K- and L-theory of graph products of groups

We compute the group homology, the algebraic $K$- and $L$-groups, and the topological $K$-groups of right-angled Artin groups, right-angled Coxeter groups, and more generally, graph products.

1. Introduction 1.1. Basic setup. Suppose that we are given the following data: • A finite simplicial graph X on the vertex set V and a collection of groups W := {W v | v ∈ V }. Denote by W = W (X, W) the associated graph product, see Section 3, and by Σ the flag complex associated to X. Examples of graph products are right-angled Artin groups and right-angled Coxeter groups; • A commutative ring with unit Λ and an equivariant homology theory H ? * with values in Λ-modules, see Definition 2.1. Our main examples will be those associated to algebraic K-and L-theory or topological K-theory, which appear in the Farrell-Jones Conjecture or the Baum-Connes Conjecture; • A non-empty class C of finite groups which is closed under isomorphisms, passage to subgroups and passage to quotient groups. Our main example will be the class of all finite groups; • A class E of Λ-modules with the property that for an exact sequence 0 → V 0 → V 1 → V 2 → 0 the Λ-module V 1 belongs to E if and only if both V 0 and V 2 belong to E.

Main result.
Fix an integer n. We obtain a covariant functor H C n : Groups → Λ-Modules, G → H G n (E C (G)), where E C (G) is the classifying space of the family of subgroups of G which belong to C, see Section 2 and (2.4).
Let S be the poset of flag subcomplexes of Σ and let P be the poset of simplices of Σ, both ordered by inclusion, where the empty subcomplex and the empty simplex are allowed. For an element L in S, we can consider the subgraph X ∩ L of X. Let W (L) be the graph product associated to X ∩ L and the collection of groups W| V ∩L = {W v | v is a vertex of L}. With this notation W (Σ) is the graph product W (X, W) and W (∅) = {1}. We obtain a covariant functor W * : S → Groups, L → W (L).

Let
I : P → S be the inclusion which sends a simplex σ of Σ to the corresponding flag subcomplex of Σ. Sometimes we identify σ in P with I(σ) in S. For instance we will often write W (σ) instead of W (I(σ)). Notice that the covariant functor W * • I : P → Groups sends a simplex σ of Σ to v∈V ∩σ W v .
We obtain a covariant functor n (E C (W (L))).
We are interested in the value at Σ, i.e., in H W n (E C (W )) for W = W (X, W). The composite H C n • W * • I : P → Λ-Modules is given by Define for a simplex σ the quotient Λ-module of H C n • W * • I(σ) by where τ runs through the simplices of I(σ) which are different from σ. The idea is to kill everything in H C n • W * • I(σ) which comes from a proper simplex τ of I(σ). For a simplex σ of Σ, let ch n (Σ, σ) be the set of n-chains σ 0 < σ 1 < · · · < σ n in P with σ 0 = σ. Define the integer n σ := n≥0 (−1) n · | ch n (Σ, σ)|.
Denote by G 0 (E) the Grothendieck group of elements in E, i.e., the abelian group with the isomorphism classes of elements in E as generators and relations is an isomorphism; (ii) For every σ ∈ P, the canonical projection p σ : H C n • W * • I(σ) → S σ H C n • W * • I has a section s σ : S σ H C n • W * • I → H C n • W * • I(σ). Any collection of such sections and the canonical maps n (E C (W (σ))).
We mention that it is both unusual and fortunate that in assertion (i) the source and target involve the same degree. In general one would expect that in the source all degrees m ≤ n occur. The reason for this simplification is that for each simplex σ the inclusion W (σ) → W is split injective. This leads also to the explicit splitting in assertion (ii).
Note that the number n σ appearing in assertion (iii) depends only on Σ. It is given by 1 − χ(Σ) for σ = ∅. It has the following geometric interpretation if σ is non-empty. Let Σ ′ be the barycentric subdivision of Σ. Then ch n (Σ, σ) can be interpreted as a collection of n-simplices in Σ ′ . Each simplex in ch n (Σ, σ) contains the vertex given by σ. The collection of the faces of all these simplices of ch n (Σ, σ) determines a simplicial subcomplex D σ of Σ ′ which can be contracted to the vertex given by σ. Its boundary ∂D σ consists of all those faces of simplices of ch n (Σ, σ) which do not contain σ. One easily checks If Σ is the triangulation of a closed manifold of dimension d, then ∂D σ is homeomorphic to S d−1−dim(σ) and hence n σ = (−1) d−dim(σ) .
1.3. Computations. We will illustrate the potential of Theorem 1.1 by computing the group homology of G, the algebraic K-and L-theory of the group ring of G, and the topological K-theory of the group C * -algebra of G in Sections 6 and 7 if G is a right-angled Artin group or a right-angled Coxeter group. These computations are based on the Baum-Connes Conjecture and the Farrell-Jones Conjecture which we will briefly recall in Section 5 and which hold for these groups. The situation in the Farrell-Jones setting is more complicated since we have to deal with the family VCY of virtually cyclic subgroups, whereas in Theorem 1.1 the family F IN of finite subgroups is considered. The passage from F IN to VCY is discussed in Subsection 5.3. This is different in the Baum-Connes setting since there the family F IN is used. In order to get full functoriality we need to consider the maximal group C * -algebra instead of the reduced C * -algebra which makes no difference for right-angled Artin groups and right-angled Coxeter groups.
Computation in this context means not only that we identify the corresponding K-and L-groups of G as abelian groups but we give explicit isomorphisms identifying them with K-and L-groups of the ground ring. For instance, we show for a right-angled Coxeter group W associated to the finite flag complex Σ that there is for every n ∈ Z an isomorphism σ K n (f σ ) : where σ runs through the simplices of Σ including the empty simplex and f σ : C → C * r (W ) is an explicit homomorphism of C * -algebras depending on σ. If σ is empty, it is given by the obvious inclusion C → C * r (W ).
(1 − t i ) for t i the generator of the i-th factor Z/2 and the homomorphism C[W (σ)] → C * r (W ) coming from the inclusion W (σ) → W , see Remark 7.17. This implies where r is the number of simplices of Σ including the empty simplex. Moreover, we can write down an explicit basis B = {b σ | σ ∈ P} for the finitely generated free Z-module K 0 (C * r (W )), namely, for σ = ∅ we take the class of the idempotent 1 in C * r (W ) and for σ = ∅ we take the class of the idempotent in C * r (W ) given by the image of the idempotent . These computations for right-angled Coxeter groups were carried out in the second author's master's thesis [22]. In this section we recall the axioms of an equivariant homology theory and the notion of a classifying space of a family of subgroups. For an amalgamated product of groups, we deduce a Mayer-Vietoris type sequence for the values of an equivariant homology theory on classifying spaces.
Fix a discrete group G and a commutative ring Λ with unit. A G-homology theory H G * with values in Λ-modules is a collection of covariant functors H G n from the category of G-CW -pairs to the category of Λ-modules indexed by n ∈ Z together with natural transformations ∂ G n (X, A) : H G n (X, A) → H G n−1 (A) := H G n−1 (A, ∅) for n ∈ Z such that the axioms concerning G-homotopy invariance, the long exact sequence of a pair, excision, and disjoint unions are satisfied, see [24, Section 1].
Let α : H → G be a group homomorphism. Given an H-space X, define the induction of X with α to be the G-space α * X = G × α X which is the quotient of G×X by the right H-action (g, x)·h := (gα(h), h −1 x) for h ∈ H and (g, x) ∈ G×X. The following definition is taken from [24, Section 1] except that the induction structure in this paper is defined for every group homomorphism α. Definition 2.1. An equivariant homology theory H ?
* with values in Λ-modules assigns to each group G a G-homology theory H G * with values in Λ-modules together with the following so called induction structure: Given a group homomorphism α : H → G and an H-CW -pair (X, A), there are for every n ∈ Z natural homomorphisms ind α : H H n (X, A) → H G n (α * (X, A)) satisfying: • Compatibility with the boundary homomorphisms: Let β : G → K be another group homomorphism. Then we have for n ∈ Z • Compatibility with conjugation: If ker(α) acts freely on X \ A, then ind α : H H n (X, A) → H G n (α * (X, A)) is bijective for all n ∈ Z.
We briefly fix some conventions concerning spectra. If X is a space, denote by X + the pointed space obtained from X by adding a disjoint base point. Let Spectra be the category of spectra in the following naive sense. A spectrum E = {(E(n), σ(n)) | n ∈ Z} is a sequence of pointed spaces {E(n) | n ∈ Z} together with pointed maps called structure maps σ(n) : E(n) ∧ S 1 −→ E(n + 1). A map of spectra f : E → E ′ is a sequence of maps f (n) : E(n) → E ′ (n) which are compatible with the structure maps σ(n), i.e., we have f (n + 1) • σ(n) = σ ′ (n) • (f (n) ∧ id S 1 ) for all n ∈ Z. Given a spectrum E and a pointed space X, we can define their smash product X ∧ E by (X ∧ E)(n) := X ∧ E(n) with the obvious structure maps.
It is a classical result that a spectrum E defines a homology theory by setting where cone denotes the reduced cone. We want to extend this to equivariant homology theories. Let Groupoids be the category of small connected groupoids with covariant functors as morphisms. Notice that a group can be considered as a groupoid with one object in the obvious way.
For the proof of the following result we refer to [ Suppose that E respects equivalences, i.e., it sends an equivalence of groupoids to a weak equivalence of spectra. Then E defines an equivariant homology theory H ? * (−; E) such that we have H G n (G/H; E) ∼ = H H n (pt; E) ∼ = π n (E(H)) for every group G, subgroup H ⊆ G and n ∈ Z. The construction is natural in E. Example 2.3 (Borel homology). Let E be a spectrum. Let H * (−; E) be the (nonequivariant) homology theory associated to E. Given a groupoid G, denote by EG its classifying space. If G has only one object and the automorphism group of this object is G, then EG is a model for EG. We obtain two covariant functors Thus we obtain two equivariant homology theories H ? * (−; c E ) and H ? * (−; b E ) from Theorem 2.2. The second one is called the equivariant Borel homology associated to H * (−; E). We get for any group G and any G-CW -complex X natural isomorphisms . Let C be a non-empty class of groups which is closed under isomorphisms, passage to subgroups and passage to quotient groups. Our main examples will be the class F IN of finite groups and the class VCY of virtually cyclic groups.
Given a group G, denote by E C (G) the classifying space of G with respect to the family of subgroups F C (G) = {H ⊆ G | H ∈ C}. It is defined to be a terminal object in the G-homotopy category of G-CW -complexes, whose isotropy groups belong to F C (G). A model for E C (G) is a G-CW -complex whose H-fixed point set is contractible if H ∈ C and is empty if H / ∈ C. With this notation E F IN (G) is the classifying space of proper actions, sometimes also denoted by EG. We sometimes denote E VCY (G) by EG. For more information about classifying spaces of families we refer for instance to [26].
Given a group homomorphism f : G → H, we denote by the up to H-homotopy unique H-map coming from the universal property of E C (H) and the fact that f * E C (G) is an H-CW -complex whose isotropy groups are of the shape f (K) for K ∈ C and hence all belong to C again. Given an equivariant homology theory H ? * with values in Λ-modules, it induces homomorphisms of Zgraded Λ-modules . One easily checks that thus we obtain a covariant functor Then we obtain a long exact Mayer-Vietoris sequence Proof. There is a 1-dimensional G-CW -complex T whose underlying space is a tree such that the 1-skeleton is obtained from the 0-skeleton by the G-pushout where q is the disjoint union of the canonical projections G/G 0 → G/G 1 and G/G 0 → G/G 2 , see [35,Theorem 7 in §4.1 on page 32]. If we take the cartesian product with E C (G) we obtain another cellular G-pushout. Its associated Mayer-Vietoris sequence yields the long exact sequence There is a G-homeomorphism (j k ) * j * k E C (G) . Using the induction structure of the equivariant homology theory H ?
* we obtain identifications for k = 0, 1, 2 ). The K-fixed point set T K is a non-empty subtree and hence contractible for every finite subgroup K ⊆ G, see [35, Theorem 15 in 6.1 on page 58 and 6.3.1 on page 60]. Hence the projection T × E C (G) → E C (G) is a G-homotopy equivalence since every element in C is finite by assumption. Hence we get an identification ). Now we obtain the desired long exact sequence from the last long exact sequence and the identifications above.

Graph products of groups
In this section we give the definition of a graph product of groups. We show that the value of an equivariant homology theory on the classifying space of a graph product is the colimit over a certain system of subgroups.
Let X be a finite simplicial graph on the vertex set V and suppose that we are given a collection of groups W : is defined as the quotient of the free product * v∈V W v of the collection of groups W by introducing the relations In other words, elements of subgroups W v and W v ′ commute if there is an edge joining v and v ′ . This notion is due to Green [18].
Let Σ be the flag complex associated to X. Denote by S = S(Σ) the poset of flag subcomplexes of Σ, ordered by inclusion, where we also allow the empty subcomplex. Then we can assign to L ∈ S the graph product group W (X ∩ L, W| V ∩L ), where X ∩ L agrees with the 1-skeleton of L and Then we obtain group homomorphisms and by the trivial homomorphism if v / ∈ L 0 . One easily checks that thus we obtain a covariant functor W * : S → Groups and a contravariant functor By construction W * and W * agree on objects and we write for an object L ∈ S.
The elementary proof of the following lemma is left to the reader.
Then we obtain a group isomorphism Notice that in particular we get from Lemma 3.1 (ii) that for any two Let H ? * be an equivariant homology theory with values in Λ-modules. Let C be a non-empty class of groups which is closed under isomorphisms, passage to subgroups and passage to quotient groups. We have defined a covariant functor H C * and studied its main properties in Section 2.
We want to study the covariant ΛS-module and are in particular interested in its value at Σ itself. Viewing a simplex as a flag subcomplex yields for every L ∈ S a map of posets For two elements L and L ′ in S with L ≤ L ′ let be the maps of posets induced by the inclusion L ⊆ L ′ . Define the Λ-module One easily checks that thus we obtain a covariant ΛS-module For every object L ∈ S there is an obvious Λ-homomorphism induced by the inclusions I L (σ) ⊆ L for σ running through the simplices of L. One easily checks that the collection of the Λ-homomorphisms T n (L) fits together to a map of covariant ΛSmodules Theorem 3.4. Suppose that each element in C is a finite group.
Then the map of ΛS-modules T n of (3.3) is an isomorphism. In particular its evaluation at Σ yields a Λ-isomorphism Proof. Notice that T n (L) is obviously an isomorphism if L lies in the image of I : P → S, since then L is a terminal object in P(L) and hence under the obvious We show for any L ∈ S that T n (L) is an isomorphism by induction over the number of vertices of L. If L is the empty subcomplex, then L is in the image of I : P → S and the claim has already been proved. The induction step is done as follows. We only have to consider the case, where L is not in the image of I : P → S. Since L is a flag complex, there must be two vertices v 1 and v 2 in L which are not connected by an edge. Let L 1 be the flag subcomplex of L spanned by v 1 and all vertices in L which are connected to v 1 by an edge. In particular v 2 is not a vertex of L 1 . Let L 0 be the flag subcomplex of L which is spanned by all vertices v for which there exists an edge whose terminal points are v and v 1 . Notice that v 1 does not belong to L 0 and L 1 is the cone over L 0 with cone point v 1 . Let L 2 be the flag subcomplex of L spanned by all vertices except v 1 . Then L = L 1 ∪ L 2 and L 0 = L 1 ∩ L 2 and the number of vertices of L 0 , L 1 and L 2 is smaller than the number of vertices of L. The induction hypothesis applies to L k and hence T n (L k ) is an isomorphism for k = 0, 1, 2. Since P(L) = P(L 1 ) ∪ P(L 2 ) and P(L 0 ) = P(L 1 )∩P(L 2 ), the sequence induced by the inclusions P(L 0 ) ⊆ P(L k ) and P(L k ) ⊆ P(L) for k = 1, 2 is split injective by functoriality. We get from Lemma 2.5 and Lemma 3.1 (i) an exact sequence One easily checks that we obtain a commutative diagram with exact rows Now the induction step follows from the Five-Lemma.
We have proven part (i) of the main Theorem 1.1.

Mackey modules
In this section we prove the remaining parts of the main Theorem 1.1. More generally, we show that the colimit of any Mackey module splits as a direct sum over its index category.
Let Σ be a finite simplicial complex. Denote by P = P(Σ) the poset of its simplices ordered by inclusion, where we allow the empty simplex as well. The dimension of the empty simplex is defined to be −1. Notice that for two elements σ and τ in P the intersection σ ∩ τ is again an element in P which is uniquely determined by the property that it is maximal among all those elements µ in P satisfying both µ ≤ σ and µ ≤ τ .
Let Λ be a commutative ring with unit. A Mackey ΛP-module M = (M * , M * ) is a bifunctor P → Λ-Modules, i.e., a covariant functor M * and a contravariant functor M * from P to Λ-Modules such that M * and M * agree on objects and for objects σ 1 , σ 2 , τ of P satisfying σ k ≤ τ for k = 1, 2, we get The name Mackey ΛP-module comes from the analogy to the classical notion of a Mackey functor, where (4.1) replaces the double coset formula, see [37, Section 6.1]. for an object τ of P.
Example 4.2 (Mackey modules coming from graph products). Our main example comes from Section 3. Let X be a finite simplicial graph on the vertex set V and suppose that we are given a collection of groups W : respectively. Define Λ-quotient modules of N (τ ) by Then we obtain a sequence of inclusions of Λ-modules and a sequence of epimorphisms of Λ-modules Proof. We compute for σ ∈ P satisfying σ < τ and dim(σ) ≤ d

This implies
We conclude for every σ ∈ P satisfying σ < τ and dim(σ) ≤ d Now the assertion follows.
Define a map   Proof. We start with injectivity. Suppose that ad( s) is not injective. Then there exists an element σ ∈ P and a non-trivial element a = (a τ ) τ ∈P ∈ τ ∈P E τ (S τ M * )(σ) such that ad( s)(σ)(a) = 0. Choose τ 0 ∈ P with a τ0 = 0 such that for all τ ∈ P with a τ = 0 we have dim(τ ) ≤ dim(τ 0 ). If τ ∈ P satisfies a τ = 0 and hence E τ (S τ M * )(σ) = 0, we use the explicit description of E τ (S τ M * ) to conclude τ ≤ σ. The composite Consider an element τ ∈ P with a τ = 0. Then dim(τ ) ≤ dim(τ 0 ) and hence we get Hence it is zero if τ = τ 0 and it is p τ0 if τ = τ 0 . This implies that the restriction of the composite (4.6) to the summand associated to τ is trivial if τ = τ 0 and is the identity under the obvious identification E τ0 (S τ0 M * )(σ) = S τ0 M * if τ = τ 0 . We conclude that the composite (4.6) sends a to a τ0 under the obvious identification E τ0 (S τ0 M * )(σ) = S τ0 M * . Since this implies a τ0 = 0, we get a contradiction. This finishes the proof that ad( s) is injective.
Fix a class of Λ-modules E with the property that for an exact sequence 0 → V 0 → V 1 → V 2 → 0 the Λ-module V 1 belongs to E if and only if both V 0 and V 2 belong to E. An example is the class E of Λ-modules whose underlying set is finite, and for a Noetherian ring Λ the class E of finitely generated Λ-modules. Denote by G 0 (E) the Grothendieck group of elements in E, i.e., the abelian group with the isomorphism classes of elements in E as generators and relations [ for every short exact sequence 0 → V 0 → V 1 → V 2 → 0 of Λ-modules belonging to E. Then we get in G 0 (E) Proof. The bar-resolution yields a finite free ΛP-resolution C * of the constant ΛPmodule Λ with value Λ such that see [11,Section 3]. Since C * is a finite free ΛP-chain complex, the Λ-chain complex C * ⊗ ΛP M * is a finite-dimensional Λ-complex whose Λ-chain modules belong to E, and we get in G 0 (E) Since

Isomorphism Conjectures in Kand L-theory
In this section we review the Isomorphism Conjectures of Baum-Cones and Farrell-Jones and recollect the most important results on the passage from F IN to VCY in the Farrell-Jones setting.
Let C be a non-empty class of groups which is closed under isomorphisms, passage to subgroups and passage to quotient groups. Recall that given a group G, we denote by E C (G) the classifying space of G with respect to the family of subgroups F C (G) = {H ⊆ G | H ∈ C}. Consider a covariant Groupoids-spectrum E : Groupoids → Spectra which respects equivalences. We obtain an equivariant homology theory H ? * (−; E) associated to E from Theorem 2.2.
Then the Meta-Isomorphism Conjecture for E and the class C predicts that the projection pr : E C (G) → G/G induces for all n ∈ Z an isomorphism H G n (pr; E) : H G n (E C (G); E) ∼ = − → H G n (G/G; E) = π n (E(G)). If we make the appropriate choices for E and C, this becomes the Baum-Connes Conjecture or it becomes the Farrell-Jones Conjecture for algebraic K-theory, for algebraic L-theory, for Waldhausen's A-theory, or for topological Hochschild homology.
5.1. The Baum-Connes Conjecture. Given a discrete group G, denote by C * r (G) and C * r (G; R) its reduced complex and reduced real group C * -algebra and by C * m (G) and C * m (G; R) its maximal complex and maximal real group C * -algebra. There are covariant functors which send equivalences of groupoids to weak equivalences of spectra and satisfy π n (K C * m (G)) = K n (C * m (G)) and π n (K C * m ;R (G)) = KO n (C * m (G; R)) for n ∈ Z. Here K * and KO * denote topological K-theory. If we consider the class of finite groups, the Meta-Isomorphism Conjecture reduces to the Baum-Connes Conjecture for the maximal group C * -algebra. It predicts the bijectivity of the assembly maps for n ∈ Z K G n (EG) → K n (C * m (G)); KO G n (EG) → KO n (C * m (G; R)), where the source is given by equivariant K-homology for which we have the identifications K G n (EG) = H G n (EG; K C * m ) and KO G n (EG) = H G n (EG; K C * m ;R ). We can apply K-theory to the natural maps of C * -algebras C * m (G) → C * r (G) and C * m (G; R) → C * r (G; R) to obtain maps f n : K n (C * m (G)) → K n (C * r (G)) and f R n : KO n (C * m (G; R)) → KO n (C * r (G; R)). The Baum-Connes Conjecture predicts that the composites There are counterexamples to the Baum-Connes Conjecture for the maximal group C * -algebra, but no counterexamples to the Baum-Connes Conjecture are known. We want to consider the Baum-Connes Conjecture for the maximal group C * -algebra since K n (C * m (G)) and KO n (C * m (G; R)) are functorial in G for all group homomorphisms, whereas K n (C * r (G)) and KO n (C * r (G; R)) are functorial for injective group homomorphisms, but not in general for any group homomorphism. Moreover the covariant functors (5.1) and (5.2) are defined on Groupoids. This ensures that the induction structure is available for all group homomorphisms and not only for injective group homomorphisms as it is the case if we replace (5.1) and (5.2) by their versions for the reduced C * -algebras. We later want to apply the induction structure also to certain split surjective group homomorphisms, see Remark 3.2.
There is a more general Baum-Connes Conjecture with coefficients, which is known to be true for a large class of groups and which has good inheritance properties. In particular, the class of groups satisfying the Baum-Connes Conjecture with coefficients is closed under taking graph products, since it is stable under finite direct products and amalgamated products, see [31] and [32]. : Groupoids → Spectra, which send equivalences of groupoids to weak equivalences of spectra and satisfy π n (K R (G)) = K n (RG) and π n (L −∞ R (G)) = L −∞ n (RG). Here K * denotes nonconnective algebraic K-theory and L −∞ * denotes algebraic L-theory with decoration −∞ . If we consider the class of virtually cyclic groups, the Meta-Isomorphism Conjecture reduces to the K-theoretic or the L-theoretic Farrell-Jones Conjecture which predicts that for all n ∈ Z the corresponding map There is a more general Full Farrell-Jones Conjecture which allows additive G-categories as coefficients. It is known to be true for a large class of groups and has good inheritance properties. In particular, the class of groups satisfying the Full Farrell-Jones Conjecture is closed under taking graph products, which is a result of Gandini-Rüping [17]. There also is a version of the Farrell-Jones Conjecture for Waldhausen's A-theory which we will not discuss here. It satisfies similar inheritance properties as the Full Farrell-Jones Conjecture, see [15] and [38]. Also the following Theorem 5.3 (i) holds in this setting, see [6].

5.3.
The passage from F IN to VCY. The Farrell-Jones Conjecture is more complicated than the Baum-Connes Conjecture since for the Farrell-Jones Conjecture the class of virtually cyclic groups has to be considered, whereas for the Baum-Connes Conjecture the class of finite groups suffices. Hence one has to understand the passage from EG to EG in the Farrell-Jones setting.

Theorem 5.3.
(i) Let G be any group and R be a ring. Then the relative assembly maps are split injective for all n ∈ Z; (ii) Let G be any group and R be a regular ring. Then the relative assembly map H G n (EG; K R ) → H G n (EG; K R ) is rationally bijective for all n ∈ Z; (iii) Let G be any group and R be a regular ring. Suppose that for any finite subgroup H ⊆ G its order |H| is invertible in R. Then the relative assembly map H G n (EG; K R ) → H G n (EG; K R ) is bijective for all n ∈ Z; (iv) Let R be a regular ring. Let W = W (X, W) be a graph product and d be a natural number. Suppose that for any vertex v ∈ V the group W v is either torsionfree or a finite group whose order divides d. Then the relative assembly map H W n (EW ; K R ) → H W n (EW ; K R ) is bijective after inverting d for all n ∈ Z; (v) Let G be any group and R be a ring with involution. Then the relative assembly map is bijective after inverting 2 for all n ∈ Z; (vi) Let G be any group and R be a ring with involution such that 2 is invertible in R. Then the relative assembly map is bijective for all n ∈ Z. (vii) Let R be a ring with involution. Let W = W (X, W) be a graph product.
Suppose that for any vertex v ∈ V the group W v is either torsionfree or a finite group of odd order. Then the relative assembly map is bijective for all n ∈ Z. Proof. (i) See [3] and [28]. (iv) Let VCY I be the class of virtually cyclic groups of type I, i.e., groups which admit a homomorphism to Z with finite kernel. Then the relative assembly map where K H ⊆ H is a finite subgroup such that H/K H is infinite cyclic and φ : K H → K H is given by conjugation with an element h ∈ H which is sent under the projection H → H/K H to a generator. Any finite subgroup of W is conjugated into a group W (σ) for some simplex σ of Σ such that W v is finite for every v ∈ V ∩ σ. (Indeed, this is obvious if Σ is a simplex. Otherwise, we can express W as an amalgamated product and use the fact that a finite subgroup of an amalgamated product is conjugated into one of the factors, see [35,Theorem 8

in 4.3 on page 36].
Then conclude via induction on the number of vertices.) Hence we can assume that there exists a simplex σ of Σ such that K H ⊆ W (σ) and W (σ) is finite. There is a group homomorphism r : W → W (σ) whose restriction to W (σ) is the identity, see Remark 3.2. Consider w ∈ K H . Then hwh −1 belongs to K H again. We compute Hence φ is given by conjugation with r(h) ∈ W (σ). The order of r(h) and thus also of φ divides d. We conclude from [28,Theorem 9.4 (v) This is proved in [25,Lemma 4.2]. (vi) The proof is analogous to the first part of the proof of (iv) using the fact that UNil-groups vanish, if 1/2 is contained in R, see [7,Corollary 3] and that the map is bijective for all n ∈ Z, see [25,Lemma 4.2].
(vii) Any finite subgroup of W is conjugated into a group W (σ) for some simplex σ of Σ such that W v is finite for every v ∈ V ∩ σ. Hence every finite subgroup has odd order and thus every infinite virtually cyclic subgroup of W is of type I. Now the claim follows from [25,Lemma 4.2]. This finishes the proof of Theorem 5.3.

Right-angled Artin groups
In this section we want to compute the group homology, the algebraic K-and L-theory, and the topological K-theory of a right-angled Artin group W . Recall that a right-angled Artin group is a graph product W = W (X, W) for which each of the groups W v is infinite cyclic. Note that W is torsionfree. Right-angled Artin groups satisfy the Baum-Connes Conjecture and the Baum-Connes Conjecture for the maximal group C * -algebra, which follows from [13] and [19]. Both the Ktheoretic and the L-theoretic Farrell-Jones Conjecture are satisfied for right-angled Artin groups, see [2] and [40]. For general information about right-angled Artin groups we refer for instance to Charney [9].
In the sequel we denote by r k the number of k-simplices in P = P(Σ) and put r = |P| = dim(Σ) k=−1 r k . Recall that the empty simplex is allowed in P and has dimension −1.
Let K * be a (non-equivariant) generalized homology theory with values in Λmodules. Let X be a CW -complex. It follows from the axioms of a generalized homology theory that there is an isomorphism, natural in X where we denote by pr : X × S 1 → X the projection, by i : S 1 = (S 1 , ∅) → (S 1 , pt) the inclusion, and by s n : K n (X × (S 1 , {pt})) ∼ = − → K n−1 (X) the suspension isomorphism.
By induction over k ≥ 0 we obtain an isomorphism where we denote by T k the k-dimensional torus k i=1 S 1 . Note that T 0 = pt. For i = 1, 2, . . . , k, let T k i ⊆ T k be the subspace consisting of elements (z 1 , z 2 , . . . , z k ) with z i = * , where * is a fixed base point in S 1 . Let j k i : T k i → T k be the inclusion. We will identify T k k = T k−1 .
Lemma 6.2. For every k ≥ 0 and n ∈ Z there is an isomorphism Its inverse is induced by the restriction of the inverse of the isomorphism B k n of (6.1) to the factor K n−k (pt) for the index i = k.
Proof. We use induction over k = 0, 1, 2, . . .. If k = 0, take c 0 n = id Kn(pt) . The induction step from (k − 1) to k ≥ 1 is done as follows. We have the following ). Since f 0 is surjective, the diagram above induces an isomorphism ) .

By induction hypothesis we have an isomorphism
This finishes the induction step and hence the proof of Lemma 6.2.
6.1. Group homology. Let K * be any generalized homology theory with values in Λ-modules. Notice that for any group G the CW -complex EG × G EG is a model for BG since EG is contractible after forgetting the G-action. We have introduced the equivariant homology theory given by the Borel construction and K * in Example 2.3. We conclude from Theorem 1.1 and Lemma 6.2 that there is an explicit Λ-isomorphism If we take for K * singular homology H * (−; Λ) with coefficients in Λ, this boils down to the well-known, see for example [21,Corollary 11], isomorphism of Λmodules Λ rn−1 ∼ = − → H n (BW ; Λ).
In particular we get the following relation for the Euler characteristics χ(BW ) = 1 − χ(Σ).

6.2.
Algebraic K-theory. Let R be a regular ring. We conclude from Theorem 1.1, Theorem 5.3 (iv), and Lemma 6.2 that there is an explicit isomorphism of abelian groups Its restriction to the summand belonging to σ is the composite of the map K n (RW (σ)) → K n (RW ) coming from the inclusion Z dim(σ)+1 = W (σ) → W with the restriction of the inverse of the iterated Bass-Heller-Swan isomorphism to the summand K n−dim(σ)−1 (R) belonging to i = dim(σ) + 1.
Since for a regular ring R its negative K-theory vanishes, we conclude K n (RW ) = 0 for n ≤ −1. If we take R = Z, we conclude that K n (ZW ) for n ≤ −1, K 0 (ZW ), and Wh(W ) vanish what is actually true if we replace W by any torsionfree group satisfying the Farrell-Jones Conjecture.
6.3. Algebraic L-theory. Let R be a ring with involution. We conclude from Theorem 1.1, Theorem 5.3 (vii), and Lemma 6.2 that there is an explicit isomorphism of abelian groups Its restriction to a summand comes from the Shaneson splitting.

Right-angled Coxeter groups
In this section we want to compute the group homology, the algebraic K-and L-theory, and the topological K-theory of a right-angled Coxeter group W . Recall that a right-angled Coxeter group is a graph product W = W (X, W) for which each of the groups W v is cyclic of order two. Right-angled Coxeter groups satisfy the Baum-Connes Conjecture and the Baum-Connes Conjecture for the maximal group C * -algebra, which follows from [13] and [19]. Both the K-theoretic and the L-theoretic Farrell-Jones Conjecture are satisfied for right-angled Coxeter groups, see [2] and [40].
In the sequel we denote by r k the number of k-simplices in P = P(Σ) and put r = |P| = dim(Σ) k=−1 r k . Recall that the empty simplex is allowed in P and has dimension −1.
During this section we denote by C 2 the cyclic group of order two. Fix an integer k ≥ 1. We will identify C k 2 = C k−1 2 × C 2 and put C 0 2 = {1}. For i = 1, 2, . . . , k, let (C k 2 ) i be the subgroup of C k 2 consisting of those elements (a 1 , a 2 , . . . , a k ) satisfying a i = 0 and denote by j k i : (C k 2 ) i → C k 2 the inclusion.
7.1. Group homology. Define for n ≥ 0 and k ≥ 0 an integer ρ n,k := where here and in the sequel we use the convention b j=a c j = 0 for a > b.
Theorem 7.1. We have for n ≥ 1 Its proof needs some preparation. Firstly, the numbers ρ n,k satisfy the following.
Proof. The assertion is obviously true for k = 0. The induction step from k − 1 to k ≥ 1 is done as follows. Recall that H n (C 2 ; Z) is Z if n = 0, C 2 if n ≥ 1 and n is odd, and {0} otherwise. The Künneth formula gives the following short exact sequence of Z-modules, which is natural in C k−1 It splits but the spitting is not natural in C k−1

2
. By rearranging the summands we obtain an isomorphism of Z-modules Using the induction hypothesis we calculate This finishes the proof of Lemma 7.5.
For n ≥ 1 and k ≥ 0 define Let the integer s n,k ≥ 0 be defined by Lemma 7.6. For n ≥ 1 and k ≥ 1 we have s n,k = ρ n,k−1 and s n,0 = 0.
Proof. Since H n ({1}; Z) = {0} for n ≥ 1, we have s n,0 = 0. The induction step from k − 1 to k ≥ 1 is done as follows. Theorem 1.1 yields an isomorphism Using the induction hypothesis and Lemma 7.5 we conclude This finishes the proof of Lemma 7.6. Now Theorem 7.1 follows from Theorem 1.1 applied to the equivariant homology theory given by Borel homology and singular homology with Z-coefficients, see Example 2.3, and from Lemma 7.6. Here we use the fact that for any group G the space EG × G EG is a model for BG.
Remark 7.7. If we replace in this subsection C 2 = Z/2 everywhere by Z/p l for some prime number p and some natural number l, then Theorem 7.1 remains true. This follows from two facts. Since Z/p l is a local ring, we conclude from [29, Lemma 1.2 on page 5] that for every natural number a, every summand of the abelian group a i=1 Z/p l is isomorphic to b j=1 Z/p l for some natural number b. The group homology H n (Z/p l ; Z) is isomorphic to Z/p l if n is odd and vanishes for even n with n ≥ 2.
7.2. Negative K-groups for R = Z. Since any finite subgroup of W is isomorphic to (Z/2) k for some natural number k, and K −1 (ZA) = 0 holds for a finite abelian group whose order is a prime power, see [4, Theorem 10.6 on page 695] or [8], the claim follows. 7.3. Projective class group for R = Z. Theorem 7.9.
(i) There is an isomorphism Proof. (i) We have for every group G the obvious splitting K 0 (ZG) ∼ = K 0 (Z) ⊕ K 0 (ZG). By [39,Theorem 12.9], is an isomorphism and we have an isomorphism
Theorem 7.11. Let R be a regular ring.
is rationally an isomorphism for all n ∈ Z; Proof. (i) Notice that any non-trivial finite cyclic subgroup C of C k 2 is isomorphic to C 2 and that the obvious composite is an isomorphism. The isomorphism appearing in [ where ε Q C : Q ⊗ Z K n (RC) → Q ⊗ Z K n (R) is induced by the projection C → {1}, the map j {1} is induced by the inclusion {1} → C, and j C is the composite of the inclusion ker(ε Q C ) → Q ⊗ Z K n (RC) with the map Q ⊗ Z K n (RC) → Q ⊗ Z K n (R[C k 2 ]) coming from the inclusion C → C k 2 . By naturality we get a commutative diagram where the vertical arrows come from the inclusions (C k 2 ) i → C k 2 . Notice that a cyclic subgroup of C k 2 belongs to (C k 2 ) i for some i ∈ {1, 2, . . . , k} if and only if it is different from the diagonal subgroup (C k 2 ) ∆ := {(a, a, . . . , a) | a ∈ C 2 } ⊆ C k 2 . Hence the composite is rationally bijective. Now assertion (i) follows from Theorem 1.1 and Theorem 5.3 (iv).
(ii) Due to Borel [5] we know for R = Z that We get from [20,Theorem 2.2] Hence we get Now assertion (ii) follows from assertion (i). 7.6. L-groups after inverting 2. The maps appearing in the result below are defined analogously to the maps appearing in Theorem 7.11. Theorem 7.12. Let R be a ring with involution.
(i) The map is an isomorphism after inverting 2 for all n ∈ Z; (ii) We have for R ∈ {Z, Q, R} otherwise. Proof. (i) Note that any non-trivial subgroup of the form C × P of C k 2 for a cyclic group C and a p-group P for an odd prime number p is isomorphic to C 2 . The isomorphism appearing in [1, (2.11)   Its proof needs some preparations. In the sequel we will write C k 2 multiplicatively and we denote by t i the generator of the i-th factor C 2 viewed as an element in C k 2 for i = 1, 2, . . . , k.
Let R be a ring in which 2 is invertible. We get a decomposition of rings, natural in R, → (a + b, a − b).
Its inverse sends (c, d) to 1 2 · (c + d) + (c − d) · t . Since algebraic K-theory is compatible with products, we obtain an isomorphism, natural in R, K n (R), which comes from the isomorphism of rings Its inverse is given by Lemma 7.15.
(i) Suppose that 2 is invertible in R. Then there is an isomorphism d k n : cok Its inverse is the composite of the homomorphism coming from the ring homomorphism . The homomorphism β agrees with the restriction of the inverse of the isomorphism S k n (R) of (7.14) to the factor K n (R) which belongs to ǫ given by ǫ(t i ) = −1 for i = 1, 2, . . . k; (ii) The same assertion holds if we replace algebraic K-theory by algebraic L-theory with the decoration −∞ ; (iii) The same assertion is true if we take R to be R or C and we replace algebraic K-theory by topological K-theory.
Note that the group W satisfies the Baum-Connes Conjecture and the Farrell-Jones Conjecture since it is a graph product of abelian groups. Hence for every regular ring R the assembly map is bijective after inverting 2 by Theorem 5.3 (iv). The proof of Theorem 7.8 applies verbatim to the group W so that we obtain K n (ZW ) = {0} for n ≤ −1.
For any equivariant homology theory we have By (8.1), we get from Theorem 7.16 for all n ∈ Z K n (C * r (W )) ∼ = Z 12 .