Colimit theorems for coarse coherence with applications
aa r X i v : . [ m a t h . K T ] J a n COLIMIT THEOREMS FOR COARSE COHERENCE WITHAPPLICATIONS
BORIS GOLDFARB AND JONATHAN L. GROSSMAN
Abstract.
We establish two versions of a central theorem, the Family ColimitTheorem, for the coarse coherence property of metric spaces. This is a coarsegeometric property and so is well-defined for finitely generated groups withword metrics. It is known that coarse coherence of the fundamental group hasimportant implications for classification of high-dimensional manifolds. TheFamily Colimit Theorem is one of the permanence theorems that give struc-ture to the class of coarsely coherent groups. In fact, all known permanencetheorems follow from the Family Colimit Theorem. We also use this theoremto construct new groups from this class. Introduction
Let Γ be a finitely generated group. Coarse coherence is a property of the groupviewed as a metric space with a word metric, defined in terms of algebraic proper-ties invariant under quasi-isometries. It has emerged in [1, 6] as a condition thatguarantees an equivalence between the K -theory of a group ring K ( R [Γ]) and its G -theory G ( R [Γ]). The K -theory is the central home for obstructions to a num-ber of constructions in geometric topology and number theory. We view K ( R [Γ])as a non-connective spectrum associated to a commutative ring R and the groupΓ whose stable homotopy groups are the Quillen K -groups in non-negative dimen-sions and the negative K -groups of Bass in negative dimensions. The corresponding G -theory, introduced in this generality in [2], is well-known as a theory with bettercomputational tools available compared to K -theory.The two spectra are related by the so-called Cartan map K ( R [Γ]) → G ( R [Γ]).It is this map that is shown in [6, Theorem 4.10] to be an equivalence if the groupΓ is coarsely coherent and belongs to Kropholler’s hierarchy LH F , and if R is aregular Noetherian ring of finite global dimension. The algebraic conditions on thering are satisfied in the most important for applications cases of R being either thering of integers or a field.In order to give precise definitions and state our theorems, we need to reviewsome background from coarse geometry and controlled algebra.A map between metric spaces f : X → Y is called uniformly expansive if thereis a function c : [0 , ∞ ) → [0 , ∞ ) such that d Y ( f ( x ) , f ( x )) ≤ c ( d X ( x , x )) for allpairs of points x , x from X . Two functions h , h : X → Y between metric spaceare close if there is a constant C ≥ d Y ( h ( x ) , h ( x )) ≤ C for all choicesof x in X . A function k : X → Y is a coarse equivalence if it is uniformly expansiveand there exists a uniformly expansive function l : Y → X so that the compositions k ◦ l and l ◦ k are close to the identity maps. In this case, we say that k and l are Date : January 30, 2020. coarse inverses to each other. If both obey the inequalities above for a function c ,we say that c is a control function for both.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. A map f is a coarse embedding if k is uniformlyexpansive and is a coarse equivalence onto its image.Next we review some notions from controlled algebra related to R -modules fil-tered by the subsets of a metric space. They are equivalent to the definitions givenin [6, section 2] and [3, section 3]. For the purposes of this paper it will be convenientto restate the definitions in terms of elements.Let X be a proper metric space. An X -filtered R -module is a module F togetherwith a covariant functor P ( X ) → I ( F ) from the set of subsets of X to the familyof R -submodules of F , both ordered by inclusion, such that the value on X is F .It is convenient to think of F as the functor above and use notation F ( S ) for thevalue of the functor on a subset S . We will assume F is reduced in the sense that F ( ∅ ) = 0.An R -linear homomorphism f : F → G of X -filtered modules is boundedly con-trolled if there is a fixed number b ≥ S of X , if x is in F ( S ), its image f ( x ) is an element of G ( S [ b ]). It is called boundedly bicontrolled ifthere exists a number b ≥ S ⊂ X , if y is in the image of f and also in G ( S ), it is the image f ( x ) issome element x from F ( S ). In this case we will say that f has filtration degree b and write fil( f ) ≤ b .There are a few properties of filtered modules that we want to consider. Definition 1.1.
Let F be an X -filtered R -module. • F is called lean or D - lean if there is a number D ≥ S of X and any element u in F ( S ), we have a decomposition u = P x ∈ S u x , where each u x is from F ( x [ D ]) and only finitely many u x are non-zero. • F is called scattered or D ′ - scattered if there is a number D ′ ≥ u in F , we have u = P x ∈ X u x , where u x is from F ( x [ D ])and only finitely many u x are non-zero.. • F is called insular or d - insular if there is a number d ≥ u simultaneously in F ( S ) and in F ( T ) is in fact an element of F ( S [ d ] ∩ T [ d ]). • F is locally finitely generated if F ( S ) is a finitely generated R -submodulefor all bounded subsets S .We will assume that all filtered modules are locally finitely generated.Clearly, being lean is a stronger property than being scattered. We refer thereader to some interesting examples of non-projective lean, insular filtered modulesin [2, Example 4.2]. Definition 1.2 (Coarse Coherence) . A metric space X is coarsely coherent if inany exact sequence 0 → E ′ f −−→ E g −−→ E ′′ → OLIMIT THEOREMS FOR COARSE COHERENCE 3 of 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.There is the following relaxation of the coarse coherence property. Definition 1.3 (Weak Coarse Coherence) . A metric space X is weakly coarselycoherent if in any exact sequence0 → E ′ f −−→ E g −−→ E ′′ → X -filtered R -modules where f and g are both bicontrolled maps, the combinationof E being both lean and insular and E ′′ being insular implies that E ′ is necessarilyscattered.The fact that groups of finite asymptotic dimension are coarsely coherent wasshown in [1]. This was generalised to groups of countable asymptotic dimension in[5]. The class of coarsely coherent groups has lots of permanence properties, such asinvariance under very general extensions, products, etc. established by the authorsin [6], so there is a lot of structure already known in this class of groups.In this paper we prove an additional permanence property, the Family ColimitTheorem. There are two versions of this theorem. One is a genuine permanencetheorem for coarse coherence which holds under some geometric assumptions onthe metric space. The other is a general theorem with a slightly weaker conclusion.Even though the conclusion is weaker, it has the same algebraic consequences in K -theory and so is as useful in that respect.In order to state our theorems, we formulate a version of coarse coherence formetric families. A metric family { X α } is simply a collection of metric spaces X α . Definition 1.4 (Coarse Coherence for Families) . A metric family { X α } 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 is a small subcategory in C . Theorem 1.5 (The Family Colimit Theorem) . Suppose { X α } is a collection ofmetric spaces that is coarsely coherent as a family and D is a diagram of metricspaces where the nodes are members of { X α } and all of the structure maps areisometric embeddings. Let X be the colimit of D . Then (1) X is weakly coarsely coherent, (2) X is coarsely coherent if there are distance-nonincreasing functions τ α : X → X α such that d ( x, τ ( x )) ≤ d ( x, X α )+ ε α holds for all x in X and some num-ber ε α ≥ . We prove the theorem in section 2. In section 3, we will apply the theorem to ar-gue that the wreath product Z ≀ Z is coarsely coherent. In section 4, we collect someconsequences of the Family Colimit Theorem. The theorem allows us to constructtwo new families of coarsely coherent groups. In addition, we include a summarystatement of known permanence properties for coarse coherence. BORIS GOLDFARB AND JONATHAN L. GROSSMAN Colimit Theorems
The proof of the Family Colimit Theorem will require several facts about modulesfiltered over the given family of metric spaces.For any filtered module F and a choice of number D ≥
0, there is new filtrationon F given by the formula e F D ( S ) = P x ∈ S F ( x [ D ]). More generally, f M D ( S ) = P x ∈ S M ∩ F ( x [ D ]) gives an X -filtration for a submodule M of F . Lemma 2.1.
The filtration f M D is always lean, for any submodule M and forany value of D ≥ . If F itself is D -lean then there is an interleaving between F and e F D in the following sense: for all subsets S , one has e F D ( S ) ⊂ F ( S [ D ]) and F ( S ) ⊂ e F D ( S ) .Proof. The equality f M D ( S ) = X x ∈ S M ∩ F ( x [ D ]) = X x ∈ S f M D ( x )shows that f M D is always 0-lean. The first interleaving inclusion follows from theinclusion x [ D ] ⊂ S [ D ] for x ∈ S . The other is the D -leanness property of F . (cid:3) Let X ′ be a metric subspace of X and let F be an X -filtered module. For anyfunction τ : X → X ′ , we will assign an X ′ -filtration to F , which we now denote F τ in order to distinguish from the X -filtration. As modules, F ( X ) = F τ ( X ′ ). The X ′ -filtration is given by the formula F τ ( S ) = F ( τ − ( S )). Lemma 2.2.
For any metric pair X ′ ⊂ X there is a function τ such that whenever F ( X ) = F ( X ′ [ B ]) for some B ≥ then (1) F τ is lean if F is lean and insular, and (2) F τ is insular if F is insular.Proof. Choose a function τ : X → X ′ such that d ( x, τ ( x )) ≤ d ( x, X ′ ) + ε for auniform choice of a number ε ≥
0. On X ′ [ B ], τ is bounded by B + ε .Suppose F is D -lean and d -insular. From the definition, F τ ( S ) = F ( τ − ( S )).From the assumption on F , F τ ( S ) ⊂ F ( X ′ [ B ]). The insularity property of F gives F τ ( S ) ⊂ F ( τ − ( S )[ d ] ∩ X ′ [ B + d ]) . Every point x in the set W = τ − ( S )[ d ] ∩ X ′ [ B + d ] is within d from a point x in τ − ( S ) ∩ X ′ [ B + 2 d ]. This means every point y in x [ D ] is within d + D from x . Let x ′ = τ ( x ), which is a point in S . We know that d ( x ′ , x ) ≤ B + 2 d + ε . Now we canestimate d ( x ′ , τ ( x )) ≤ d ( x ′ , x ) + d ( x, y ) + d ( y, τ ( y )) ≤ ( B + 2 d + ε ) + ( d + D ) + ( B + d + D + ε ) . This means y is contained in τ − ( x ′ [2 B + 4 d + 2 D + 2 ε ]), so the metric ball x [ D ]in X is contained in τ − ( x ′ [2 B + 4 d + 2 D + 2 ε ]). So we finally have F τ ( S ) ⊂ F ( τ − ( S )[ d ] ∩ X ′ [ B + d ]) ⊂ X x ∈ W F ( x [ D ]) ⊂ X x ′ ∈ S F τ ( x ′ [2 B + 4 d + 2 D + 2 ε ]) . This shows F τ is (2 B + 4 d + 2 D + 2 ε )-lean. OLIMIT THEOREMS FOR COARSE COHERENCE 5
Insularity follows from a similar estimate. Suppose S and T are subsets of X ′ .Then using d -insularity of F , we have F τ ( S ) ∩ F τ ( T ) ⊂ F ( τ − ( S )[ d ] ∩ τ − ( T )[ d ]).We also assume F ⊂ F ( X ′ [ B ]), so( ∗ ) F τ ( S ) ∩ F τ ( T ) ⊂ F ( τ − ( S )[2 d ] ∩ τ − ( T )[2 d ] ∩ X ′ [ B + d ]) . Take a point x in τ − ( S )[2 d ] ∩ τ − ( T )[2 d ] ∩ X ′ [ B + d ], then there is x in τ − ( S ) suchthat d ( x, x ) ≤ d . We know that d ( x, τ ( x )) ≤ B + d + ε , because x is in X ′ [ B + d ],and d ( τ ( x ) , x ) ≤ ( B + d ) + 2 d + ε = B + 3 d + ε . Now d ( τ ( x ) , τ ( x )) ≤ d ( τ ( x ) , x ) + d ( x, x ) + d ( x, τ ( x )) ≤ ( B + 3 d + ε ) + 2 d + ( B + d + ε ) = 2 B + 6 d + 2 ε. Let us use d ′ for this constant 2 B + 6 d + 2 ε . The last inequality means that x isin τ − ( S [ d ′ ] ∩ X ′ ) because τ ( x ) is in S . Using the same estimates applied withrespect to T rather than S , we have that x is also in τ − ( T [ d ′ ] ∩ X ′ ), and so in τ − ( S [ d ′ ] ∩ T [ d ′ ] ∩ X ′ ). This means that from ( ∗ ) above F τ ( S ) ∩ F τ ( T ) ⊂ F τ ( S [ d ′ ] ∩ T [ d ′ ] ∩ X ′ ) . We conclude that F τ is (2 B + 6 d + 2 ε )-insular. (cid:3) In a special geometric situation where there is a distance-nonincreasing function τ : X → X ′ , stronger facts can be proved more easily. Our main applications in thispaper will in fact be made in this kind of situation. Lemma 2.3.
Suppose X ′ ⊂ X is a metric pair with a distance-nonincreasingfunction τ : X → X ′ . Then F τ is lean if F is lean and is insular if F is insular.Proof. The property of τ guarantees that for any x in X and any D ≥ x [ D ] ⊂ τ − ( τ ( x )[ D ]). So F ( x [ D ]) is contained in F τ ( τ ( x )[ D ]). Now given S ⊂ X ′ , F τ ( S ) = F ( τ − ( S )) ⊂ X x ∈ τ − ( S ) F ( x [ D ]) ⊂ X x ′ ∈ S F τ ( x ′ [ D ]) . This shows F τ is D -lean if F is D -lean. A very similar argument gives that F τ is d -insular if F is d -insular. (cid:3) These lemmas will allow us to show Theorem 1.5. As the reader must be expect-ing, the special case (2) will have a more direct proof than (1).
Proof of Theorem . (1) Let( † ) 0 → E ′ f −−→ E g −−→ E ′′ → X -filtered R -modules such that both f and g are b -bicontrolled, E is D -lean, and both E and E ′′ are d -insular. We will demonstratethat E ′ is scattered.Let k be an element of E ′ . As E is D -lean, and in particular D -scattered, f ( k )is a finite sum P x k x for some elements k x ∈ E ( x [ D ]). Notice that the elements k x are not necessarily in the kernel of g . However, if we denote by C the finite set of all x with non-zero k x , this establishes that f ( k ) ∈ E ( C [ D ]). Let α be an index suchthat f ( k ) is an element of E ( X α ). It is therefore an element of a larger submodule E = e E D ( X α ) defined in Lemma 2.1, where it is given a 0-lean X -filtration.We define a new X -filtered module E ′′ as the image of e E D ( X α ) under the ho-momorphism g with the canonical filtration E ′′ ( S ) = E ′′ ∩ E ′′ ( S ). It is immediate BORIS GOLDFARB AND JONATHAN L. GROSSMAN that E ′′ is d -insular because E ′′ is d -insular. It is also b -lean as an image of a0-lean module under a boundedly controlled homomorphism. Note also that since e E D ( X α ) ⊂ E ( X α [ D ]), we have E ′′ ⊂ E ′′ ( X α [ D + b ]). Choose a function τ : X → X α such that d ( x, τ ( x )) ≤ d ( x, X α )+ ε for a fixed number ε ≥
0. Define an X α -filtrationof the X -filtered submodule E ′′ according to the formula E ′′ α ( S ) = E ′′ ( τ − ( S )).The X α -filtered module E ′′ α is lean and insular by Lemma 2.2. Similarly, the X α -filtration E α ( S ) = E ( τ − ( S )) of E is lean and insular. Now k is in the kernel of abicontolled epimorphism g | : E α → E ′′ α from a lean X α -filtered module to an insular X α -filtered module. The assumption that X α is coarsely coherent gives a decom-position k = P x k α,x in the kernel of g | . The supports of k α,x depend only on theconstants involved and so independent of the choice of k itself and from α becauseof the family condition. This shows that E ′ is scattered.(2) Now suppose X possesses distance-nonincreasing functions τ α as in the state-ment, and we consider an exact sequence ( † ) where E is lean and E ′′ is insular.The same constructions as in part (1) allow to construct an exact sequence of X α -modules. This time E α and E ′′ α are, respectively, lean and insular by applyingLemma 2.3. The conclusion is the same: E ′ is scattered. (cid:3) Coarse Coherence of Z ≀ Z This example illustrates the use of Family Colimit Theorems. It is then easilygeneralized to other groups with subexponential dimension growth [4] in the nextsection.Recall that the restricted wreath product of finitely generated groups G and H , denoted G ≀ H , is the semi-direct product L h ∈ H G ⋊ θ H . There is a bijectionbetween the set of elements of L h ∈ H G and the set of functions f : H → G offinite support given by identifying any element ( g , g , . . . ) of L h ∈ H G with theassignment function f g : H → G sending h g , h g , . . . . The twisting actionof the semi-direct product may then be described by ( θ h ( f g ))( h ′ ) = f g ( h − h ′ ). Let g = ( g , g , . . . ) ∈ L h ∈ H G correspond to the assignment function f g , as above.Then ( g, h ) · ( g ′ , h ′ ) = ( f g ◦ θ h ( f g ′ ) , hh ′ ) = ( f g ◦ f g ′ ( h − ) , hh ′ ) = ( g ˜ g ′ , hh ′ )where ˜ g ′ = (˜ g ′ , ˜ g ′ , . . . ) is the permutation of g ′ given by taking each g ′ i = f g ′ ( h i ) andreplacing it with ˜ g ′ i = f g ′ ( h − h i ). The wreath product L h ∈ H G ⋊ θ H is generatedby Σ G × { H } ∪ { G } × Σ H , where Σ G , Σ H are finite generating sets for G and H ,respectively. The wreath product is therefore finitely generated.We refer to the word metric associated to the generating set Σ G × { H } ∪ { G } × Σ H as the wreath metric and denote it d ≀ . We now set forth some facts regarding aspecific wreath product that we propose to study: Z ≀ Z .Consider Z ≀ Z generated by {± } × { } ∪ { } × {± } . Let γ denote the generator1 G from the first factor Z in the wreath product, and σ denote the generator 1 H from the copy of Z that is the second factor in the wreath product.Observe that X n := h γ k σγ − k | k = 1 , . . . , n i ∼ = Z n ≤ Z ≀ Z and that X ∞ := h γ k σγ − k | k ∈ N i ∼ = Z ∞ ≤ Z ≀ Z . OLIMIT THEOREMS FOR COARSE COHERENCE 7
If we use e i to denote the product γ i σγ − i , we obtain the usual basis-like generatorsfor Z n and Z ∞ . Denote this standard word metric on Z n = h e i | i = 1 , . . . , n i by d n and denote the word metric on Z ∞ = h e i | i ∈ N i by d ∞ . Each e i has length 2 i + 1 in( Z ≀ Z , d ≀ ), as is clear from their definition, and the distance between any pair e i , e j is 2( i + j + 1) from d ≀ ( e i , e j ) = k e − i e j k = k γ − i σ − γ i γ j σγ − j k = k γ − i σ − γ i + j σγ − j k = 2( i + j + 1) . The benefit of employing the wreath metric d ≀ inherited from Z ≀ Z rather than themetric given by the infinite generating set { e i } i ∈ N is that ( Z ∞ , d ≀ ) is a subgroupof a finitely generated group ( Z ≀ Z , d ≀ ), and thus both ( Z ∞ , d ≀ ) and ( Z ≀ Z , d ≀ ) arediscrete, proper, bounded geometry metric spaces.Further, the isomorphism φ : ( X n , d ≀ ) → ( Z n , d n ) is a coarse equivalence, sincethe word metric on both spaces yields for any ω ∈ X n , k ω k ≀ = k φ ( ω ) k n + 2 n. Denote the inverse isomorphism ψ : ( Z n , d n ) → ( X n , d ≀ ). Lemma 3.1.
The subgroup ( Z ∞ , d ≀ ) of ( Z ≀ Z , d ≀ ) is coarsely coherent.Proof. Let 0 → E ′ f −→ E g −→ E ′′ → Z ∞ -filtered R -modules with E D -lean, E ′′ d -insular,and f, g b -boundedly bicontrolled. Without loss of generality, assume that all con-stants are integers.Observe that for any nonnegative integer k , there exists a nonnegative integer n k such that the ball of radius k about the identity element of ( Z ∞ , d ≀ ) is containedin ( Z n k , d ≀ ), which we will denote by X . The decomposition of Z ∞ into the cosetsof X is k -disjoint. Choose k = 2 D + 2 b + 2 d . Since we can write Z ∞ as the unionof (2 D + 2 b + 2 d )-disjoint sets, it follows from the main theorem of [5] that E ′ ⊆ X z ∈ Z ∞ E ′ (( z · X )[ D ]) = X z ∈ Z ∞ E ′ ( z [ D ] · X ) . The family { z · X } z is isometric to X , and thus is a coarsely coherent family. Con-sequently, { z [ D ] · X } z is a coarsely coherent family, as is { z [ D + b ] · X } z .We present an argument analogous to that in [6]. Define K z = z [ D + b ] · X andconsider the D -leanly constructed K z -filtered module E K z ,D = X x ∈ K z E ( x [ D ])where for any S ⊆ K z , E K z ,D ( S ) = X x ∈ S ∩ K z E ( x [ D ]) . Denote this module by E z , and the submodule associated to S by E z ( S ), for brevity.By design, E z is lean, contains f ( E ′ ( z [ D ] · X )). Define E ′′ z = g ( E z ) equipped withthe standard submodule filtration from E ′′ , which is also a K z -filtration, and de-fine E ′ z = f − ( E z ) equipped with the standard submodule filtration as well. The BORIS GOLDFARB AND JONATHAN L. GROSSMAN resulting short exact sequence of K z -filtered modules0 → E ′ z f | E′ z −−−→ E z g | E z −−−→ E ′′ z → E ′ z is δ -scattered for some constant δ ≥
0. Therefore, E ′ ⊆ X z ∈ Z ∞ E ′ ( z [ D ] · X ) ⊆ X z ∈ Z ∞ X x ∈ K z E ′ z ( x [ δ ])= X x ∈ Z ∞ E ′ ( x [ δ ]) ∩ E ′ ( K z ) ⊆ X x ∈ Z ∞ E ′ ( x [ δ ]) , and ( Z ∞ , d ≀ ) is coarsely coherent. (cid:3) This argument utilizes that ( Z n , d ≀ ) is coarsely coherent for all n , but does notrequire that they be so as a family. However, since ( Z ∞ , d ≀ ) is coarsely coherent, itthen follows by the subspace permanence property of coarse coherence that the set { ( Z n , d ≀ ) : n ∈ N } is in fact a coarsely coherent family.Since the family { ( Z n , d ≀ ) : n ∈ N } is coarsely coherent, the family of subgroups { ( Z n ⋊ Z , d ) : n ∈ N } is coarsely coherent (where d is any metric that yields d ≀ and d when restricted to Z n and Z , respectively). From part (2) of Theorem 1.5 weobtain the following result. Corollary 3.2.
The wreath product ( Z ≀ Z , d ≀ ) is coarsely coherent. Applications of the Colimit Theorem
We view the Family Colimit Theorem as one from the collection of permanenceresults for coarse coherence. We will use other results of this type in our applications,so we want to state a summary of known permanence results.In this section we consider generalized metric spaces where ∞ is a possible valueof the metric. In this setting one has metric disjoint unions.Suppose p : X → Y is a uniformly expansive map of metric spaces. Then X issaid to be a fibering with base Y and coarse fibers which are preimages of metricballs in Y . Theorem 4.1 (Permanence Properties of Coarse Coherence) . The collection ofcoarsely coherent metric spaces is closed under the following operations: (1) passage to metric subspaces, (2) passage to commensurable metric overspaces, (3) disjoint unions, (4) uniformly expansive fiberings with the base and the coarse fibers in the col-lection.The subcollection of finitely generated groups with word metrics is closed underthe following constructions: (5) passage to subgroups of finite index, (6) passage to commensurable overgroups, (7) finite semi-direct products, including finite direct products, (8) free products with amalgamation and HNN extensions.
OLIMIT THEOREMS FOR COARSE COHERENCE 9
Proof.
Parts (5) and (6) follow from (1) and (2). Part (4) is Theorem 2.12 of [6].All of the remaining properties are proved in loc. cit. as a basis for stating a similarclosure theorem for the coarse regular coherence property. The proof of Theorem4.12 of [6] gives precise references to results in that paper. (cid:3)
For a countable discrete group with a fixed countable generating set, the worddistance d ( γ , γ ) is defined as the minimal length of words in the given countablealphabet that represent γ − γ .Given a countable ordered generating set A = { a i } for a group Γ, one mayconsider the sequence of subgroups H i generated by the prefix of size i in A . Inother words, H i = h a , a , . . . , a i i . One may also assume, without affecting thesequence of subgroups, that each a t +1 is not contained in H t . This guarantees thateach inclusion H i → H j for i < j is a proper isometric embedding for the wordmetrics induced from the listed generating sets. Theorem 4.2. If { H i } is a coarsely coherent family, Γ is weakly coarsely coherent.If Γ possesses distance-nonincreasing functions τ i : Γ → H i such that d ( γ, τ ( γ )) ≤ d ( γ, H i ) + ε i for all γ in Γ and some ε i ≥ , then Γ is coarsely coherent.Proof. This is precisely Theorem 1.5 applied to the present situation. (cid:3)
Corollary 4.3.
A countable discrete group is weakly coarsely coherent if and onlyif all of its finitely generated subgroups are coarsely coherent as a family.
Corollary 4.4. If G and H are countable, discrete, coarsely coherent groups, then G ≀ H , the reduced wreath product of G with H , is coarsely coherent.Proof. We can identify G ≀ H with the semi-direct product (cid:18) M h ∈ H G (cid:19) ⋊ H = (cid:18) colim −−−−→ n M h ∈ H n G (cid:19) ⋊ H, where H n = h h , h , . . . , h n i , the group generated by the first n elements of H under any fixed ordering. From Corollary 4.3, L h ∈ H G is coarsely coherent. Sincethe collection of coarsely coherent groups is closed under semi-direct products, and H is coarsely coherent, the result follows. (cid:3) Example 4.5.
Lamplighter groups L n , for all natural numbers n , are the restrictedwreath products given as L n = Z n ≀ Z . Both Z n and Z are coarsely coherent, so L n is coarsely coherent from Corollary 4.4. Corollary 4.6. If G is a countable, discrete, coarsely coherent group and α : H → K is an isomorphism of subgroups of G , then the HNN extension G ∗ α is coarselycoherent.Proof. HNN extensions are constructed by taking colimits, amalgamated free prod-ucts, and semi-direct products with Z , so the result follows from the precedingcorollaries and Theorem 4.1. (cid:3) References [1] G. Carlsson and B. Goldfarb,
On homological coherence of discrete groups , J. Algebra (2004), 502–514.[2] G. Carlsson and B. Goldfarb,
On modules over infinite group rings (with G. Carlsson), Int.J. Algebra Comput. (2016), 451–466. [3] G. Carlsson and B. Goldfarb, Bounded G -theory with fibred control , J. Pure Appl. Algebra (2019), 5360–5395.[4] A. Dranishnikov, Groups with a polynomial dimension growth , Geom. Dedicata (2006),1–15.[5] B. Goldfarb,
Weak coherence and the K -theory of groups with finite decomposition complexity ,preprint. arXiv:1307.5345 [6] B. Goldfarb and J.L. Grossman, Coarse coherence of metric spaces and groups and its per-manence properties , Bull. Aust. Math. Soc. (2018), 422–433.(2018), 422–433.