Coarse coherence of metric spaces and groups and its permanence properties
aa r X i v : . [ m a t h . K T ] S e p COARSE COHERENCE OF METRIC SPACES AND GROUPSAND ITS PERMANENCE PROPERTIES
BORIS GOLDFARB AND JONATHAN L. GROSSMAN
Abstract.
We introduce properties of metric spaces and, specifically, finitelygenerated groups with word metrics which we call coarse coherence and coarseregular coherence . They are geometric counterparts of the classical algebraicnotion of coherence and the regular coherence property of groups defined andstudied by F. Waldhausen. The new properties can be defined in the generalcontext of coarse metric geometry and are coarse invariants. In particular, theyare quasi-isometry invariants of spaces and groups.We show that coarse regular coherence implies weak regular coherence,a weakening of regular coherence by G. Carlsson and the first author. Thelatter was introduced with the same goal as Waldhausen’s, in order to performcomputations of algebraic K -theory of group rings. However, all groups knownto be weakly regular coherent are also coarsely regular coherent. The newframework allows us to prove structural results by developing permanenceproperties, including the particularly important fibering permanence property,for coarse regular coherence. Precursors of coarse coherence
Let A be an associative ring. All modules over A we consider are left A -modules. Definition 1.1. A presentation of an A -module E is an exact sequence F → F → E → F and F free A -modules. It is a finite presentation if the freemodules are finitely generated. More generally, one has the notion of a projectiveresolution of E which is an exact sequence . . . −→ P n −→ . . . −→ P −→ P −→ E −→ P i are projective A -modules. The projective resolution is of finite type ifthe projective modules are finitely generated. It is called finite if there is a number n such that the modules P i = 0 for i > n .A module is said to be of type FP ∞ or have finite projective dimension if it hasa projective resolution of finite type. The ring A has finite global dimension if thereis a number n such that every finitely generated A -module has a finite projectiveresolution of length n .The ring A is called coherent if every finitely presented A -module is of typeFP ∞ . A coherent ring A is called regular coherent if each projective resolution offinite type over A is chain homotopy equivalent to a finite projective resolution.Restricting further, a regular Noetherian ring is a regular coherent ring which isNoetherian in the usual sense—a submodule of any finitely generated module over A is finitely generated. Date : September 11, 2018.
The most important regular Noetherian ring for applications in geometric topol-ogy is the ring of integers Z . Definition 1.2 (Waldhausen [14]) . A group Γ is regular coherent if the groupalgebra R [Γ] is regular coherent for any choice of a regular Noetherian ring R .The collection X of regular coherent groups includes free groups, free abeliangroups, torsion-free one relator groups, fundamental groups of submanifolds ofthe three-dimensional sphere, and their various amalgamated products and HNNextensions and so, in particular, the fundamental groups of submanifolds of thethree-dimensional sphere. Waldhausen used this property to compute the algebraic K -theory of regular coherent groups.Two remarks regarding Waldhausen’s regular coherence are in order.(1) The regular coherence property seems to be very special: simply constructingindividual non-projective finite dimensional modules over group rings is hard.(2) The collection X is not well-understood structurally beyond the portion iden-tified by Waldhausen. For example, it is unknown whether X is closed under prod-ucts. While all groups in X are necessarily torsion-free, it is unknown if there is atorsion-free group outside of it.Waldhausen asked if a weaker property of the group ring would suffice in hisargument, see for example the paragraph after the proof of Theorem 11.2 in [14].This paper is a response to that question.A weakening of regular coherence ( weak regular coherence ) was introduced in[2, 3, 6, 8] with essentially the same goal as Waldhausen’s which was a computationof the K -theory of weakly regular coherent groups. However, in contrast with thesituation for X , the class of weakly regular coherent groups is known to be verylarge. It includes groups that admit a finite classifying space and have straight finitedecomposition complexity and so, in particular, groups that have finite asymptoticdimension. The notion of weak coherence is more technical to define; we do thatas part of the narrative in section 4. Unfortunately one realizes quickly that eventhough the notion of weak regular coherence can be verified for a large family ofgroups, it is not amenable to proving structural permanence results. The goal and structure of the paper.
Our goal is to define a genuinely coarsegeometric property of metric spaces that ensures that finitely generated groups withword metrics that have this property also have the weak regular coherence property.Then we show that the class of coarsely regular coherent groups Y is closed undermany natural geometric operations. This collection of results has became knownas permanence properties [10] in the literature. For example, invariance of Y underfinite products is a simple consequence of this paper.We start by defining the general non-equivariant property of metric spaces called coarse coherence in section 2 and stating main technical results. Section 3 containsproofs of permanence results for coarse coherence. In the last section 4 of the paper,we define coarse regular coherence which is a property of groups. We relate thisproperty to weak regular coherence and finally state and prove theorems aboutcoarse regular coherence together with some immediate applications. Acknowledgements.
We would like to thank Daniel Kasprowski, Marco Varisco,and the referee for comments that improved the strength of the results and the pre-cision of the narrative.
OARSE COHERENCE OF METRIC SPACES 3 Definition of coarse coherence
Let X be a metric space, R be a ring. Definition 2.1. An X -filtered R -module is a covariant functor F : P ( X ) → Mod R from the power set of X ordered by inclusion to the category of R -modules andinjective homomorphisms. It will be convenient to view F as the value F ( X ) filteredby submodules associated to subsets S ⊂ X . We will always assume that the valueof F on the empty subset is 0.The notation S [ r ] stands for the metric r -enlargement of S in X . So, in particular, x [ r ] is the closed metric ball of radius r centered at x .(1) F is called lean or D - lean if there is a number D ≥ F ( S ) ⊂ X x ∈ S F ( x [ D ])for every subset S of X .(2) F is called scattered or δ - scattered if there is a number δ ≥ F ( X ) ⊂ X x ∈ X F ( x [ δ ]) . (3) F is called insular or d - insular if there is a number d ≥ F ( S ) ∩ F ( U ) ⊂ F ( S [ d ] ∩ U [ d ])for every pair of subsets S , U of X .(4) F is locally finitely generated if F ( S ) is a finitely generated R -module forevery bounded subset S ⊂ X . Remark 2.2.
We note that being scattered is a consequence of being lean but notthe other way around.
Definition 2.3. An R -homomorphism f : F → F ′ of X -filtered modules is con-trolled if there is a fixed number b ≥ f ( F ( S )) is a submoduleof F ′ ( S [ b ]) for all subsets S of X .The filtered modules that are lean and insular form a category LI ( X, R ), wherethe morphisms are the controlled R -homomorphisms. The category B ( X, R ) is thefull subcategory of LI ( X, R ) on the locally finitely generated objects. Basic proper-ties of this category can be found in section 3.1 of [5].Let us point out that the geometric features we define are independent of theassumption that R is Noetherian which is used throughout [5]. Example 2.4.
Given any ring R and any metric space X , one can consider a specialkind of filtrations which go back to the original geometric modules of Pedersen-Weibel. The geometric modules are a collection of choices F x which are free finitelygenerated R -modules associated to each point x in X , with the requirement thatonly finitely many F x are non-zero for x from a bounded subset S . Now the module F ( X ) = L F x is assigned the filtration given by F ( S ) = L x ∈ S F x . Let us denotethe full subcategory of geometric modules in B ( X, R ) by B ( X, R ).Within LI ( X, R ) there is a special kind of morphisms which were used in thedescription of the exact structure in LI ( X, R ). BORIS GOLDFARB AND JONATHAN L. GROSSMAN
Definition 2.5.
A homomorphism f is bicontrolled if, for some fixed b ≥
0, inaddition to inclusions of submodules f ( F ( S )) ⊂ F ′ ( S [ b ]), there are inclusions f ( F ) ∩ F ′ ( S ) ⊂ f F ( S [ b ]) for all subsets S ⊂ X .Here is a list of facts about lean and insular modules. Theorem 2.6.
Let → E ′ f −−→ E g −−→ E ′′ → be an exact sequence of X -filtered R -modules where f and g are bicontrolled. (1) If the object E is lean then E ′′ is lean. (2) If E is insular then E ′ is insular. (3) If E is insular and E ′ is lean then E ′′ is insular. (4) If both E ′ and E ′′ are lean then E is lean. (5) If both E ′ and E ′′ are insular then E is insular.Proof. This is a summary of results from section 3.1 in [5]. (cid:3)
There are several viable conditions on X that enforce a version of the “missing”item in this theorem. All of them can be viewed as relaxations of the algebraiccoherence property when X is a group Γ with a word metric. In this case thefiltered R -modules are Γ-equivariant, thus becoming R [Γ]-modules, with the maps f and g being R [Γ]-homomorphisms.The first condition is the “missing” item itself. Definition 2.7 (Coherence of Metric Spaces) . A metric space X is coherent if inany exact sequence 0 → E ′ f −−→ E g −−→ E ′′ → X -filtered R -modules where f and g are both bicontrolled maps, the combinationof E being lean and E ′′ being insular implies that E ′ is necessarily lean.For example, it is shown in Grossman [9] that the real line with the standardmetric is a coherent metric space. It is likely that all groups from Waldhausen’sclass X are coherent. Loc. cit. also contains basic properties of coherence such ascoarse, and therefore quasi-isometry, invariance. It turns out however that for themost desired permanence properties this notion is too restrictive.The following definition isolates the most important property in terms of itsalgebraic impact which also happens to be amenable to fibering permanence results. Definition 2.8 (Coarse Coherence) . A metric space X is coarsely coherent if inany exact sequence 0 → E ′ f −−→ E g −−→ E ′′ → X -filtered R -modules where f and g are both bicontrolled maps, the combinationof E being lean and E ′′ being insular implies that E ′ is necessarily scattered. Notation . We will use the notation C for the class of all coarsely coherent metricspaces.There is a version of this condition for metric families. For this definition we usethe terminology from Guentner [10]. We find the notion of the total space of thefamily especially convenient for our purposes. OARSE COHERENCE OF METRIC SPACES 5
Definition 2.10. A metric family { X α } is simply a collection of metric spaces X α .In all situations in this paper, a metric family will be a collection of subspaces ofa given metric space, each equipped with the subspace metric. The total space X of the family { X α } is the disjoint union of the metric spaces X α with the extendedmetric with values ∞ between points from X α and X β for α = β . Definition 2.11 (Coarse Coherence for Families) . A metric family { X α } is coarselycoherent if the total space X of the family is coarsely coherent.It might be instructive to spell out what this entails. The total space is coarselycoherent if for a collection of exact sequences0 → E ′ α f α −−−→ E α g α −−−→ E ′′ α → X α -filtered R -modules where all E α are D -lean, all E ′′ α are d -insular, all f α and g α are all b -bicontrolled maps for some fixed constants D , d , b ≥
0, it follows thatall E ′ α are ∂ - scattered for some uniform constant ∂ ≥ C , we mean that the family is coarsely coherent.This is definitely a stronger assumption than each space in the family being in C .The following is the main permanence result of the paper.A map between metric spaces f : X → Y is uniformly expansive if there is afunction φ : [0 , ∞ ) → [0 , ∞ ) such that d Y ( f ( x ) , f ( x )) ≤ φ ( d X ( x , x )) for allpairs of points x , x from X . Theorem 2.12 (Fibering Permanence for Coarse Coherence) . Assume π : X → Y is a uniformly expansive map with Y in C . If for any r > the family { f − ( y [ r ]) | y ∈ Y } is in C , then X is in C . The proof of this theorem will require establishing other permanence theoremswhich we do in the next section. We will also point out that there is a strongertheorem that can be formulated in terms of families which is a consequence ofTheorem 2.12 and which has many other permanence theorems as corollaries.3.
Permanence properties and applications
Let us first introduce additional terminology from coarse geometry.Two functions h , h : X → Y between metric space are close if there is aconstant C ≥ d Y ( h ( x ) , h ( x )) ≤ C for all choices of x in X . A function k : X → Y is a coarse equivalence if it is uniformly expansive and there exists auniformly expansive function l : Y → X so that the compositions k ◦ l and l ◦ k areclose to the identity maps.An example of a coarse equivalence is the notion of quasi-isometry. This is simplya coarse equivalence k for which the uniformly expansive functions for k and itscoarse inverse can be chosen to be linear polynomials. In geometric group theory,it is very useful that any two choices for a finite generating set of a group producequasi-isometric word metrics.The following are basic permanence properties of coarse coherence. Theorem 3.1 (Coarse Invariance of Coarse Coherence) . If X and Y are coarselyequivalent then X is coarsely coherent if and only if Y is coarsely coherent. Theorem 3.2 (Subspace Permanence for Coarse Coherence) . If X is a subspaceof Y , and Y is coarsely coherent, then X is coarsely coherent. BORIS GOLDFARB AND JONATHAN L. GROSSMAN
From Lemma 6.1 of Guentner [10], the combination of the two statements is trueif and only if a different coarse geometric condition holds. We recall that a map k : X → Y is a coarse embedding if k is a uniformly expansive map which is a coarseequivalence onto its image. In this situation, the uniformly expansive counterpart l : im( k ) → X is called a coarse inverse . Now the combination of Theorems 3.1 and3.2 is true if and only if whenever Y is coarsely coherent and X coarsely embeds in Y then X is coarsely coherent, so it suffices to prove the latter statement. Proof.
Suppose k : X → Y is a coarse embedding controlled by the function ℓ . Weassume that Y is coarsely coherent, and we are given an exact sequence0 → E ′ X f −−→ E X g −−→ E ′′ X → X -filtered modules where f and g are both b -bicontrolled, E X is D -lean and E ′′ X is d -insular. We aim to show that E ′ X is scattered. Consider the Y -filtrations of themodules induced by k as follows: E Y ( S ) = E X ( k − ( S )). It follows that E Y is ℓ ( D )-lean, and E ′′ Y is ℓ ( d )-insular. Also f and g are bicontrolled as morphisms between Y -filtered modules. We can conclude that E ′ Y is δ -scattered for some δ ≥
0, since Y is coarsely coherent. Now using the same kind of estimate, E ′ X is ℓ ( δ )-scattered. (cid:3) There are two types of natural filtrations that can be assigned to submodules offiltered modules.
Definition 3.3. (1) Suppose F is a filtered module and F ′ is any submodule of F .Then F ′ can be given the canonical filtration F ′ ( S ) = F ( S ) ∩ F ′ . It is clear that if F is an insular filtered module then F ′ is also insular.(2) Suppose T is a subset of X . For any choice of a number D ≥
0, one hasthe submodules F T,D ( S ) = P x ∈ S ∩ T F ( x [ D ]) for all S ⊂ X . They give a filtrationof F T,D = P x ∈ T F ( x [ D ]). It follows easily that F T,D is always a lean X -filteredmodule.Notice that, as defined, both filtrations are X -filtrations.We are ready to prove the Fibering Permanence Theorem 2.12. Proof of Theorem . First, observe that in view of Theorem 3.2 we may assumethat π : X → Y is surjective. Given an exact sequence0 → E ′ f −−→ E g −−→ E ′′ → X -filtered R -modules where f and g are b -bicontrolled maps, E is D -lean, and E ′′ is d -insular, we want to show E ′ is scattered. There is a Y -filtration of E ′ givenby E ′ Y ( T ) = E ′ ( π − ( T )). It is easy to see that the exact sequence0 → E ′ Y f −−→ E Y g −−→ E ′′ Y → Y -filtered modules has the same properties with respect to the new filtrations: f and g are bicontrolled, E Y is lean, and E ′′ Y is insular. This allows to conclude that E ′ Y is δ Y -scattered for some number δ Y ≥
0, so every k ∈ E ′ Y is a sum k = P k y where k y ∈ E ′ Y ( y [ δ Y ]) = E ′ ( π − ( y [ δ Y ])).Suppose ℓ is a uniform expansion control function for π . Let E y be the filteredmodule E π − ( y [ δ Y + ℓ ( b )]) ,D , described as option (2) in Definition 3.3, which is a leanmodule containing f E ′ Y ( y [ δ Y ]). The kernel element k y is in the kernel of the re-striction map g : E y → E ′′ Y ( y [ δ Y + 2 ℓ ( b ) + ℓ ( D )]). The image E ′′ y = g ( E y ) is giventhe canonical filtration induced from the insular filtration of E ′′ which makes E ′′ y OARSE COHERENCE OF METRIC SPACES 7 insular. By the family assumption, we conclude that the kernel of g : E y → E ′′ y is δ -scattered for a constant δ ≥ y . So k y ∈ X x ∈ π − ( y [ δ Y ]) E ′ ( x [ δ ])because of our assumption that π is surjective. The conclusion is that E ′ is δ -scattered. (cid:3) Recall that straight finite decomposition complexity (sFDC) is a property ofmetric spaces introduced by A. Dranishnikov and M. Zarichnyi [7]. The class ofgroups with sFDC is remarkably broad. It includes groups with finite asymptoticdimension, all elementary amenable groups, and all countable subgroups of almostconnected Lie groups.
Theorem 3.4.
A metric space X with sFDC is coarsely coherent.Proof. The proof of the main result from [8] applies literally and verifies coarsecoherence of the metric space. (cid:3)
It is known that a metric space with finite asymptotic dimension has sFDC.
Corollary 3.5.
A metric space X with finite asymptotic dimension is coarselycoherent. The following simple corollary to Theorem 2.12 shows that coarse coherence offinitely generated groups is preserved by group extensions.
Corollary 3.6.
Let π : G → H be a surjective homomorphism from a finitely gen-erated group G . We assume that the groups are given word metrics with respect tofinite generating sets and that the kernel K is given the subspace metric. If K iscoarsely coherent and H is coarsely coherent then G is coarsely coherent.Proof. If S is a finite generating set for G , π ( S ) can be used as a finite generatingset of H , and the resulting word metric space is known to be coarsely coherentby quasi-isometry invariance of coarse coherence. The isometric action of G on H is transported from the left action of G on itself, so the action is by isometries.In this situation, the quasi-stabilizers W r ( e ) from the proof of [1, Theorem 7] are π − ( e [ r ]) = K [ r ]. Since K is coarsely equivalent to K [ r ], we have π − ( e [ r ]) iscoarsely coherent for any r and so all π − ( x [ r ]) are coarsely coherent by quasi-isometry invariance. This means that for a fixed r > { π − ( x [ R ]) } x ∈ X is uniformly coarsely coherent. We conclude that G is coarsely coherent by applyingthe Fibering Permanence Theorem 2.12. (cid:3) Using more of the recent technology developed by D. Kasprowski, A. Nicas, andD. Rosenthal in [11] we can deduce many other permanence properties for coarsecoherence. As a consequence, coarse coherence is preserved under many other grouptheoretic constructions.First we observe that Theorem 3.2 together with the Subspace Permanence fromTheorem 3.2 confirm the following General Fibering Permanence property of coarsecoherence. For the sake of brevity we don’t review the extension of uniformly ex-pansive maps of metric families. The details can be found in [11].
Theorem 3.7.
Assume π : X → Y is a uniformly expansive map with Y in C . Iffor any uniformly bounded subfamily B of Y we have that the family f − ( B ) isnecessarily in C , then X is in C . BORIS GOLDFARB AND JONATHAN L. GROSSMAN
Remark 3.8.
The remarkable sequence of Theorems 5.4, 5.6, 5.8, 5.10 and 5.12from [11] establishes a number of permanence properties as formal consequencesof Theorem 3.7 for a collection of metric families C which also contains all metricfamilies with finite asymptotic dimension. We have seen that this is so for coarsecoherence from Corollary 3.5. The new properties that follow are the Finite Amal-gamation Permanence, Finite Union Permanence, Union Permanence and LimitPermanence. We refer to [11] for the precise definitions and details.4. Coarse regular coherence
We want to define the new property coarse regular coherence of finitely generatedgroups and leverage our permanence theorems for coarse coherence to say what weknow about the class of coarsely regular coherent groups. We also want to recallthe notion of weak regular coherence from [3, 6, 8] and explain the relationship tothis paper.In the rest of this section, R will be a commutative Noetherian ring.Let Γ be a finitely generated group with a word metric. There is an isometricaction of Γ on itself by left multiplication. Definition 4.1.
A Γ-filtered R -module is Γ- equivariant or simply a Γ- module if F ( γS ) = γF ( S ) for all choices of γ ∈ Γ and S ⊂ Γ.It is clear that a Γ-module has the structure of an R [Γ]-module. It turns outthat any R [Γ]-module can be given a Γ-filtration specific to a finite generating setin Γ as follows. Given a left R [Γ]-module F with a finite generating set Σ, it is alsoan R -module with the generating set Σ ′ = { γσ ∈ F | γ ∈ Γ , σ ∈ Σ } . Now one canassociate to every subset S of Γ the left R -submodule F ( S ) generated by γσ ∈ Σ ′ such that γ ∈ S and σ ∈ Σ. This gives a functor F : P (Γ) → Mod R ( F ), from thepower set of Γ to the R -submodules of F such that F (Γ) = F , F ( ∅ ) = 0, and for abounded subset T ⊂ Γ, F ( T ) is a finitely generated R -module. This shows that F with the given filtration, which will be denoted by F Σ , is a Γ-filtered R -module. Itis easy to see that F Σ is Γ-equivariant, so F Σ is a Γ-module. Clearly, F Σ is 0-leanby design.As an example, a finitely generated free R [Γ]-module with a finite set of freegenerators Σ can be given the structure of a Γ-module as above. In this case Σ ′ =Γ × Σ. It is easy to see that in this case the Γ-filtration is 0-lean and 0-insular. Itfollows from Theorem 2.6 that images of idempotents of lean, insular, locally finitelygenerated modules also have these three properties. This allows us to generateexamples of such modules from idempotents of free geometric modules. Examplesof harder to construct non-projective lean, insular R [Γ]-modules can be found inthe last section of [6].Coarse coherence of the group Γ implies that the kernel of any R [Γ]-equivariantsurjection between finitely generated, lean, insular Γ-modules is finitely generated.We refer to [3] for an explanation. So we have the following consequence. Theorem 4.2.
Suppose R is a commutative Noetherian ring and Γ is a finitelygenerated group viewed as a metric space with respect to the word metric associatedto some fixed choice of a generating set. If Γ is coarsely coherent then any finitelygenerated Γ -module F has a resolution of finite type by finitely generated free Γ -modules. OARSE COHERENCE OF METRIC SPACES 9
Proof.
Given F , select a finite generating set Σ and consider the Γ-filtered module F Σ as above. Now the R -submodule h Σ i generated by Σ has a free finitely generated R -module F e with a surjection onto h Σ i . Similarly, all submodules h γ Σ i have freefinitely generated R -modules F γ isomorphic to F e and surjections onto each h γ Σ i which serve as components of a Γ-equivariant surjection L γ ∈ S F γ → F . Coarsecoherence of Γ gives that the kernel of this surjection is finitely generated and is, infact, lean when given the canonical filtration as a submodule of the free geometricΓ-module L F γ . This allows to iterate the construction inductively to produce aresolution of F of finite type. (cid:3) Now here is finally the definition.
Definition 4.3 (Coarse Regular Coherence) . A finitely generated group Γ is coarselyregular coherent relative to a ring R if every R [Γ]-module F which is lean, insular,and locally finitely generated when viewed as a Γ-filtered R -module has a finiteprojective resolution over R [Γ]. It is called simply coarsely regular coherent if itis coarsely regular coherent relative to any regular coherent ring of finite globaldimension. Notation . We will use the notation Y for the class of all coarsely regular coherentgroups.Now let us address the relationship of coarse coherence to weak coherence. Definition 4.5.
A Γ-module F has an admissible presentation if there is an exactsequence F → F → F → F and F are finitely generated free Γ-modules, and the homomorphism f : F → F is bicontrolled. Definition 4.6 (Weak Regular Coherence) . A finitely generated group Γ is weaklycoherent relative to R if every Γ-module with an admissible presentation has a pro-jective resolution of finite type over R [Γ]. It is weakly regular coherent relative to R if every Γ-module with an admissible presentation has a finite projective resolutionover R [Γ]. We define Γ to be simply weakly coherent , resp. weakly regular coher-ent , if Γ is weakly coherent, resp. weakly regular coherent, relative to any regularNoetherian ring of finite global dimension.Weak regular coherence was introduced and studied in [2, 3, 6, 8]. Theorem 4.7.
Coarse coherence of a group implies weak coherence. Coarse regularcoherence implies weak regular coherence.
The proof follows from the next two lemmas.
Lemma 4.8.
A module with an admissible presentation is lean, insular, and lo-cally finitely generated with respect to the Γ -filtration. If Γ is a coarsely coherentgroup then conversely, every lean, insular, finitely generated R [Γ] -module has anadmissible presentation.Proof of Lemma . If F is the quotient of a boundedly bicontrolled homomor-phism F → F with the image I filtered by I ( S ) = I ∩ F ( S ), then the filtration F ( S ) = F ( S ) /I ( S ) makes the quotient map F → F bicontrolled. It is knownthat boundedly bicontrolled homomorphisms are balanced (Proposition 2.8 of [4]), so the kernel K of this quotient map is lean as the image of a lean module F .Therefore F is lean and insular by parts (1) and (3) of Theorem 2.6 applied to theshort exact sequence K → F → F .Now suppose F is a D -lean, insular, finitely generated R [Γ]-module. Since the R -submodule F ( e [ D ]) is finitely generated, there is a finitely generated free R -module F e and an R -linear epimorphism φ e : F e → F ( e [ D ]). One similarly hasan epimorphisms φ γ : F γ → F ( γ [ D ]) using isomorphic copies F γ of F e . A newgeometric Γ-filtered module F is defined by assigning F ( S ) = L γ ∈ S F γ , so F is a lean insular Γ-module. There is an R [Γ]-homomorphism φ : F → F inducedby sending F ( γ ) onto F ( γ [ D ]) via φ γ . This is a D -bicontrolled R -homomorphism.Since Γ is coarsely coherent, the kernel K of φ is a scattered Γ-module. It is insularby part (2) of Theorem 2.6. One similarly constructs a filtered module F and abicontrolled R [Γ]-homomorphism φ : F → K using the Γ-equivariant scattering of K . Since K is finitely generated over R [Γ], F can be chosen to be finitely generatedas well. The composition of φ and the inclusion of K in F gives a bicontrolled ψ : F → F with the cokernel F , as required. (cid:3) Lemma 4.9. If Γ be a coarsely coherent group then every lean, insular, finitelygenerated R [Γ] -module is of type FP ∞ .Proof of Lemma . Applying the construction from the proof of Lemma 4.8 to φ : F → K and proceeding inductively, one obtains a projective R [Γ]-resolutionby finitely generated R [Γ]-modules . . . → F n → F n − → . . . → F → F → F → ψ n : F n → F n − factors through an epimor-phism φ n onto a scattered, insular Γ-submodule K n − of F n − . (cid:3) To state the next theorem which provides examples of coarsely regular coherentgroups, recall that P. Kropholler [12, 13] defined the class of groups LH F whichincludes, in particular, all groups with finite K (Γ , Theorem 4.10.
A coarsely coherent group that belongs to Kropholler’s hierarchy LH F is coarsely regular coherent.Proof. The main result of [8] was stated for weak regular coherence. From inspectingthe argument, it is evident that the key property of the syzygies analyzed in theproof is that they are scattered rather than lean. So the argument applies verbatimunder this new assumption. (cid:3)
Corollary 4.11.
A finitely generated group with sFDC that belongs to Kropholler’shierarchy LH F is coarsely regular coherent. Finally, we list some closure properties of the class Y . Theorem 4.12.
The class of coarsely regular coherent groups is closed under • passage to subgroups, • passage to supergroups of finite index, • extensions such as finite semi-direct products of groups, including finitedirect products, • direct unions, • amalgamated free products and HNN extensions. OARSE COHERENCE OF METRIC SPACES 11
Proof.
The corresponding closure properties for LH F are in sections 2.2 and 2.4 of[12]. For the class C with the coarse coherence property, these are consequences ofTheorem 4.10 in conjunction with Theorem 3.7 and Corollary 3.5 as explained inRemark 3.8. (cid:3) Note an elementary consequence of this fact: the family of regular coherentgroups that Waldhausen constructed in [14], which are essentially multiple amal-gamated free products and HNN extensions of finitely many copies of Z , are in Y . References [1] G.C. Bell and A.N. Dranishnikov,
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