The K-theoretic Farrell-Jones conjecture for CAT(0)-groups

We prove the K-theoretic Farrell-Jones conjecture with (twisted) coefficients for CAT(0)-groups.


Introduction
The K-theoretic Farrell-Jones conjecture with coefficients for a group G says that the K-theoretic assembly map is an isomorphism for all m ∈ Z and every additive category A with a strict right G-action. Here E VCyc G denotes the classifying space of the group G with respect to the family of virtually cyclic subgroups. Any additive category A with a right Gaction induces a covariant functor K A from the orbit category of G to the category of spectra with (strict) maps of spectra as morphisms (see [BR07,Definition 3.1]). We denote the associated G-homology theory by H G * (−; K A ) (see [DL98,sections 4 and 7]). The assembly map is the map induced by the projection E VCyc G → pt onto the space consisting of one point.
The K-and L-theoretic Farrell-Jones conjecture plays an important role in the classification and geometry of manifolds. Moreover, it implies a variety of wellknown conjectures, e.g. the Bass-, Borel-, Kaplansky-and Novikov-conjecture. For more information on the Farrell-Jones conjecture we refer to the survey article [LR05].
In this paper we prove the K-theoretic Farrell-Jones conjecture with coefficients for CAT(0)-groups. By a CAT(0)-group we mean a group which admits a cocompact proper action by isometries on a finite dimensional CAT(0)-space. The proof is based on methods from [BLR08], [BL09] and [BL10]. In [BLR08] Bartels, Lück and Reich show the bijectivity of the K-theoretic assembly map for hyperbolic groups. In [BL09] and [BL10] Bartels and Lück investigate the K-theoretic assembly map for CAT(0)-groups and prove bijectivity in degree m ≤ 0 and surjectivity in degree m = 1.
The general strategy to prove the K-theoretic Farrell-Jones conjecture is to study the obstruction category O G (E F G, pt; A) whose K-theory gives the homotopy fiber of the K-theoretic assembly map. Then a transfer map has to be constructed which allows to replace the one-point-space by a suitable metric space which gives room for certain constructions. This metric space has to be carefully chosen since we need contracting properties afterwards. In the case of hyperbolic groups this space is a compactification of the Rips complex of the group G. In the case of CAT(0)-groups we use large closed balls in the associated CAT(0)-space. Finally, contractible maps on the metric spaces are used to gain control and to prove the vanishing of the K-theory groups of the obstruction category.
The main difficulty in enlarging the result of Bartels and Lück comes from the fact that the closed balls in the CAT(0)-space are no G-spaces. They only admit a homotopy G-action. This is sufficient to define the transfer map for K 1 since this map only requires homotopy chain actions. But for higher K-theory we have to take account of higher homotopies. A useful tool to tackle this problem are strong homotopy actions which we introduce in section 2. They describe in a simple way a homotopy action together with all higher homotopies. We use them in section 3 where we define the notion of strong transfer reducibility for groups. This definition specifies the requirements that we have on the metric space which replaces the onepoint-space. We show that hyperbolic groups and CAT(0)-groups are strongly transfer reducible over the family of virtually cyclic subgroups (see Example 3.2 and Theorem 3.4).
The following sections are dedicated to the proof of the K-theoretic Farrell-Jones conjecture with coefficients for groups which are strongly transfer reducible. More precisely, we prove Theorem 1.1. Let G be a group which is strongly transfer reducible over a family F of subgroups of G. Let A be an additive G-category, i.e. an additive category with a strict right G-action by functors of additive categories. Then the K-theoretic assembly map is an isomorphism for all m ∈ Z.
In section 4 we give a short review of controlled algebra which is a crucial tool in the proof. In particular, we define the obstruction category. An outline of the proof of Theorem 1.1 is given in section 5. The last two sections deal with the transfer map and finish the proof of Theorem 1.1.
Following the proof of [BL10, Lemma 2.3] we see that Theorem 1.1 and Theorem 3.4 imply Corollary 1.2. Let G 1 , G 2 be groups which satisfy the K-theoretic Farrell-Jones conjecture with coefficients. Then the groups G 1 × G 2 and G 1 * G 2 satisfy the K-theoretic Farrell-Jones conjecture with coefficients, too. This paper was supported by the SFB 878 -Groups, Geometry & Actions.

Strong homotopy actions
Let G be a CAT(0)-group, i.e. a group which admits a cocompact proper action by isometries on a finite dimensional CAT(0)-space Y . We would like to replace the CAT(0)-space Y by a compact space, namely a large ball in Y . The price we have to pay for this replacement is that we only retain a G-action on the ball up to homotopy. To control these homotopies we introduce the notion of a strong homotopy action.
Remark 2.2. It is not true in general that a topological space X which is homotopy equivalent to a G-space admits a strong homotopy action by conjugation with the homotopy equivalence. Strong homotopy actions appear in the following situation: Let Y be a G-space For example, we can consider the CAT(0)-space together with a deformation retraction on a ball by projecting along geodesics. (We will make this more precise in the proof of Theorem 3.4.) In this situation we define Ω : inductively by Ω(g 0 , x) := g 0 · x and Ω(g j , t j , g j−1 , . . . ) := g j · H tj (Ω(g j−1 , . . .)) for j ≥ 1. Then Ψ := H 0 • Ω is a strong homotopy action. On the other hand, a strong homotopy action Ψ induces a subspace M of the space of continuous mappings ∞ j=0 (G × [0, 1]) j × G → X. It is defined by M := Ψ(?, t α , g α−1 , . . . , g 0 , x) α ∈ N 0 , t i ∈ [0, 1], g i ∈ G, x ∈ X and has a G-action given by c g (f ) := f (?, 1, g). Moreover, we obtain a deformation retraction H : If we start with a strong homotopy action and construct the associated deformation retraction then the strong homotopy action associated to this deformation retraction coincides with the original strong homotopy action. In general, the other composition of the two constructions is not the identity. Nevertheless, both constructions are inverse to each other if the interiorX of X = H 0 (Y ) satisfies G ·X = Y . This condition is satisfied in the case of our CAT(0)-group as long as the ball, on which we project, is large enough. Anyhow, we will not make use of this fact.
In analogy to [BL09, Definition 1.4 and Definition 3.4] we make the following definition.
Definition 2.3. Let Ψ be a strong homotopy G-action on a metric space (X, d X ). Let S ⊆ G be a finite symmetric subset which contains the trivial element e ∈ G. Let k ∈ N be a natural number.
(1) For g ∈ G we define F g (Ψ, S, k) ⊂ map(X, X) by (2) For (g, x) ∈ G × X we define S 1 Ψ,S,k (g, x) ⊂ G × X as the subset consisting of all (h, y) ∈ G × X with the following property: There are a, b ∈ S, (3) For Λ ∈ R >0 we define the quasi-metric d Ψ,S,k,Λ on G × X as the largest quasi-metric on G × X satisfying . We remind the reader that the difference between a metric and a quasi-metric is that in the later case the distance ∞ is allowed. Notice that the quasi-metric (1) The subset S generates G if and only if d Ψ,S,k,Λ is a metric.
(3) The topology on G × X induced by d Ψ,S,k,Λ coincides with the product topology.

Strong transfer reducibility
In this section we introduce the notion of strong transfer reducibility for groups which is an analogue of the notion of transfer reducibility defined in [BL09, Definition 1.8].
Definition 3.1. Let F be a family of subgroups of G. The group G is called strongly transfer reducible over F if there exists a natural number N ∈ N with the following property: For every finite symmetric subset S ⊆ G containing the trivial element e ∈ G and every natural numbers k, n ∈ N there are • a compact contractible controlled N -dominated metric space X, • a strong homotopy G-action Ψ on X and • a cover U of G × X by open sets such that (1) U is an open F -cover, Hyperbolic groups are strongly transfer reducible over the family of virtually cyclic subgroups. As metric space X we choose the compactification of the Rips complex. The strong homotopy action Ψ on X is given by the action of the hyperbolic group on X: Ψ(g j , t j , . . . , g 0 , x) := g j · . . . · g 0 · x. For more details we refer to the proof of [BL09, Proposition 2.1].
Remark 3.3. Let G be a group which is strongly transfer reducible over a family of subgroups F . Let H < G be a subgroup. We set By restricting the strong homotopy G-action Ψ on X we obtain a strong homotopy H-action. Moreover, we can restrict the cover U of G × X to a cover of H × X. We conclude that H is strongly transfer reducible over F H .

Following [BL10] we obtain
Theorem 3.4. Every CAT(0)-group is strongly transfer reducible over the family of virtually cyclic subgroups.
Proof. Let Y be a finite dimensional CAT(0)-space on which G acts cocompactly and properly. Fix a base point x 0 ∈ Y . As metric space X we will choose a (large) closed ball B R (x 0 ) ⊆ Y around the base point. By [BL10, Lemma 6.2] B R (x 0 ) is a compact contractible controlled (2 · dim(Y ) + 1)-dominated metric space.
For R > 0 we define a strong homotopy action We define Ψ R as the associated strong homotopy action (see Remark 2.2).
For the construction of the cover U we have to introduce the flow space F S(Y ) which is the G-space consisting of all generalized geodesics c : R → Y . The metric on F S(Y ) is given by 2 · e |t| dt.
We define a G-equivariant flow Φ : We set N := max{ N , 2 · dim(Y ) + 1}. The construction of the cover U is based on the contracting property described in Lemma 3.5 below. We fix α > 0 as in the assertion of Lemma 3.5. Let V be an open VCyc-cover of F S(Y ) of dimension at most N and let ǫ be a positive real number such that the two properties mentioned above are satisfied. For this ǫ > 0 we obtain R, T > 0 from Lemma 3.5. Then the cover with ι : G × B R (x 0 ) → F S(Y ), (g, y) → c gx0,gy has the desired properties.
In the proof of Theorem 3.4 we used the following lemma which is a modification of [BL10, Proposition 3.8] resp. [BL10, Lemma 5.12].
Lemma 3.5. Let S ⊆ G be a finite symmetric subset containing the trivial element e ∈ G. Let k, n ∈ N. Then there exists α > 0 with the following property: For all ǫ > 0 there are R, T > 0 such that for every (g, (We use the same notation as in the proof of Theorem 3.4.) Proof. We set α := 2n · (k + 1) · α ′ with α ′ := max{d Y (gx 0 , hx 0 ) | g, h ∈ S k+1 }. Let ǫ > 0. By [BL10, Proposition 3.5] there exist R, T > 0 such that for x, We fix such positive real numbers R, T .
For τ := σ n we obtain This finishes the proof of Lemma 3.5.
The proof of Theorem 1.1 is based on the following proposition which is a modification of [BL09, Proposition 3.9].
Proposition 3.6. Let G be a group which is strongly transfer reducible over a family F of subgroups. Let N be the number appearing in the definition of "strongly transfer reducible". Let S ⊆ G be a finite symmetric subset containing the trivial element e ∈ G. Then for every k ∈ N there exist • a compact contractible controlled N -dominated metric space X, • a strong homotopy G-action Ψ on X, • a positive real number Λ, • a simplicial complex Σ of dimension ≤ N with a simplicial cell preserving G-action and • a G-equivariant map f : G × X → Σ such that • the isotropy groups of Σ belong to F and • k · d 1 f (g, x), f (h, y) ≤ d Ψ,S,k,Λ (g, x), (h, y) for all (g, x), (h, y) ∈ G × X.
Proof. We choose a strong homotopy G-action Ψ on a metric space X and a cover U of G×X which satisfy the properties stated in Definition 3.1 for S, k and n := 4N k. Using Lemma 2.4 (2) we conclude as in the proof of [BL09, Proposition 3.7] that for every x ∈ X there exists Λ x > 0 and U x ∈ U such that the n-ball around (e, x) with respect to the quasi-metric d Ψ,S,k,Λx lies in U x . Moreover, since X is compact, there exists Λ > 0 such that every n-ball with respect to the quasi-metric d Ψ,S,k,Λ lies in some U ∈ U (see the proof of [BL09, Proposition 3.7]). Let Σ := |U| be the realization of the nerve of U and let f be the map induced by U, i.e. , x), (h, y)) | (h, y) / ∈ U } and s(g, x) := U a U (g, x). Notice that |a U (g, x) − a U (h, y)| ≤ d Ψ,S,k,Λ ((g, x), (h, y)) and hence U∈U |a U (g, x) − a U (h, y)| ≤ 2N · d Ψ,S,k,Λ (g, x), (h, y) for all (g, x), (h, y) ∈ G × X. We calculate For the last inequality we used the fact that the n-ball around (g, x) lies in some U ∈ U and hence s(g, x) ≥ n.

The obstruction category
In this section we recall the definition of the obstruction category. In the following A denotes a small additive category (with strictly associative direct sum) which is provided with a strict right G-action.
Definition 4.1. Let X be a G-space and let (Y, d Y ) be a metric space with an isometric G-action. We consider the G-space G × X × Y × [1, ∞) with the Gaction given by h(g, x, y, t) := (hg, hx, hy, t). We define the obstruction category O G (X, (Y, d Y ); A) as follows. An object in O G (X, (Y, d Y ); A) is a collection A = (A g,x,y,t ) (g,x,y,t)∈G×X×Y ×[1,∞) of objects in A with the following properties: • A is locally finite, i.e. for every z 0 ∈ G×X ×Y ×[1, ∞) there exists an open neighborhood U such that the set ∞)) with the following properties: Composition is given by matrix multiplication, i.e.
The obstruction category O G (X, (Y, d Y ); A) inherits the structure of an additive category from A. We use the same notation as in [BL09,subsection 4.4] which slightly differs from the notation used in [BLR08] (see [BL09,Remark 4.10]).
The construction is functorial in Y : Let f : Y → Y ′ be a G-equivariant map with the property that for every r > 0 there exists R > 0 such that d Y ′ (f (y 1 ), f (y 2 )) < R whenever d Y (y 1 , y 2 ) < r. Then the map f induces a functor f * : We are mostly interested in O G (E F G, pt; A) because of the following proposition which is proven in [BLR08, Proposition 3.8].
Proposition 4.2. Let G be a group and m 0 ∈ Z such that for all m ≥ m 0 and all additive G-categories A. Then the assembly map (1.1) is an isomorphism for all m ∈ Z and all additive G-categories A.
The reason why we study the category O G (E F G, (Y, d Y ); A) not only for Y := pt is that we need room for certain constructions. Moreover, we want to consider simultaneously metric spaces (Y n , d n ) with isometric G-action (n ∈ N). In analogy to [BLR08,subsection 3.4] we define the additive subcategory by requiring additional conditions on the morphisms. A morphism φ = (φ(n)) n∈N is allowed if there are R > 0 and a finite subset F ⊆ G (not depending on n) such that φ(n) (g,x,y,t),(g ′ ,x ′ ,y ′ ,t ′ ) = 0 whenever is a Karoubi filtration and we denote the quotient by O G (E F G, (Y n , d n ) n∈N ; A) >⊕ . Notice that a sequence of G-equivariant maps (f n : if for every r > 0 there exists R > 0 such that d ′ n (f n (y 1 ), f n (y 2 )) < R whenever d n (y 1 , y 2 ) < r.

5.
Outline of the proof of Theorem 1.1 In this section we sketch the proof of Theorem 1.1. Since the class of groups satisfying the K-theoretic Farrell-Jones conjecture is closed under directed colimits, it suffices to prove the bijectivity of the K-theoretic assembly map (1.1) for every finitely generated subgroup H of G (with respect to the family F H := {F ∩H | F ∈ F }). Moreover, strong transfer reducibility is stable under taking subgroups (see Remark 3.3). This shows that it is enough to prove Theorem 1.1 for finitely generated groups. Therefore we can and will assume that G is finitely generated.
We fix a finite symmetric generating subset S ⊆ G which contains the trivial element e ∈ G. We apply Proposition 3.6 to S n := {s 1 · s 2 · . . . · s n | s i ∈ S} ⊆ G and k := n and obtain • compact contractible controlled N -dominated metric spaces X n , • strong homotopy G-actions Ψ n on X n , • positive real numbers Λ n , • simplicial complexes Σ n of dimension ≤ N with simplicial cell preserving G-actions and • G-equivariant maps f n : G × X n → Σ n such that • the isotropy groups of Σ n belong to F and • n · d 1 (f n (g, x), f n (h, y)) ≤ d Ψn,S n ,n,Λn ((g, x), (h, y)) for all (g, x), (h, y) ∈ G × X n . We abbreviate d n := d Ψn,S n ,n,Λn .
By Proposition 4.2 it suffices to show where the middle row comes from the Karoubi filtration. (Notice that the composition n∈N pr n • diag * in the middle column is the diagonal map.) The transfer map will be constructed in section 7. The maps pr * are induced by the projections pr : G × X n → pt resp. pr : Σ n → pt. The equation

Preparations for the transfer
We will define the transfer map as the map induced by a functor In this section we give a quite short review of the construction of the category ch hfd O G (E F G, (G × X n , d n ) n∈N ; A) >⊕ . For more details we refer to [BLR08, subsection 6.2].
For a metric space (Y, d Y ) with an isometric G-action we define the category O G (E F G, (Y, d Y ); A κ ) in the same way as in section 4 but we replace A by A κ for a fixed (suitably chosen) infinite cardinal κ and drop the assumption that the support of objects is locally finite. Moreover, instead of requiring for a morphism φ = (φ z,z ′ ) z,z ′ ∈G×EF G×Y ×[1,∞) that the sets {z | φ z,z ′ = 0} and {z | φ z ′ ,z = 0} are finite, we define a morphism to be a morphism z ′ ∈G×EF G×Y ×[1,∞) B z ′ → z∈G×EF G×Y ×[1,∞) A z in the category A κ . For a sequence (Y n , d n ) n∈N of metric spaces with isometric G-action we define by requiring additional conditions on the morphisms precisely as in section 4. The inclusion is a Karoubi filtration and we denote the quotient by O G (E F G, (Y n , d n ) n∈N ; A κ ) >⊕ .
For the rest of this section we abbreviate One should think of the inclusion O ⊂ O as an inclusion of a full additive subcategory on objects satisfying finiteness conditions into a large category which gives room for constructions. Let C be an additive category (e.g. O or O). We write Idem(C) for its idempotent completion. We define ch f (C) to be the category of chain complexes in C that are bounded above and below and ch ≥ (C) to be the category of chain complexes that are bounded below. We write ch hf (Idem(O) ⊂ Idem(O)) for the full subcategory of ch ≥ (Idem(O)) consisting of chain complexes which are chain homotopy equivalent to a chain complex in ch f (Idem(O)). We write ch hfd (O) for the full subcategory of ch ≥ Idem(O) consisting of objects C which are homotopy retracts of objects in ch f (O), i.e. there exists a diagram C i − → D r − → C with D ∈ ch f (O) such that the composition r • i is chain homotopic to the identity on C.
The category ch hfd (O) is a Waldhausen category: The notion of chain homotopy leads to a notion of weak equivalence, and we define cofibrations to be those chain maps which are degree-wise the inclusion of a direct summand. The following lemma is proven in [BLR08, Lemma 6.5].
Lemma 6.1. The inclusion O ⊂ ch hfd (O) induces an equivalence on K m for all m ≥ 1.
We recall from [BR05, subsection 8.2] that for a given Waldhausen category W there exists a Waldhausen category W whose objects are sequences where the c α are morphisms in W that are both cofibrations and weak equivalences. A morphism f in W is represented by a sequence of morphisms (f α , f α+1 , f α+2 , · · · )