some high enough syzygy of it occurs as a kernel in a doubly infinite acyclic complex of R-projectives) (see Remark 1.3 for a clarification of the term "totally acyclic"); we call M a Gorenstein projective Rmodule iff M occurs as a kernel in a doubly infinite totally acyclic complex of R-projectives.
The Gorenstein projective dimension of M over R, denoted Gpd R ðMÞ, is the minimal length of a resolution of Gorenstein projective R-modules admitted by M.
When R ¼ AC, for some commutative ring A and some group C, Gcd A ðCÞ :¼ Gpd AC ðAÞ: We say a group C admits complete resolutions over A iff the trivial AC-module A admits complete resolutions. Remark 1.3. (See Remark 1.7 of [7]) Following standard terminology from homological algebra, note that the phrase "totally acyclic complex" of R-modules, for any ring R, refers to an acylic complex of R-projectives, P Ã , such that Hom R ðP Ã , QÞ is acylic for any R-projective Q.
Also, it is easy to note that for any ring R, an R-module M admitting complete resolutions is equivalent to Gpd R ðMÞ < 1: The last invariant that we need to introduce is defined as the projective dimension of a particular module: Definition 1.4. For any commutative ring A and any group C, denote by BðC, AÞ the module of those functions C ! A that are only allowed to take finitely many values in A. The AC-module structure on BðC, AÞ is given the following way: for any f 2 BðC, AÞ, ðc 1 Á f ÞðcÞ :¼ f ðc À1 1 cÞ, for all c, c 1 2 C: Following [4], we define an AC-module M to be a Benson's cofibrant if M A BðC, AÞ is a projective AC-module.
We first define groups of type U as those groups will play a crucial role in our treatment. Definition 1.5. (made over Z in [36]) For any commutative ring A, a group C is said to be of type U over A if, for any AC-module M, the following two statements are equivalent.
(a) proj: dim AC M < 1: (b) proj: dim AG M < 1, for all finite G C: We denote the class of all groups of type U over A by F /, A : When A ¼ Z, we write F / :¼ F /, Z : Examples of groups of type U over all commutative rings of finite global dimension are groups of finite virtual cohomological dimension, groups acting on trees with finite stabilizers. (see [28] or [34] for more examples).
Another important class of groups comes from Kropholler's hierarchy: Definition 1.6. ( [24]) Let X be a class of groups. Define a hierarchy of groups in the following way: H 0 X :¼ X, and for any successor ordinal (like an integer) a, a group C 2 H a X iff there exists a finite dimensional contractible CW-complex on which C acts by permuting the cells with all the cell stabilizers in H aÀ1 X: If a is a limit ordinal, H a X :¼ [ b<a H b X: A group is said to be in HX iff it is in H a X for some ordinal a. Also, for any ordinal a, H <a X : The class LX is defined to be the class of all groups C such that every finitely generated subgroup of C is in X: Throughout this article, F denotes the class of all finite groups.
Part of the following conjecture, which appears in this form in [7], was originally made over A ¼ Z in [36], and one of the crucial conjectured equivalent statements comes from Conjecture 43.1 of [11]. Conjecture 1.7. (Conjecture 2.5 of [7]) For any group C and any commutative ring A of finite global dimension, the following are equivalent.
The following is one of our main results from [7]. Theorem 1.9. (Theorem 3.1 of [7]) Let C 2 LHF /, A with A being a commutative ring of global dimension t. Then, proj: dim AC BðC, AÞ ¼ Gcd A ðCÞ and proj: dim AC BðC, AÞ fin: dimðACÞ ¼ silpðACÞ ¼ spliðACÞ ¼ kðACÞ proj: dim AC BðC, AÞ þ t In addition to Conjecture 1.7, there is the following conjecture regarding the coincidence of two important classes of modules which will come handy for us in Sec. 3. Notation 1.10. For any ring R, denote by GProjðRÞ the class of Gorenstein projective R-modules, and for any group ring AC, denote by CoFðACÞ the class of all Benson's cofibrant AC-modules. Conjecture 1.11. ([8], made over Z in [14], Conjecture 3.2 of [7]) For any group C and any commutative ring A of finite global dimension, GProjðACÞ ¼ CoFðACÞ: Using more or less the same methods as [14], we were able to prove the following result in relation to Conjecture 1.11: Theorem 1.12. (see Theorem 3.4 of [7], originally from [8]) Let A be a commutative ring of finite global dimension and let C 2 LHF /, A . Then, (a) Any AC-module admits a complete resolution iff it admits a weak complete resolution. (b) GProjðACÞ ¼ CoFðACÞ: The way Conjecture 1.11 is related to the invariants introduced earlier is through a generation property in the module category explored in [9]. We will not be using this notion of generation in this paper, but we briefly introduce it nonetheless to end this section so that the above-mentioned connection is clear. Definition 1.13. (Definition 3.5 of [9]) Let R be a ring. Let T be a class of R-modules. An Rmodule M is generated in zero steps from T iff it is in T and in n steps iff there is an exact sequence 0 ! M 2 ! M 1 ! M ! 0, where M i is generated from T in a i steps, and a 1 þ a 2 n À 1: The class of all R-modules generated in finitely many steps from T is denoted hTi: In the language of Definition 1.13 and in light of Conjecture 1.11, one can expect that, for any group C and any commutative ring A of finite global dimension, hGProjðACÞi ¼ hCoFðACÞi: A special case would be when either of hGProjðACÞi and hCoFðACÞi is the whole module category. In this regard, we were able to show the following: Theorem 1.14. (follows from Proposition 3.5 of [8]) Let A be a commutative ring of finite global dimension. Let C be a group such that Gcd A ðCÞ ¼ proj: dim AC BðC, AÞ (this is stated as a separate conjecture in Conjecture 1.17 of [7] and proved for the case proj: dim AC BðC, AÞ < 1 (Theorem 1.18 of [7]) and for C 2 LHF /, A (by Theorem 1.9)). Then, the following are equivalent.
(a) hCoFðACÞi ¼ ModðACÞ: (b) hGProjðACÞi ¼ ModðACÞ: Notation 1.15. For any ring R, we denote by ModðRÞ the standard abelian category of R-modules. This notation is used in Theorem 1.14 above and also later.

Part 1. Applications in derived and stable categories
Derived and stable module categories are two of the very frequently arising triangulated categories in representation theory. In [6], we studied various generation properties of a range of derived categories of modules over groups in Kropholler's hierarchy (Definition 1.6); so, our handling of derived categories here in Sec. 2 can be considered as providing more information about these derived categories (see Question 2.12).
Stable module categories are usually studied for finite groups, so we make clear in Sec. 3 what definitions we are using for infinite groups. Upon close reading of the applications in Part 1, one can see that the properties of cohomological invariants (or in the case of the stable module categories, of a result (Theorem 1.12) closely related to the cohomological invariants) that are being used are only being applied at one or two key steps.

Injective generation of derived categories
A very important conjecture in the area of finite dimensional algebras is the finitistic dimension conjecture which goes as follows: Conjecture 2.1. Let R be a finite dimensional algebra over a field. Then, fin: dimðRÞ < 1: Recently, Rickard [32] showed that proving Conjecture 2.1 for a given R can be connected to a generation property of the unbounded derived category of cochain complexes of R-modules. Throughout this section, whenever we will write "complexes," we will mean "cochain complexes." Before we can recall Rickard's result, we need to introduce the following definition.
Definition 2.2. Let T be a triangulated category admitting arbitrary coproducts. For any class of objects U in T, denote by hUi the smallest triangulated subcategory of T containing U and closed under arbitrary coproducts. In other words, hUi denotes the smallest localizing subcategory of T containing U: Notation 2.3. For any ring R, denote by DðMod-RÞ and D b ðMod-RÞ, respectively, the derived unbounded category and the derived bounded category of complexes of R-modules. Denote also by ProjðRÞ and InjðRÞ the class of R-projectives and R-injectives respectively. When we will consider these classes as classes of complexes in a derived category, it will be understood that we are considering the modules as complexes concentrated in degree zero.
In the language of Definition 2.2, what Rickard proved was the following: It is noted by Rickard in [32] that we do not know of a finite dimensional algebra R over a field for which DðMod-RÞ ¼ hInjðRÞi: In general, checking generation properties for derived unbounded categories of modules over a given algebra can be easier to handle than computing the finitistic dimension of that algebra-this direction in research can be traced back to some groundbreaking work by Happel [20] in the eighties. So, in the statement of Theorem 2.4, if the injective generation property and the finiteness of the finitistic dimension were actually equivalent instead of one implying the other, that would be much more convenient: Question 2.5. Let R be a finite dimensional algebra over a field. Then, is DðMod-RÞ ¼ hInjðRÞi () fin: dimðRÞ < 1?
Noting that for group rings, the finitistic dimension appears as one of the invariants in Conjecture 1.7 and Theorem 1.9, it is an interesting question to ask whether any of the implications in Question 2.5 holds for group rings. Before proving our main original result in this regard, we need the following basic lemma whose proof we will omit. Lemma 2.6. (Lemma 4.2 of [6], Proposition 2:1:f of [32]) Let R be a ring and let T be a triangulated subcategory of DðMod-RÞ or D b ðMod-RÞ. Then, any complex, X Ã , of the form 0 ! X 0 ! X 1 ! ::: ! X n ! 0 is in T if each X i , when considered as a complex concentrated in degree zero, is in T: Proposition 2.7. Let R be a ring and let T ¼ D b ðMod-RÞ or DðMod-RÞ. For any class of objects U in T, denote by D T ðUÞ the smallest triangulated subcategory of T containing U: Then, (a) silpðRÞ < 1 ) D T ðProjðRÞÞ D T ðInjðRÞÞ: (b) spliðRÞ < 1 ) D T ðInjðRÞÞ D T ðProjðRÞÞ: Therefore, silpðRÞ, spliðRÞ < 1 ) D T ðInjðRÞÞ ¼ D T ðProjðRÞÞ: Proof. We prove (a). The proof for (b) is similar. If silpðRÞ < 1, every projective module, as a chain complex concentrated in degree zero, is quasi-isomorphic to a bounded complex of injectives. So, by Lemma 2.6, ProjðRÞ D T ðInjðRÞÞ, and therefore D T ðProjðRÞÞ D T ðInjðRÞÞ: w Remark 2.8. Note that if R does not have finite injective dimension over itself, i.e. if it is not of finite self-injective dimension, then D T ðInjðRÞÞ 6 ¼ D T ðProjðRÞÞ, where T ¼ D b ðMod-RÞ, because R cannot be quasi-isomorphic to a bounded complex of injectives.
The following lemma shows us that the finiteness of the silp-invariant can be quite strong and useful for handling generation of the unbounded derived category in relation to Question 2.5.  Note that if ðf Þ ) ðbÞ in Conjecture 1.7, then by Lemma 2.9, fin: dimðACÞ < 1 ) DðMod-ACÞ ¼ hInjðACÞi, for any group C and any commutative ring A of finite global dimension. However, we can get the same result for groups in LHF /, A : Proposition 2.10. Let C 2 LHF /, A , with A of finite global dimension. Then, fin: dimðACÞ < 1 ) DðMod-ACÞ ¼ hInjðACÞi: Proof. This follows directly from Theorem 1.9 which gives us that if fin: dimðACÞ < 1, then silpðACÞ ¼ fin: dimðACÞ < 1, and Lemma 2.9.a. w Since Proposition 2.10 forces generation results in the derived unbounded category with just the finiteness of an invariant as the hypothesis, it is relevant to state the following interesting generation property admitted by derived unbounded categories of modules over groups in Kropholler's hierarchy. We end this section with the following couple of questions that can be easily seen to be related to Conjecture 1.7, Proposition 2.10 and all the results discussed in this section including Theorem 2.11. Question 2.12. (a) If C is an HF-group satisfying DðMod-ACÞ ¼ hInjðACÞi, for some commutative ring A, then is C 2 H 1 F? The answer is unlikely to be in the affirmative for any A, but in light of Conjecture 1.7, we can expect this to be the case when A ¼ Z: (b) It follows from Lemma 2.9 and Theorem 2.11 that if C is an H 1 F-group, then hIðC, Also, can we find a group C such that for some A, hIðC, FÞi ¼ hInj-ACi but hIðC, FÞi, hInjðACÞi 6 ¼ DðMod-ACÞ? It follows from Theorem 2.11 that such a C cannot be in H n F for any integer n; whether it can still be in H a F for some higher ordinal a is unclear.

Generating stable module categories of infinite groups
It is well-known that for a finite group G and a field k, the class of finitely generated kG-modules forms a triangulated subcategory of St:ModðkGÞ, usually denoted st:modðkGÞ: Since we will be dealing with stable module categories of not necessarily finite groups in this section, we briefly recall the definition that we will be using. Throughout this section, we fix a commutative ring A of finite global dimension and a group C of type U over A because the stable module category constructed in [28] applies to this class of groups. StabðACÞ is a triangulated category with the inverse syzygy functor X À1 as the suspension functor.
Remark 3.2. Note that although in [28], Mazza and Symonds require A to be Noetherian in addition to having finite global dimension, we do not need the Noetherian condition. That is because in [28], the Noetherian condition is only used to conclude that silpðACÞ < 1 for C 2 F /, A , and we know this holds without the Noetherian condition (by Theorem 1.8.g. and Theorem 1.9).
For the rest of this section, we make the extra assumption that C is an LHF-group. Remark 3.3. Note that, in Conjecture 1.7, groups that are of type U over the integers are conjectured to be in H 1 F (note that H 1 F & LHF); H 1 F-groups are of type U over any commutative ring of finite global dimension (this follows from Proposition 2.5 of [28]). We do not need to assume that C is in H 1 F; we can just make the weaker assumption that C 2 LHF because the only reason this assumption is useful is that we want to use Theorem 3.9. The only groups that are known to be outside LHF do not admit complete resolutions over any commutative ring of finite global dimension. Groups admitting complete resolutions over any given commutative ring of finite global dimension is important to us for getting the X À1 functor. So, keeping the extra LHF-assumption in mind, we can just state that C is an LHF-group admitting complete resolutions over A. Such groups are of type U over A (see Proposition 4.15), so the Mazza-Symonds stable module category construction still works. Note that if a group is of type U over A, it admits complete resolutions over A (see Remark 4.11), but the reverse is not known to be true for all groups (it is true for groups in LHF /, A ) and also it is not known if type U groups over any given commutative ring of finite global dimension is necessarily in LHF: In light of Remark 3.3, let C be an LHF-group that admits complete resolutions over a commutative ring A of finite global dimension, and consider the class of all AC-modules of type FP 1 (i.e. those modules that admit a projective resolution by finitely generated projectives), denoted FPðACÞ: Then, we look at the smallest triangulated subcategory of StabðACÞ containing FPðACÞ, denoted stabðACÞ, and prove a generation property admitted by it. Stable module categories of infinite groups with some additional finiteness properties on the modules were partially considered in [5], however we are not using the definitions of the stable category used in [5].
Note that the objects in stabðACÞ can be given an easy characterization: (b) Any module M of type FP 1 is in M as X 0 ðMÞ is isomorphic to M in StabðACÞ: Therefore, the smallest triangulated subcategory of StabðACÞ containing all modules that are of type FP 1 in the module category is contained in M, i.e. stabðACÞ M: Now, take a module M 2 M: Then, for some non-negative n, X n ðMÞ 2 stabðACÞ: Since stabðACÞ is a triangulated subcategory of StabðACÞ, by repeated applications of the suspension functor X À1 , we get that X 0 ðMÞ 2 stabðACÞ: Thus, M 2 stabðACÞ as M is isomorphic to X 0 ðMÞ in the stable category.
w Before going forward, we need to define two classes of modules-completely finitary modules and polybasic modules. However, since we need to invoke complete Ext-groups to define these classes, we start with the definition of complete Ext-groups.  [19]) Let C be a group. An AC-module is said to be basic if it is of the form U AG AC where G is a finite subgroup of C and U is a completely The following result will be crucial for us. It is easy to note that the class of polybasics, as defined in Definition 3.7, allows us to just deal with basic modules and capture all polybasics by triangles in the the stable category:  For any class of objects U in T and any object M 2 T, we say M is properly generated by U in T if M is in the smallest thick subcategory of T containing U: Remark 3.12. We have seen generation in triangulated categories using localizing in Sec. 2. Generation using thick subcategories is also a very useful concept (one can consult [33] to see more about the theory surrounding this) in general-to be clear, in this concept, one can say a class of objects U in a triangulated subcategory T "generates" T iff the smallest thick subcategory of T containing U is all of T: So, the definition of "proper" generation that we provide in Definition 3.11 is not very unnatural.
Take an LHF-group that admits complete resolutions over commutative ring A of finite global dimension (such a group is in F /, A by Proposition 4.15). Note that if we take any FP 1 module M, some high enough syzygy of it, say X n ðMÞ, is Gorenstein projective (¼Benson's cofibrant in this case, see Theorem 1.12), and also of type FP 1 : Recall that FP 1 modules are completely finitary by Remark 3.8. Now, by Theorem 3.9, Lemma 3.10 and Definition 3.11, X n ðMÞ is in the smallest thick subcategory of StabðACÞ containing the basics (note that projectives are isomorphic to zero in the stable category). Like we saw in the proof of Lemma 3.4.b., it is straightforward to note that whenever X n ðMÞ is in a triangulated subcategory T StabðACÞ, then, by repeated application of the suspension functor X À1 , X 0 ðMÞ (which is isomorphic to M in StabðACÞ) is in T: Thus, we have the following result.
Theorem 3.13. Let C be an LHF-group that admits complete resolutions over a commutative ring A of finite global dimension, and let B be the class of all basic AC-modules. Then, in the language of Definition 3.11, every object in stabðACÞ is properly generated by B in StabðACÞ:

Part 2. Applications in cohomology and representation theory
In Part 2, we look at some representation theoretic applications. In Sec. 4, we explore the properties admitted by groups in Ikenaga's classes (see Definition 4.3), which is a close analogue of the main hierarchy of groups (Kropholler's hierarchy-Definition 1.6) from where we get most of our groups. To study how closely Ikenaga's classes admit similar properties as groups in Kropholler's hierarchy, after proving some of our results, we draw up a list of conjectured relations (Conjecture 4.16) and show how those conjectures interact with each other (Proposition 2.7). This is helpful because until now, such comparative study of Ikenaga's and Kropholler's classes had not been carried out in such detail.
Earlier, in Sec. 2, we dealt with the finitistic dimension of group rings. Since, over integral group rings, it is conjectured (see Conjecture 4.13) that groups whose integral group rings have finite finitistic dimension have finite dimensional models for their classifying space of proper actions (see Definition 4.12), it is a natural question to ask for similar algebraic properties of groups implying the same conclusion. In this regard, we use a result by L€ uck [26] and apply a result from [7] on cohomological invariants to get a new result (Proposition 5.5). In Secs. 6 and 7, we deal with very general questions on projectivity of modules and on groups with periodic cohomology respectively. Although there is no apparent connection between them, it is interesting to see how at key moments, one can invoke properties of cohomological invariants and related questions to extend the scope of some existing results from the literature. Finally, in Sec. 8, we look at some groups that are known to lie beyond Kropholler's hierarchy with the class of finite groups as the base class. We investigate whether one can prove these results for the case where the base class is the much larger class of type U groups (See Definition 1.5). It is worth noting that although one should expect HF /, Z ¼ HF (as Conjecture 1.7 claims that H 1 F ¼ F /, Z ), for any arbitrary commutative ring A of finite global dimension in place of Z, we do not have the same expectation. We do not however know of a concrete example of a group in HF /, A nHF or LHF /, A nLHF, for some commutative A of finite global dimension.

Ikenaga's hierarchy
About 10 years before Peter Kropholler introduced his hierarchy of groups, Bruce Ikenaga used similar geometric ideas to inroduce his classes of groups which we define below. We need to provide the definition of a new invariant for a group, called the generalized cohomological dimension, first. (made over Z in [21]) For any commutative ring A and any group C, define the generalized cohomological dimension of C with respect to A, denoted cd A ðCÞ, to be supfn 2 Z !0 : Ext n AC ðM, FÞ 6 ¼ 0, for some A-free M and some AC-free Fg: The following are some useful facts regarding the generalized cohomological dimension: Ikenaga's classes of groups were defined in the following way. Definition 4.3. (based on Sec. 5, [21]) Let X be a class of groups. Define C 0 ðXÞ :¼ X, and a group C 2 C n ðXÞ iff there exists an acyclic simplicial complex X on which C acts by permuting the simplices such that C r 2 C nÀ1 ðXÞ, for each simplex r 2 X, where C r denotes the stabilizer of r, and sup r2R fdimðrÞ þ cd Z ðCÞg < 1, where R is a set of representatives of X modulo the C-action.
For groups in C 1 ðFÞ, the following was proved in [21]. Although it was not noted in [21], groups in C 1 ðFÞ actually admit complete resolutions, which we can show using the following result.   One can form Ikenaga's classes of groups starting with the class of all groups of type U as the base class. Whether or not we get any groups that we do not get when we start with the class of all finite groups as the base class is part of a conjecture (See Conjecture 4.13) that we make later.
Remark 4.7. Since both the definitions of Ikenaga's classes and Kropholler's hierarchy involve a kind of iteration on the definition of a level to get to the next level, it is natural to wonder whether one can do something similar with type U groups by iterating Definition 1.5. It turns out we can't as we explain below.
Let's fix an A of finite global dimension, and call type U groups type U 1 : For all n ! 1, define a group C to be of type U n if, for any AC-module M, M is of finite projective dimension as an AC-module iff it is of finite projective dimension over all type U nÀ1 subgroups.
If C is type U n , and M is of finite projective dimension over finite subgroups, then M is of finite projective dimension over type U subgroups, and by the iterative definition above, it is of finite projective dimension over type U 2 groups, and going on like this, it is of finite projective dimension over type U nÀ1 groups, from which it follows from the iterative definition above again, that proj: dim AC M < 1: Thus, C is of type U: Remark 4.8. By the main result of [22], it is now known that for every integer n, H nþ1 F is a strictly bigger class than H n F: No such result is known for Ikenaga's classes and that is why we feature this as a conjecture in Conjecture 4.16.
The following are some handy connections between the classes of groups we have introduced. Proof. Take A to be any commutative ring of finite global dimension. The only part of Lemma 4.9 that is new is the claim that C 1 ðFÞ F /, A : Let C 2 C 1 ðFÞ: So, by Corollary 4.6, C admits complete resolutions over A, and therefore Gcd A ðCÞ < 1: Since C 1 ðFÞ HF by Lemma 4.9.a., it follows from Theorem 1.9 that kðACÞ < 1: By Theorem 1.8.g., it now follows that C is of type U over A.
w It is noteworthy that the operator L is quite powerful in that when applied to classes of groups like C 1 ðFÞ, F /, A (for any A of finite global dimension) and H 1 F, it gives a strictly larger class of groups: Proof. Take C to be a free abelian group of infinite rank. Then any finitely generated subgroup of it, say a free abelian groups of finite rank n, acts on an n-dimensional CW-complex with R n as the underlying space, and therefore C 2 LH 1 F and by Lemma 4.9.a., is in LC 1 ðFÞ and LF /, A : C does not admit complete resolutions over A, so it is not in H 1 F, C 1 ðFÞ or F /, A : w It follows from Theorem 1.8.g. and Theorem 1.9 that for type U groups all of our invariants are finite and well-behaved (see Remark 4.11 below). Remark 4.11. Note that it follows from Theorem 1.8.g., Theorem 1.9, Theorem 4.2 and Remark 1.3 that if C 2 F /, A with A of finite global dimension, then C admits complete resolutions over A and all the invariants-cd A ðCÞ, Gcd A ðCÞ, proj: dim AC BðC, AÞ, silpðACÞ, spliðACÞ, fin: dimðACÞ, kðACÞ-are finite. We are recording this here because we will be making repeated use of this in the proof of Proposition 4.17.
We will be making repeated use of Remark 4.11 in proving the connections between the various conjectures in Proposition 4.9.
Before stating our conjectures, we need to state a close restatement of Conjecture 1.7, with the difference being that we include a statement on the classifying space of proper actions. We need to define the classifying space of proper actions of a group first. Definition 4.12. For any group C, EC denotes a CW-complex on which C acts cellularly with finite stabilizers such that for any finite subgroup G of C, the fixed point subcomplex EC G is contractible. (it is known that for any group, such a complex exists) Conjecture 4.13. Let A be a commutative ring of finite global dimension. For any group C, the following are equivalent.
Remark 4.14. Conjecture 4.13 looks very similar to Conjecture 1.7, except (g) of Conjecture 1.7 is the statement that C 2 H 1 F, but here (g) of Conjecture 4.13 is the statement that C admits a finite dimensional model for EC: Although it is clear that if C satisfies the latter it is definitely in H 1 F (see Sec. 4 of [29]), whether the converse holds is still open to conjecture (see Conjecture 43.1 of [11]).
It seems a sensible question to ask whether one could place all groups with complete resolutions within a known hierarchical class. The following result sheds some light in that direction. Proof. Let C be a group in LHF /, A that admits complete resolutions over A. Then, Gcd A ðCÞ < 1 by Theorem 1.9, and therefore by Theorem 1.9, kðACÞ is finite. So, by Theorem 1.8.g., C 2 F /, A : w Whether or not Proposition 4.15 can in any way be stated with the base class F instead of F /, A , i.e. whether we can say that any HF-group with complete resolutions has to be in a particular level of Kropholler's hierarchy, is an interesting question and it forms one of our conjectured statements below. In Conjecture 4.16 below, most of the statements are expectations based on evidence of a lack of examples to indicate otherwise. 4.16.b., for example, is a standard question to ask once all the different hierarchical classes have been defined in any hierarchy in general. The same logic applies to asking 4.16.c/d/f. For the following conjecture, we denote by CRðZÞ the class of all groups that admit complete resolutions over the integers. (a) HF \ CRðZÞ ¼ H 1 F: (b) C 1 ðFÞ ¼ C 2 ðFÞ ¼ ::: In the following result, whenever we say p 1 ) p 2 p 3 , or p 1 () p 2 p 3 , for some statements p 1 , p 2 , p 3 , we mean p 1 ) p 3 , or resp. p 1 () p 3 , if p 2 is assumed to be true.    [30], it follows that C 1 ðF / Þ ¼ H x F / (the proof in [30] is for F as the base class but it translates to the case where F / is the base class). Therefore, (g) It follows from Theorem 1.8.a., Theorem 1.8.d. and Theorem 1.8.f. that we can streamline our hypothesis to 4:13:e: ) 4:13:g: (we denote this statement by ðÃÞ). We assume ðÃÞ is true. ðÃÞ ) 4:16:c: : Now, if C 2 C 1 ðFÞ, then fin: dimðZCÞ < 1 by Lemma 4.9 and Remark 4.11, so there is a finite dimensional model for EC, so clearly C 2 H 1 F (see Remark 4.14). Thus 4.16.c. holds, and so 4.16.a-c. hold as well by part (a) of this proposition. ðÃÞ ) 4:16:d: : If C 2 F / , fin: dimðZCÞ < 1 by Remark 4.11, and since ðÃÞ holds, there is a finite dimensional model for EC, therefore C 2 H 1 F ¼ C 1 ðFÞ C 1 ðFÞ: So, 4.16.d. holds as we already know that C 1 ðFÞ F / by Lemma 4.9.b. ðÃÞ ) 4:16:e: : If C be a group such that cd Z ðCÞ < 1, then by Theorem 4.2, Theorem 1.8.d. and Theorem 1.8.a., fin: dimðZCÞ < 1, and therefore there is a finite dimensional model for EC, so C 2 H 1 F ¼ C 1 ðFÞ: Again note that we already know that groups in Ikenaga's classes have finite generalized cohomological dimension over the integers (Theorem 4.4).

A small result on classifying spaces
As we saw in Proposition 4.17.g., the finiteness of almost any cohomological invariant for C implying the existence of a finite dimensional model for EC is quite strong. In this section, we show using a key result from [26] that some of the classes of groups that we have dealt with admit finite dimensional models for their classifying space of proper actions if an additional condition is satisfied. To introduce this additional condition, we need the following definition.
Definition 5.1. For a finite group G, define the length of G, denoted l(G), as the supremum over n such that there is a nested sequence H 0 ( H 1 ( :: ( H n where each H i is a subgroup of G. Definition 5.2. For any integer d, a group C is said to be of type b(d) if for every ZC-module M that is projective over finite subgroups, proj: dim ZC M d: C is said to be of type B(d) if, for every finite G C, W C ðGÞ : It is easy to note that groups of type U over the integers are of type b(d) for some d ! 0 : The following is the key result from [26] that we will be using in this section. Proof. From Theorem 5.4, it follows that we will be done if we show that C is of type B(d) for some d ! 0: From Remark 4.11, it follows that Gcd Z ðCÞ < 1: For any finite subgroup G C, W C ðGÞ is in LHF (this follows from the fact that HF is Weyl group closed-see Proposition 7.1 of [25]). So, it follows from Theorem 1.9 that kðZW C ðGÞÞ Gcd Z ðW C ðGÞÞ þ 1 Gcd Z ðCÞ þ 1 (the last inequality is by Proposition 2.5 of [16]). Thus, W C ðGÞ is of type bðGcd Z ðCÞ þ 1Þ: So we have shown that C is of type BðGcd Z ðCÞ þ 1Þ, and we are done. w It is interesting to note that we can replace the hypothesis C 2 LHF \ F / in the statement of Proposition 5.5 by C 2 C 1 ðFÞ: Corollary 5.6. Let C be in C 1 ðFÞ with a bound on the length of its finite subgroups. Then, there is a finite dimensional model for EC: Proof. This follows directly from Proposition 5.5 using Lemma 4.9.
w The hypothesis of Corollary 5.6 is weaker than that of Proposition 5.5, but we state Corollary 5.6 separately because it is an interesting question as to what the connection is between groups in Ikenaga's hierarchy and groups admitting finite dimensional models for their classifying space of proper actions.

Two general questions on projectivity
It is well-known that for a finite group G, a ZG-module M is projective iff M is Z-free and of finite projective dimension as a ZG-module. In [23], the following question was asked: Question 6.1. (Question A of [23]) Fix Z to be the base ring. Is it only for finite groups G that a ZG-module is projective iff it is Z-free and of finite projective dimension as a ZG-module?
In making some progress on Question 6.1, the following theorem was proved in [23]. Theorem 6.2. (Theorem 2.4 of [23]) Let C be a group such that every Z-free ZC-module of finite projective dimension is projective. If C 2 HF, then C is finite.
One can prove the statement of Theorem 6.2 replacing "C 2 HF" with "C 2 LHF / ": Theorem 6.3. Let C be a group such that every Z-free ZC-module of finite projective dimension is projective. If C 2 LHF / , then C is finite.
w The second question on projectivity that [23] tackles deals with stably flat modules as defined in Definition 6.4. Stably flat modules arise in the study of complete cohomology for infinite groups. This is again a concept that we have not dealt with elsewhere, so we provide a definition below: We can prove the statement of Theorem 6.5 replacing Z with a coherent commutative ring A of finite global dimension and the condition "C 2 LHF" with "C 2 LHF /, A :" To do this proof, we need two results about stably flats that come from Alcock. Alcock [2] considered the question of whether, for some class of infinite groups and with certain conditions on the base ring, one can provide a complete characterization of stably flat modules in terms of projective dimension and proved the following: Theorem 6.6. (Theorem A of [2]) Let A be a coherent commutative ring (i.e. every finitely generated ideal is finitely presented as a module) with finite global dimension and let C 2 H 1 F. Then, for any AC-module N, the following are equivalent.
(a) N is stably flat as an AC-module. (b) proj: dim AC N < 1: We have seen before that H 1 F F /, A : We can now prove the statement of Theorem 6.6 replacing the condition C 2 H 1 F with C 2 F /, A : Theorem 6.7. Let A be a coherent commutative ring of finite global dimension and let C 2 F /, A . Then, for any AC-module N, the following are equivalent: (a) N is stably flat as an AC-module. (b) proj: dim AC N < 1: Proof. ðbÞ ) ðaÞ: This is obvious as if proj: dim AC N < 1, then c Ext 0 AC ðM, NÞ ¼ 0 for all M of type FP 1 because complete cohomology vanishes on modules with finite projective dimension.
ðaÞ ) ðbÞ: If N is stably flat as an AC-module, then by Corollary 3.4 of [2], N is stably flat as an AGmodule for all finite G C, and by Theorem A 0 of [2] (or even just by Theorem 6.6), proj: dim AG N < 1 for all finite G C, and therefore proj: dim AC N < 1 as C is of type U over A. w Theorem 6.8. Let A be a coherent commutative ring of finite global dimension and let C 2 LHF /, A . For any AC-module M, if M is stably flat as an AC-module and also a Benson's cofibrant, then it is projective.
Proof. First, we deal with the case when C 2 HF /, A : We proceed by transfinite induction on the smallest ordinal a such that C 2 H a F /, A : If a ¼ 0, then by Theorem 6.7, proj: dim AC M < 1: So, by Proposition 5.4 of [4], M is projective. Now, as our induction hypothesis, assume that the statement of the theorem holds for all C 2 H b F /, A for all ordinals b < a: If, now, C 2 H a F /, A , then C acts on a finite dimensional contractible CW-complex with stabilizers in H <a F /, A , and by tensoring the augmented cellular complex with M, we get a finite length resolution of M with modules that are direct sums of modules of the form Ind C C 0 ðRes C C 0 ðMÞÞ for some C 0 2 H <a F /, A (Here Ind and Res denote the induction and restriction functors respectively). As an AC 0 -module, Res C C 0 M is stably flat by Corollary 3.4 of [2] and also Benson's cofibrant as cofibrants remain cofibrant upon restriction to subgroups (see Remark 6.9 below), and so by our induction hypothesis, it is projective as an AC 0 -module and Ind C C 0 ðRes C C 0 ðMÞÞ is projective as an AC-module. Therefore, M has finite projective dimension as an AC-module. Again by Proposition 5.4 of [4], M is projective. This ends our proof for the case where C 2 HF /, A : Now, let C 2 LHF /, A : We can assume that C is uncountable because if it is countable then since every countable group admits an action on a tree with finitely generated vertex and edge stabilizers (see Lemma 2.5 of [22]), it follows that C 2 HF /, A : Assume, as an induction hypothesis, that the theorem has been proved for all groups with cardinality strictly smaller than C. We can express C as an ascending union of subgroups fC k : k 2 Kg where each C k is of strictly smaller cardinality than C. By the induction hypothesis, M is projective over each C k (note that again, to go down to subgroups here, we are using Corollary 3.4 of [2] and the fact that cofibrants remain cofibrants when restricted to subgroups), and so by Lemma 5.6 of [4], proj: dim AC M 1: And again, this means M is Gorenstein projective with finite projective dimension, so it must be projective. w Remark 6.9. (See Lemma 4.8 of [8]) For any group C and any commutative ring A, take an AC-module M such that M A BðC, AÞ is projective as an AC-module. This implies that Res C C 0 ðM A BðC, AÞÞ is projective as an AC 0 -module for any subgroup C 0 of C. BðC 0 , AÞ is a direct summand of Res C C 0 ðBðC, AÞÞ-this is proved in [27] for A ¼ Z and it follows over any commutative ring A because BðC, AÞ ¼ BðC, ZÞ Z A (see the proof of Lemma 3.4 of [4]). So, Res C C 0 M A BðC 0 , AÞ is a summand of Res C C 0 M A Res C C 0 BðC, AÞ ¼ Res C C 0 ðM A BðC, AÞÞ which is projective. Therefore, Res C C 0 M is cofibrant as an AC 0 -module. Our proof of Theorem 6.8 here is quite independent of the way Theorem 6.5 is proved in [23].

Periodic cohomology and complete resolutions
We have seen before how for a group the property of admitting complete resolutions is quite helpful in dealing with many questions. A good indicator of whether a group admits (weak) complete resolutions or not, is checking whether it has periodic cohomology after a finite number of steps (Proposition 3.1 of [35]). Since we didn't work much with periodic cohomology elsewhere, we provide its definition here. Definition 7.1. (see [30,35]) A group C is said to have periodic cohomology of period q after k steps iff the functors H i ðC, ?Þ and H iþq ðC, ?Þ are naturally equivalent for all i > k: The following important conjecture was made for groups with periodic cohomology in [35]. [35]) A group C has periodic cohomology after some steps iff C admits a finite-dimensional free C-CW-complex, homotopy equivalent to a sphere.
Talelli settled Conjecture 7.2 for HF-groups in Corollary 3.5 of [35]. Almost the same proof works for LHF / -groups. Proof. We assume that C has periodic cohomology of period q after k steps. By Proposition 3.1 of [35], C admits a weak complete resolution, and since C 2 LHF / , this implies that C admits complete resolutions by Theorem 1.12, and therefore by Theorem 1.8.e. and Remark 1.3, silpðZCÞ < 1: So, by Theorem 3.2 and Corollary 3.3 of [35], the periodicity isomorphisms are induced by the cup product in H q ðG, ZÞ, and as noted in [35], Adem and Smith [1] proved that Conjecture 7.2 holds when this happens. It was shown in [24] that Thompson's group F :¼ hx 0 , x 1 , x 2 , :: : x À1 k x n x k ¼ x nþ1 , k < ni is not in LHF: Using basically the same argument, we can say that F is not in LHF / , as we show here.
First, we quote the following theorem from [24] which is one of the main results of that paper.
Theorem 8.1. (Theorem A of [24]) Let X be a class of groups and let A be a commutative ring. Take an LHX-group C and an AC-module M. Assume that Ext n AC ðM, ?Þ commutes with direct limits for infinitely many non-negative n. Then, the following statements are equivalent.
(a) proj: dim AC M < 1: (b) proj: dim AC 0 M < 1, for all C 0 C such that C 0 2 X: Corollary 8.2. Thompson's group F is not in LHF / : Proof. Note that for any group C and any commutative ring A, an AC-module M is of type FP 1 iff the functors Ext Ã AC ðM, ?Þ commute with direct limits. Now take X ¼ F / and A ¼ Z in the statement of Theorem 8.1, and let M be of type FP 1 : Then, if C 2 LHF / , then proj: dim ZC M < 1 iff proj: dim ZC 0 M < 1 for all type U subgroups C 0 C, which in turn can happen iff proj: dim ZG M < 1 for all finite subgroups G C (this last bit follows from the definition of type U groups).
It follows from Corollary 5.4 of [10] that F is of type FP 1 , i.e. the trivial module Z is of type FP 1 as a ZF-module, and it follows from Corollary 1.5 of [10] that F is torsion-free. So, the only finite subgroup of F is the trivial subgroup. It therefore follows from the preceding paragraph that if F 2 LHF / , then proj: dim ZF Z < 1, i.e. F has finite cohomological dimension over Z, which is not possible as F contains a free abelian group of infinite rank which has infinite cohomological dimension.
w For a long time, since F was the most well-known group outside LHF, the most common way to show a group was not in HF or LHF was to show that it had a subgroup isomorphic to F. In [3], the authors introduce a different set of methods that give examples of groups outside Kropholler's hierarchy. We quote below one of the main theorems of [3]. Theorem 8.3. (part of Theorem 1.1 of [3]) There exists an infinite finitely generated group Q which cannot act on any finite dimensional CW-complex without a global fixed point. For any countable group C, Q can be chosen so that Q is simple, has Kazhdan's property (T) and contains an isomorphic copy of C.
It is natural to ask if the groups constructed in proving Theorem 8.3 in [3] are in HF / : Remark 8.4. It has been noted in [3], and it is also easy to see, that if Q is a group satisfying the statement of Theorem 8.3, then Q 2 HX, for any class X, iff Q 2 X: Taking X ¼ F / , we get that if Q 2 HF / , then Q is of type U: But if in the statement of Theorem 8.3, we take C to be the free abelian group of rank @ 0 , then Q cannot admit complete resolutions as C does not admit complete resolutions, and therefore Q cannot be of type U as groups of type U admit complete resolutions (see Remark 4.11). It is noteworthy that to reach this conclusion, we are having to choose a convenient countable group for C.
Another known concrete example of a group outside HF is the first Grigorchuk group (Theorem 4.11 of [17]). A major ingredient in the proof of Theorem 4.11 of [17] is the following result of Petrosyan [31].
Theorem 8.5. (Theorem 3.2 of [31]) Take A to be a commutative ring, and let C be a discrete group with no A-torsion such that it has jump cohomology of height k over A, which means that for any subgroup C 1 C, cd A ðC 1 Þ < 1 implies cd A ðC 1 Þ k. If C 2 HF, then cd A ðCÞ k. So, an HF-group C can have jump cohomology of height k over Q if and only if cd Q ðCÞ k: It is easy to see that the statement of Theorem 8.5 holds with HF replaced by HF /, A , for any commutative ring A of finite global dimension. Corollary 8.6. Let A be a commutative ring and let C be a discrete group with no A-torsion. Assume that there exists a non-negative integer k such that for any subgroup C 1 C, cd A ðC 1 Þ < 1 implies cd A ðC 1 Þ k. Now, if C 2 HF /, A , then cd A ðCÞ k: Proof. Theorem 8.5 is proved in [31] by first proving it for the base case, i.e. when C is finite, and then proving it by transfinite induction on the level of C in HF: We reproduce that proof for our case.
For our base case of type U groups, note that if C is of type U, then cd A ðGÞ ¼ 0 for all finite G C since G needs to be A-torsion-free as per the hypothesis of Theorem 8.5. Thus, cd A ðCÞ < 1 by definition of type U groups. Now, we make the following induction hypothesis: for some fixed ordinal a, cd A ðC 0 Þ < k for any H <a F /, A -subgroup C 0 of C (note that from the hypothesis of Corollary 8.6, it follows that cd A ðC 0 Þ < 1 iff cd A ðC 0 Þ < k). Let C 00 be an H a F /, A -subgroup of C. Then, C 00 acts on a finite dimensional contractible CW-complex X with stabilizers in H <a F /, A : By our induction hypothesis, all these stabilizers have cohomological dimension at most k over A. If the dimension of X is n, then using Lemma 3.3 of [31], we get that cd A ðC 00 Þ k þ n: Again from the hypothesis of Corollary 8.6 as noted in parentheses above, it now follows that cd A ðC 00 Þ k: We have thus proved that any HF /, A -subgroup of C has cohomological dimension at most k over A. So, if C 2 HF /, A , then cd A ðCÞ k: w Corollary 8.7. The first Grigorchuk group is not in HF /, Q : Proof. It is shown in Theorem 4.11 of [17] that the first Grigorchuk group has jump rational cohomology of height 1 and has infinite cohomological dimension over the rationals, so by Corollary 8.6, it is not in HF /, Q : w