aa r X i v : . [ m a t h . G T ] S e p ON PROOFS OF THE FARRELL-JONES CONJECTURE
BARTELS, A.
Abstract.
These notes contain an introduction to proofs of Farrell-JonesConjecture for some groups and are based on talks given in Ohio, Oxford,Berlin, Shanghai, M¨unster and Oberwolfach in 2011 and 2012.
Introduction
Let R be a ring and G be a group. The Farrell-Jones Conjecture [25] is concernedwith the K - and L -theory of the group ring R [ G ]. Roughly it says that the K - and L -theory of R [ G ] is determined by the K - and L -theory of the rings R [ V ] where V varies over the family of virtually cyclic subgroups of G and group homology.The conjecture is related to a number of other conjectures in geometric topologyand K -theory, most prominently the Borel Conjecture. Detailed discussions ofapplications and the formulation of this conjecture (and related conjectures) canbe found in [10, 32, 33, 34, 35].These notes are aimed at the reader who is already convinced that the Farrell-Jones Conjecture is a worthwhile conjecture and is interested in recent proofs [3, 6,9] of instances of this Conjecture. In these notes I discuss aspects or special casesof these proofs that I think are important and illustrating. The discussion is basedon talks given over the last two years. It will be much more informal than theactual proofs in the cited papers, but I tried to provide more details than I usuallydo in talks. I took the liberty to express opinion in some remarks; the reader isencouraged to disagree with me. The cited results all build on the seminal workof Farrell and Jones surrounding their conjecture, in particular, their introductionof the geodesic flow as a tool in K - and L -theory [23]. Nevertheless, I will notassume that the reader is already familiar with the methods developed by Farrelland Jones.A brief summary of these notes is as follows. Section 1 contains a brief discussionof the statement of the conjecture. The reader is certainly encouraged to consult [10,32, 33, 34, 35] for much more details, motivation and background. Section 2 containsa short introduction to geometric modules that is sufficient for these notes. Threeaxiomatic results, labeled Theorems A, B and C, about the Farrell-Jones Conjectureare formulated in Section 3. Checking for a group G the assumptions of these resultsis never easy. Nevertheless, the reader is encouraged to find further applicationsof them. In Section 4 an outline of the proof of Theorem A is given. Section 5describes the role of flows in proofs of the Farrell-Jones Conjecture. It also containsa discussion of the flow space for CAT(0)-groups. Finally, in Section 6 an applicationof Theorem C to some groups of the form Z n ⋊ Z is discussed. Acknowledgement.
I had the good fortune to learn from and work with greatcoauthors on the Farrell-Jones Conjecture; everything discussed here is taken fromthese cooperations. I thank Daniel Kasprowski, Sebastian Meinert, Adam Mole,
Date : March 2013.1991
Mathematics Subject Classification.
Key words and phrases.
Farrell-Jones Conjecture, K - and L -theory of group rings. Holger Reich, Mark Ullmann and Christoph Winges for helpful comments on anearlier version of these notes. The work described here was supported by theSonderforschungsbereich 878 – Groups, Geometry & Actions.1.
Statement of the Farrell-Jones Conjecture
Classifying spaces for families.
Let G be a group. A family of subgroups of G is a non-empty collection F of subgroups of G that is closed under conjugationand taking subgroups. Examples are the family Fin of finite subgroups, the familyCyc of cyclic subgroups, the family of virtually cyclic subgroups VCyc, the familyAb of abelian subgroups, the family { } consisting of only the trivial subgroupand the family All of all subgroups. If F is a family, then the collection V F of all V ⊆ G which contain a member of F as a finite index subgroup is also a family. Allthese examples are closed under abstract isomorphism, but this is not part of thedefinition. If G acts on a set X then { H ≤ G | X H = ∅} is a family of subgroups. Definition 1.1. A G - CW -complex E is called a classifying space for the family F , if E H is non-empty and contractible for all H ∈ F and empty otherwise.Such a G - CW -complex always exists and is unique up to G -equivariant homotopyequivalence. We often say such a space E is a model for E F G ; less precisely wesimply write E = E F G for such a space. Example . Let F be a family of subgroups. Consider the G -set S := ` F ∈F G/F .The full simplicial complex ∆( S ) spanned by S (i.e., the simplicial complex thatcontains a simplex for every non-empty finite subset of S ) carries a simplicial G -action. The isotropy groups of vertices of ∆( S ) are all members of F , but for anarbitrary point of ∆( S ) the isotropy group will only contain a member of F as afinite index subgroup. The first barycentric subdivision of ∆( S ) is a G - CW -complexand it is not hard to see that it is a model for E V F G .This construction works for any G -set S such that F = { H ≤ G | S H = ∅} .More information about classifying spaces for families can be found in [31]. Statement of the conjecture.
The original formulation of the Farrell-Jones Con-jecture [25] used homology with coefficients in stratified and twisted Ω-spectra. Wewill use the elegant formulation of the conjecture developed by Davis and L¨uck [21].Given a ring R and a group G Davis-L¨uck construct a homology theory for G -spaces X H G ∗ ( X ; K R )with the property that H G ∗ ( G/H ; K R ) = K ∗ ( R [ H ]). Definition 1.3.
Let F be a family of subgroups of G . The projection E F G ։ G/G to the one-point G -space G/G induces the F -assembly map α F : H G ∗ ( E F G ; K R ) → H G ∗ ( G/G ; K R ) = K ∗ ( R [ G ]) . Conjecture 1.4 (Farrell-Jones Conjecture) . For all groups G and all rings R theassembly map α VCyc is an isomorphism.Remark . Farrell-Jones really only conjectured this for R = Z . Moreover, theywrote (in 1993) that they regard this and related conjectures only as estimateswhich best fit the known data at this time . It still fits all known data today.For arbitrary rings the conjecture was formulated in [2]. The proofs discussedin this article all work for arbitrary rings and it seems unlikely that the conjectureholds for R = Z and all groups, but not for arbitrary rings. N PROOFS OF THE FARRELL-JONES CONJECTURE 3
Remark . Let F be a family of subgroups of G . If R is a ring such that K ∗ R [ F ] =0 for all F ∈ F , then H G ∗ ( E F G ; K R ) = 0.In particular, the Farrell-Jones Conjecture predicts the following: if R is a ringsuch that K ∗ ( R [ V ]) = 0 for all V ∈ VCyc then K ∗ ( R [ G ]) = 0 for all groups G . Transitivity principle.
The family in the Farrell-Jones Conjecture is fixed to bethe family of virtually cyclic groups. Nevertheless, it is beneficial to keep the familyflexible, because of the following transitivity principle [25, A.10].
Proposition 1.7.
Let
F ⊆ H be families of subgroups of G . Write F ∩ H for thefamily of subgroups of H that belong to F . Assume that(a) α H : H G ∗ ( E H G ; K R ) → K ∗ ( R [ G ]) is an isomorphism,(b) α F∩ H : H H ∗ ( E F∩ H H ; K R ) → K ∗ ( R [ H ]) is an isomorphism for all H ∈ H .Then α F : H G ∗ ( E F G ; K R ) → K ∗ ( R [ G ]) is an isomorphism.Remark . The following illustrates the transitivity principle.Assume that R is a ring such that K ∗ ( R [ F ]) = 0 for all F ∈ F . Assume more-over that the assumptions of Proposition 1.7 are satisfied. Combining Remark 1.6with (b) we conclude K ∗ ( R [ H ]) = 0 for all H ∈ H . Then combining Remark 1.6with (a) it follows that K ∗ ( R [ G ]) = 0. Remark . The transitivity principle can be used to prove the Farrell-Jones Con-jecture for certain classes by induction. For example the proof of the Farrell-JonesConjecture for GL n ( Z ) uses an induction on n [11]. Of course the hard part is stillto prove in the induction step that α F n − is an isomorphism for GL n ( Z ) where thefamily F n − contains only groups that can be build from GL n − ( Z ) and poly-cyclicgroups. The induction step uses Theorem B from Section 3. See also Remark 5.21. More general coefficients.
Farrell and Jones also introduced a generalization oftheir conjecture now called the fibered Farrell-Jones Conjecture. This version of theconjecture is often not harder to prove than the original conjecture. Its advantageis that it has better inheritance properties. An alternative to the fibered conjectureis to allow more general coefficients where the group can act on the ring. As K -theory only depends on the category of finitely generated projective modules andnot on the ring itself, it is natural to also replace the ring by an additive category.We briefly recall this generalization from [13].Let A be an additive category with a G -action. There is a construction of anadditive category A [ G ] that generalizes the twisted group ring for actions of G on aring R . (In the notation of [13, Def. 2.1] this category is denoted as A∗ G G/G ; A [ G ]is a more descriptive name for it.) There is also a homology theory H G ∗ ( − ; K A )for G -spaces such that H G ∗ ( G/H ; K A ) = K ∗ ( A [ H ]). Therefore there are assemblymaps α F : H G ∗ ( E F G ; K A ) → H G ∗ ( G/G ; K A ) = K ∗ ( A [ G ]) . Conjecture 1.10 (Farrell-Jones Conjecture with coefficients) . For all groups G and all additive categories A with G -action the assembly map α VCyc is an isomor-phism.
An advantage of this version of the conjecture is the following inheritance prop-erty.
Proposition 1.11.
Let N G ։ Q be an extension of groups. Assume that Q and all preimages of virtually cyclic subgroups under G ։ Q satisfies the Farrell-Jones Conjecture with coefficients 1.10. Then G satisfies the Farrell-Jones Conjec-ture with coefficients 1.10. BARTELS, A.
Remark . Proposition 1.11 can be used to prove the Farrell-Jones Conjecturewith coefficients for virtually nilpotent groups using the conjecture for virtuallyabelian groups, compare [10, Thm. 3.2].It can also be used to reduce the conjecture for virtually poly-cyclic groups toirreducible special affine groups [3, Sec. 4]. The latter class consists of certaingroups G for which there is an exact sequence ∆ → G → D , where D is infinitecyclic or the infinite dihedral group and ∆ is a crystallographic group. Remark . For additive categories with G -action the consequence from Re-mark 1.6 becomes an equivalent formulation of the conjecture: A group G sat-isfies the Farrell-Jones Conjecture with coefficients 1.10 if and only if for additivecategories B with G -action we have K ∗ ( B [ V ]) = 0 for all V ∈ VCyc = ⇒ K ∗ ( B [ G ]) = 0 . (This follows from [9, Prop. 3.8] because the obstruction category O G ( E F G ; A ) isequivalent to B [ G ] for some B with K ∗ ( B [ F ]) = 0 for all F ∈ F .)In particular, surjectivity implies bijectivity for the Farrell-Jones Conjecture withcoefficients. Remark . The Farrell-Jones Conjecture 1.4 should be viewed as a conjectureabout finitely generated groups. If it holds for all finitely generated subgroups ofa group G , then it holds for G . The reason for this is that the conjecture is stableunder directed unions of groups [27, Thm. 7.1].With coefficients the situation is even better. This version of the conjecture isstable under directed colimits of groups [4, Cor. 0.8]. Consequently the Farrell-Jones Conjecture with coefficients holds for all groups if and only if it holds for all finitely presented groups, compare [1, Cor. 4.7]. It is therefore a conjecture aboutfinitely presented groups.Despite the usefulness of this more general version of the conjecture I will mostlyignore it in this paper to keep the notation a little simpler. L -theory. There is a version of the Farrell-Jones Conjecture for L -theory. For someapplications this is very important. For example the Borel Conjecture assertingthe rigidity of closed aspherical topological manifolds follows in dimensions ≥ K - and L -theory. The L -theoryversion of the conjecture is very similar to the K -theory version. Everything saidso far about the K -theory version also holds for the L -theory version.For some time proofs of the L -theoretic Farrell-Jones conjecture have been con-siderably harder than their K -theoretic analoga. Geometric transfer argumentsused in L -theory are considerably more involved than their counterparts in K -theory. A change that came with considering arbitrary rings as coefficients in [2],is that transfers became more algebraic. It turned out [6] that this more algebraicpoint of view allowed for much easier L -theory transfers. (In essence, because theworld of chain complexes with Poincar´e duality is much more flexible than the worldof manifolds.) This is elaborated at the end of Section 4.I think that it is fair to say that, as far as proofs are concerned, there is as atthe moment no significant difference between the K -theoretic and the L -theoreticFarrell-Jones Conjecture. For this reason L -theory is not discussed in much detailin these notes. 2. Controlled topology
The thin h -cobordism theorem. An h -cobordism W is a compact manifoldwhose boundary is a disjoint union ∂W = ∂ W ∐ ∂ W of closed manifolds suchthat the inclusions ∂ W → W and ∂ W → W are homotopy equivalences. If N PROOFS OF THE FARRELL-JONES CONJECTURE 5 M = ∂ W , then we say W is an h -cobordism over M . If W is homeomorphic to M × [0 , W is called trivial. Definition 2.1.
Let M be a closed manifold with a metric d . Let ε ≥ h -cobordism W over M is said to be ε -controlled over M if there exists aretraction p : W → M for the inclusion M → W and a homotopy H : id W → p such that for all x ∈ W the track { p ( H ( t, x )) | t ∈ [0 , } ⊆ M has diameter at most ε . Remark . Clearly, the trivial h -cobordism is 0-controlled. Thus it is natural tothink of being ε -controlled for small ε as being close to the trivial h -cobordism.The following theorem is due to Quinn [39, Thm. 2.7]. See [18, 19, 28] for closelyrelated results by Chapman and Ferry. Theorem 2.3 (Thin h -cobordism theorem) . Assume dim M ≥ . Fix a metric d on M (generating the topology of M ).Then there is ε > such that all ε -controlled h -cobordisms over M are trivial.Remark . Farrell-Jones used the thin h -cobordism Theorem 2.3 and generaliza-tions thereof to study K ∗ ( Z [ G ]), ∗ ≤
1. For example in [23] they used the geodesicflow of a negatively curved manifold M to show that any element in Wh( π M )could be realized by an h -cobordism that in turn had to be trivial by an applica-tion of (a generalization of) the thin h -cobordism Theorem. Thus Wh( π M ) = 0.In later papers they replaced the thin h -cobordism theorem by controlled surgerytheory and controlled pseudoisotopy theory.The later proofs of the Farrell-Jones Conjecture that we discuss here do notdepend on the thin h -cobordism Theorem, controlled surgery theory or controlledpseudoisotopy theory, but on a more algebraic control theory that we discuss in thenext subsection. An algebraic analog of the thin h -cobordism theorem. Geometric groups(later also called geometric modules) were introduced by Connell-Hollingsworth [20].The theory was developed much further by, among others, Quinn and Pedersen andis sometimes referred to as controlled algebra. A very pleasant introduction to thistheory is given in [37].Let R be a ring and G be a group. Definition 2.5.
Let X be a free G -space and p : X → Z be a G -map to a metricspace with an isometric G -action.(a) A geometric R [ G ] -module over X is a collection ( M x ) x ∈ X of finitely gener-ated free R -modules such that the following two conditions are satisfied. − M x = M gx for all x ∈ X , g ∈ G . − { x ∈ X | M x = 0 } = G · S for some finite subset S of X .(b) Let M and N be geometric R [ G ]-modules over X . Let f : L x ∈ X M x → L x ∈ X N x be an R [ G ]-linear map (for the obvious R [ G ]-module structures).Write f x ′′ ,x ′ for the composition M x ′ M x ∈ X M x f −→ M x ∈ X N x ։ N x ′′ . The support of f is defined as supp f := { ( x ′′ , x ′ ) | f x ′′ ,x ′ = 0 } ⊆ X × X .Let ε ≥
0. Then f is said to be ε -controlled over Z if d Z ( p ( x ′′ ) , p ( x ′ )) ≤ ε for all ( x ′′ , x ′ ) ∈ supp f. BARTELS, A. (c) Let M be a geometric R [ G ]-module over X . Let f : L x ∈ X M x → L x ∈ X M x be an R [ G ]-automorphism. Then f is said to be an ε -automorphism over Z if both f and f − are ε -controlled over Z . Remark . Geometric R [ G ]-modules over X are finitely generated free R [ G ]-modules with an additional structure, namely an G -equivariant decomposition into R -modules indexed by points in X . This additional structure is not used to changethe notion of morphisms which are still R [ G ]-linear maps. But this structure pro-vides an additional point of view for R [ G ]-linear maps: the set of morphisms be-tween two geometric R [ G ]-modules now carries a filtration by control.A good (and very simple) analog is the following. Consider finitely generatedfree R -modules. An additional structure one might be interested in are bases forsuch modules. This additional information allows us to view R -linear maps betweenthem as matrices.Controlled algebra is really not much more than working with (infinite) matriceswhose index set is a (metric) space. Nevertheless this theory is very useful andflexible.It is a central theme in controlled topology that sufficiently controlled obstruc-tions (for example Whitehead torsion) are trivial. Another related theme is thatassembly maps can be constructed as forget-control maps. In this paper we will usea variation of this theme for K of group rings over arbitrary rings. Before we canstate it we briefly fix some conventions for simplicial complexes. Convention . Let F be a family of subgroups of G . By a simplicial ( G, F )-complex we shall mean a simplicial complex E with a simplicial G -action whoseisotropy groups G x = { g ∈ G | g · x = x } belong to F for all x ∈ E . Convention . We will always use the l -metric on simplicial complexes. Let Z (0) be the vertex set of the simplicial complex Z . Then every element z ∈ Z can beuniquely written as z = P v ∈ Z (0) z v · v where z v ∈ [0 , z v arezero and P v ∈ Z (0) z v = 1. The l -metric on Z is given by d Z ( z, z ′ ) = X v ∈ V | z v − z ′ v | . Remark . If E is a simplicial complex with a simplicial G -action such that theisotropy groups G v belong to F for all vertices v ∈ E (0) of E , then E is a simplicial( G, V F )-complex, where V F consists of all subgroups H of G that admit a subgroupof finite index that belongs to F . Theorem 2.10 (Algebraic thin h -cobordism theorem) . Given a natural number N , there is ε N > such that the following holds: Let(a) Z be a simplicial ( G, F ) -complex of dimension at most N ,(b) p : X → Z be a G -map, where X is a free G -space,(c) M be a geometric R [ G ] -module over X ,(d) f : M → M be an ε N -automorphism over Z (with respect to the l -metricon Z ).Then the K -class [ f ] of f belongs to the image of the assembly map α F : H G ( E F G ; K R ) → K ( R [ G ]) . Remark . I called Theorem 2.10 the algebraic thin h -cobordism theorem here,because it can be used to prove the thin h -cobordism theorem. Very roughly, thisworks as follows. Let W be an ε -thin h -cobordism over M . Let G = π M = π W . The Whitehead torsion of W can be constructed using the singular chaincomplexes of the universal covers f W and f M . This realizes the Whitehead torsion N PROOFS OF THE FARRELL-JONES CONJECTURE 7 τ W ∈ Wh( G ) of W by an e ε -automorphism f W over f M , i.e. [ f W ] maps to τ W under K ( Z [ G ]) → Wh( G ). Moreover, e ε can be explicitly bounded in terms of ε , such that e ε → ε →
0. Because f M is a free G = π M -space it follows from Theorem 2.10that [ f W ] belongs to the image of the assembly map α : H G ( EG, K Z ) → K ( Z [ G ]).But Wh( G ) is the cokernel of α and therefore τ W = 0. This reduces the thin h -cobordism theorem to the s -cobordism theorem.I believe that – at least in spirit – this outline is very close to Quinn’s proofin [39]. Remark . If f : M → M ′ is ε -controlled over Z and and f ′ : M ′ → M ′′ is ε ′ -controlled over Z , then their composition f ′ ◦ f is ε + ε ′ -controlled. In particular,there is no category whose objects are geometric modules and whose morphisms are ε -controlled for fixed (small) ε . However, there are very elegant substitutes for thisill-defined category. These are built by considering a variant of the theory over anopen cone over Z and taking a quotient category. In this quotient category everymorphisms has for every ε > ε -controlled representative. Pedersen-Weibel [38]used this to construct homology of a space E with coefficients in the K -theoryspectrum as the K -theory of an additive category. Similar constructions can beused to describe the assembly maps as forget-control maps [2, 17]. This also leadsto a category (called the obstruction category in [9]), whose K -theory describes thefiber of the assembly map. A minor drawback of these constructions is that theyusually involve a dimension shift.A very simple version of such a construction is discussed at the end of thissection. See in particular Theorem 2.20. Remark . It is not hard to deduce Theorem 2.10 from [6, Thm. 5.3]. Thelatter result is a corresponding result for the obstruction category mentioned inRemark 2.12. In fact this result about the obstruction is stronger and can be usedto prove that the assembly map is an isomorphism and not just surjective, see [6,Thm. 5.2]. I have elected to state the weaker Theorem 2.10 because it is much easierto state, but still grasps the heart of the matter. On the other hand, I think it isnot at all easier to prove Theorem 2.10 than to prove the corresponding statementfor the obstruction category. (The result in [6] deals with chain complexes, but thisis not an essential difference.)
Remark . Results like Theorem 2.10 are very useful to prove the Farrell-JonesConjecture. But it is not clear to me, that it really provides the best possibledescription of the image of the assembly map. For g ∈ G we know that [ g ] liesin the image of the assembly map. But its most natural representative (namelythe isomorphism of R [ G ] given by right multiplication by g ) is not ε -controlled forsmall ε .It may be beneficial to find other, maybe more algebraic and less geometric,characterizations of the image of the assembly map. But I do not know how toapproach this. Remark . The use of the l -metric in Theorem 2.10 is of no particular impor-tance. In order for ε N to only depend on N and not on Z , one has to commit tosome canonical metric. Remark . If F is closed under finite index supergroups, i.e., if F = V F thenthere is no loss of generality in assuming that Z is the N -skeleton of the modelfor E F G discussed in Example 1.2. This holds because there is always a G -map Z (0) → S := ` F ∈F G/F and this map extends to a simplicial map Z → ∆( S ) ( N ) .Barycentric subdivision only changes the metric on the N -skeleton in a controlled(depending on N ) way. BARTELS, A.
Remark . There is also a converse to Theorem 2.10. If a ∈ K ( R [ G ]) lies inthe image of the assembly map α F then there is some N such that it can for any ε > ε -automorphism over an N -dimensional simplicial complex Z with a simplicial G -action all whose isotropy groups belong to F . The simplicialcomplex can be taken to be the N -skeleton of a simplicial complex model for E F G .This is a consequence of the description of the assembly map as a forget-controlmap as for example in [2, Cor. 6.3]. Remark . It is not hard to extend the theory of geometric R [ G ]-modules fromrings to additive categories. In this case one considers collections ( A x ) x ∈ X whereeach A x is an object from A . In fact [6, Thm. 5.3], which implies Theorem 2.10, isformulated using additive categories as coefficients. Remark . Results for K often imply results for K i , i ≤
0, using suspensionrings. For a ring R , there is a suspension ring Σ R with the property that K i ( R ) = K i +1 (Σ R ) [44]. This construction can be arranged to be compatible with grouprings: Σ( R [ G ]) = (Σ R )[ G ]. A consequence of this is that for a fixed group G the surjectivity of α F : H G ( E F G ; K R ) → K ( R [ G ]) for all rings R implies thesurjectivity of α F for all i ≤
1, compare [2, Cor. 7.3].Because of this trick there is no need for a version of Theorem 2.10 for K i , i ≤ Higher K -theory. We end this section by a brief discussion of a version of The-orem 2.10 for higher K -theory. Because there is no good concrete description ofelements in higher K -theory it will use slightly more abstract language.Let p n : X n → Z n be a sequence of G -maps where each X n is a free G -spaceand each Z n is a simplicial ( G, F )-complex of dimension N . Define a category C as follows. Objects of C are sequences ( M n ) n ∈ N where for each n , M n is ageometric R [ G ]-module over X n . A morphism ( M n ) n ∈ N → ( N n ) n ∈ N in C is givenby a sequence ( f n ) n ∈ N of R [ G ]-linear maps f n : L x ∈ X n ( M n ) x → L x ∈ X n ( N n ) x satisfying the following condition: there is α > n , f n is αn -controlled over Z n . For each k ∈ N ,( M n ) n ∈ N M x ∈ X k ( M k ) x defines a functor π k from C to the category of finitely generated free R [ G ]-modules.The following is a variation of [14, Cor. 4.3]. It can be proven using [9, Thm. 7.2]. Theorem 2.20.
Let a ∈ K ∗ ( R [ G ]) . Suppose that there is A ∈ K ∗ ( C ) such that forall k ( π k ) ∗ ( A ) = a. Then a belongs to the image of α F : H G ∗ ( E F G ; K R ) → K ∗ ( R [ G ]) . Conditions that imply the Farrell-Jones Conjecture
In [6, 9] the Farrell-Jones Conjecture is proven for hyperbolic and CAT(0)-groups.Both papers take a somewhat axiomatic point of view. They both contain careful(and somewhat lengthy) descriptions of conditions on groups that imply the Farrell-Jones conjecture. The conditions in the two papers are closely related to eachother. A group satisfying them is said to be transfer reducible over a given familyof subgroups in [6]. Further variants of these conditions are introduced in [11, 45].The existence of all these different versions of these conditions seem to me to suggestthat we have not found the ideal formulation of them yet. The notion of transferreducible groups (and all its variations) can be viewed as an axiomatization ofthe work of Farrell-Jones using the geodesic flow that began with [23]. Somewhatdifferent conditions – related to work of Farrell-Hsiang [22] – are discussed in [5].
N PROOFS OF THE FARRELL-JONES CONJECTURE 9
Transfer reducible groups – strict version.
Let R be a ring and G be a group. Definition 3.1. An N -transfer space X is a compact contractible metric spacesuch that the following holds.For any δ > K of dimension at most N andcontinuous maps and homotopies i : X → K , p : K → X , and H : p ◦ i → id X suchthat for any x ∈ X the diameter of { H ( t, x ) | t ∈ [0 , } is at most δ . Example . Let T be a locally finite simplicial tree. The compactification T of T by equivalence classes of geodesic rays is a 1-transfer space. Theorem A.
Suppose that G is finitely generated by S . Let F be a family ofsubgroups of G . Assume that there is N ∈ N such that for any ε > there are(a) an N -transfer space X equipped with a G -action,(b) a simplicial ( G, F ) -complex E of dimension at most N ,(c) a map f : X → E that is G -equivariant up to ε : d ( f ( s · x ) , s · f ( x )) ≤ ε forall s ∈ S , x ∈ X .Then α F : H G ∗ ( E F G ; K R ) → K ∗ ( RG ) is an isomorphism.Remark . It follows from [8] that Theorem A (with F the family of virtuallycyclic subgroups VCyc) applies to hyperbolic groups. Example . Let G be a group and K be a finite contractible simplicial com-plex with a simplicial G -action. Then for the family F := F K the assembly map α F : H G ∗ ( E F G ; K R ) → K ∗ ( RG ) is an isomorphism. This follows from Theorem Aby setting N := dim K and X := K , E := K , f := id K (for all ε > K isfinite, the group of simplicial automorphisms of K is also finite. It follow that forall x ∈ K the isotropy group G x has finite index in G .The assumptions of Theorem A should be viewed as a weakening of this example.The properties of K are reflected in requirements on X or on E and the existenceof the map f yields a strong relationship between X and E . Remark . Rufus Willet and Guoliang Yu pointed out that the assumption ofTheorem A implies that the group G has finite asymptotic dimension, providedthere is a uniform bound on the asymptotic dimension of groups in F . The latterassumptions is of course satisfied for the family of virtually cyclic groups VCyc. Remark . Martin Bridson pointed out that the assumptions of Theorem A areformally very similar to the concept of amenability for actions on compact spaces.The main difference is that in the latter context E is replaced by the (infinitedimensional) space of probability measures on G . Remark . Theorem A is a minor reformulation of [9, Thm. 1.1]. In this referenceinstead of the existence of f the existence of certain covers U of G × X are postulated.But the first step in the proof is to use a partition of unity to construct a G -mapfrom G × X to the nerve |U| of U . Under the assumptions formulated in Theorem Athis map is simply ( g, x ) g · f ( g − x ).Avoiding the open covers makes the theorem easier to state, but there is no realmathematical difference. Remark . The proof of Theorem A in [9] really shows a little bit more: there is M (depending on N ) such that the restriction of α F to H G ∗ ( E F G ( M ) ; K R ) is surjective.For arbitrary groups and rings with non-trivial K -theory in infinitely many negativedegrees there will be no such M . It is reasonable to expect that groups satisfyingthe assumptions of Theorem A will also admit a finite dimensional model for thespace E F G . Remark . Let E be a simplicial complex of dimension N . Let g be a simplicialautomorphism of E . Let x = P v ∈ E (0) x v · v be a point of E . Let supp x := { v ∈ E (0) | x v = 0 } . It is a disjoint union of the sets P x := { v ∈ supp x | ∀ n ∈ N : g n ∈ supp x } ,D x := { v ∈ supp x | ∃ n ∈ N : g n supp x } . Observe that for v ∈ D x , we have d ( x, gx ) ≥ x v . Assume now that d ( x, gx ) < N +1 . As P v x v = 1 there is a vertex v with v ≥ N +1 . Such a vertex v belongsthen to P x and it follows that { g n v | n ∈ N } is finite and spans a simplex of E whose barycenter is fixed by g .Assume now that f : X → E is as in assumption (c) of Theorem A. If G x is theisotropy group for x ∈ X (and if G x is finitely generated by S x say) then if ε issufficiently small it follows that d ( f ( x ) , gf ( x )) < N +1 . The previous observationimplies then G x ∈ F .On the other hand one can apply the Lefschetz fixed point theorem to the sim-plicial dominations to X and finds for fixed g ∈ G and each ε > x ε ∈ X such that d ( gx ε , x ε ) ≤ ε . The compactness of X implies that there is a fixed pointin X for each element of G . Altogether, it follows that F will necessarily containthe family of cyclic subgroups. Remark . Frank Quinn has shown that one can replace the family of virtuallycyclic groups in the Farrell-Jones Conjecture by the family of (possibly infinite)hyper-elementary groups [40].It is an interesting question whether one can (maybe using Smith theory) buildon the argument from Remark 3.9 to conclude that in order for the assumptions ofTheorem A to be satisfied it is necessary for F to contain this family of (possiblyinfinite) hyper-elementary groups. Remark . One can ask for which N -transfer spaces X with a G -action it ispossible to find for all ε > f : X → E as in assumptions (b) and (c).Remark 3.9 shows that a necessary condition is G x ∈ F for all x ∈ X , but it isnot clear to me that this condition is not sufficient.In light of the observation of Willet and Yu from Remark 3.5 a related questionis whether there is a group G of infinite asymptotic dimension for which there is an N -transfer space with a G -action such that the asymptotic dimension of G x , x ∈ X is uniformly bounded. Remark . The reader is encouraged to try to check that finitely generatedfree groups satisfy the assumptions of Theorem A with respect to the family of(virtually) cyclic subgroups. In this case one can use the compactification ¯ T of theusual tree by equivalence classes of geodesic rays as the transfer space. I am keento see a proof of this that is easier than the one coming out of [8] and avoids flowspaces. Maybe a clever application of Zorn’s Lemma could be useful here.I am not completely sure whether or not it is possible to write down the maps f : ¯ T → E in assumption (c) explicitly for finitely generated free groups. Transfer reducible groups – homotopy version.
Let R be a ring. Definition 3.13.
Let G = h S | R i be a finitely presented group. A homotopyaction of G on a space X is given by − for all s ∈ S ∪ S − maps ϕ s : X → X , − for all r = s · s · · · s l ∈ R homotopies H r : ϕ s ◦ ϕ s ◦ · · · ◦ ϕ s l → id X Theorem B.
Suppose that G = h S | R i is a finitely presented group. Let F bea family of subgroups of G . Assume that there is N ∈ N such that for any ε > there are N PROOFS OF THE FARRELL-JONES CONJECTURE 11 (a) an N -transfer space X equipped with a homotopy G -action ( ϕ, H ) ,(b) a simplicial ( G, F ) -complex E of dimension at most N ,(c) a map f : X → E that is G -equivariant up to ε : for all x ∈ X , s ∈ S ∪ S − , r ∈ R − d ( f ( ϕ s ( x )) , s · f ( x )) ≤ ε , − { H r ( t, x ) | t ∈ [0 , } has diameter at most ε .Then α F : H Gi ( E F G ; K R ) → K i ( RG ) is an isomorphism for i ≤ and surjectivefor i = 1 .Remark . It follows from [7] that Theorem B applies to CAT(0)-groups (where F is the family of virtually cyclic groups). We will sketch the proof of this fact inSection 5.Wegner introduced the notion of a strong homotopy action and proved a versionof Theorem B where the conclusion is that α F is an isomorphism in all degrees [45].This result also applies to CAT(0)-groups. Remark . Theorem B is a reformulation of [6, Thm. 1.1] (just as in Remark 3.7).The assumptions of Theorem A feel much cleaner than the assumptions of The-orem B. It would be very interesting if one could show, maybe using some kind oflimit that promotes a (strong) homotopy action to an actual action, such that thelatter (or Wegner’s variation of them) do imply the former.In light of Remark 3.5 this would imply in particular that CAT(0)-groups havefinite asymptotic dimension and is therefore probably a difficult (or impossible)task.
Remark . I do not know whether semi-direct products of the form Z n ⋊Z satisfythe assumptions of Theorem B, for example if F is the family of abelian groups.On the other hand the Farrell-Jones Conjecture is known to hold for such groupsand more general for virtually poly-cyclic groups [3]. Remark . Remark 3.8 also applies to Theorem B.
Remark . There is an L -theory version of Theorem B, see [6, Thm. 1.1(ii)].There, the conclusion is that the assembly map α F is an isomorphism in L -theorywhere F is the family of subgroups that contain a member of F as a subgroupof index at most 2. Of course VCyc = VCyc . There is no restriction on thedegree i in this L -theoretic version and so this also provides an L -theory version ofTheorem A. Farrell-Hsiang groups.Definition 3.19.
A finite group H is said to be hyper-elementary if there exists ashort exact sequence C H ։ P where C is a cyclic group and the order of P is a prime power.Quinn generalized this definition to infinite groups by allowing the cyclic groupto be infinite [40].Hyper-elementary groups play a special role in K -theory because of the followingresult of Swan [43]. For a group G we denote by Sw( G ) the Swan group of G . It canbe defined as K of the exact category of Z [ G ]-modules that are finitely generatedfree as Z -modules. This group encodes information about transfer maps in algebraic K -theory. Theorem 3.20 (Swan) . For a finite group F the induction maps combine to asurjective map M H ∈H ( F ) Sw( H ) ։ Sw( F ) , where H ( F ) denotes the family of hyper-elementary subgroups of F . Let R be a ring and G be a group. Theorem C.
Suppose that G is finitely generated by S . Assume that there is N ∈ N such that for any ε > there are(a) a group homomorphism π : G → F where F is finite,(b) a simplicial ( G, F ) -complex E of dimension at most N (c) a map f : ` H ∈H ( F ) G/π − ( H ) → E that is G -equivariant up to ε : d ( f ( sx ) , s · f ( x )) ≤ ε for all s ∈ S , x ∈ ` H ∈H ( F ) G/π − ( H ) .Then α F : H G ∗ ( E F G ; K R ) → K ∗ ( RG ) is an isomorphism.Remark . Theorem C is proven in [5] building on work of Farrell-Hsiang [22].The main difference to Theorems A and B is that the transfer space X is replacedby the discrete space ` H ∈H ( F ) G/π − ( H ). It is Swan’s theorem 3.20 that replacesthe contractibility of X .I have no conceptual understanding of Swan’s theorem. For this reason Theo-rem C is to me not as conceptually satisfying as Theorem A. Moreover, I expectthat a version of Theorem C for Waldhausen’s A -theory will need a larger familythan the family of hyper-elementary subgroups. Remark . Groups satisfying the assumption of Theorem C are called
Farrell-Hsiang groups with respect to F in [5]. Remark . Theorem C can be used to prove the Farrell-Jones Conjecture forvirtually poly-cyclic groups [3, Sec. 3 and 4]. We will discuss some semi-directproducts of the form Z n ⋊ Z in Section 6. Remark . Remark 3.8 also applies to Theorem C.
Remark . Theorem C holds without change in L -theory as well [5]. Remark . It would be good to find a natural common weakening of the assump-tions in Theorems A,B and C that still implies the Farrell-Jones Conjecture. Ideallysuch a formulation should have similar inheritance properties as the Farrell-JonesConjecture, see Propositions 1.7 and 1.11.
Injectivity.
It is interesting to note that injectivity of the assembly map α { } or α Fin is known for seemingly much bigger classes of groups, than the class ofgroups known to satisfy the Farrell-Jones Conjecture. Rational injectivity of the L -theoretic assembly map α { } is of particular interest, as it implies Novikov’sconjecture on the homotopy invariance of higher signatures. Yu [46] proved theNovikov conjecture for all groups admitting a uniform embedding into a Hilbert-space. This class of groups contains all groups of finite asymptotic dimension.Integral injectivity of the assembly map α { } for K - and L -theory is known forall groups that admit a finite CW -complex as a model for BG and are of finitedecomposition complexity [30, 41]. The latter property is a generalization of finiteasymptotic dimension. Rational injectivity of the K -theoretic assembly map α { } for the ring Z is proven by B¨okstedt-Hsiang-Madsen [15] for all groups G satisfyingthe following homological finiteness condition: for all n the rational group-homology H ∗ ( G ; Q ) is finite dimensional.4. On the Proof of Theorem A
Using the results from controlled topology discussed in Section 2 we will outlinea proof of the surjectivity of α F : H G ( E F G ; K R ) → K ( R [ G ])under the assumptions of Theorem A. N PROOFS OF THE FARRELL-JONES CONJECTURE 13
Step 1: preparations.
Let G be a finitely generated group and F be a familyof subgroups of G . Let N ∈ N be the number appearing in Theorem A. Let a ∈ K ( R [ G ]). Then a = [ ψ ] where ψ : R [ G ] n → R [ G ] n is an R [ G ]-right linearautomorphism. We write R [ G ] n = Z [ G ] ⊗ Z R n . There is a finite subset T ⊆ G andthere are R -linear maps ψ g : R n → R n , ψ − g : R n → R n , g ∈ T such that ψ ( h ⊗ v ) = X g ∈ T hg − ⊗ ψ g ( v ) and ψ − ( h ⊗ v ) = X g ∈ T hg − ⊗ ψ − g ( v ) . Because of Theorem 2.10 it suffices to find − a G -space Y , − a ( G, F )-complex E of dimension at most N , − a G -map Y → E , − a geometric R [ G ]-module M over Y , − an ε N -automorphism over E , ϕ : M → M ,such that [ ϕ ] = a ∈ K ( R [ G ]). Here ε N is the number depending on N , whoseexistence is asserted in Theorem 2.10.Let L be a (large) number. We will later specify L ; it will only depend on N .From the assumption of Theorem A we easily deduce that there are(a) an N -transfer space X equipped with a G -action,(b) a simplicial ( G, F )-complex E of dimension at most N ,(c) a map f : X → E such that d ( f ( g · x ) , g · f ( x )) ≤ ε N / x ∈ X andall g ∈ G that can be written as g = g . . . g L with g , . . . , g L ∈ T .By compactness of X there is δ > d ( f ( x ) , f ( x ′ )) ≤ ε N / x, x ′ ∈ X with d ( x, x ′ ) ≤ Lδ . We will use Y := G × X with the G -action definedby g · ( h, x ) := ( gh, x ). We will also use the G -map G × X → E , ( g, x ) gf ( x ).The action of G on X will be used later. Step 2: a chain complex over X . To simplify the discussion let us assumethat for X the maps i and p appearing in Definition 3.1 can be arranged to be δ -homotopy equivalences. This means that in addition to H there is also a homotopy H ′ : i ◦ p → id K such that for any y ∈ K the diameter of { H ′ ( t, y ) | t ∈ [0 , } withrespect to the l -metric on K is at most δ .Let C ∗ be the simplicial chain complex of the l -fold simplicial subdivision of K .Using p : K → X we can view C ∗ as a chain complex of geometric Z -modules over X . If we choose l sufficiently large, then we can arrange that the boundary maps ∂ C ∗ of C ∗ are δ -controlled over X . Moreover, using the action of G on X and a δ -homotopy equivalence between K and X (for 0 < δ ≪ δ ) and enlarging l we canproduce chain maps ϕ g : C ∗ → C ∗ , g ∈ G , chain homotopies H g,h : ϕ g ◦ ϕ h → ϕ gh satisfying the following control conditions − if g ∈ T and ( x ′ , x ) ∈ supp ϕ g then d ( x ′ , gx ) ≤ δ (recall that we view C ∗ as a chain complex over X ), − if g, h ∈ T and ( x ′ , x ) ∈ supp H g,h then d ( x ′ , ghx ) ≤ δ . Remark . If we drop the additional assumption on X (i.e., if we no longer assumethe existence of the homotopy H ′ ), then it is only possible to construct the chaincomplex C ∗ in the idempotent completion of geometric Z -modules over X . This isa technical but – I think – not very important point. Remark . A construction very similar to this step 2 is carried out in great detailin [6, Sec. 8].
Step 3: transfer to a chain homotopy equivalence.
Recall our automorphism ψ of R [ G ] n = Z [ G ] ⊗ Z R n . We will now replace the R -module R n by the R -chaincomplex C ∗ ⊗ Z R n to obtain the chain complex D ∗ := Z [ G ] ⊗ Z C ∗ ⊗ Z R n . As C ∗ is achain complex of geometric Z -modules over X , D ∗ is naturally a geometric R [ G ]-module over G × X . Here ( D ∗ ) h,x = { h ⊗ w ⊗ v | v ∈ R n , w ∈ ( C ∗ ) x } for h ∈ G , x ∈ X . Recall that we use the left action defined by g · ( h, x ) = ( gh, x ) on G × X .We can now use the data from Step 2 to transfer ψ to a chain homotopy equivalenceΨ = P g ∈ T g ⊗ ϕ g ⊗ ψ g : D ∗ → D ∗ . Similarly, there is a chain homotopy inverse Ψ ′ for Ψ and there are chain homotopies H : Ψ ◦ Ψ ′ → id D ∗ and H ′ : Ψ ′ ◦ Ψ → id D ∗ .In more explicit formulas these are defined byΨ( h ⊗ w ⊗ v ) = X g ∈ T hg − ⊗ ϕ g ( w ) ⊗ ψ g ( v ) , Ψ ′ ( h ⊗ w ⊗ v ) = X g ∈ T hg − ⊗ ϕ g ( w ) ⊗ ψ − g ( v ) , H ( h ⊗ w ⊗ v ) = X g,g ′ ∈ T h ( gg ′ ) − ⊗ H g,g ′ ( w ) ⊗ ψ g ◦ ψ − g ′ ( v ) , H ′ ( h ⊗ w ⊗ v ) = X g,g ′ ∈ T h ( gg ′ ) − ⊗ H g,g ′ ( w ) ⊗ ψ − g ◦ ψ g ′ ( v ) , for h ∈ G , w ∈ C ∗ , v ∈ R n . Digression on torsion.
Let S be a ring. If Φ is a self-homotopy equivalence of abounded chain complex D ∗ of finitely generated free S -modules then its self-torsion τ (Φ) ∈ K ( S ) is the K -theory class of an automorphism ˜ τ (Φ) of L n ∈ Z D n . Thereis an explicit formula for ˜ τ (Φ) that involves the boundary map of D ∗ , Φ, a chainhomotopy inverse Φ ′ for Φ and chain homotopies Φ ◦ Φ ′ → id D ∗ , Φ ′ ◦ Φ → id D ∗ .The ingredients for such a formula can be found for example in [2, Sec. 12.1]. Akey property is that given a commutative diagram D ∗ Φ / / q (cid:15) (cid:15) D ∗ q (cid:15) (cid:15) D ′∗ Φ / / D ′∗ where Φ , Φ and q are chain homotopy equivalences one has τ (Φ ) = τ (Φ ) ∈ K ( S ). Remark . It is possible to formulate Theorem 2.10 directly for self-chain homo-topy equivalences of chain complexes of geometric modules of bounded dimension.Then the discussion of torsion can be avoided here. This is the point of view takenin [6, Thm. 5.3].
Step 4: τ (Ψ) = a . Because X is contractible, the augmentation map C ∗ → Z induces a homotopy equivalence q : D ∗ = Z [ G ] ⊗ Z C ∗ ⊗ Z R n → Z [ G ] ⊗ Z Z ⊗ Z R n = Z [ G ] ⊗ Z R n . Moreover, q ◦ Ψ = ψ ◦ q . It follows that a = [ ψ ] = τ ( ψ ) = τ (Ψ) = [˜ τ (Ψ)] N PROOFS OF THE FARRELL-JONES CONJECTURE 15
Step 5: control of ˜ τ (Ψ) . In order to understand the support of ˜ τ (Ψ) we firstneed to understand the support of its building blocks. If (( h ′ , x ′ ) , ( h, x )) ∈ ( G × X ) belongs to the support of ∂ D ∗ , then h ′ = h and d ( x ′ , x ) ≤ δ . If (( h ′ , x ′ ) , ( h, x ))belongs to the support of Ψ or of its homotopy inverse Ψ ′ , then there is g ∈ T such that h ′ = hg − and d ( x ′ , gx ) ≤ δ . If (( h ′ , x ′ ) , ( h, x )) belongs to the supportof the chain homotopy H or H ′ then there are g, g ′ ∈ T such that h ′ = h ( gg ′ ) − and d ( x ′ , gg ′ x ) ≤ δ . From the explicit formula for ˜ τ (Ψ) one can then read off thatthere is a number K , depending only on the dimension of D ∗ (which is in our casebounded by N ), such that the support of ˜ τ (Ψ) satisfies the following condition: if(( h ′ , x ′ ) , ( h, x )) ∈ supp ˜ τ (Ψ) then there are g , . . . , g K ∈ T such that h ′ = h ( g . . . g K ) − and d ( x ′ , g . . . g K x ) ≤ Kδ . Note that we specified K in this step; note also that K does only depend on N . Remark . The actual value of K is of course not important. It is not very large;for example K := 10 N works – I think. Step 6: applying f . Using the map f : X → E we define the G -map F : G × X → E by F ( h, x ) := hf ( x ). Combining the estimates from the end of step 2 and theanalysis of supp(˜ τ (Ψ)) it is not hard to see that ˜ τ (Ψ) is an ε N -automorphism over E (with respect to F ).This finishes the discussion of the surjectivity of α F : H G ( E F G ; K R ) → K ( R [ G ])under the assumptions of Theorem A. Surjectivity of this map under the assump-tions of Theorem B follows from a very similar argument; mostly step 2 is slightlymore complicated. For Theorem C the transfer can no longer be constructed usinga chain complex associated to a space; instead Swan’s Theorem 3.20 is used toconstruct a transfer. Otherwise the proof is again very similar. L -theory transfer. The proof of the L -theory version of Theorems A and B followsthe same outline. Now elements in L -theory are given by quadratic forms. Theanalog of chain homotopy self-equivalences in L -theory are ultra-quadratic Poincar´ecomplexes [42]. These are chain complex versions of quadratic forms. The maindifference is that to construct a transfer it is no longer sufficient to have just thechain complex C , in addition we need a symmetric structure on this chain complex.Moreover, this symmetric structure needs to be controlled (just as the boundarymap ∂ is controlled). While there may be no such symmetric structure on C , thereis a symmetric structure on the product of C with its dual D := C ⊗ C −∗ . Thissymmetric structure is given (up to signs) by h a ⊗ α, b ⊗ β i = α ( b ) β ( a ) and turnsout to be suitably controlled. This is the only significant change from the proof in K -theory to the proof in L -theory. Transfer for higher K -theory. We end this section with a very informal dis-cussion of one aspect of the proof of Theorem A for higher K -theory. Again, wefocus on surjectivity. In this case we use Theorem 2.20 in place of Theorem 2.10.Thus we need to produce an element in K ∗ ( C ). Recall that objects of C are se-quences ( M ν ) ν ∈ N of geometric R [ G ]-modules and that morphisms are sequences of R [ G ]-linear maps that become more controlled with ν → ∞ . The general idea is toapply the transfer from Step 3 to each ν to produce a functor from R [ G ]-modulesto C . The problem is, however, that the construction from Step 3 is not functorial.The reason for this in turn is that the group G only acts up to homotopy on thechain complex C ∗ . The remedy for this failure is to use the singular chain complexof C sing ∗ ( X ) in place of C ∗ . It is no longer finite, but it is homotopy finite, whichis finite enough. For the control consideration from Step 5 it was important, thatthe boundary map of C ∗ is δ -controlled. This is no longer true for C sing ∗ ( X ). Onemight be tempted to use the subcomplex C sing ,δ ( X ) spanned by singular simplices in X of diameter ≤ δ . However, the action of G on X is not isometric and there-fore there is no G -action on C sing ,δ ( X ). Finally, the solution is to use C sing ∗ ( X )together with its filtration by the subcomplexes ( C sing ,δ ∗ ( X )) δ> . Using this ideait is possible to construct a transfer functor from the category of R [ G ]-modules toa category f ch hfd C . The latter is a formal enlargement of the Waldhausen category ch hfd C of homotopy finitely dominated chain complexes over the category C [12,Appendix]. Both the higher K -theory of ch hfd C and of f ch hfd C coincide with thehigher K -theory of C . Similar constructions are used in [9, 45].5. Flow spaces
Convention . A CAT(0) -group is a group that admits a cocompact, proper andisometric action on a finite dimensional CAT(0)-space.The goal of this section is to outline the proof of the fact [7] that CAT(0)-groups satisfy the assumptions of Theorem B. Note that CAT(0)-groups are finitelypresentable [16, Thm. III.Γ.1.1(1), p.439].
Proposition 5.2.
Let G be a CAT(0) -group. Exhibit G as a finitely presentedgroup h S | R i . Then there is N ∈ N such that for any ε > there are(a) an N -transfer space X equipped with a homotopy G -action ( ϕ, H ) ,(b) a simplicial ( G, VCyc) -complex E of dimension at most N ,(c) a map f : X → E that is G -equivariant up to ε : for all x ∈ X , s ∈ S ∪ S − , r ∈ R − d ( f ( ϕ s ( x )) , s · f ( x )) ≤ ε , − { H r ( t, x ) | t ∈ [0 , } has diameter at most ε . An ( α, ε ) -version of the assumptions of Theorem B. Let G be a group. Definition 5.3. An N -flow space FS for G is a metric space with a continuousflow φ : FS × R → FS and an isometric proper action of G such that(a) the flow is G -equivariant: φ t ( gx ) = gφ t ( x ) for all x ∈ X , t ∈ R and g ∈ G ;(b) FS \ { x | φ t ( x ) = x for all t ∈ R } is locally connected and has coveringdimension at most N . Notation . Let α, ε ≥
0. For x, y ∈ FS we write d folFS ( x, y ) ≤ ( α, ε )if there is t ∈ [ − α, α ] such that d ( φ t ( x ) , y ) ≤ ε .Of course, ε will usually be a small number while α will often be much larger. Proposition 5.5.
Let G be a CAT(0) -group. Exhibit G as a finitely presentedgroup h S | R i . Then there exists N ∈ N and a cocompact N -flow space for G and α > such that for all ε > there are(a) an N -transfer space X equipped with a homotopy G -action ( ϕ, H ) ,(b) a map f : X → FS that is G -equivariant up to ( α, ε ) : for all x ∈ FS , s ∈ S ∪ S − , r ∈ R , t ∈ [0 , − d folFS ( f ( ϕ s ( x )) , s · f ( x )) ≤ ( α, ε ) , − d folFS ( f ( H r ( t, x )) , f ( x )) ≤ ( α, ε ) . The proof of Proposition 5.5 will be discussed in a later subsection. The keyingredient that allows to deduce Proposition 5.2 from Proposition 5.5 are the longand thin covers for flow spaces from [8], that in turn generalize the long and thincell structures of Farrell-Jones [23, Sec. 7].
N PROOFS OF THE FARRELL-JONES CONJECTURE 17
Definition 5.6.
Let
R >
0. A collection U of open subsets of FS is said to be an R -long cover of A ⊆ FS if for all x ∈ A there is U ∈ U such that φ [ − R,R ] ( x ) := { φ t ( x ) | t ∈ [ − R, R ] } ⊆ U. Notation . (Periodic orbits) Let γ >
0. Write FS ≤ γ for the subset of FS consist-ing of all points x for which there are 0 < τ ≤ γ and g ∈ G such that φ τ ( x ) = gx . Theorem 5.8 (Existence of long thin covers) . Let FS be a cocompact N -flow spacefor G . Then there is ˜ N such that for all R > there exists γ > and a collection U of open subsets of FS such that(a) dim U ≤ ˜ N : any point of FS is contained in at most ˜ N + 1 members of U ,(b) U is an R -long cover of FS \ FS ≤ γ ,(c) U is G -invariant: for g ∈ G , U ∈ U we have g ( U ) ∈ U ,(d) U has finite isotropy: for all U ∈ U the group G U := { g ∈ G | g ( U ) = U } isfinite.Example . Let G := Z . Consider FS := R with the usual Z -action and the flowdefined by φ t ( x ) := x + t . If U R is an R -long Z -invariant cover of R of finite isotropythen the dimension of U R grows linearly with R .Theorem 5.8 states that this is the only obstruction to the existence of uniformlyfinite dimensional arbitrary long G -invariant covers of FS of finite isotropy. Remark . Theorem 5.8 is more or less [8, Thm. 1.4], see also [7, Thm. 5.6].The proof depends only on fairly elementary constructions, but is nevertheless verylong. (It would be nice to simplify this proof – but I do not know where to begin.)In these references in addition an upper bound for the order of finite subgroupsof G is assumed. This assumption is removed in recent (and as of yet unpublished)work of Adam Mole and Henrik R¨uping. Remark . For the flow spaces, that have been relevant for the Farrell-Jonesconjecture so far, it is possible to extend the cover U from FS \ FS ≤ γ to all of FS .The only price one has to pay for this extension is that in assertion (d) one hasto allow virtually cyclic groups instead of only finite groups. Note that with thischange Example 5.9 is no longer problematic; we can simply set U R := { R } .It is really at this point where the family of virtually cyclic subgroups plays aspecial role and appears in proofs of the Farrell-Jones Conjecture. Remark . In the case of CAT(0) groups the extension of the cover from FS \ FS γ to FS is really the most technical part of the arguments in [7].It would be more satisfying to have a result that provides this extension (afterallowing virtually cyclic groups) for general cocompact flow spaces. Remark . One may think of Theorem 5.8 as a (as of now quite difficult!)parametrized version of the very easy fact that Z has finite asymptotic dimension. Sketch of proof for Proposition 5.2 using Proposition 5.5.
The idea is easy.We produce a map F : FS → E that is suitably contracting along the flow lines of φ . Then we can compose f : X → FS from Proposition 5.5 with F to produce therequired map F ◦ f : X → E .Let G be a CAT(0)-group. Let ε > FS be the cocompact N -flowspace for G from Proposition 5.5. As discussed in Remark 5.11 there is ˜ N suchthat for all R > U of open subsets of FS such that(a) dim U ≤ ˜ N ,(b) U is an R -long cover of FS ,(c) U is G -invariant, (d) U has virtually cyclic isotropy: for all U ∈ U the group G U := { g ∈ G | g ( U ) = U } is virtually cyclic.Let now E := |U| be the nerve of the cover U . The vertex set of this simplicial com-plex is U and we have |U| = { P U ∈U t U U | t U ∈ [0 , , P U ∈U t U = 1 and T t U =0 U = ∅} . Note that |U| is a simplicial ( G, VCyc)-complex. To construct the desired map F : FS → E we first change the metric on FS . For (large) Λ > φ , and corresponds to timealong flow lines. More precisely, d Λ ( x, y ) := inf n n X i =1 α i + Λ ε i | ∃ x = x , x , . . . , x n such that d folFS ( x i − , x i ) ≤ ( α i , ε i ) for i = 1 , . . . , n o For U ∈ U , x ∈ FS let a U ( x ) := d Λ ( x, FS \ U ) and define F : FS → |U| by F ( x ) := X U ∈U a U ( x ) P V ∈U a V ( x ) U. As U is G -invariant, F is G -equivariant. If R > ε ), then there are Λ > δ > d folFS ( x, x ′ ) ≤ ( α, δ ) = ⇒ d ( F ( x ) , F ( x ′ )) ≤ ε. (More details for similar calculations can be found in [9, Sec. 4.3, Prop. 5.3].) Thuswe can compose with F and conclude that Proposition 5.5 implies Proposition 5.2. The flow space for a
CAT(0) -space.
This subsection contains an introductionto the flow space for CAT(0)-groups from [7]. Let Z be a CAT(0)-space. Definition 5.14. A generalized geodesic in Z is a continuous map c : R → Z forwhich there exists an interval ( c − , c + ) such that c | ( c − ,c + ) is a geodesic and c | ( −∞ ,c − ) and c | ( c + , + ∞ ) are constant. (Here c − = −∞ and/or c + = + ∞ are allowed.) Definition 5.15.
The flow space for Z is the space FS ( Z ) of all generalizedgeodesics c : R → Z . It is equipped with the metric d FS ( c, c ′ ) := Z R d ( c ( t ) , c ′ ( t ))2 e | t | dt and the flow φ τ ( c )( t ) := c ( t + τ ) . Remark . The fixed point space for the flow FS ( Z ) R := { c | φ t ( c ) = c for all t } is via c c (0) canonically isometric to Z .The flow space FS ( Z ) is somewhat singular around Z = FS ( Z ) R . For examplethere are well defined maps c c ( ±∞ ) from FS ( Z ) to the bordification [16, Ch.II.8]¯ Z of Z , but these maps fail to be continuous at Z . Remark . The metric d FS ( c, c ′ ) cares most about d ( c ( t ) , c ′ ( t )) for t close to0. For example if c (0) = c ′ (0) then d FS ( c, c ′ ) is bounded by R ∞ te t dt . For thisreason one can think of c (0) as marking the generalized geodesic c . If c (0) isdifferent from both c ( c − ) and c ( c + ) (equivalently if c − < < c + ) then the triple( c ( c − ) , c (0) , c ( c + )) uniquely determines c . Remark . An isometric action of G on Z induces an isometric action on FS ( Z )via ( g · c )( t ) := g · c ( t ). If the action of G on Z is in addition cocompact, properand Z has dimension at most N , then FS ( Z ) is a cocompact 3 N + 2-flow space for G in the sense of Definition 5.3, see [7, Sec. 2]. N PROOFS OF THE FARRELL-JONES CONJECTURE 19
For cocompactness it is important that we allowed c − = −∞ and c + = + ∞ inthe definition of generalized geodesics. Remark . For hyperbolic groups there is a similar flow space constructed byMineyev [36]. This space is an essential ingredient for the proof that hyperbolicgroups satisfy the assumptions of Theorem A. Mineyev’s construction motivatedthe flow space for CAT(0) groups.However, for hyperbolic groups the construction is really much more difficult.A priori, there is really no local geometry associated to a hyperbolic group, hy-perbolicity is just a condition on the large scale geometry and Mineyev extractslocal information from this in the construction of his flow space. In contrast, for aCAT(0)-group the corresponding CAT(0)-space provides local and global geometryright from the definition.
Sketch of proof for Proposition 5.5.
Let Z be a finite dimensional CAT(0)-space with an isometric, cocompact, proper action of the group G . Let G = h S | R ′ i be a finite presentation of G . Pick a base point x ∈ Z . For R > B R ( x ) bethe closed ball in Z of radius R around x . This will be our transfer space. Let ρ R : Z → B R ( x ) be the closest point projection. For x, x ′ ∈ Z , t ∈ [0 ,
1] we write t (1 − t ) · x + t · x ′ for the straight line from x to x ′ parametrized by constantspeed d ( x, x ′ ). For g, h ∈ G , t ∈ [0 , x ∈ B R ( x ) let ϕ Rg ( x ) := ρ R ( g · x ) ,H Rg,h ( t, x ) := ρ R ((1 − t ) · gϕ Rh ( x ) + t · ghx ) . Then H Rg,h is a homotopy ϕ Rg ◦ ϕ Rh → ϕ Rgh . This data also specifies a homotopy action( ϕ R , H R ) on B R ( x ). We will use the map ι R : B R ( x ) → FS ( Z ) where ι R ( x ) is theunique generalized geodesic c in Z with c − = 0, c + = d ( x, x ), c ( c − ) = c (0) = x and c ( c + ) = x , i.e., the generalized geodesic from x to x that starts at time 0 at x . For T ≥ f T,R := φ T ◦ ι R : B R ( x ) → FS ( x ). Proposition 5.5 follows fromthe next Lemma; this will conclude the sketch of proof for Proposition 5.5. Lemma 5.20.
Let α := max s ∈ S d ( x , sx ) . For any ε > there are T, R > suchthat for all x ∈ FS , s ∈ S ∪ S − , r ∈ R ′ , t ∈ [0 , we have − d folFS ( f T,R ( ϕ Rs ( x )) , s · f T,R ( x )) ≤ ( α, ε ) , − d folFS ( f T,R ( H Rr ( t, x )) , f T,R ( x )) ≤ ( α, ε ) .Sketch of proof. We will only discuss the first inequality; the second inequality in-volves essentially no additional difficulties.Let us first visualize the generalized geodesics c := f T,R ( ϕ Rs ( x )) and c ′ := s · f T,R ( x ). The generalized geodesic c starts at c ( c − ) = x and ends at c ( c + ) = ϕ Rs ( x ).If d ( x , sx ) ≤ R , then the endpoint ϕ Rs ( x ) coincides with sx ; otherwise we canprolong c (as a geodesic) until it hits sx . If T ≤ d ( x , ϕ Rs ( x ) then c (0) is the uniquepoint on the image of c of distance T from x , otherwise c (0) = c ( c + ) = ϕ Rs ( x ) = ρ R ( sx ). The generalized geodesic c ′ starts at c ′ ( c ′− ) = sx and ends at c ′ ( c ′− ) = sx .If T ≤ d ( sx , sx ), then c ′ (0) is the unique point on the image of c ′ of distance T from sx , otherwise c ′ (0) = c ′ ( c ′ + ) = sx . We draw this as sx x sxρ R ( sx ) c ′ (0) c (0) . There are two basic cases to consider.
Case I: d ( sx, x ) is small.Then ρ R ( sx ) = sx , and both c and c ′ converge to the constant geodesic at sx with T → ∞ . Consequently d FS ( c, c ′ ) is small for large T . Case II: d ( sx, x ) is large.Then we may have ρ R ( sx ) = sx . Note that d ( ρ R ( sx ) , sx ) ≤ d ( x , sx ) ≤ α . Let t := d ( c (0) , sx ) − d ( c ′ (0) , sx ) ∈ [ − α, α ]. Using the CAT(0)-condition one can thencheck that d FS ( φ t ( c ) , c ′ ) will be small provided that T , R − T , RR − T are large.A more careful analysis of the two cases shows that it is possible to pick R and T (depending only on ε ) such that for any x one of the two cases applies and therefore d folFS ( c, c ′ ) ≤ ( α, ε ). (cid:3) Remark . The assumption that the action of G on the CAT(0)-space Z is co-compact is important for the proof of Proposition 5.2, because it implies that theaction of G on the flow space FS ( Z ) is also cocompact. This in turn is impor-tant for the construction of R -long covers: Theorem 5.8 otherwise only allows theconstruction of R -long covers for a cocompact subspace of the flow space.Nevertheless, there are situations where it is possible to construct R -long coversfor flow spaces that are not cocompact. For example GL n ( Z ) acts properly andisometrically but not cocompactly on a CAT(0) space. But it is possible to construct R -long covers for the corresponding flow space [11]. This uses as an additional inputa construction of Grayson [29] and enforces a larger family of isotropy groups forthe cover. This is the family F n − mentioned in Remark 1.9.There are very general results of Farrell-Jones [26] without a cocompactnessassumption, but I have no good understanding of these methods.6. Z n ⋊ Z as a Farrell-Hsiang group For A ∈ GL n ( Z ) let Z n ⋊ A Z be the corresponding semi-direct product. We fixa generator t ∈ Z . Then for v ∈ Z n we have t · vt − = Av in Z n ⋊ A Z . The goalof this section is to outline a proof of the following fact from [3]. Recall that Abdenotes the family of abelian subgroups. In the case of Z n ⋊ A Z these are all finitelygenerated free abelian. Proposition 6.1.
Suppose that no eigenvalue of A over C is a root of unity. Thenthe group Z n ⋊ A Z is a Farrell-Hsiang group with respect to the family Ab of abeliangroups, i.e., there is N such that for any ε > there are(a) a group homomorphism π : Z n ⋊ A Z → F where F is finite,(b) a simplicial ( Z n ⋊ A Z , Ab) -complex E of dimension at most N (c) a map f : ` H ∈H ( F ) Z n ⋊ A Z /π − ( H ) → E that is Z n ⋊ A Z -equivariant upto ε : d ( f ( sx ) , s · f ( x )) ≤ ε for all s ∈ S , x ∈ ` H ∈H ( F ) G/π − ( H ) .Here S is any fixed generating set for G . N PROOFS OF THE FARRELL-JONES CONJECTURE 21
Remark . The Farrell-Jones Conjecture holds for abelian groups. Thus usingTheorem C and the transitivity principle 1.7 we deduce from Proposition 6.1 thatthe Farrell-Jones Conjecture holds for the group Z n ⋊ A Z from Proposition 6.1. Finite quotients of Z n ⋊ A Z . We write Z /s for the quotient ring (and underlyingcyclic group) Z /s Z . Let A s denote the image of A in GL n ( Z /s ). Choose r, s ∈ N such that the order | A s | of A s in GL n ( Z /s ) divides r . Then we can form ( Z /s ) n ⋊ A s Z /r and there is canonical surjective group homomorphism π : Z n ⋊ A Z ։ ( Z /s ) n ⋊ A s Z /r. Hyper-elementary subgroups of ( Z /s ) n ⋊ A s Z /r .Lemma 6.3. Let s = p · p be the product of two primes. Let r := s · | A s | . If H isa hyper-elementary subgroup of ( Z /s ) n ⋊ A s Z /r then there is q ∈ { p , p } such that(a) π − ( H ) ∩ Z n ⊆ ( q Z ) n or(b) the image of π − ( H ) under Z n ⋊ A s Z ։ Z is contained in q Z . To prove Lemma 6.3 we recall [3, Lem. 3.20].
Lemma 6.4.
Let s be any natural number. Let r := s · | A s | . Let C be a cyclicsubgroup of Z /s n ⋊ A s Z /r that has nontrivial intersection with ( Z /s ) n .Then there is a prime power q N ( N ≥ ) such that − q N divides r = r ′ s , − q N does not divide the order of the image of C in Z /r , − q divides the order of C ∩ ( Z /s ) n .Proof of Lemma 6.3. Let H ⊆ ( Z /s ) n ⋊ A s Z /r be hyper-elementary. There is ashort exact sequence C H ։ P with P a p -group and C a cyclic group. Thecyclic group C can always be arranged to be of order prime to p .( Z /s ) n / / / / ( Z /s ) n ⋊ A s Z /r pr / / / / Z /rH ∩ ( Z /s ) n / / / / O O O O H pr / / / / O O O O pr( H ) O O O O C ∩ ( Z /s ) n / / / / O O O O C pr / / / / O O O O pr( C ) O O O O There are two cases.
Case I: C ∩ ( Z /s ) n is trivial.Then H ∩ ( Z /s ) n is a p -group. Let q be the prime from { p , p } that is differentfrom p . Then (a) will hold. Case II: C ∩ ( Z /s ) n is nontrivial.Then there is a prime q as in Lemma 6.4. As q divides | C ∩ ( Z /s ) n | we have q ∈ { p , p } and q = p . It follows that q divides [ Z /r : pr( H )]. This implies (b). (cid:3) Contracting maps.
Example . Consider the standard action of Z n on R n . Let ¯ H := ( l Z ) n ⊆ Z n and ϕ : ¯ H → Z n be the isomorphism v vl . Let res ϕ R n be the ¯ H -space obtainedby restricting the action of Z n on R n with ϕ l . Then x xl defines an ¯ H -map F : Z n → res ϕ R n . This map is contracting. In fact by increasing l we can make F as contracting as we like, while we can keep the metric on R n fixed.A variant of this simple construction will be used to produce maps as in (c) ofProposition 6.1. This will finish the discussion of the proof of Proposition 6.1. Proposition 6.6.
Let S ⊆ Z n ⋊ A Z be finite. For any ε > there is l such thatfor all l ≥ l the following holds.Let ¯ H := Z n ⋊ A ( l Z ) ⊆ Z n ⋊ A Z . Then there is a simplicial ( Z n ⋊ A Z , Ab) -complex E of dimension and an ¯ H -equivariant map F : Z n ⋊ A Z → E such that d ( F ( g ) , F ( h )) ≤ ε whenever g − h ∈ S .Proof. We apply the construction of Example 6.5 to the quotient Z of Z n ⋊ A Z .Let E := R . We use the standard way of making E = R a simplicial complexin which Z ⊆ R is the set of vertices. Let ¯ H act on E via ( vt k ) · ξ := kl ξ ; this is asimplicial action. Finally define F : Z n ⋊ A Z → E by F ( vt k ) := kl . It is very easyto check that F has the required properties for sufficiently large l . (cid:3) Proposition 6.7.
Let S ⊆ Z n ⋊ A Z be finite. There is N ∈ N depending only on n such that for any ε > there is l such that for all l ≥ l the following holds.Let ¯ H := ( l Z ) n ⋊ A Z ⊆ Z n ⋊ A Z . Then there is a simplicial ( Z n ⋊ A Z , Cyc) -complex E of dimension at most N and an ¯ H -equivariant map F : Z n ⋊ A Z → E such that d ( F ( g ) , F ( h )) ≤ ε whenever g − h ∈ S .Sketch of proof. As in the proof of Proposition 6.6 we start with the constructionfrom Example 6.5, now applied to the subgroup Z n ⊆ Z n ⋊ A Z . However, unlikethe quotient Z , there is no homomorphism from Z n ⋊ A Z to the subgroup and itwill be more difficult to finish the proof.Let Z n ⋊ A Z act on R n × R via vt k · ( x, ξ ) := ( v + A k ( x ) , k + ξ ). Let ϕ : ¯ H → Z n ⋊ A Z be the isomorphism vt k vl t k . The map F : Z n ⋊ A Z → res ϕ R n × R ,( vt k ) ( v/l, k ) is ¯ H -equivariant and contracting in the Z n -direction, but not in the Z -direction. In order to produce a map that is also contracting in the Z -directionwe use the flow methods from Section 5.There is Z n ⋊ A Z -equivariant flow on R n × R defined by φ τ ( x, ξ ) = ( x, τ + ξ ). Itis possible to produce a simplicial ( Z n ⋊ A Z , Cyc)-complex E of uniformly boundeddimension (depending only on n ) and Z n ⋊ A Z -equivariant map F : R n × R → E that is contracting in the flow direction (but expanding in the transversal R n -direction). To do so one uses Theorem 5.8; E will be the nerve of a suitable longcover of R n × R .The fact that F is expanding in the R n -direction can be countered by thecontracting property of F . All together, the composition F ◦ F : Z n ⋊ A Z → res ϕ E has the desired properties. (cid:3) Remark . As many other things, the idea of using a flow space in the proof ofProposition 6.7 originated in the work of Farrell and Jones [24]. I found this trickvery surprising when I first learned about it.
Lemma 6.9.
Let ¯ H be a subgroup of Z n ⋊ A Z and l, k ∈ N such that(a) ¯ H ∩ Z n ⊆ l Z ,(b) ¯ H maps to k Z under the projection Z n ⋊ A Z → Z ,(c) the index [ Z n : (id − A k ) Z n ] is finite and l ≡ modulo [ Z n : (id − A k ) Z n ] .Then ¯ H is subconjugated to ( l Z ) n ⋊ A Z .Proof. Consider the image ¯ H l of ¯ H under Z n ⋊ A Z → ( Z /l ) n ⋊ A Z . Then (a)implies that the restriction of the projection ( Z /l ) n ⋊ A Z → Z to ¯ H l is injective.In particular ¯ H l is cyclic. By (b) there is v ∈ Z n such that vt k ∈ Z n ⋊ A Z mapsto a generator of ¯ H l . Assumption (c) implies that there is w ∈ Z n such that N PROOFS OF THE FARRELL-JONES CONJECTURE 23 v ≡ (id − A k ) w modulo ( l Z ) n . A calculation shows that w conjugates ¯ H to asubgroup of ( l Z ) n ⋊ A Z . (cid:3) Proof of Proposition 6.1.
Let L be a large number. Since A has no roots of unity aseigenvalues, the index i k := [ Z n : (id − A k ) Z n ] is finite for all k . Let K := i · i · · · i L .By a theorem of Dirichlet there are infinitely many primes congruent to 1 modulo K . Let s = p · p be the product of two such primes, both ≥ L , and set r := s ·| A s | .We use the group homomorphism π : Z n ⋊ A Z ։ ( Z /s ) n ⋊ A s Z /r . Because ofLemma 6.3 we find for any hyper-elementary subgroup H of ( Z /s ) n ⋊ A s Z /r an q ∈ { p , p } such that π − ( H ) ⊆ Z n ⋊ A ( q Z ) or π − ( H ) ∩ Z n ⊆ ( q Z ) n . In thefirst case we set l := q . In the second case we have either π − ( H ) ⊆ Z n ⋊ A ( l Z )for some l > L or we can apply Lemma 6.9 to deduce that (up to conjugation) π − ( H ) ⊆ ( q Z ) n ⋊ A Z and we again set l := q .Therefore it suffices to find simplicial ( Z n ⋊ A Z , Ab)-complexes E , E whosedimension is bounded by a number depending only on n (and not on l ) and maps f : Z n ⋊ A Z / ( l Z ) n ⋊ A Z → E f : Z n ⋊ A Z / Z n ⋊ A ( l Z ) → E that are G -equivariant up to ε . If f : Z n ⋊ A Z → E is the map from Proposi-tion 6.7, then we can set E := ( Z n ⋊ A Z ) × ( l Z n ) ⋊ A Z E and define f by f ( vt k ) := (cid:0) ( vt k ) , f (( vt k ) − ) (cid:1) . Similarly, we can produce f using Proposition 6.6. (cid:3) References [1] A. Bartels, S. Echterhoff, and W. L¨uck. Inheritance of isomorphism conjectures under colim-its. In Cortinaz, Cuntz, Karoubi, Nest, and Weibel, editors,
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