K- and L-theory of group rings over GL n (Z)

Arthur Bartels; Wolfgang Lück; Holger Reich; Henrik Rüping

doi:10.1007/s10240-013-0055-0

K- and L-theory of group rings over GL n (Z)

Bartels, Arthur; Lück, Wolfgang; Reich, Holger; Rüping, Henrik 2013-05-25 00:00:00 K- AND L-THEORY OF GROUP RINGS OVER GL (Z) by ARTHUR BARTELS , WOLFGANG LÜCK , HOLGER REICH , and HENRIK RÜPING ABSTRACT We prove the K- and L-theoretic Farrell-Jones Conjecture (with coefﬁcients in additive categories) for GL (Z). Introduction The Farrell-Jones Conjecture predicts a formula for the K- and L-theory of group rings R[G]. This formula describes these groups in terms of group homology and K- and L-theory of group rings RV, where V varies over the family VCyc of virtually cyclic subgroups of G. Main Theorem. — Both the K-theoretic and the L-theoretic Farrell-Jones Conjecture (see Deﬁ- nitions 0.1 and 0.2) hold for GL (Z). We will generalize this theorem in the General Theorem below. In particular it also holds for arithmetic groups deﬁned over number ﬁelds, compare Example 0.4,and extends to the more general version “with wreath products”. For cocompact lattices in almost connected Lie groups this result holds by Bartels- Farrell-Lück [1]. The lattice GL (Z) has ﬁnite covolume but is not cocompact. It is a long standing question whether the Baum-Connes Conjecture holds for GL (Z). For torsion free discrete subgroups of GL (R), or more generally, for fundamental groups of A-regular complete connected non-positive curved Riemannian manifolds, the Farrell-Jones Conjecture with coefﬁcients in Z has been proven by Farrell-Jones [14]. The formulation of the Farrell-Jones Conjecture. — Deﬁnition 0.1 (K-theoretic FJC). — Let G be agroup andletF be a family of subgroups. Then G satisﬁes the K-theoretic Farrell-Jones Conjecture with respect toF if for any additive G-categoryA the assembly map G G H (E (G); K ) → H (pt; K ) = KA FAA n n n induced by the projection E (G) → pt is bijective for all n ∈ Z. If this map is bijective for all n ≤ 0 and surjective for n = 1, then we say G satisﬁes the K-theoretic Farrell-Jones Conjecture up to dimension 1 with respect toF . If the familyF is not mentioned, it is by default the family VCyc of virtually cyclic subgroups. DOI 10.1007/s10240-013-0055-0 98 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING If one choosesA to be (a skeleton of) the category of ﬁnitely generated free R- modules with trivial G-action, then K (A) is just the algebraic K-theory K (RG) of n n the group ring RG. If G is torsion free, R is a regular ring, andF is VCyc, then the claim boils down to the more familiar statement that the classical assembly map H (BG; K ) → K (RG) n R n from the homology theory associated to the (non-connective) algebraic K-theory spec- trum of R applied to the classifying space BG of G to the algebraic K-theory of RG is a bijection. If we restrict further to the case R = Z and n ≤ 1, then this implies the vanish- ing of the Whitehead group Wh(G) of G, of the reduced projective class group K (ZG), and of all negative K-groups K (ZG) for n ≤−1. Deﬁnition 0.2 (L-theoretic FJC). — Let G be a group and letF be a family of subgroups. Then G satisﬁes the L-theoretic Farrell-Jones Conjecture with respect toF if for any additive G-category with involutionA the assembly map −∞ −∞ G G −∞ H E (G); L → H pt; L = LA n n n AA induced by the projection E (G) → pt is bijective for all n ∈ Z. If the familyF is not mentioned, it is by default the family VCyc of virtually cyclic subgroups. Given a group G, a family of subgroupsF is a collection of subgroups of G that is closed under conjugation and taking subgroups. For the notion of a classifying space E (G) for a familyF we refer for instance to the survey article [20]. The natural choice forF in the Farrell-Jones Conjecture is the family VC yc of virtually cyclic subgroups but for inductive arguments it is useful to consider other families as well. Remark 0.3 Relevance of the additive categories as coefﬁcients. — The versions of the Farrell-Jones Conjecture appearing in Deﬁnitions 0.1 and 0.2 are formulated and an- alyzed in [2], [7]. They encompass the versions for group rings RG over arbitrary rings R, where one can built in a twisting into the group ring or treat more generally crossed product rings R ∗ G and one can allow orientation homomorphisms w : G →{±1} in the L-theory case. Moreover, inheritance properties, e.g., passing to subgroups, ﬁnite prod- ucts, ﬁnite free products, and directed colimits, are built in and one does not have to pass to ﬁbered versions anymore. The original source for the (Fibered) Farrell-Jones Conjecture is the paper by Farrell-Jones [13, 1.6 on p. 257 and 1.7 on p. 262]. For more information about the Farrell-Jones Conjecture, its relevance and its various applications to prominent conjec- tures due to Bass, Borel, Kaplansky, Novikov and Serre, we refer to [6], [21]. We will often abbreviate Farrell-Jones Conjecture to FJC. K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 99 Extension to more general rings and groups. — We will see that it is not hard to generalize the Main Theorem as follows. General Theorem. — Let R be a ring whose underlying abelian group is ﬁnitely generated. Let G be a group which is commensurable to a subgroup of GL (R) for some natural number n. Then G satisﬁes both the K-theoretic and the L-theoretic Farrell-Jones Conjecture with wreath products Deﬁnition 6.1. Two groups G and G are called commensurable if they contain subgroups G ⊆ G 1 2 1 and G ⊆ G of ﬁnite index such that G and G are isomorphic. In this case G satisﬁes 2 1 2 1 2 the FJC with wreath products if and only if G does, see Remark 6.2. Example 0.4 (Ring of integers). — Let K be an algebraic number ﬁeld andO be its ring of integers. ThenO considered as abelian group is ﬁnitely generated free (see [22, Chapter I, Proposition 2.10 on p. 12]). Hence by the General Theorem any group G which is commensurable to a subgroup of GL (O ) for some natural number n satisﬁes n K both the K-theoretic and L-theoretic FJC with wreath products. This includes in partic- ular arithmetic groups over number ﬁelds. Discussion of the proof. — The proof of the FJC for GL (Z) will use the transitivity principle [13, Theorem A.10], that we recall here. Proposition 0.5 Transitivity principle. — LetF ⊂H be families of subgroups of G. Assume that G satisﬁes the FJC with respect toH and that each H ∈H satisﬁes the FJC with respect toF . Then G satisﬁes the FJC with respect toF . This principle applies to all versions of the FJC discussed above. In this form it can be found for example in [1, Theorem 1.11]. Themainstepinproving theFJC for GL (Z) is to prove that GL (Z) satisﬁes the n n FJC with respect to a familyF . This family is deﬁned at the beginning of Section 3. This family is larger than VCyc andcontains forexample GL (Z) for k < n.Wecan then use induction on n to prove that every group fromF satisﬁes the FJC. At this point we also use the fact that virtually poly-cyclic groups satisfy the FJC. To prove that GL (Z) satisﬁes the FJC with respect toF we will apply two results n n from [4], [24]. Originally these results were used to prove that CAT(0)-groups satisfy the FJC. Checking that they are applicable to GL (Z) is more difﬁcult. While GL (Z) is not n n aCAT(0)-group, it does act on a CAT(0)-space X. This action is proper and isometric, but not cocompact. Our main technical step is to show that the ﬂow space associated to this CAT(0)-space admits longF -covers at inﬁnity, compare Deﬁnition 3.7. In Section 1 we analyze the CAT(0)-space X. On it we introduce, following Grayson [16], certain volume functions and analyze them from a metric point of view. These functions will be used to cut off a suitable well-chosen neighborhood of inﬁnity so 100 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING that the GL (Z)-action on the complement is cocompact. In Section 2 we study sublat- tices in Z . This will be needed to ﬁnd the neighborhood of inﬁnity mentioned above. Here we prove a crucial estimate in Lemma 2.1. As outlined this proof works best for the K-theoretic FJC up to dimension 1; this case is contained in Section 3. The modiﬁcations needed for the full K-theoretic FJC are discussed in Section 4 and use results of Wegner [24]. For L-theory the induction does not work quite as smoothly. The appearance of index 2 overgroups in the statement of [4, Theorem 1.1(ii)] force us to use a stronger induction hypothesis: we need to assume that ﬁnite overgroups of GL (Z), k < n satisfy the FJC. (It would be enough to consider overgroups of index 2, but this seems not to simplify the argument.) A good formalism to accommodate this is the FJC with wreath products (which implies the FJC). In Section 5 we provide the necessary extensions of the results from [4] for this version of the FJC. In Section 6 we then prove the L-theoretic FJC with wreath products for GL (Z). In Section 7 we give the proof of the General Theorem. 1. The space of inner products and the volume function Throughout this section let V be an n-dimensional real vector space. Let X(V) be the set of all inner products on V. We want to examine the smooth manifold X(V) and equip it with an aut(V)-invariant complete Riemannian metric with non-positive sectional curvature. With respect to this structure we will examine a certain volume func- tion. We try to keep all deﬁnitions as intrinsic as possible and then afterward discuss what happens after choices of extra structures (such as bases). 1.1. The space of inner products. — We can equip V with the structure of a smooth manifold by requiring that any linear isomorphism V → R is a diffeomorphism with respect to the standard smooth structure on R . In particular V carries a preferred struc- ture of a (metrizable) topological space and we can talk about limits of sequences in V. We obtain a canonical trivialization of the tangent bundle TV (1.1) φ : V × V → TV which sends (x,v) to the tangent vector in T V represented by the smooth path R → V, t → x + t · v. The inverse sends the tangent vector in TV represented by a path w : (−, ) → Vto (w(0), w (0)).If f : V → W is a linear map, the following diagram commutes V × V TV f ×f Tf W × W TW = K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 101 ∗ ∗ Let hom(V, V ) be the real vector space of linear maps V → V from V to the dual ∗ ∗ ∗ V of V. In the sequel we will always identify V and (V ) by the canonical isomorphism ∗ ∗ ∗ V → (V ) which sends v ∈ V to the linear map V → R,α → α(v).Hence for s ∈ ∗ ∗ ∗ ∗ ∗ ∗ hom(V, V ) its dual s : (V ) = V → V belongs to hom(V, V ) again. Let Sym(V) ⊆ ∗ ∗ ∗ hom(V, V ) be the subvector space of elements s ∈ hom(V, V ) satisfying s = s.Wecan identify Sym(V) with the set of all bilinear symmetric pairings V × V → R, namely, given s ∈ Sym(V) we obtain such a pairing by (v, w) → s(v)(w). We will often write s(v, w) := s(v)(w). Under the identiﬁcation above the set X(V) of inner products on V becomes the open subset of Sym(V) consisting of those elements s ∈ Sym(V) for which s : V → V is bijec- tive and s(v, v) ≥ 0 holds for all v ∈ V, or, equivalently, for which s(v, v) ≥ 0 holds for all v ∈ V and we have s(v, v) = 0 ⇔ v = 0. In particular X(V) inherits from the vector space Sym(V) the structure of a smooth manifold. Given a linear map f : V → W, we obtain a linear map Sym(f ) : Sym(W) → ∗ ∗ Sym(V) by sending s : W → W to f ◦ s ◦ f . A linear isomorphism f : V → W induces a bijection X(f ) : X(W) → X(V). Obviously this is a contravariant functor, i.e., X(g ◦ f ) = X(f ) ◦ X(g).If aut(V) is the group of linear automorphisms of V, we obtain a right aut(V)-action on X(V). If f : V → W is a linear map and s and s are inner products on V and W, then V W −1 the adjoint of f with respect to these inner products is s ◦ f ◦ s : W → V. Consider a natural number m ≤ n := dim(V). There is a canonical isomorphism = ∗ m ∗ m β (V) : V − → V which maps α ∧ α ∧ ··· ∧ α to the map V → R sending v ∧ v ∧ ··· ∧ v to 1 2 m 1 2 m sign(σ ) · α (v ).Let s : V → V be an inner product on V. We obtain an i σ(i) σ ∈S i=1 inner product s on V by the composite β (V) s m ∗ m m ∗ m s : V −→ V −−→ V . One easily checks by a direct calculation for elements v ,v ,...,v ,w ,w ,...,w in V 1 2 m 1 2 m (1.2) s m (v ∧ ··· ∧ v ,w ∧ ··· ∧ w ) = det s(v ,w ) , 1 m 1 m i j i,j where (s(v ,w )) is the obvious symmetric (m, m)-matrix. i j i,j Next we want to deﬁne a Riemannian metric g on X(V). Since X(V) is an open subset of Sym(V) and we have a canonical trivialization φ of T Sym(V) (see (1.1)), Sym(V) we have to deﬁne for every s ∈ X(V) an inner product g on Sym(V).Itisgiven by −1 −1 g (u,v) := tr s ◦ v ◦ s ◦ u , s 102 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING for u,v ∈ Sym(V). Here tr denotes the trace of endomorphisms of V. Obviously g (−, −) −1 is bilinear and symmetric since the trace is linear and satisﬁes tr(ab) = tr(ba). Since s ◦ −1 ∗ −1 −1 (s ◦ u) ◦ s = s ◦ u holds, the endomorphism s ◦ u : V → V is selfadjoint with respect −1 −1 −1 −1 to the inner product s.Hence tr(s ◦ u ◦ s ◦ u) ≥ 0 holds and we have tr(s ◦ u ◦ s ◦ −1 u) = 0 if and only if s ◦ u = 0, or, equivalently, u = 0. Hence g is an inner product on Sym(V). We omit the proof that g depends smoothly on s. Thus we obtain a Riemannian metric g on X(V). Lemma 1.1. — The Riemannian metric on X is aut(V)-invariant. Proof. — We have to show for an automorphism f : V → V, an element s ∈ X(V) and two elements u,v ∈ T X(V) = T Sym(V) = Sym(V) that s s g T X(f )(u), T X(f )(v) = g (u,v) s s s X(f )(s) holds. This follows from the following calculation: g T X(f )(u), T X(f )(v) s s X(f )(s) −1 −1 = tr X(f )(s) ◦ T X(f )(v) ◦ X(f )(s) ◦ T X(f )(u) s s −1 −1 ∗ ∗ ∗ ∗ = tr f ◦ s ◦ f ◦ f ◦ v ◦ f ◦ f ◦ s ◦ f ◦ f ◦ u ◦ f −1 −1 −1 −1 ∗ ∗ −1 −1 ∗ ∗ = tr f ◦ s ◦ f ◦ f ◦ v ◦ f ◦ f ◦ s ◦ f ◦ f ◦ u ◦ f −1 −1 −1 = tr f ◦ s ◦ v ◦ s ◦ u ◦ f −1 −1 −1 = tr s ◦ v ◦ s ◦ u ◦ f ◦ f −1 −1 = tr s ◦ v ◦ s ◦ u = g (u,v). Recall that so far we worked with the natural right action of aut(V) on X(V).In the sequel we prefer to work with the corresponding left action obtained by precomposing −1 with f → f in order to match with standard notation, compare in particular diagram (1.3)below. Fix a base point s ∈ X(V). Choose a linear isomorphism R − → V which is isomet- ric with respect to the standard inner product on R and s . It induces an isomorphism GL (R) − → aut(V) and thus a smooth left action ρ : GL (R) × X(V) → X(V). Since for n n any two elements s , s ∈ X(V) there exists an automorphism of V which is an isom- 1 2 etry (V, s ) → (V, s ), this action is transitive. The stabilizer of s ∈ X(V) is the com- 1 2 0 pact subgroup O(n) ⊆ GL (R). Thus we obtain a diffeomorphism φ : GL (R)/O(n) − → n n −1 >0 X(V). Deﬁne smooth maps p : X(V) → R , s → det(s ◦ s) and q : GL (R)/O(n) → >0 2 R , A · O(n) → det(A) . Both maps are submersions. In particular the preimages of K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 103 >0 −1 1 ∈ R under p and q are submanifolds of codimension 1. Denote by X(V) := p (1) and let i : X(V) → X(V) be the inclusion. Set SL (R) ={A ∈ GL (R) | det(A) =±1}. 1 n The smooth left action ρ : GL (R) × X(V) → X(V) restricts to an action ρ : SL (R) × n 1 −1 ∗ ∗ 2 X(V) → X(V) . Since f ◦ s ◦ f ∈ X(V) implies 1 = det(s ◦ f ◦ s ◦ f ) = det(f ) 1 1 0 1 0 this action is still transitive. The stabilizer of s is still O(n) ⊆ SL (R) and thus we ob- tain a diffeomorphism φ : SL (R)/O(n) → X(V) . Note that the inclusion induces a 1 1 diffeomorphism SL (R)/SO(n) = SL (R)/O(n) butwepreferthe righthandsidebe- cause we are interested in the GL (Z)-action. The inclusion SL (R) → GL (R) induces n n −1 an embedding j : SL (R)/O(n) → GL (R)/O(n) with image q (1). One easily checks that the following diagram commutes >0 (1.3) X(V) X(V) R ˜ ˜ φ φ id j q >0 SL (R)/O(n) GL (R)/O(n) n R Equip X(V) with the Riemannian metric g obtained from the Riemannian metric g 1 1 ± ± on X(V). We conclude from Lemma 1.1 that g is SL (R)-invariant. Since SL (R) is n n a semisimple Lie group with ﬁnite center and O(n) ⊆ SL (R) is a maximal compact subgroup, X(V) is a symmetric space of non-compact type and its sectional curvature is non-positive (see [12, Section 2.2 on p. 70],[18, Theorem 3.1 (ii) in V.3 on p. 241]). Alternatively [8, Chapter II, Theorem 10.39 on p. 318 and Lemma 10.52 on p. 324] show that X(V) and X(V) are proper CAT(0) spaces. >0 We call two elements s , s ∈ X(V) equivalent if there exists r ∈ R with r · s = s . 1 2 1 2 Denote by X(V) the set of equivalence classes under this equivalence relation. Let pr : X(V) → X(V) be the projection and equip X(V) with the quotient topology. The composite pr ◦i : X(V) → X(V) is a homeomorphism. In the sequel we equip X(V) with the structure of a Riemannian manifold for which pr ◦i is an isometric diffeomor- phism. In particular X(V) is a CAT(0)-space. The GL (R)-action on X(V) descends to an action on X(V) and the diffeomorphism pr ◦i is SL (R)-equivariant. 1.2. The volume function. — Fix an integer m with 1 ≤ m ≤ n = dim(V). In this section we investigate the following volume function. Deﬁnition 1.2 (Volume function). — Consider ξ ∈ V with ξ = 0.Deﬁne the volume function associated to ξ by vol : X(V) → R, s → (s m )(ξ , ξ ), i.e., the function vol sends an inner product s on V to the length of ξ with respect to s m . ξ 104 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING Fix ξ = 0in Vof the form ξ = v ∧ v ··· ∧ v for the rest of this section. 1 2 m This means that the line ξ lies in the image of the Plücker embedding, compare [17, Chapter 1.5 p. 209–211]. Then there is precisely one m-dimensional subvector space m m V ⊆ V such that the image of the map V → V induced by the inclusion is the ξ ξ 1-dimensional subvector space spanned by ξ . The subspace V is the span of the vectors v ,v ,...,v . It can be expressed as V ={v ∈ V|v ∧ ξ = 0}. This shows that V depends 1 2 m ξ ξ on the line ξ but is independent of the choice of v ,...,v . 1 m Given an inner product s on V, we obtain an orthogonal decomposition V = V ⊕ V of V with respect to s and we deﬁne s ∈ Sym(V) to be the element which satisﬁes ⊥ ⊥ ⊥ ⊥ ⊥ s (v + v ,w + w ) = s(v, w) for all v, w ∈ V and v ,w ∈ V . ξ ξ Theorem 1.3 (Gradient of the volume function). — The gradient of the square vol of the volume function vol is given for s ∈ X(V) by 2 2 ∇ vol = vol (s) · s ∈ Sym(V) = T X(V). s ξ s ξ ξ Proof.—Let A be any (m, m)-matrix. Then det(I + t · A) − det(I ) m m (1.4) lim = tr(A). t→0 This follows because det(I + t · A) = 1 + t tr(A) mod t by the Leibniz formula for the determinant. Consider s ∈ X(V) and u ∈ T (X(V)) = Sym(V). Notice that there exists > 0 such that s + t · u lies in X(V) for all t ∈ (−, ).Fix v ,v ,...,v ∈ Vwith ξ = v ∧ 1 2 m 1 ··· ∧ v . We compute using (1.2)and (1.4) 2 2 vol (s + t · u) − vol (s) ξ ξ lim t→0 det(((s + t · u)(v ,v )) ) − det((s(v ,v )) ) i j i,j i j i,j = lim t→0 det((s(v ,v )) + t · (u(v ,v )) ) − det((s(v ,v )) ) i j i,j i j i,j i j i,j = lim t→0 = det s(v ,v ) i j i,j −1 det(I + t · (s(v ,v )) · (u(v ,v )) ) − det(I ) m i j i j i,j m i,j · lim t→0 −1 = det s(v ,v ) · tr s(v ,v ) · u(v ,v ) . i j i j i j i,j i,j i,j K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 105 The gradient ∇ (vol ) is uniquely determined by the property that for all u ∈ Sym(V) we have 2 2 vol (s + t · u) − vol (s) ξ ξ g ∇ vol , u = lim . s s t→0 Hence it remains to show for every u ∈ Sym(V) −1 vol (s) · s , u = det s(v ,v ) · tr s(v ,v ) · u(v ,v ) . s ξ i j i j i j i,j i,j i,j −1 −1 Since vol (s) = det((s(v ,v )) ) by (1.2)and g (s , u) = tr(s ◦ s ◦ s ◦ u) by deﬁnition, i j i,j s ξ ξ it remains to show −1 −1 −1 tr s ◦ s ◦ s ◦ u = tr s(v ,v ) · u(v ,v ) . ξ i j i j i,j i,j s 0 ⊥ ⊥ ⊥ ∗ We obtain a decomposition s = ⊥ for an inner product s : V → (V ) ,where 0 s ξ ξ ξ ∗ ⊥ ∗ ⊥ ∗ we will here and in the sequel identify (V ) ⊕ (V ) and (V ⊕ V ) by the canonical ξ ξ ξ ξ isomorphism. We decompose u u ∗ ξ ⊥ ∗ ⊥ u = : V ⊕ V → (V ) ⊕ V ξ ξ ξ ξ u u for a linear map u : V → V . One easily checks ξ ξ −1 −1 −1 tr s ◦ s ◦ s ◦ u = tr s ◦ u . ξ ξ ∗ ∗ ∗ ∗ The set {v ,v ,...,v } is a basis for V .Let {v ,v ,...,v } be the dual basis of V . 1 2 m ξ 1 2 m ξ Then the matrix of u with respect to these basis is (u(v ,v )) and the matrix of s with ξ i j i,j ξ −1 respect to these basis is (s(v ,v )) .Hence thematrixof s ◦ u : V → V with respect i j i,j ξ ξ ξ −1 to the basis {v ,v ,...v } is (s(v ,v )) · (u(v ,v )) . This implies 1 2 m i j i j i,j i,j −1 −1 tr s ◦ u = tr s(v ,v ) · u(v ,v ) . ξ i j i j i,j i,j This ﬁnishes the proof of Theorem 1.3. Corollary 1.4. — The gradient of the function ln ◦ vol : X(V) → R at s ∈ X(V) is given by the tangent vector s ∈ T X(V) = Sym(V). In particular its norm with ξ s respect to the Riemannian metric g on X(V) is independent of s, namely, m. 106 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING −1 Proof. — Since the derivative of ln(x) is x , the chain rule implies together with Theorem 1.3 for s ∈ X(V) 1 1 2 2 2 ∇ ln ◦ vol =∇ vol(ξ ) · = vol (s) · s · = s . s s ξ ξ ξ ξ 2 2 vol (s) vol (s) ξ ξ We use the orthogonal decomposition V = V ⊕ V with respect to s to compute 2 2 −1 −1 ∇ ln ◦ vol = ( s ) = g (s , s ) = tr s ◦ s ◦ s ◦ s s ξ s ξ ξ ξ ξ s s ⊥ ⊥ = tr id ⊕ 0 : V ⊕ V → V ⊕ V V ξ ξ ξ ξ ξ = dim(V ) = m. 2. Sublattices of Z n n n A sublattice Wof Z is a Z-submodule W ⊆ Z such that Z /Wis a projective Z- n n module. Equivalently, W is a Z-submodule W ⊆ Z such that for x ∈ Z for which k · x belongs to W for some k ∈ Z, k = 0we have x ∈ W. LetL be the set of sublattices L of Z . m m Consider W ∈L.Let m be its rank as an abelian group. Let W → Z → Z Z m n m n R be the obvious map. Let ξ(W) ∈ R be the image of a generator of the inﬁnite cyclic group W. We have deﬁned the map vol : X(R ) → R above. Obviously it ξ(W) does not change if we replace ξ(W) by −ξ(W). Hence it depends only on W and not on the choice of generator ξ(W). Notice that for W = 0 we have by deﬁnition W = Z and 0 n R = R and ξ(W) is ±1 ∈ R.Inthatcase vol is the constant function with value 1. We will abbreviate n n n X = X R , X = X R , X = X R and vol = vol 1 W ξ(W) for W ∈L. Given a chain W W of elements W , W ∈L, we deﬁne a function 0 1 0 1 ˜ c : X → R W W 0 1 by ln(vol (s)) − ln(vol (s)) W W 1 0 ˜ c (s) := . W W 0 1 rk (W ) − rk (W ) Z 1 Z 0 Obviously this can be rewritten as 2 2 1 ln(vol (s) ) − ln(vol (s) ) W W 1 0 ˜ c (s) = · . W W 0 1 2 rk (W ) − rk (W ) Z 1 Z 0 K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 107 Hence we get for the norm of the gradient of this function at s ∈ X ∇ (˜ c ) s W W 0 1 2 2 ∇ (ln ◦ vol ) −∇ (ln ◦ vol ) 1 s s W W 1 0 = · 2 rk (W ) − rk (W ) Z 1 Z 0 2 2 ∇ (ln ◦ vol )+∇ (ln ◦ vol ) 1 s s W W 1 0 ≤ · 2 rk (W ) − rk (W ) Z 1 Z 0 2 2 ≤ · ∇ ln ◦ vol + ∇ ln ◦ vol . s s W W 1 0 We conclude from Corollary 1.4 for all s ∈ X. √ √ rk (W ) + rk (W ) Z 1 Z 0 ∇ (˜ c ) ≤ ≤ n. s W W 0 1 If f : M → R is a differentiable function on a Riemannian manifold M and C = sup{∇ f | x ∈ M} we always have f (x ) − f (x ) ≤ Cd (x , x ), 1 2 M 1 2 where d denotes the metric associated to the Riemannian metric. In particular we get for any two elements s , s ∈ X 0 1 ˜ c (s ) −˜ c (s ) ≤ n · d(s , s ), W W 1 W W 0 0 1 0 1 0 1 X where d is the metric on X coming from the Riemannian metric g on X. Recall that the Riemannian metric g on X is obtained by restricting the metric g.Let d be the 1 1 X metric on X coming from the Riemannian metric g on X .Recallthat i : X → Xis 1 1 1 1 the inclusion. Then we get for s , s ∈ X 0 1 1 d i(s ), i(s ) ≤ d (s , s ). 0 1 0 1 X X Indeed X is a geodesic submanifold of X([8, Chapter II, Lemma 10.52 on p. 324]). So both sides are even equal. We obtain for s , s ∈ X 0 1 1 (2.1) ˜ c ◦ i(s ) −˜ c ◦ i(s ) ≤ n · d (s , s ). W W 1 W W 0 X 0 1 0 1 0 1 1 Put L = W ∈L | W = 0, W = Z . Deﬁne for W ∈L functions i s ˜ c , ˜ c : X → R W W 108 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING by ˜ c (x) := inf ˜ c (x) | W ∈L, W W ; WW 2 2 W 2 ˜ c (x) := sup ˜ c (x) | W ∈L, W W . W W 0 0 W 0 In order to see that inﬁmum and supremum exist we use the fact that for ﬁxed x ∈ X there are at most ﬁnitely many W ∈L with vol (x) ≤ 1, compare [16, Lemma 1.15]. Put i s (2.2) d : X → R, x → exp ˜ c (x) −˜ c (x) . W W rk (W) >0 Since vol (r · s) = r · vol (s) holds for r ∈ R ,W ∈L and s ∈ X, we have ˜ c (r · W W W W 0 1 s) =˜ c (s) + ln(r) and the function d factorizes over the projection pr : X → Xto W W W 0 1 afunction d : X → R. >0 Lemma 2.1. —Consider x ∈ X, W ∈L ,and α ∈ R . Then we get for all y ∈ X with d (x, y) ≤ α √ √ 2 nα 2 nα d (y) ∈ d (x)/e , d (x) · e . W W W Proof. — By the deﬁnition of the structure of a smooth Riemannian manifold on X, it sufﬁces to show for all s , s ∈ X with d (s , s ) ≤ α 0 1 1 0 1 √ √ 2 nα 2 nα ˜ ˜ ˜ d (s ) ∈ d (s )/e , d (s ) · e . W 1 W 0 W 0 One concludes from (2.1)for s , s ∈ X with d (s , s ) ≤ α and W ∈L 0 1 1 X 0 1 √ √ s s s ˜ c (s ) ∈ c˜ (s ) − n · α, ˜ c (s ) + n · α , 1 0 0 W W W √ √ i i i ˜ c (s ) ∈ c˜ (s ) − n · α, ˜ c (s ) + n · α . 1 0 0 W W W Now the claim follows. In particular the function d : X → R is continuous. Deﬁne the following open subset of X for W ∈L and t ≥ 1 X(W, t) := x ∈ X | d (x)> t . There is an obvious GL (Z)-action onL andL and in the discussion following Lemma 1.1 we have already explained how GL (Z) ⊂ SL (R) acts on X and hence n 1 on X (choosing the standard inner product on R as the basepoint s ). Lemma 2.2. —For any t ≥ 1 we get: K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 109 (i) X(gW, t) = gX(W, t) for g ∈ GL (Z), W ∈L ; (ii) The complement of the GL (Z)-invariant open subset W (t) := X(W, t) W∈L in X is a cocompact GL (Z)-set; (iii) If X(W , t) ∩ X(W , t) ∩··· ∩ X(W , t) = ∅ for W ∈L ,thenwecan ﬁnda 1 2 k i permutation σ ∈ such that W ⊆ W ⊆ ··· ⊆ W holds. k σ(1) σ(2) σ(k) Proof. — This follows directly from Grayson [16, Lemma 2.1, Cor. 5.2] as soon as we have explained how our setup corresponds to the one of Grayson. We are only dealing with the caseO = Z and F = Q of [16]. In particular there is only one archimedian place, namely, the absolute value on Q and for it Q = R.So an element s ∈ X corresponds to the structure of a lattice which we will denote by (Z , s) with underlying Z-module Z in the sense of [16]. Given s ∈ X, an element W ∈L deﬁnes n n a sublattice of the lattice (Z , s) in the sense of [16] which we will denote by (Z , s) ∩ W. n n Thevolumeofasublattice (Z , s) ∩ Wof (Z , s) in the sense of [16]is vol (s). Given W ∈L ,weobtaina Q-subspace in the sense of [16, Deﬁnition 2.1] which we denote again by W, and vice versa. It remains to explain why our function d of (2.2) agrees with the function d of [16, Deﬁnition 2.1] which is given by n n n d (s) = exp min Z , s / Z , s ∩ W − max Z , s ∩ W , see [16, Deﬁnition 1.23, 1.9]. This holds by the following observation. Consider s ∈ Xand W ∈L . Consider the canonical plot and the canonical poly- gon of the lattice (Z , s) ∩ W in the sense of [16, Deﬁnition 1.10 and Discussion 1.16]. The slopes of the canonical polygon are strictly increasing when going from the left to the right because of [16, Corollary 1.30]. Hence max((Z , s) ∩ W) in the sense of [16,Def- inition 1.23] is the slope of the segment of the canonical polygon ending at (Z , s) ∩ W. Consider any W ∈L with W W. Obviously the slope of the line joining the plot point 0 0 n n of (Z , s) ∩ W and (Z , s) ∩ W is less than or equal to the slope of the segment of the n n canonical polygon ending at (Z , s) ∩ W. If (Z , s) ∩ W happens to be the starting point of this segment, then this slope agrees with the slope of the segment of the canonical plot ending at (Z , s) ∩ W. Hence max Z , s ∩ W ln(vol (s)) − ln(vol (s)) W W 0 s (2.3) = max W ∈L, W W =˜ c (s). 0 0 rk (W) − rk (W ) Z Z 0 We have the formula n n n n vol Z , s ∩ W = vol Z , s ∩ W · vol Z , s ∩ W / Z , s ∩ W 2 2 110 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING for any W ∈L with W W (see [16, Lemma 1.8]). Hence 2 2 ln(vol (s)) − ln(vol (s)) ln(vol (s)) W W W /W 2 2 = . rk (W ) − rk (W) rk (W /W) Z 2 Z Z 2 There is an obvious bijection of the set of direct summands in Z /W and the set of direct summand in Z containing W. Now analogously to the proof of (2.3)one shows n n (2.4) min Z , s / Z , s ∩ W ln(vol (s)) − ln(vol (s)) W W 2 i (2.5) = min W ∈L, W W =˜ c (s). 2 2 rk (W ) − rk (W) Z 2 Z Now the equality of the two versions for d follows from (2.3)and (2.5). This ﬁnishes the poof of Lemma 2.2. 3. Transfer reducibility of GL (Z) LetF be the family of those subgroups H of GL (Z), which are virtually cyclic or n n for which there exists a ﬁnitely generated free abelian group P, natural numbers r and n with n < n and an extension of groups 1 → P → K → GL (Z) → 1 i=1 such that H is isomorphic to a subgroup of K. In this section we prove the following theorem, which by [4, Theorem 1.1] implies the K-theoretic FJC up to dimension 1 for GL (Z) with respect to the familyF .The n n notion of transfer reducibility has been introduced in [4, Deﬁnition 1.8]. Transfer reducibility asserts the existence of a compact space Z and certain equivariant covers of G×Z. A slight modiﬁcation of transfer reducibility is discussed in Section 5, see Deﬁnition 5.3. Here our main work is to verify the conditions formulated in Deﬁnition 3.7 for our situation, see Lemma 3.8. Once this has been done the veriﬁcation of transfer reducibility proceedsasin[3], as is explained after Lemma 3.8. Theorem 3.1. — The group GL (Z) is transfer reducible overF . n n To prove this we will use the space X = X(R ) and its subsets X(W, t) considered in Sections 1 and 2. For a G-space X and a family of subgroupsF , a subset U ⊆ X is called anF - subset if G := {g ∈ G | g(U) = U} belongs toF and gU ∩ U =∅ for all g ∈ G \ G .An U U open G-invariant cover consisting ofF -subsets is called anF -cover. K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 111 Lemma 3.2. — Consider for t ≥ 1 the collection of subsets of X W (t) = X(W, t) | W ∈L . It is a GL (Z)-invariant set of openF -subsets of X whose covering dimension is at most (n − 2). n n Proof. — The setW (t) is GL (Z)-invariant because of Lemma 2.2 (i) and its cov- ering dimension is bounded by (n − 2) because of Lemma 2.2 (iii) since for any chain of n n sublattices {0} W W ··· W Z of Z we have r ≤ n − 2. 0 1 r It remains to show thatW (t) consists ofF -subsets. Consider W ∈L and g ∈ GL (Z). Lemma 2.2 implies: X(W, t) ∩ gX(W, t) = ∅ ⇐⇒ X(W, t) ∩ X(gW, t) = ∅ ⇐⇒ gW = W. n n Put GL (Z) ={g ∈ GL (Z) | gW = W}. Choose V ⊂ Z with W ⊕ V = Z . Under this n W n identiﬁcation every element φ ∈ GL (Z) is of the shape n W φ φ W V,W 0 φ for Z-automorphism φ of V, φ of W and a Z-homomorphism φ : V → W. Deﬁne V W V,W φ φ W V,W p : GL (Z) → aut (W) × aut (V), → (φ ,φ ) n W Z Z W V 0 φ and id ψ i : hom (V, W) → GL (Z) ,ψ → . Z n W 0id Then we obtain an exact sequence of groups i pr 1 → hom (V, W) − → GL (Z) − → aut (W) × aut (V) → 1, Z n W Z Z where hom (V, W) is the abelian group given by the obvious addition. Since both V ∼ ∼ and W are different from Z ,weget aut (V) GL (Z) and aut (W) GL (Z) for = = Z n(V) Z n(W) n(V), n(W)< n. Hence each element inW (t) is anF -subset with respect to the GL (Z)- n n action. >0 For a subset A of a metric space X and α ∈ R we denote by B (A) := x ∈ X | d (x, a)<α for some a ∈ A α X the α-neighborhood of A. Lemma 3.3. —For every α> 0 and t ≥ 1 there exists β> 0 such that for every W ∈L we have B (X(W, t + β)) ⊆ X(W, t). α 112 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING 2 nα Proof. — This follows from Lemma 2.1 if we choose β> (e − 1) · t. Let FS(X) be the ﬂow space associated to the CAT(0)-space X = X(R ) in [3, Section 2]. It consists of generalized geodesics, i.e., continuous maps c : R → X for which there is a closed subinterval I of R such that c| is a geodesic and c| is locally constant. I R\I The ﬂow on FS(X) is deﬁned by the formula ( (c))(t) = c(τ + t). Lemma 3.4. — Consider δ, τ > 0 and c ∈ FS(X).Thenweget for d ∈ B ( (c)) δ [−τ,τ ] d d(0), c(0) < 4 + δ + τ. Proof. — Choose s ∈[−τ, τ ] with d (d, (c)) < δ. We estimate using [3, FS(X) s Lemma 1.4 (i)] and a special case of [3, Lemma 1.3] d d(0), c(0) ≤ d d(0), (c)(0) + d (c)(0), c(0) X X s X s ≤ d d, (c) + 2 + d (c), c + 2 FS(X) s FS(X) s <δ + 2 +|s|+ 2 ≤ 4 + δ + τ. Let ev : FS(X) → X, c → c(0) be the evaluation map at 0. It is GL (Z)- 0 n equivariant, uniform continuous and proper [3, Lemma 1.10]. Deﬁne subsets of FS(X) by −1 Y(W, t) := ev X(W, t) for t ≥ 1, W ∈L , and setV (t) := {Y(W, t) | W ∈L }, |V (t)|:= Y(W, t) ⊆ FS(X). W∈L Lemma 3.5. —Let τ, δ > 0 and t ≥ 1 and set α := 4 + δ + τ.If β> 0 is such that for every W ∈L we have B (X(W, t + β)) ⊆ X(W, t) then the following holds (i) The setV (t) is GL (Z)-invariant; (ii) Each element inV (t) is an openF -subset with respect to the GL (Z)-action; n n (iii) The dimension ofV (t) is less or equal to (n − 2); (iv) The complement of |V (t + β)| in FS(X) is a cocompact GL (Z)-subspace. (v) For every c ∈|V (t + β)| there exists W ∈L with B (c) ⊂ Y(W, t). δ [−τ,τ ] The existence of a suitable β is the assertion of Lemma 3.3. Proof. — (i), (ii) and (iii) follow from Lemma 3.2 since ev is GL (Z)-equivariant. 0 n −1 (iv) follows from Lemma 2.2 (ii) since the map ev : FS(X) → Xis proper and ev (X − |W (t + β)|) = FS(X) −|V (t + β)|. K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 113 For (v) consider c ∈|V (t + β)|. Choose W ∈L with c ∈ Y(W, t + β).Then c(0) ∈ X(W, t + β).Consider d ∈ B ( (c)). Lemma 3.4 implies d (d(0), c(0)) < α.Hence δ [−τ,τ ] X d(0) ∈ B (c(0)). We conclude d(0) ∈ B (X(W, t + β))). This implies d(0) ∈ X(W, t) and α α hence d ∈ Y(W, t). This shows B ( (c)) ⊆ Y(W, t). δ [−τ,τ ] Let FS (X) be the subspace of FS(X) of those generalized geodesics c for which ≤γ there exists for every > 0a number τ ∈ (0,γ + ] and g ∈ Gsuch that g · c = (c) holds. As an instance of [3, Theorem 4.2] we obtain the following in our situation. Theorem 3.6. — There is a natural number M such that for every γ> 0 and every compact subset L ⊆ X there exists a GL (Z)-invariant collectionU of subsets of FS(X) satisfying: (i) Each element U ∈U is an open VC yc-subset of the GL (Z)-space FS(X); (ii) We have dimU ≤ M; (iii) There is > 0 with the following property: for c ∈ FS (X) such that c(t) ∈ GL (Z) · L ≤γ n for some t ∈ R there is U ∈U such that B ( (c)) ⊆ U. [−γ,γ ] Deﬁnition 3.7 ([3, Deﬁnition 5.5]). — Let G be a group,F be a family of subgroups of G, (FS, d ) be a locally compact metric space with a proper isometric G-action and : FS × R → FS FS be a G-equivariant ﬂow. We say that FS admits longF -covers at inﬁnity and at periodic ﬂow lines if the following holds: There is N > 0 such that for every γ> 0 there is a G-invariant collection of openF -subsetsV of FS and ε> 0 satisfying: (i) dimV ≤ N; (ii) there is a compact subset K ⊆ FS such that (a) FS ∩ G · K =∅; ≤γ (b) for z ∈ FS − G · K there is V ∈V such that B ( (z)) ⊂ V. ε [−γ,γ ] We remark that it is natural to think of this deﬁnition as requiring two conditions, the ﬁrst dealing with everything outside some cocompact subset (“at inﬁnity”) and the second dealing with (short) periodic orbits of the ﬂow that meet a given cocompact subset (“at periodic ﬂow lines”). In proving that this condition is satisﬁed in our situation in the next lemma we deal with these conditions separately. For the ﬁrst condition we use the sets Y(W, t) introduced earlier; for the second the theorem cited above. Lemma 3.8. — The ﬂow space FS(X) admits longF -covers at inﬁnity and at periodic ﬂow lines. Proof.—Fix γ> 0. Choose t ≥ 1. Put δ := 1and τ := γ .Let β> 0be the number appearing in Lemma 3.5 and let M ∈ N be the number appearing in Theorem 3.6. Since 114 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING by Lemma 2.2 (ii) the complement of |W (t + β)| in X is cocompact, we can ﬁnd a compact subset L of this complement such that GL (Z) · L = X \W (t + β) . For this compact subset L we obtain a real number > 0 and a setU of subsets of FS(X) from Theorem 3.6. We can arrange that ≤ 1. ConsiderV :=U ∪V (t),whereV (t) is the collection of open subsets deﬁned before Lemma 3.5. We want to show thatV satisﬁes the conditions appearing in Deﬁnition 3.7 with respect to the number N := M + n − 1. Since the covering dimension ofU is less or equal to M by Theorem 3.6 (ii) and the covering dimension ofV (t) is less or equal to n − 2 by Lemma 3.5 (iii), the covering dimension ofU ∪V (t) is less or equal to N. SinceU andV (t) are GL (Z)-invariant by Theorem 3.6 and Lemma 3.5 (i),U ∪ V (t) is GL (Z)-invariant. Since each element ofU is an open VCyc-set by Theorem 3.6 (i) and each element ofV (t) is an openF -subset by Lemma 3.5 (ii), each element ofU ∪V (t) is an open F -subset, as VC yc ⊂F .Deﬁne n n S := c ∈ FS(X) |∃Z ∈U ∪V (t) with B (c) ⊆ Z . [−γ,γ ] This set S contains FS(X) ∪|V (t + β)| by the following argument. If c ∈|V (t + β)|, ≤γ then c ∈ S by Lemma 3.5(v). If c ∈ FS(X) and c ∈| /V (t + β)|,then c ∈ FS(X) and ≤γ ≤γ c(0) ∈ GL (Z) · L and hence c ∈ Sby Theorem 3.6 (iii). The subset S ⊆ FS(X) is GL (Z)-invariant becauseU ∪V (t) is GL (Z)-invariant. n n Next we prove that S is open. Assume that this is not the case. Then there exists c ∈ S and a sequence (c ) of elements in FS(X) − Ssuch that d (c, c )< 1/k holds k k≥1 FS(X) k for k ≥ 1. Choose Z ∈U ∪V (t) with B ( (c)) ⊆ Z. Since FS(X) is proper as metric [−γ,γ ] space by [3, Proposition 1.9] and B ( (c)) has bounded diameter, B ( (c)) [−γ,γ ] [−γ,γ ] is compact. Hence we can ﬁnd μ> 0withB ( (c)) ⊆ Z. We conclude from [3, +μ [−γ,γ ] Lemma 1.3] for all s ∈[−γ, γ ] |s| τ d (c), (c ) ≤ e · d (c, c )< e · 1/k. FS(X) s s k FS(X) k Hence we get for k ≥ 1 B (c ) ⊆ B (c) [−γ,γ ] k +e ·1/k [−γ,γ ] Since c does not belong to S, we conclude that B ( (c)) is not contained in k +e ·1/k [−γ,γ ] Z. This implies e · 1/k ≥ μ for all k ≥ 1, a contradiction. The GL (Z)-set FS(X) −|V (t + β)| is cocompact by Lemma 3.5 (iv). Since S is an open GL (Z)-subset of FS(X) and contains |V (t + β)|,the GL (Z)-set FS(X) − Sis n n cocompact. Hence we can ﬁnd a compact subset K ⊆ FS(X) − S satisfying GL (Z) · K = FS(X) − S. n K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 115 Obviously FS(X) ∩ GL (Z) · K =∅. ≤γ n Proof of Theorem 3.1.—Thegroup GL (Z) and the associated ﬂow space FS(X) satisfy [3, Convention 5.1] by the argument of [3, Section 6.2]. Notice that in [3,Conven- tion 5.1] it is not required that the action is cocompact. The argument of [3, Section 6.2] showing that there is a constant k such that the order of any ﬁnite subgroup of G is bounded by k uses that the action is cocompact. But such a number exists for GL (Z) as G n well, since GL (Z) is virtually torsion free (see [9, Exercise II.3 on p. 41]). Because of [3, Proposition 5.11] it sufﬁces to show that FS(X) admits longF - covers at inﬁnity and at periodic ﬂow lines in the sense of Deﬁnition 3.7 and admits contracting transfers in the sense of [3, Deﬁnition 5.9]. For the ﬁrst condition this has been done in Lemma 3.8, while the second condition follows from the argument given in [3, Section 6.4]. Proposition 3.9. —The K-theoretic FJC up to dimension 1 holds for GL (Z). Proof. — We proceed by induction over n.As GL (Z) is ﬁnite, the initial step of the induction is trivial. Since GL (Z) is transfer reducible overF by Theorem 3.1 it follows from [4,The- n n orem 1.1] that GL (Z) satisﬁes the K-theoretic FJC up to dimension 1 with respect to F . It remains to replaceF by the family VCyc. Because of the Transitivity Principle 0.5 n n it sufﬁces to show that each H ∈F satisﬁes the FJC up to dimension 1 (with respect to VCyc). Combining the induction assumption for GL (Z), k < n with well known inher- itance properties for direct products, exact sequences of groups and subgroups (see for example [1, Theorem 1.10, Corollary 1.13, Theorem 1.9]) it is easy to reduce the K- theoretic FJC up to dimension 1 for members ofF to the class of virtually poly-cyclic groups. Finally, for virtually poly-cyclic groups the FJC holds by [1]. 4. Strong transfer reducibility of GL (Z) In this section we will discuss the modiﬁcations needed to extend Proposition 3.9 to higher K-theory. The necessary tools for this extension have been developed by Weg- ner [24]. Theorem 4.1. — The group GL (Z) is strongly transfer reducible overF in the sense of [24, n n Deﬁnition 3.1]. Wegner proves in [24, Theorem 3.4] that CAT(0)-groups are strongly transfer reducible over VC yc. As GL (Z) does not act cocompactly on X, we cannot use Wegner’s result directly. However, in combination with Lemma 3.8 his method yields a proof of Theorem 4.1. 116 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING Proof of Theorem 4.1. — The only place where Wegner uses cocompactness of the action is when he veriﬁes the assumptions of [3, Theorem 5.7], see [24, Proof of Theo- rem 3.4]. We know by Lemma 3.8 that FS(X) admits longF -covers at inﬁnity and periodic ﬂow lines. Wegner cites [3, Section 6.3] for this assumption. In the cocompact setting the family can even be chosen to be VC yc. That the assumptions of [3, Convention 5.1], which are used implicitly in [3,The- orem 5.7], are satisﬁed has already been explained in the proof of Theorem 3.1. Theorem 4.2. —The K-theoretic FJC holds for GL (Z). Proof.—Theorem 4.1 together with [24, Theorem 1.1] imply that GL (Z) satisﬁes the K-theoretic FJC with respect to the familyF . Using the induction from the proof of Proposition 3.9 the familyF can be re- placed by VC yc. 5. Wreath products and transfer reducibility Our main result in this section is the following variation of [4, Theorem 1.1]. Theorem 5.1. —LetF be a family of subgroups of the group G and let F be a ﬁnite group. Denote byF the family of subgroups H of G F that contain a subgroup of ﬁnite index that is isomorphic to a subgroup of H ×···×H for some n and H ,..., H ∈F . 1 n 1 n (i) If G is transfer reducible overF , then the wreath product G F satisﬁes the K-theoretic FJC up to dimension 1 with respect toF and the L-theoretic FJC with respect toF ; (ii) If G is strongly transfer reducible overF , then G F satisﬁes the K-theoretic and L-theoretic FJC in all dimensions with respect toF . The idea of the proof of this result is very easy. We only need to show that G F is transfer reducible overF and apply [4, Theorem 1.1]. However, it will be easier to verify a slightly weaker condition for G F. Deﬁnition 5.2 (Homotopy S-action; [4, Deﬁnition 1.4]). — Let S be a ﬁnite subset of a group G (containing the identity element e ∈ G). Let X be a space. (i) A homotopy S-action (ϕ, H) on X consists of continuous maps ϕ : X → X for g ∈ S and homotopies H : X×[0, 1]→ X for g, h ∈ S with gh ∈ S such that H (−, 0) = g,h g,h ϕ ◦ ϕ and H (−, 1) = ϕ holds for g, h ∈ S with gh ∈ S. Moreover, it is required g h g,h gh that H (−, t) = ϕ = id for all t ∈[0, 1]; e,e e X (ii) For g ∈ S let F (ϕ, H) be the set of all maps X → X of the form x → H (x, t) where g r,s t ∈[0, 1] and r, s ∈ S with rs = g; K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 117 (iii) Given a subset A ⊂ G × X let S (A) ⊂ G × X denote the set ϕ,H −1 ga b, y |∃x ∈ X, a, b ∈ S, f ∈ F (ϕ, H), f ∈ F (ϕ, H) a b satisfying (g, x) ∈ A, f (x) = f (y) . n−1 n 1 Then deﬁne inductively S (A) := S (S (A)); ϕ,H ϕ,H ϕ,H (iv) Let (ϕ, H) be a homotopy S-action on X andU be an open cover of G × X.Wesay thatU is S-long with respect to (ϕ, H) if for every (g, x) ∈ G × X there is U ∈U |S| containing S (g, x) where |S| is the cardinality of S. ϕ,H We will use the following variant of [4, Deﬁnition 1.8]. Deﬁnition 5.3 (Almost transfer reducible). — Let G be a group andF be a family of subgroups. We will say that G is almost transfer reducible overF if there is a number N such that for any ﬁnite subset S of G we can ﬁnd (i) a contractible, compact, controlled N-dominated, metric space X ([3, Deﬁnition 0.2]), equipped with a homotopy S-action (ϕ, H) and (ii) a G-invariant coverU of G × X of dimension at most N that is S-long. Moreover we require that for all U ∈U the subgroup G := {g ∈ G | gU = U} belongs toF.(Here we use the G-action on G×X given by g · (h, x) = (gh, x).) The original deﬁnition for transfer reducibility requires in addition that gUand U are disjoint if U ∈U and g ∈ / G , in other words each U is required to be anF - subset. One can also drop this condition from the notion of “strongly transfer reducible” introduced in [24, Deﬁnition 3.1]. A group satisfying this weaker version will be called almost strongly transfer reducible. The result corresponding to [4, Theorem 1.1] (respectively [24, Theorem 1.1]) is as follows. Proposition 5.4. —LetF be a family of subgroups of a group G and letF be the family of subgroups of G that contain a member ofF as a ﬁnite index subgroup. (i) If G is almost transfer reducible overF , it satisﬁes the K-theoretic FJC up to dimension 1 with respect toF and the L-theoretic FJC with respect toF . (ii) If G is almost strongly transfer reducible overF,itsatisﬁesthe K-theoretic FJC with respect toF and the L-theoretic FJC with respect toF . Proof. — (i) The proof can be copied almost word for word from the proof of [4, Theorem 1.1]. The only difference is that we no longer know that the isotropy groups of the action of G on the geometric realization of the nerve ofU belong toF,and this action may no longer be cell preserving. But it is still simplicial and therefore we can 118 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING just replace by its barycentric subdivision by the following Lemma 5.5 provided we replaceF byF . (The precise place where this makes a difference is in the proof of [4, Proposition 3.9].) (ii) The L-theory part follows from part (i) since almost strongly transfer reducible implies almost transfer reducible. For the proof of the K-theory part we have to adapt Wegner’s proof of [24,The- orem 1.1]. The necessary changes concern [24, Proposition 3.6] and are similar to the changes discussed above, but as Wegner’s argument is somewhat differently organized we have to be a little more careful. First we observe that for any ﬁxed M > 0 the second assertion in [24, Proposition 3.6] can be strengthened to (5.1)Mk · d f (g, x), f (h, y) ≤ d (g, x), (h, y) for all(g, x), (h, y) ∈ G×X. ,S,k, To do so we only need to set n := M · 4Nk instead of 4Nk in the second line of Wegner’s proof; then k can be replaced by M · k in the denominator of the ﬁnal expression on p.786. (Of course X, , , and f depend now also on M.) Then we can replace by its barycentric subdivision . Using Lemma 5.5 we conclude from (5.1) with sufﬁciently large M > 0that • d (f (g, x), f (h, y)) ≤ for all (g, x), (h, y) ∈ G×X satisfying the inequality d ((g, x), (h, y)) ≤ k. ,S,k, This assertion still guarantees that the maps (f ) induce a functor as needed on the right hand side of the diagram on p. 789 of [24]. With this change Wegner’s argument proves the K-theory part of (ii). (Alternatively one can strengthen Lemma 5.5 below and check that the l -metric under barycentric subdivision changes only up to Lipschitz equivalence. Thus [24,Propo- sition 3.6] remains in fact true for .) F be the family of subgroups of G.LetF be the Lemma 5.5. —Let G be a group, and let family of subgroups of G that contain a member ofF as a ﬁnite index subgroup. Let be a simplicial complex with a simplicial G-action such that the isotropy group of each vertex is contained inF.Let 1 1 1 1 be the barycentric subdivision. Denote by d the l -metric on and by d the l -metric on (i) The group G acts cell preserving on . All isotropy groups of lie inF . In particular is a G-CW-complex whose isotropy groups belong toF ; (ii) Given a number > 0 and a natural number N, there exists a number > 0 depending only on and N such that the following holds: If dim( ) ≤ N and x, y ∈ satisfy 1 1 d (x, y)<,then d (x, y)< . Proof. — The elementary proof of assertion (i) is left to the reader. The proof of assertion (ii) is an obvious variation of the proof of [4, Lemma 9.4 (ii)]. Namely, any simplicial complex can be equipped with the l -metric. With those metrics the inclusion of K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 119 a subcomplex is an isometric embedding. Let be the simplicial complex with the same vertices as , such that any ﬁnite subset of the vertices spans a simplex. The inclusions i : and i : are isometric embeddings. The images i(x), i(y) of any two points x, y ∈ are contained in a closed simplex of dimension at most 2N + 1. Thus it sufﬁces to consider the case where is replaced by the standard (2N + 1)-simplex. A compactness argument gives the result in that case. Remark 5.6. —Often thefamilyF is closed under ﬁnite overgroups. In this case F =F and it is really easier to work with the weaker notion of almost transfer reducible instead of transfer reducible. However, there are situations in which a family that is not closed under ﬁnite overgroups is important. For example, in [10] the family of virtually cyclic groups of type I is considered. Let G be a group and F be ﬁnite group. We will think of elements of the F-fold F F F product G as functions g : F → G. For a ∈ F, g ∈ G we write l (g) ∈ G for the function b → g(ba); this deﬁnes a left action of F on G and the corresponding semi-direct product is the wreath product G F with multiplication gag a = gl (g )aa for g, g ∈ G and a, a ∈ F. If G acts on a set X, then we obtain an action of G FonX . In formulas this action is given by (g · x)(b) := g(b) · x(b); (a · x)(b) := x(ba), and hence (ga · x)(b) = g(b) · x(ba) F F for g ∈ G , x ∈ X and a, b ∈ F. We will sometimes also write l (x) for a · x.Let now S ⊂ G and (ϕ, H) be a homotopy S-action on a space X. Set S F := {sa | s ∈ S , a ∈ F}⊂ G F. Then we obtain a homotopy S F-action (ϕ, ˆ H) on X . In formulas this is given by ϕ ˆ (x) (b) := ϕ x(ba) ; sa s(b) H (x, t) (b) := H x baa , t , sa,s a s(b),s (ba) for t ∈[0, 1], s, s ∈ S , a, a , b ∈ Fwith sas a = sl (s )aa ∈ S F. Hence (sl (s )(b) = a a s(b)s (ba) ∈ S and the right hand side is deﬁned. It is easy to check that if X is a con- tractible, compact, controlled N-dominated, metric space, then the same is true for X provided we replace N by N ·|F|. Proof of Theorem 5.1. — Let us postpone the “strong”-case until the end of the proof. Because of Proposition 5.4 it sufﬁces to show that G F is almost transfer reducible over F .Let S be a ﬁnite subset of G F. By enlarging it we can assume that it has the form S F 120 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING for some ﬁnite subset S ⊂ G. Pick a ﬁnite subset S ⊂ Gsuch thatS ⊂ S and |S |≥|S F|. (If G is ﬁnite, then G F ∈F and there is nothing to prove.) As G is transfer reducible and hence in particular almost transfer reducible, there is a number N, (depending only on G, not on S or S) a compact, contractible, controlled N-dominated, metric space X, a homotopy S -action (ϕ, H) on X, and a G-invariant S -long open coverU of G×Xof dimension at most N such that for all U ∈U we have G ∈F . As pointed out before, X is a compact, contractible, controlled N ·|F|-dominated, metric space. For u : F →U , u F F let V := {(g, x) ∈ G ×X | (g(b), x(b)) ∈ u(b) for all b ∈ F}.Weobtainanopencover u F F |F| V := {V | u : F →U } of G ×X of dimension at most (N + 1) − 1. This cover is invariant for the G F-action deﬁned by g · (h, x) (b) := g(b)h(b), x(b) ; a · (h, x) (b) := h(ba), x(ba) , F F F for g, h ∈ G , x ∈ X and a, b ∈ F. As we have (G ) u = G , it follows that (G V u(b) b∈F −1 F F) ∈F for all V ∈V . Now we pull backV to a coverU := {p (V) | V ∈V } of G F×X F F F along the G F-equivariant map p : G F×X → G ×X , (ga, x) → (g, a · x).Here G F F F F operates on G F × X via left multiplication on the ﬁrst factor and on G × X via the operation deﬁned above. The deﬁnition of the homotopy S F-action on X gives H (−, t) = H (−, t) ◦ l = l ◦ H (−, t) sa,s a s,l (s ) aa aa l (s),l (s ) a −1 −1 −1 a a a for s, s ∈ S , a, a ∈ F. For ¯ s := sl (s ), a ¯ := aa we have ¯ sa ¯ = sas a and consequently F F F F (5.2)F (ϕ, ˆ H) = F (ϕ ˆ| , H| ) ◦ l = l ◦ F (ϕ ˆ| , H| ). ¯ sa ¯ ¯ s G G a ¯ a ¯ l (¯ s) G G −1 a ¯ 1 F F Let us insert this into the deﬁnition of (S F) (hb, x) with h ∈ G , b ∈ F, x ∈ X . ϕ, ˆ H 1 F Pick any (h b , x ) ∈ (S F) (hb, x) with h ∈ G , b ∈ F. Then there are elements ϕ, ˆ H s, s ∈ S , a, a ∈ Fand f ∈ F (ϕ, ˆ H), f ∈ F (ϕ, ˆ H) such that sa s a −1 f (x) = f x , and h b = hb(sa) s a . ¯ ¯ F F Using the ﬁrst equality in (5.2)weﬁnd f ∈ F (ϕ ˆ| , H| ) such that f = f ◦ l . Using s G G a F F −1 −1 the second equality in (5.2)weﬁnd f ∈ F (ϕ ˆ| , H| ) such that f ◦ l = l ◦ f . l (s) G G ab ab −1 ba −1 −1 −1 Then f ◦ l = l ◦ f ; equivalently l ◦ f = f ◦ l . Similarly, using both equations b ab ba ˜ ˜ ˜ F F −1 −1 in (5.2) again, we ﬁnd f ∈ F (ϕ ˆ| , H| ) such that l ◦ f = f ◦ l . l (s ) G G ba ba a −1 ba F 1 We claim that p(h b , x ) belongs to (S ) (p(hb, x)).Wehave p(h b , x ) = ϕ ˆ| ,H| F F G G −1 −1 (h , b · x ) and p(hb, x) = (h, b · x).From h b = hb(sa) s a we conclude h = hl −1 (s s ) ba −1 and b = ba a . Now the equations −1 ˜ ˜ ˜ f (b · x) = l −1 f (x) = l −1 f x = f ba a · x = f b x ba ba K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 121 −1 −1 h = hl −1 s s = h l −1 (s) l −1 s ba ba ba prove our claim. Summarizing we have shown that for any A ⊂ G F × X we have 1 F p (S F) (A) ⊂ S p(A) . ϕ, ˆ H ϕ ˆ| ,H| F F G G By induction then for all n n F p (S F) (A) ⊂ S p(A) . ϕ, ˆ H ϕ ˆ| ,H| F F G G F F Since the S -homotopy action on X is deﬁned componentwise we have F n S (g, x) ⊂ S g(a), x(a) . ϕ,H ϕ ˆ| ,H| K K G G a∈F Recall the deﬁnition of S from the beginning of the proof. Since the coverU of G × Xis |S | S -long, thereisfor each a ∈ Fa u(a) ∈U with (S ) (h(a), x(ab)) ⊂ u(a).Thusweobtain ϕ,H |SF| |SF| |SF| p (S F) (hb, x) ⊂ S (h, b · x) ⊂ S h(a), x(ab) ϕ,H ϕ, ˆ H ϕ ˆ| ,H| F F G G a∈F |S | ⊂ S h(a), x(ab) ⊂ u(a) = V . ϕ,H a∈F a∈F |SF| −1 u So (S F) (hb, x) ⊂ p (V ). Hence the coverU is S F-long. ϕ, ˆ H If X is equipped with a strong homotopy action (see [24, Section 2]), we obtain a strong homotopy action on X .Informulasitisgiven by (g a , t ,..., t , g a , x)(b) n n n 1 0 0 := g (b), t , g (ba ), t , g (ba a ), ..., n n n−1 n n−1 n−2 n n−1 g (ba ... a ), x(ba ... a ) 0 n 1 n 0 F F for g ,..., g ∈ G , a ,..., a , b ∈ F, t ,..., t ∈[0, 1], x ∈ X .Wealsohavehere 0 n 0 n 1 n (g a , t ,..., t , g a , −) n n n 1 0 0 = g , t , l (g ), t , l (g ), t ,..., l (g ), − ◦ l n n a n−1 n−1 a a n−2 n−2 a ...a 0 a ...a n n n−1 n 1 n 0 = l ◦ l −1 g , t , l −1 (g ), t ,..., l −1 (g ), − . a ...a (a ...a ) n n (a ...a ) n−1 n−1 0 n 0 n 0 n−1 0 a For g¯ := g l (g )l (g )... l (g ) ∈ G , a ¯ := a ... a ∈ F, n ∈ N we have g¯a ¯ = n a n−1 a a n−2 a ...a 0 n 0 n n n−1 n 1 g a g a ... g a and consequently we have analogously to (5.2) n n n−1 n−1 0 0 F F F F F (, S F, n) = F | , S , n ◦ l = l ◦ F | , S , n . g¯a ¯ g¯ a ¯ a ¯ l (g¯) G G −1 a ¯ With this observation the proof can be carried out in exactly the same way as in the case of a homotopy action. 122 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING Remark 5.7. — The proof of Theorem 5.1 given above only uses that G is almost (strongly) transfer reducible overF , not that G is (strongly) transfer reducible. Conse- quently, Theorem 5.1 remains true if we replace the assumption “(strongly) transfer re- ducible” by the weaker assumption “almost (strongly) transfer reducible”. 6. The Farrell-Jones conjecture with wreath products Deﬁnition 6.1. —Agroup G is said to satisfy the L-theoretic Farrell-Jones Conjecture with wreath products with respect to the familyF if for any ﬁnite group F the wreath product G F satisﬁes the L-theoretic Farrell-Jones Conjecture with respect to the familyF (in the sense of Deﬁnition 0.2). If the familyF is not mentioned, it is by default the family VCyc of virtually cyclic subgroups. There are similar versions with wreath products of the K-theoretic Farrell-Jones Conjecture and the K-theoretic Farrell-Jones Conjecture up to dimension 1. The FJC with wreath products has ﬁrst been used in [15] to deal with ﬁnite exten- sions, see also [23, Deﬁnition 2.1]. Remark 6.2. — The inheritance properties of the FJC for direct products, sub- groups, exact sequences and directed colimits hold also for the FJC with wreath products and can be deduced from the corresponding properties of the FJC itself. See for exam- ple [19, Lemma 3.2, 3.15, 3.16, Satz 3.5]. Foragroup G andtwo ﬁnitegroups F and F we have (H F ) F ⊂ H (F F ) 1 2 1 2 1 2 and F F is ﬁnite. In particular, if G satisﬁes the FJC with wreath products, then the 1 2 same is true for any wreath product G F with F ﬁnite. The main advantage of the FJC with wreath products is that in addition it passes to overgroups of ﬁnite index. Let G be an overgroup of G of ﬁnite index, i.e., G ⊂ G , [G : G] < ∞. Let S denote a system of representatives of the cosets G /G. Then −1 N := sGs is a ﬁnite index, normal subgroup of G .Now G can be embedded in s∈S N G /N (see [11, Section 2.6], [15, Section 2]). This implies that G satisﬁes the FJC with wreath products, because N G /N does by the inheritance properties discussed before. Theorem 6.3. —The L-theoretic FJC with wreath products holds for GL (Z). Proof. — We proceed by induction over n.As GL (Z) is ﬁnite, the induction begin- ning is trivial. Let F be a ﬁnite group. Since GL (Z) is transfer reducible overF by Theorem 3.1 n n it follows from Theorem 5.1 (i) that GL (Z) F satisﬁes the L-theoretic FJC with respect to (F ) . It remains to replace (F ) by the family VCyc. By the Transitivity Principle 0.5 n n it sufﬁces to prove the L-theoretic FJC (with respect to VCyc) for all groups H ∈ (F ) . Because the FJC with wreath products passes to products and ﬁnite index over- groups, see Remark 6.2 it sufﬁces to consider H ∈F . n K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 123 Combining the induction assumption for GL (Z), k < n with the inheritance prop- erties for direct products, exact sequences of groups and subgroups (see Remark 6.2)itis easy to reduce the FJC with wreath products for members ofF to the class of virtually poly-cyclic groups. Wreath products of virtually poly-cyclic groups with ﬁnite groups are again virtually poly-cyclic. Thus the result follows in this case from [1]. Because for ﬁnite F the wreath product GL (Z) F can be embedded into GL (Z) n m for some m > n, there is really no difference between the FJC and the FJC with wreath products for the collection of groups GL (Z), n ∈ N. Nevertheless, as discussed in the introduction, for L-theory the induction only works for the FJC with wreath products. Remark 6.4. — We also conclude that a hyperbolic group G satisﬁes the K- and L-theoretic FJC with wreath products. For K-theory (without ﬁnite wreath products) this has already been proved in [5]. A hyperbolic group is strongly transfer reducible over VC yc by [24, Example 3.2] and in particular transfer reducible over VC yc. Hence it satisﬁes the K-theoretic FJC in all dimensions and the L-theoretic FJC with respect to the family VC yc by Theorem 5.1 (ii). By the transitivity principle 0.5 it sufﬁces to show the FJC for all groups from VC yc . Since those groups are virtually polycyclic the FJC holds for them by [1, Theorem 0.1]. Notice that a group is hyperbolic if a subgroup of ﬁnite index is hyperbolic. Never- theless, it is desirable to have the wreath product version for hyperbolic groups also since it inherits to colimits of hyperbolic groups and many constructions of groups with exotic properties occur as colimits of hyperbolic groups. 7. Proof of the general theorem Lemma 7.1. —Let R be a ring whose underlying abelian group is ﬁnitely generated. Then both the K-theoretic and the L-theoretic FJC hold for GL (R) and SL (R). n n Proof. — Since the FJC passes to subgroups (by [7]) we only need to treat GL (R). n k Choose an isomorphism of abelian groups h : R − → Z × T for some natural number k and a ﬁnite abelian group T. We obtain an injection of groups f h n n k GL (R) aut R − → aut R − → aut Z × T , n R Z Z where f is the forgetful map and h comes by conjugation with h . Since the FJC passes to subgroups, it sufﬁces to prove the FJC for aut (Z × T). There is an obvious exact sequence of groups k k 1 → hom Z , T → aut Z × T → GL (Z) × aut (T) → 1. Z Z k Z Since hom (Z , T) and aut (T) are ﬁnite and GL (Z) satisﬁes the FJC, Lemma 7.1 fol- Z Z k lows from [1, Corollary 1.12]. 124 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING Proof of General Theorem. — Let G be a group which is commensurable to a subgroup H ⊆ GL (R) for some natural number n. We have to show that G satisﬁes the FJC with wreath products. We have explained in Remark 6.2 that the FJC with wreath products passes to overgroups of ﬁnite index and all subgroups. Therefore it sufﬁces to show that the FJC with wreath products holds for GL (R). Consider a ﬁnite group H. Let R H be the twisted group ring of H with coefﬁcients in R where the H-action on R is given by permuting the factors. Since R is ﬁnitely H H generated as abelian group by assumption, the same is true for R H. Hence GL (R H) satisﬁes the FJC by Lemma 7.1. There is an obvious injective group homomorphism GL (R) H = GL (R) n n H H H → GL (R H). 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HARRIS, Principles of Algebraic Geometry, Wiley-Interscience/Wiley, New York, 1978 Pure and Applied Mathematics. 18. S. HELGASON, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, vol. 80, Academic Press/Harcourt Brace Jovanovich, New York, 1978. 19. P. KÜHL, Isomorphismusvermutungen und 3-Mannigfaltigkeiten. Preprint, arXiv:0907.0729v1 [math.KT], 2009. 20. W. LÜCK, Survey on classifying spaces for families of subgroups, in Inﬁnite Groups: Geometric, Combinatorial and Dynamical Aspects. Progress in Mathematics, vol. 248, pp. 269–322, Birkhäuser, Basel, 2005. 21. W. LÜCK and H. REICH, The Baum-Connes and the Farrell-Jones conjectures in K- and L-theory, in Handbook of K-Theory, vols.1,2, pp. 703–842, Springer, Berlin, 2005. 22. J. NEUKIRCH, Algebraic Number Theory, Springer, Berlin, 1999 Translated from the 1992 German original and with a note by N. Schappacher, With a foreword by G. Harder. 23. S. K. ROUSHON, The Farrell-Jones isomorphism conjecture for 3-manifold groups, K-Theory, 1 (2008), 49–82. 24. C. WEGNER, The K-theoretic Farrell-Jones conjecture for CAT(0)-groups, Proc.Am. Math.Soc., 140 (2012), 779–793. A. B. Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 60, 48149 Münster, Germany bartelsa@math.uni-muenster.de W. L.,H.R. Mathematisches Institut, Rheinische Wilhelms-Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany wolfgang.lueck@him.uni-bonn.de H. R. henrik.rueping@hcm.uni-bonn.de H. R. Institut für Mathematik, Freie Universität Berlin, Arnimallee 7, 14195 Berlin, Germany holger.reich@fu-berlin.de Manuscrit reçu le 11 avril 2012 Manuscrit accepté le 6 mai 2013 publié en ligne le 25 mai 2013. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals http://www.deepdyve.com/lp/springer-journals/k-and-l-theory-of-group-rings-over-gl-n-z-SF9KdKPssw

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Publisher: Springer Journals
Copyright: Copyright © 2013 by IHES and Springer-Verlag Berlin Heidelberg
Subject: Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN: 0073-8301
eISSN: 1618-1913
DOI: 10.1007/s10240-013-0055-0
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Abstract

K- AND L-THEORY OF GROUP RINGS OVER GL (Z) by ARTHUR BARTELS , WOLFGANG LÜCK , HOLGER REICH , and HENRIK RÜPING ABSTRACT We prove the K- and L-theoretic Farrell-Jones Conjecture (with coefﬁcients in additive categories) for GL (Z). Introduction The Farrell-Jones Conjecture predicts a formula for the K- and L-theory of group rings R[G]. This formula describes these groups in terms of group homology and K- and L-theory of group rings RV, where V varies over the family VCyc of virtually cyclic subgroups of G. Main Theorem. — Both the K-theoretic and the L-theoretic Farrell-Jones Conjecture (see Deﬁ- nitions 0.1 and 0.2) hold for GL (Z). We will generalize this theorem in the General Theorem below. In particular it also holds for arithmetic groups deﬁned over number ﬁelds, compare Example 0.4,and extends to the more general version “with wreath products”. For cocompact lattices in almost connected Lie groups this result holds by Bartels- Farrell-Lück [1]. The lattice GL (Z) has ﬁnite covolume but is not cocompact. It is a long standing question whether the Baum-Connes Conjecture holds for GL (Z). For torsion free discrete subgroups of GL (R), or more generally, for fundamental groups of A-regular complete connected non-positive curved Riemannian manifolds, the Farrell-Jones Conjecture with coefﬁcients in Z has been proven by Farrell-Jones [14]. The formulation of the Farrell-Jones Conjecture. — Deﬁnition 0.1 (K-theoretic FJC). — Let G be agroup andletF be a family of subgroups. Then G satisﬁes the K-theoretic Farrell-Jones Conjecture with respect toF if for any additive G-categoryA the assembly map G G H (E (G); K ) → H (pt; K ) = KA FAA n n n induced by the projection E (G) → pt is bijective for all n ∈ Z. If this map is bijective for all n ≤ 0 and surjective for n = 1, then we say G satisﬁes the K-theoretic Farrell-Jones Conjecture up to dimension 1 with respect toF . If the familyF is not mentioned, it is by default the family VCyc of virtually cyclic subgroups. DOI 10.1007/s10240-013-0055-0 98 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING If one choosesA to be (a skeleton of) the category of ﬁnitely generated free R- modules with trivial G-action, then K (A) is just the algebraic K-theory K (RG) of n n the group ring RG. If G is torsion free, R is a regular ring, andF is VCyc, then the claim boils down to the more familiar statement that the classical assembly map H (BG; K ) → K (RG) n R n from the homology theory associated to the (non-connective) algebraic K-theory spec- trum of R applied to the classifying space BG of G to the algebraic K-theory of RG is a bijection. If we restrict further to the case R = Z and n ≤ 1, then this implies the vanish- ing of the Whitehead group Wh(G) of G, of the reduced projective class group K (ZG), and of all negative K-groups K (ZG) for n ≤−1. Deﬁnition 0.2 (L-theoretic FJC). — Let G be a group and letF be a family of subgroups. Then G satisﬁes the L-theoretic Farrell-Jones Conjecture with respect toF if for any additive G-category with involutionA the assembly map −∞ −∞ G G −∞ H E (G); L → H pt; L = LA n n n AA induced by the projection E (G) → pt is bijective for all n ∈ Z. If the familyF is not mentioned, it is by default the family VCyc of virtually cyclic subgroups. Given a group G, a family of subgroupsF is a collection of subgroups of G that is closed under conjugation and taking subgroups. For the notion of a classifying space E (G) for a familyF we refer for instance to the survey article [20]. The natural choice forF in the Farrell-Jones Conjecture is the family VC yc of virtually cyclic subgroups but for inductive arguments it is useful to consider other families as well. Remark 0.3 Relevance of the additive categories as coefﬁcients. — The versions of the Farrell-Jones Conjecture appearing in Deﬁnitions 0.1 and 0.2 are formulated and an- alyzed in [2], [7]. They encompass the versions for group rings RG over arbitrary rings R, where one can built in a twisting into the group ring or treat more generally crossed product rings R ∗ G and one can allow orientation homomorphisms w : G →{±1} in the L-theory case. Moreover, inheritance properties, e.g., passing to subgroups, ﬁnite prod- ucts, ﬁnite free products, and directed colimits, are built in and one does not have to pass to ﬁbered versions anymore. The original source for the (Fibered) Farrell-Jones Conjecture is the paper by Farrell-Jones [13, 1.6 on p. 257 and 1.7 on p. 262]. For more information about the Farrell-Jones Conjecture, its relevance and its various applications to prominent conjec- tures due to Bass, Borel, Kaplansky, Novikov and Serre, we refer to [6], [21]. We will often abbreviate Farrell-Jones Conjecture to FJC. K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 99 Extension to more general rings and groups. — We will see that it is not hard to generalize the Main Theorem as follows. General Theorem. — Let R be a ring whose underlying abelian group is ﬁnitely generated. Let G be a group which is commensurable to a subgroup of GL (R) for some natural number n. Then G satisﬁes both the K-theoretic and the L-theoretic Farrell-Jones Conjecture with wreath products Deﬁnition 6.1. Two groups G and G are called commensurable if they contain subgroups G ⊆ G 1 2 1 and G ⊆ G of ﬁnite index such that G and G are isomorphic. In this case G satisﬁes 2 1 2 1 2 the FJC with wreath products if and only if G does, see Remark 6.2. Example 0.4 (Ring of integers). — Let K be an algebraic number ﬁeld andO be its ring of integers. ThenO considered as abelian group is ﬁnitely generated free (see [22, Chapter I, Proposition 2.10 on p. 12]). Hence by the General Theorem any group G which is commensurable to a subgroup of GL (O ) for some natural number n satisﬁes n K both the K-theoretic and L-theoretic FJC with wreath products. This includes in partic- ular arithmetic groups over number ﬁelds. Discussion of the proof. — The proof of the FJC for GL (Z) will use the transitivity principle [13, Theorem A.10], that we recall here. Proposition 0.5 Transitivity principle. — LetF ⊂H be families of subgroups of G. Assume that G satisﬁes the FJC with respect toH and that each H ∈H satisﬁes the FJC with respect toF . Then G satisﬁes the FJC with respect toF . This principle applies to all versions of the FJC discussed above. In this form it can be found for example in [1, Theorem 1.11]. Themainstepinproving theFJC for GL (Z) is to prove that GL (Z) satisﬁes the n n FJC with respect to a familyF . This family is deﬁned at the beginning of Section 3. This family is larger than VCyc andcontains forexample GL (Z) for k < n.Wecan then use induction on n to prove that every group fromF satisﬁes the FJC. At this point we also use the fact that virtually poly-cyclic groups satisfy the FJC. To prove that GL (Z) satisﬁes the FJC with respect toF we will apply two results n n from [4], [24]. Originally these results were used to prove that CAT(0)-groups satisfy the FJC. Checking that they are applicable to GL (Z) is more difﬁcult. While GL (Z) is not n n aCAT(0)-group, it does act on a CAT(0)-space X. This action is proper and isometric, but not cocompact. Our main technical step is to show that the ﬂow space associated to this CAT(0)-space admits longF -covers at inﬁnity, compare Deﬁnition 3.7. In Section 1 we analyze the CAT(0)-space X. On it we introduce, following Grayson [16], certain volume functions and analyze them from a metric point of view. These functions will be used to cut off a suitable well-chosen neighborhood of inﬁnity so 100 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING that the GL (Z)-action on the complement is cocompact. In Section 2 we study sublat- tices in Z . This will be needed to ﬁnd the neighborhood of inﬁnity mentioned above. Here we prove a crucial estimate in Lemma 2.1. As outlined this proof works best for the K-theoretic FJC up to dimension 1; this case is contained in Section 3. The modiﬁcations needed for the full K-theoretic FJC are discussed in Section 4 and use results of Wegner [24]. For L-theory the induction does not work quite as smoothly. The appearance of index 2 overgroups in the statement of [4, Theorem 1.1(ii)] force us to use a stronger induction hypothesis: we need to assume that ﬁnite overgroups of GL (Z), k < n satisfy the FJC. (It would be enough to consider overgroups of index 2, but this seems not to simplify the argument.) A good formalism to accommodate this is the FJC with wreath products (which implies the FJC). In Section 5 we provide the necessary extensions of the results from [4] for this version of the FJC. In Section 6 we then prove the L-theoretic FJC with wreath products for GL (Z). In Section 7 we give the proof of the General Theorem. 1. The space of inner products and the volume function Throughout this section let V be an n-dimensional real vector space. Let X(V) be the set of all inner products on V. We want to examine the smooth manifold X(V) and equip it with an aut(V)-invariant complete Riemannian metric with non-positive sectional curvature. With respect to this structure we will examine a certain volume func- tion. We try to keep all deﬁnitions as intrinsic as possible and then afterward discuss what happens after choices of extra structures (such as bases). 1.1. The space of inner products. — We can equip V with the structure of a smooth manifold by requiring that any linear isomorphism V → R is a diffeomorphism with respect to the standard smooth structure on R . In particular V carries a preferred struc- ture of a (metrizable) topological space and we can talk about limits of sequences in V. We obtain a canonical trivialization of the tangent bundle TV (1.1) φ : V × V → TV which sends (x,v) to the tangent vector in T V represented by the smooth path R → V, t → x + t · v. The inverse sends the tangent vector in TV represented by a path w : (−, ) → Vto (w(0), w (0)).If f : V → W is a linear map, the following diagram commutes V × V TV f ×f Tf W × W TW = K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 101 ∗ ∗ Let hom(V, V ) be the real vector space of linear maps V → V from V to the dual ∗ ∗ ∗ V of V. In the sequel we will always identify V and (V ) by the canonical isomorphism ∗ ∗ ∗ V → (V ) which sends v ∈ V to the linear map V → R,α → α(v).Hence for s ∈ ∗ ∗ ∗ ∗ ∗ ∗ hom(V, V ) its dual s : (V ) = V → V belongs to hom(V, V ) again. Let Sym(V) ⊆ ∗ ∗ ∗ hom(V, V ) be the subvector space of elements s ∈ hom(V, V ) satisfying s = s.Wecan identify Sym(V) with the set of all bilinear symmetric pairings V × V → R, namely, given s ∈ Sym(V) we obtain such a pairing by (v, w) → s(v)(w). We will often write s(v, w) := s(v)(w). Under the identiﬁcation above the set X(V) of inner products on V becomes the open subset of Sym(V) consisting of those elements s ∈ Sym(V) for which s : V → V is bijec- tive and s(v, v) ≥ 0 holds for all v ∈ V, or, equivalently, for which s(v, v) ≥ 0 holds for all v ∈ V and we have s(v, v) = 0 ⇔ v = 0. In particular X(V) inherits from the vector space Sym(V) the structure of a smooth manifold. Given a linear map f : V → W, we obtain a linear map Sym(f ) : Sym(W) → ∗ ∗ Sym(V) by sending s : W → W to f ◦ s ◦ f . A linear isomorphism f : V → W induces a bijection X(f ) : X(W) → X(V). Obviously this is a contravariant functor, i.e., X(g ◦ f ) = X(f ) ◦ X(g).If aut(V) is the group of linear automorphisms of V, we obtain a right aut(V)-action on X(V). If f : V → W is a linear map and s and s are inner products on V and W, then V W −1 the adjoint of f with respect to these inner products is s ◦ f ◦ s : W → V. Consider a natural number m ≤ n := dim(V). There is a canonical isomorphism = ∗ m ∗ m β (V) : V − → V which maps α ∧ α ∧ ··· ∧ α to the map V → R sending v ∧ v ∧ ··· ∧ v to 1 2 m 1 2 m sign(σ ) · α (v ).Let s : V → V be an inner product on V. We obtain an i σ(i) σ ∈S i=1 inner product s on V by the composite β (V) s m ∗ m m ∗ m s : V −→ V −−→ V . One easily checks by a direct calculation for elements v ,v ,...,v ,w ,w ,...,w in V 1 2 m 1 2 m (1.2) s m (v ∧ ··· ∧ v ,w ∧ ··· ∧ w ) = det s(v ,w ) , 1 m 1 m i j i,j where (s(v ,w )) is the obvious symmetric (m, m)-matrix. i j i,j Next we want to deﬁne a Riemannian metric g on X(V). Since X(V) is an open subset of Sym(V) and we have a canonical trivialization φ of T Sym(V) (see (1.1)), Sym(V) we have to deﬁne for every s ∈ X(V) an inner product g on Sym(V).Itisgiven by −1 −1 g (u,v) := tr s ◦ v ◦ s ◦ u , s 102 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING for u,v ∈ Sym(V). Here tr denotes the trace of endomorphisms of V. Obviously g (−, −) −1 is bilinear and symmetric since the trace is linear and satisﬁes tr(ab) = tr(ba). Since s ◦ −1 ∗ −1 −1 (s ◦ u) ◦ s = s ◦ u holds, the endomorphism s ◦ u : V → V is selfadjoint with respect −1 −1 −1 −1 to the inner product s.Hence tr(s ◦ u ◦ s ◦ u) ≥ 0 holds and we have tr(s ◦ u ◦ s ◦ −1 u) = 0 if and only if s ◦ u = 0, or, equivalently, u = 0. Hence g is an inner product on Sym(V). We omit the proof that g depends smoothly on s. Thus we obtain a Riemannian metric g on X(V). Lemma 1.1. — The Riemannian metric on X is aut(V)-invariant. Proof. — We have to show for an automorphism f : V → V, an element s ∈ X(V) and two elements u,v ∈ T X(V) = T Sym(V) = Sym(V) that s s g T X(f )(u), T X(f )(v) = g (u,v) s s s X(f )(s) holds. This follows from the following calculation: g T X(f )(u), T X(f )(v) s s X(f )(s) −1 −1 = tr X(f )(s) ◦ T X(f )(v) ◦ X(f )(s) ◦ T X(f )(u) s s −1 −1 ∗ ∗ ∗ ∗ = tr f ◦ s ◦ f ◦ f ◦ v ◦ f ◦ f ◦ s ◦ f ◦ f ◦ u ◦ f −1 −1 −1 −1 ∗ ∗ −1 −1 ∗ ∗ = tr f ◦ s ◦ f ◦ f ◦ v ◦ f ◦ f ◦ s ◦ f ◦ f ◦ u ◦ f −1 −1 −1 = tr f ◦ s ◦ v ◦ s ◦ u ◦ f −1 −1 −1 = tr s ◦ v ◦ s ◦ u ◦ f ◦ f −1 −1 = tr s ◦ v ◦ s ◦ u = g (u,v). Recall that so far we worked with the natural right action of aut(V) on X(V).In the sequel we prefer to work with the corresponding left action obtained by precomposing −1 with f → f in order to match with standard notation, compare in particular diagram (1.3)below. Fix a base point s ∈ X(V). Choose a linear isomorphism R − → V which is isomet- ric with respect to the standard inner product on R and s . It induces an isomorphism GL (R) − → aut(V) and thus a smooth left action ρ : GL (R) × X(V) → X(V). Since for n n any two elements s , s ∈ X(V) there exists an automorphism of V which is an isom- 1 2 etry (V, s ) → (V, s ), this action is transitive. The stabilizer of s ∈ X(V) is the com- 1 2 0 pact subgroup O(n) ⊆ GL (R). Thus we obtain a diffeomorphism φ : GL (R)/O(n) − → n n −1 >0 X(V). Deﬁne smooth maps p : X(V) → R , s → det(s ◦ s) and q : GL (R)/O(n) → >0 2 R , A · O(n) → det(A) . Both maps are submersions. In particular the preimages of K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 103 >0 −1 1 ∈ R under p and q are submanifolds of codimension 1. Denote by X(V) := p (1) and let i : X(V) → X(V) be the inclusion. Set SL (R) ={A ∈ GL (R) | det(A) =±1}. 1 n The smooth left action ρ : GL (R) × X(V) → X(V) restricts to an action ρ : SL (R) × n 1 −1 ∗ ∗ 2 X(V) → X(V) . Since f ◦ s ◦ f ∈ X(V) implies 1 = det(s ◦ f ◦ s ◦ f ) = det(f ) 1 1 0 1 0 this action is still transitive. The stabilizer of s is still O(n) ⊆ SL (R) and thus we ob- tain a diffeomorphism φ : SL (R)/O(n) → X(V) . Note that the inclusion induces a 1 1 diffeomorphism SL (R)/SO(n) = SL (R)/O(n) butwepreferthe righthandsidebe- cause we are interested in the GL (Z)-action. The inclusion SL (R) → GL (R) induces n n −1 an embedding j : SL (R)/O(n) → GL (R)/O(n) with image q (1). One easily checks that the following diagram commutes >0 (1.3) X(V) X(V) R ˜ ˜ φ φ id j q >0 SL (R)/O(n) GL (R)/O(n) n R Equip X(V) with the Riemannian metric g obtained from the Riemannian metric g 1 1 ± ± on X(V). We conclude from Lemma 1.1 that g is SL (R)-invariant. Since SL (R) is n n a semisimple Lie group with ﬁnite center and O(n) ⊆ SL (R) is a maximal compact subgroup, X(V) is a symmetric space of non-compact type and its sectional curvature is non-positive (see [12, Section 2.2 on p. 70],[18, Theorem 3.1 (ii) in V.3 on p. 241]). Alternatively [8, Chapter II, Theorem 10.39 on p. 318 and Lemma 10.52 on p. 324] show that X(V) and X(V) are proper CAT(0) spaces. >0 We call two elements s , s ∈ X(V) equivalent if there exists r ∈ R with r · s = s . 1 2 1 2 Denote by X(V) the set of equivalence classes under this equivalence relation. Let pr : X(V) → X(V) be the projection and equip X(V) with the quotient topology. The composite pr ◦i : X(V) → X(V) is a homeomorphism. In the sequel we equip X(V) with the structure of a Riemannian manifold for which pr ◦i is an isometric diffeomor- phism. In particular X(V) is a CAT(0)-space. The GL (R)-action on X(V) descends to an action on X(V) and the diffeomorphism pr ◦i is SL (R)-equivariant. 1.2. The volume function. — Fix an integer m with 1 ≤ m ≤ n = dim(V). In this section we investigate the following volume function. Deﬁnition 1.2 (Volume function). — Consider ξ ∈ V with ξ = 0.Deﬁne the volume function associated to ξ by vol : X(V) → R, s → (s m )(ξ , ξ ), i.e., the function vol sends an inner product s on V to the length of ξ with respect to s m . ξ 104 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING Fix ξ = 0in Vof the form ξ = v ∧ v ··· ∧ v for the rest of this section. 1 2 m This means that the line ξ lies in the image of the Plücker embedding, compare [17, Chapter 1.5 p. 209–211]. Then there is precisely one m-dimensional subvector space m m V ⊆ V such that the image of the map V → V induced by the inclusion is the ξ ξ 1-dimensional subvector space spanned by ξ . The subspace V is the span of the vectors v ,v ,...,v . It can be expressed as V ={v ∈ V|v ∧ ξ = 0}. This shows that V depends 1 2 m ξ ξ on the line ξ but is independent of the choice of v ,...,v . 1 m Given an inner product s on V, we obtain an orthogonal decomposition V = V ⊕ V of V with respect to s and we deﬁne s ∈ Sym(V) to be the element which satisﬁes ⊥ ⊥ ⊥ ⊥ ⊥ s (v + v ,w + w ) = s(v, w) for all v, w ∈ V and v ,w ∈ V . ξ ξ Theorem 1.3 (Gradient of the volume function). — The gradient of the square vol of the volume function vol is given for s ∈ X(V) by 2 2 ∇ vol = vol (s) · s ∈ Sym(V) = T X(V). s ξ s ξ ξ Proof.—Let A be any (m, m)-matrix. Then det(I + t · A) − det(I ) m m (1.4) lim = tr(A). t→0 This follows because det(I + t · A) = 1 + t tr(A) mod t by the Leibniz formula for the determinant. Consider s ∈ X(V) and u ∈ T (X(V)) = Sym(V). Notice that there exists > 0 such that s + t · u lies in X(V) for all t ∈ (−, ).Fix v ,v ,...,v ∈ Vwith ξ = v ∧ 1 2 m 1 ··· ∧ v . We compute using (1.2)and (1.4) 2 2 vol (s + t · u) − vol (s) ξ ξ lim t→0 det(((s + t · u)(v ,v )) ) − det((s(v ,v )) ) i j i,j i j i,j = lim t→0 det((s(v ,v )) + t · (u(v ,v )) ) − det((s(v ,v )) ) i j i,j i j i,j i j i,j = lim t→0 = det s(v ,v ) i j i,j −1 det(I + t · (s(v ,v )) · (u(v ,v )) ) − det(I ) m i j i j i,j m i,j · lim t→0 −1 = det s(v ,v ) · tr s(v ,v ) · u(v ,v ) . i j i j i j i,j i,j i,j K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 105 The gradient ∇ (vol ) is uniquely determined by the property that for all u ∈ Sym(V) we have 2 2 vol (s + t · u) − vol (s) ξ ξ g ∇ vol , u = lim . s s t→0 Hence it remains to show for every u ∈ Sym(V) −1 vol (s) · s , u = det s(v ,v ) · tr s(v ,v ) · u(v ,v ) . s ξ i j i j i j i,j i,j i,j −1 −1 Since vol (s) = det((s(v ,v )) ) by (1.2)and g (s , u) = tr(s ◦ s ◦ s ◦ u) by deﬁnition, i j i,j s ξ ξ it remains to show −1 −1 −1 tr s ◦ s ◦ s ◦ u = tr s(v ,v ) · u(v ,v ) . ξ i j i j i,j i,j s 0 ⊥ ⊥ ⊥ ∗ We obtain a decomposition s = ⊥ for an inner product s : V → (V ) ,where 0 s ξ ξ ξ ∗ ⊥ ∗ ⊥ ∗ we will here and in the sequel identify (V ) ⊕ (V ) and (V ⊕ V ) by the canonical ξ ξ ξ ξ isomorphism. We decompose u u ∗ ξ ⊥ ∗ ⊥ u = : V ⊕ V → (V ) ⊕ V ξ ξ ξ ξ u u for a linear map u : V → V . One easily checks ξ ξ −1 −1 −1 tr s ◦ s ◦ s ◦ u = tr s ◦ u . ξ ξ ∗ ∗ ∗ ∗ The set {v ,v ,...,v } is a basis for V .Let {v ,v ,...,v } be the dual basis of V . 1 2 m ξ 1 2 m ξ Then the matrix of u with respect to these basis is (u(v ,v )) and the matrix of s with ξ i j i,j ξ −1 respect to these basis is (s(v ,v )) .Hence thematrixof s ◦ u : V → V with respect i j i,j ξ ξ ξ −1 to the basis {v ,v ,...v } is (s(v ,v )) · (u(v ,v )) . This implies 1 2 m i j i j i,j i,j −1 −1 tr s ◦ u = tr s(v ,v ) · u(v ,v ) . ξ i j i j i,j i,j This ﬁnishes the proof of Theorem 1.3. Corollary 1.4. — The gradient of the function ln ◦ vol : X(V) → R at s ∈ X(V) is given by the tangent vector s ∈ T X(V) = Sym(V). In particular its norm with ξ s respect to the Riemannian metric g on X(V) is independent of s, namely, m. 106 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING −1 Proof. — Since the derivative of ln(x) is x , the chain rule implies together with Theorem 1.3 for s ∈ X(V) 1 1 2 2 2 ∇ ln ◦ vol =∇ vol(ξ ) · = vol (s) · s · = s . s s ξ ξ ξ ξ 2 2 vol (s) vol (s) ξ ξ We use the orthogonal decomposition V = V ⊕ V with respect to s to compute 2 2 −1 −1 ∇ ln ◦ vol = ( s ) = g (s , s ) = tr s ◦ s ◦ s ◦ s s ξ s ξ ξ ξ ξ s s ⊥ ⊥ = tr id ⊕ 0 : V ⊕ V → V ⊕ V V ξ ξ ξ ξ ξ = dim(V ) = m. 2. Sublattices of Z n n n A sublattice Wof Z is a Z-submodule W ⊆ Z such that Z /Wis a projective Z- n n module. Equivalently, W is a Z-submodule W ⊆ Z such that for x ∈ Z for which k · x belongs to W for some k ∈ Z, k = 0we have x ∈ W. LetL be the set of sublattices L of Z . m m Consider W ∈L.Let m be its rank as an abelian group. Let W → Z → Z Z m n m n R be the obvious map. Let ξ(W) ∈ R be the image of a generator of the inﬁnite cyclic group W. We have deﬁned the map vol : X(R ) → R above. Obviously it ξ(W) does not change if we replace ξ(W) by −ξ(W). Hence it depends only on W and not on the choice of generator ξ(W). Notice that for W = 0 we have by deﬁnition W = Z and 0 n R = R and ξ(W) is ±1 ∈ R.Inthatcase vol is the constant function with value 1. We will abbreviate n n n X = X R , X = X R , X = X R and vol = vol 1 W ξ(W) for W ∈L. Given a chain W W of elements W , W ∈L, we deﬁne a function 0 1 0 1 ˜ c : X → R W W 0 1 by ln(vol (s)) − ln(vol (s)) W W 1 0 ˜ c (s) := . W W 0 1 rk (W ) − rk (W ) Z 1 Z 0 Obviously this can be rewritten as 2 2 1 ln(vol (s) ) − ln(vol (s) ) W W 1 0 ˜ c (s) = · . W W 0 1 2 rk (W ) − rk (W ) Z 1 Z 0 K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 107 Hence we get for the norm of the gradient of this function at s ∈ X ∇ (˜ c ) s W W 0 1 2 2 ∇ (ln ◦ vol ) −∇ (ln ◦ vol ) 1 s s W W 1 0 = · 2 rk (W ) − rk (W ) Z 1 Z 0 2 2 ∇ (ln ◦ vol )+∇ (ln ◦ vol ) 1 s s W W 1 0 ≤ · 2 rk (W ) − rk (W ) Z 1 Z 0 2 2 ≤ · ∇ ln ◦ vol + ∇ ln ◦ vol . s s W W 1 0 We conclude from Corollary 1.4 for all s ∈ X. √ √ rk (W ) + rk (W ) Z 1 Z 0 ∇ (˜ c ) ≤ ≤ n. s W W 0 1 If f : M → R is a differentiable function on a Riemannian manifold M and C = sup{∇ f | x ∈ M} we always have f (x ) − f (x ) ≤ Cd (x , x ), 1 2 M 1 2 where d denotes the metric associated to the Riemannian metric. In particular we get for any two elements s , s ∈ X 0 1 ˜ c (s ) −˜ c (s ) ≤ n · d(s , s ), W W 1 W W 0 0 1 0 1 0 1 X where d is the metric on X coming from the Riemannian metric g on X. Recall that the Riemannian metric g on X is obtained by restricting the metric g.Let d be the 1 1 X metric on X coming from the Riemannian metric g on X .Recallthat i : X → Xis 1 1 1 1 the inclusion. Then we get for s , s ∈ X 0 1 1 d i(s ), i(s ) ≤ d (s , s ). 0 1 0 1 X X Indeed X is a geodesic submanifold of X([8, Chapter II, Lemma 10.52 on p. 324]). So both sides are even equal. We obtain for s , s ∈ X 0 1 1 (2.1) ˜ c ◦ i(s ) −˜ c ◦ i(s ) ≤ n · d (s , s ). W W 1 W W 0 X 0 1 0 1 0 1 1 Put L = W ∈L | W = 0, W = Z . Deﬁne for W ∈L functions i s ˜ c , ˜ c : X → R W W 108 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING by ˜ c (x) := inf ˜ c (x) | W ∈L, W W ; WW 2 2 W 2 ˜ c (x) := sup ˜ c (x) | W ∈L, W W . W W 0 0 W 0 In order to see that inﬁmum and supremum exist we use the fact that for ﬁxed x ∈ X there are at most ﬁnitely many W ∈L with vol (x) ≤ 1, compare [16, Lemma 1.15]. Put i s (2.2) d : X → R, x → exp ˜ c (x) −˜ c (x) . W W rk (W) >0 Since vol (r · s) = r · vol (s) holds for r ∈ R ,W ∈L and s ∈ X, we have ˜ c (r · W W W W 0 1 s) =˜ c (s) + ln(r) and the function d factorizes over the projection pr : X → Xto W W W 0 1 afunction d : X → R. >0 Lemma 2.1. —Consider x ∈ X, W ∈L ,and α ∈ R . Then we get for all y ∈ X with d (x, y) ≤ α √ √ 2 nα 2 nα d (y) ∈ d (x)/e , d (x) · e . W W W Proof. — By the deﬁnition of the structure of a smooth Riemannian manifold on X, it sufﬁces to show for all s , s ∈ X with d (s , s ) ≤ α 0 1 1 0 1 √ √ 2 nα 2 nα ˜ ˜ ˜ d (s ) ∈ d (s )/e , d (s ) · e . W 1 W 0 W 0 One concludes from (2.1)for s , s ∈ X with d (s , s ) ≤ α and W ∈L 0 1 1 X 0 1 √ √ s s s ˜ c (s ) ∈ c˜ (s ) − n · α, ˜ c (s ) + n · α , 1 0 0 W W W √ √ i i i ˜ c (s ) ∈ c˜ (s ) − n · α, ˜ c (s ) + n · α . 1 0 0 W W W Now the claim follows. In particular the function d : X → R is continuous. Deﬁne the following open subset of X for W ∈L and t ≥ 1 X(W, t) := x ∈ X | d (x)> t . There is an obvious GL (Z)-action onL andL and in the discussion following Lemma 1.1 we have already explained how GL (Z) ⊂ SL (R) acts on X and hence n 1 on X (choosing the standard inner product on R as the basepoint s ). Lemma 2.2. —For any t ≥ 1 we get: K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 109 (i) X(gW, t) = gX(W, t) for g ∈ GL (Z), W ∈L ; (ii) The complement of the GL (Z)-invariant open subset W (t) := X(W, t) W∈L in X is a cocompact GL (Z)-set; (iii) If X(W , t) ∩ X(W , t) ∩··· ∩ X(W , t) = ∅ for W ∈L ,thenwecan ﬁnda 1 2 k i permutation σ ∈ such that W ⊆ W ⊆ ··· ⊆ W holds. k σ(1) σ(2) σ(k) Proof. — This follows directly from Grayson [16, Lemma 2.1, Cor. 5.2] as soon as we have explained how our setup corresponds to the one of Grayson. We are only dealing with the caseO = Z and F = Q of [16]. In particular there is only one archimedian place, namely, the absolute value on Q and for it Q = R.So an element s ∈ X corresponds to the structure of a lattice which we will denote by (Z , s) with underlying Z-module Z in the sense of [16]. Given s ∈ X, an element W ∈L deﬁnes n n a sublattice of the lattice (Z , s) in the sense of [16] which we will denote by (Z , s) ∩ W. n n Thevolumeofasublattice (Z , s) ∩ Wof (Z , s) in the sense of [16]is vol (s). Given W ∈L ,weobtaina Q-subspace in the sense of [16, Deﬁnition 2.1] which we denote again by W, and vice versa. It remains to explain why our function d of (2.2) agrees with the function d of [16, Deﬁnition 2.1] which is given by n n n d (s) = exp min Z , s / Z , s ∩ W − max Z , s ∩ W , see [16, Deﬁnition 1.23, 1.9]. This holds by the following observation. Consider s ∈ Xand W ∈L . Consider the canonical plot and the canonical poly- gon of the lattice (Z , s) ∩ W in the sense of [16, Deﬁnition 1.10 and Discussion 1.16]. The slopes of the canonical polygon are strictly increasing when going from the left to the right because of [16, Corollary 1.30]. Hence max((Z , s) ∩ W) in the sense of [16,Def- inition 1.23] is the slope of the segment of the canonical polygon ending at (Z , s) ∩ W. Consider any W ∈L with W W. Obviously the slope of the line joining the plot point 0 0 n n of (Z , s) ∩ W and (Z , s) ∩ W is less than or equal to the slope of the segment of the n n canonical polygon ending at (Z , s) ∩ W. If (Z , s) ∩ W happens to be the starting point of this segment, then this slope agrees with the slope of the segment of the canonical plot ending at (Z , s) ∩ W. Hence max Z , s ∩ W ln(vol (s)) − ln(vol (s)) W W 0 s (2.3) = max W ∈L, W W =˜ c (s). 0 0 rk (W) − rk (W ) Z Z 0 We have the formula n n n n vol Z , s ∩ W = vol Z , s ∩ W · vol Z , s ∩ W / Z , s ∩ W 2 2 110 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING for any W ∈L with W W (see [16, Lemma 1.8]). Hence 2 2 ln(vol (s)) − ln(vol (s)) ln(vol (s)) W W W /W 2 2 = . rk (W ) − rk (W) rk (W /W) Z 2 Z Z 2 There is an obvious bijection of the set of direct summands in Z /W and the set of direct summand in Z containing W. Now analogously to the proof of (2.3)one shows n n (2.4) min Z , s / Z , s ∩ W ln(vol (s)) − ln(vol (s)) W W 2 i (2.5) = min W ∈L, W W =˜ c (s). 2 2 rk (W ) − rk (W) Z 2 Z Now the equality of the two versions for d follows from (2.3)and (2.5). This ﬁnishes the poof of Lemma 2.2. 3. Transfer reducibility of GL (Z) LetF be the family of those subgroups H of GL (Z), which are virtually cyclic or n n for which there exists a ﬁnitely generated free abelian group P, natural numbers r and n with n < n and an extension of groups 1 → P → K → GL (Z) → 1 i=1 such that H is isomorphic to a subgroup of K. In this section we prove the following theorem, which by [4, Theorem 1.1] implies the K-theoretic FJC up to dimension 1 for GL (Z) with respect to the familyF .The n n notion of transfer reducibility has been introduced in [4, Deﬁnition 1.8]. Transfer reducibility asserts the existence of a compact space Z and certain equivariant covers of G×Z. A slight modiﬁcation of transfer reducibility is discussed in Section 5, see Deﬁnition 5.3. Here our main work is to verify the conditions formulated in Deﬁnition 3.7 for our situation, see Lemma 3.8. Once this has been done the veriﬁcation of transfer reducibility proceedsasin[3], as is explained after Lemma 3.8. Theorem 3.1. — The group GL (Z) is transfer reducible overF . n n To prove this we will use the space X = X(R ) and its subsets X(W, t) considered in Sections 1 and 2. For a G-space X and a family of subgroupsF , a subset U ⊆ X is called anF - subset if G := {g ∈ G | g(U) = U} belongs toF and gU ∩ U =∅ for all g ∈ G \ G .An U U open G-invariant cover consisting ofF -subsets is called anF -cover. K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 111 Lemma 3.2. — Consider for t ≥ 1 the collection of subsets of X W (t) = X(W, t) | W ∈L . It is a GL (Z)-invariant set of openF -subsets of X whose covering dimension is at most (n − 2). n n Proof. — The setW (t) is GL (Z)-invariant because of Lemma 2.2 (i) and its cov- ering dimension is bounded by (n − 2) because of Lemma 2.2 (iii) since for any chain of n n sublattices {0} W W ··· W Z of Z we have r ≤ n − 2. 0 1 r It remains to show thatW (t) consists ofF -subsets. Consider W ∈L and g ∈ GL (Z). Lemma 2.2 implies: X(W, t) ∩ gX(W, t) = ∅ ⇐⇒ X(W, t) ∩ X(gW, t) = ∅ ⇐⇒ gW = W. n n Put GL (Z) ={g ∈ GL (Z) | gW = W}. Choose V ⊂ Z with W ⊕ V = Z . Under this n W n identiﬁcation every element φ ∈ GL (Z) is of the shape n W φ φ W V,W 0 φ for Z-automorphism φ of V, φ of W and a Z-homomorphism φ : V → W. Deﬁne V W V,W φ φ W V,W p : GL (Z) → aut (W) × aut (V), → (φ ,φ ) n W Z Z W V 0 φ and id ψ i : hom (V, W) → GL (Z) ,ψ → . Z n W 0id Then we obtain an exact sequence of groups i pr 1 → hom (V, W) − → GL (Z) − → aut (W) × aut (V) → 1, Z n W Z Z where hom (V, W) is the abelian group given by the obvious addition. Since both V ∼ ∼ and W are different from Z ,weget aut (V) GL (Z) and aut (W) GL (Z) for = = Z n(V) Z n(W) n(V), n(W)< n. Hence each element inW (t) is anF -subset with respect to the GL (Z)- n n action. >0 For a subset A of a metric space X and α ∈ R we denote by B (A) := x ∈ X | d (x, a)<α for some a ∈ A α X the α-neighborhood of A. Lemma 3.3. —For every α> 0 and t ≥ 1 there exists β> 0 such that for every W ∈L we have B (X(W, t + β)) ⊆ X(W, t). α 112 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING 2 nα Proof. — This follows from Lemma 2.1 if we choose β> (e − 1) · t. Let FS(X) be the ﬂow space associated to the CAT(0)-space X = X(R ) in [3, Section 2]. It consists of generalized geodesics, i.e., continuous maps c : R → X for which there is a closed subinterval I of R such that c| is a geodesic and c| is locally constant. I R\I The ﬂow on FS(X) is deﬁned by the formula ( (c))(t) = c(τ + t). Lemma 3.4. — Consider δ, τ > 0 and c ∈ FS(X).Thenweget for d ∈ B ( (c)) δ [−τ,τ ] d d(0), c(0) < 4 + δ + τ. Proof. — Choose s ∈[−τ, τ ] with d (d, (c)) < δ. We estimate using [3, FS(X) s Lemma 1.4 (i)] and a special case of [3, Lemma 1.3] d d(0), c(0) ≤ d d(0), (c)(0) + d (c)(0), c(0) X X s X s ≤ d d, (c) + 2 + d (c), c + 2 FS(X) s FS(X) s <δ + 2 +|s|+ 2 ≤ 4 + δ + τ. Let ev : FS(X) → X, c → c(0) be the evaluation map at 0. It is GL (Z)- 0 n equivariant, uniform continuous and proper [3, Lemma 1.10]. Deﬁne subsets of FS(X) by −1 Y(W, t) := ev X(W, t) for t ≥ 1, W ∈L , and setV (t) := {Y(W, t) | W ∈L }, |V (t)|:= Y(W, t) ⊆ FS(X). W∈L Lemma 3.5. —Let τ, δ > 0 and t ≥ 1 and set α := 4 + δ + τ.If β> 0 is such that for every W ∈L we have B (X(W, t + β)) ⊆ X(W, t) then the following holds (i) The setV (t) is GL (Z)-invariant; (ii) Each element inV (t) is an openF -subset with respect to the GL (Z)-action; n n (iii) The dimension ofV (t) is less or equal to (n − 2); (iv) The complement of |V (t + β)| in FS(X) is a cocompact GL (Z)-subspace. (v) For every c ∈|V (t + β)| there exists W ∈L with B (c) ⊂ Y(W, t). δ [−τ,τ ] The existence of a suitable β is the assertion of Lemma 3.3. Proof. — (i), (ii) and (iii) follow from Lemma 3.2 since ev is GL (Z)-equivariant. 0 n −1 (iv) follows from Lemma 2.2 (ii) since the map ev : FS(X) → Xis proper and ev (X − |W (t + β)|) = FS(X) −|V (t + β)|. K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 113 For (v) consider c ∈|V (t + β)|. Choose W ∈L with c ∈ Y(W, t + β).Then c(0) ∈ X(W, t + β).Consider d ∈ B ( (c)). Lemma 3.4 implies d (d(0), c(0)) < α.Hence δ [−τ,τ ] X d(0) ∈ B (c(0)). We conclude d(0) ∈ B (X(W, t + β))). This implies d(0) ∈ X(W, t) and α α hence d ∈ Y(W, t). This shows B ( (c)) ⊆ Y(W, t). δ [−τ,τ ] Let FS (X) be the subspace of FS(X) of those generalized geodesics c for which ≤γ there exists for every > 0a number τ ∈ (0,γ + ] and g ∈ Gsuch that g · c = (c) holds. As an instance of [3, Theorem 4.2] we obtain the following in our situation. Theorem 3.6. — There is a natural number M such that for every γ> 0 and every compact subset L ⊆ X there exists a GL (Z)-invariant collectionU of subsets of FS(X) satisfying: (i) Each element U ∈U is an open VC yc-subset of the GL (Z)-space FS(X); (ii) We have dimU ≤ M; (iii) There is > 0 with the following property: for c ∈ FS (X) such that c(t) ∈ GL (Z) · L ≤γ n for some t ∈ R there is U ∈U such that B ( (c)) ⊆ U. [−γ,γ ] Deﬁnition 3.7 ([3, Deﬁnition 5.5]). — Let G be a group,F be a family of subgroups of G, (FS, d ) be a locally compact metric space with a proper isometric G-action and : FS × R → FS FS be a G-equivariant ﬂow. We say that FS admits longF -covers at inﬁnity and at periodic ﬂow lines if the following holds: There is N > 0 such that for every γ> 0 there is a G-invariant collection of openF -subsetsV of FS and ε> 0 satisfying: (i) dimV ≤ N; (ii) there is a compact subset K ⊆ FS such that (a) FS ∩ G · K =∅; ≤γ (b) for z ∈ FS − G · K there is V ∈V such that B ( (z)) ⊂ V. ε [−γ,γ ] We remark that it is natural to think of this deﬁnition as requiring two conditions, the ﬁrst dealing with everything outside some cocompact subset (“at inﬁnity”) and the second dealing with (short) periodic orbits of the ﬂow that meet a given cocompact subset (“at periodic ﬂow lines”). In proving that this condition is satisﬁed in our situation in the next lemma we deal with these conditions separately. For the ﬁrst condition we use the sets Y(W, t) introduced earlier; for the second the theorem cited above. Lemma 3.8. — The ﬂow space FS(X) admits longF -covers at inﬁnity and at periodic ﬂow lines. Proof.—Fix γ> 0. Choose t ≥ 1. Put δ := 1and τ := γ .Let β> 0be the number appearing in Lemma 3.5 and let M ∈ N be the number appearing in Theorem 3.6. Since 114 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING by Lemma 2.2 (ii) the complement of |W (t + β)| in X is cocompact, we can ﬁnd a compact subset L of this complement such that GL (Z) · L = X \W (t + β) . For this compact subset L we obtain a real number > 0 and a setU of subsets of FS(X) from Theorem 3.6. We can arrange that ≤ 1. ConsiderV :=U ∪V (t),whereV (t) is the collection of open subsets deﬁned before Lemma 3.5. We want to show thatV satisﬁes the conditions appearing in Deﬁnition 3.7 with respect to the number N := M + n − 1. Since the covering dimension ofU is less or equal to M by Theorem 3.6 (ii) and the covering dimension ofV (t) is less or equal to n − 2 by Lemma 3.5 (iii), the covering dimension ofU ∪V (t) is less or equal to N. SinceU andV (t) are GL (Z)-invariant by Theorem 3.6 and Lemma 3.5 (i),U ∪ V (t) is GL (Z)-invariant. Since each element ofU is an open VCyc-set by Theorem 3.6 (i) and each element ofV (t) is an openF -subset by Lemma 3.5 (ii), each element ofU ∪V (t) is an open F -subset, as VC yc ⊂F .Deﬁne n n S := c ∈ FS(X) |∃Z ∈U ∪V (t) with B (c) ⊆ Z . [−γ,γ ] This set S contains FS(X) ∪|V (t + β)| by the following argument. If c ∈|V (t + β)|, ≤γ then c ∈ S by Lemma 3.5(v). If c ∈ FS(X) and c ∈| /V (t + β)|,then c ∈ FS(X) and ≤γ ≤γ c(0) ∈ GL (Z) · L and hence c ∈ Sby Theorem 3.6 (iii). The subset S ⊆ FS(X) is GL (Z)-invariant becauseU ∪V (t) is GL (Z)-invariant. n n Next we prove that S is open. Assume that this is not the case. Then there exists c ∈ S and a sequence (c ) of elements in FS(X) − Ssuch that d (c, c )< 1/k holds k k≥1 FS(X) k for k ≥ 1. Choose Z ∈U ∪V (t) with B ( (c)) ⊆ Z. Since FS(X) is proper as metric [−γ,γ ] space by [3, Proposition 1.9] and B ( (c)) has bounded diameter, B ( (c)) [−γ,γ ] [−γ,γ ] is compact. Hence we can ﬁnd μ> 0withB ( (c)) ⊆ Z. We conclude from [3, +μ [−γ,γ ] Lemma 1.3] for all s ∈[−γ, γ ] |s| τ d (c), (c ) ≤ e · d (c, c )< e · 1/k. FS(X) s s k FS(X) k Hence we get for k ≥ 1 B (c ) ⊆ B (c) [−γ,γ ] k +e ·1/k [−γ,γ ] Since c does not belong to S, we conclude that B ( (c)) is not contained in k +e ·1/k [−γ,γ ] Z. This implies e · 1/k ≥ μ for all k ≥ 1, a contradiction. The GL (Z)-set FS(X) −|V (t + β)| is cocompact by Lemma 3.5 (iv). Since S is an open GL (Z)-subset of FS(X) and contains |V (t + β)|,the GL (Z)-set FS(X) − Sis n n cocompact. Hence we can ﬁnd a compact subset K ⊆ FS(X) − S satisfying GL (Z) · K = FS(X) − S. n K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 115 Obviously FS(X) ∩ GL (Z) · K =∅. ≤γ n Proof of Theorem 3.1.—Thegroup GL (Z) and the associated ﬂow space FS(X) satisfy [3, Convention 5.1] by the argument of [3, Section 6.2]. Notice that in [3,Conven- tion 5.1] it is not required that the action is cocompact. The argument of [3, Section 6.2] showing that there is a constant k such that the order of any ﬁnite subgroup of G is bounded by k uses that the action is cocompact. But such a number exists for GL (Z) as G n well, since GL (Z) is virtually torsion free (see [9, Exercise II.3 on p. 41]). Because of [3, Proposition 5.11] it sufﬁces to show that FS(X) admits longF - covers at inﬁnity and at periodic ﬂow lines in the sense of Deﬁnition 3.7 and admits contracting transfers in the sense of [3, Deﬁnition 5.9]. For the ﬁrst condition this has been done in Lemma 3.8, while the second condition follows from the argument given in [3, Section 6.4]. Proposition 3.9. —The K-theoretic FJC up to dimension 1 holds for GL (Z). Proof. — We proceed by induction over n.As GL (Z) is ﬁnite, the initial step of the induction is trivial. Since GL (Z) is transfer reducible overF by Theorem 3.1 it follows from [4,The- n n orem 1.1] that GL (Z) satisﬁes the K-theoretic FJC up to dimension 1 with respect to F . It remains to replaceF by the family VCyc. Because of the Transitivity Principle 0.5 n n it sufﬁces to show that each H ∈F satisﬁes the FJC up to dimension 1 (with respect to VCyc). Combining the induction assumption for GL (Z), k < n with well known inher- itance properties for direct products, exact sequences of groups and subgroups (see for example [1, Theorem 1.10, Corollary 1.13, Theorem 1.9]) it is easy to reduce the K- theoretic FJC up to dimension 1 for members ofF to the class of virtually poly-cyclic groups. Finally, for virtually poly-cyclic groups the FJC holds by [1]. 4. Strong transfer reducibility of GL (Z) In this section we will discuss the modiﬁcations needed to extend Proposition 3.9 to higher K-theory. The necessary tools for this extension have been developed by Weg- ner [24]. Theorem 4.1. — The group GL (Z) is strongly transfer reducible overF in the sense of [24, n n Deﬁnition 3.1]. Wegner proves in [24, Theorem 3.4] that CAT(0)-groups are strongly transfer reducible over VC yc. As GL (Z) does not act cocompactly on X, we cannot use Wegner’s result directly. However, in combination with Lemma 3.8 his method yields a proof of Theorem 4.1. 116 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING Proof of Theorem 4.1. — The only place where Wegner uses cocompactness of the action is when he veriﬁes the assumptions of [3, Theorem 5.7], see [24, Proof of Theo- rem 3.4]. We know by Lemma 3.8 that FS(X) admits longF -covers at inﬁnity and periodic ﬂow lines. Wegner cites [3, Section 6.3] for this assumption. In the cocompact setting the family can even be chosen to be VC yc. That the assumptions of [3, Convention 5.1], which are used implicitly in [3,The- orem 5.7], are satisﬁed has already been explained in the proof of Theorem 3.1. Theorem 4.2. —The K-theoretic FJC holds for GL (Z). Proof.—Theorem 4.1 together with [24, Theorem 1.1] imply that GL (Z) satisﬁes the K-theoretic FJC with respect to the familyF . Using the induction from the proof of Proposition 3.9 the familyF can be re- placed by VC yc. 5. Wreath products and transfer reducibility Our main result in this section is the following variation of [4, Theorem 1.1]. Theorem 5.1. —LetF be a family of subgroups of the group G and let F be a ﬁnite group. Denote byF the family of subgroups H of G F that contain a subgroup of ﬁnite index that is isomorphic to a subgroup of H ×···×H for some n and H ,..., H ∈F . 1 n 1 n (i) If G is transfer reducible overF , then the wreath product G F satisﬁes the K-theoretic FJC up to dimension 1 with respect toF and the L-theoretic FJC with respect toF ; (ii) If G is strongly transfer reducible overF , then G F satisﬁes the K-theoretic and L-theoretic FJC in all dimensions with respect toF . The idea of the proof of this result is very easy. We only need to show that G F is transfer reducible overF and apply [4, Theorem 1.1]. However, it will be easier to verify a slightly weaker condition for G F. Deﬁnition 5.2 (Homotopy S-action; [4, Deﬁnition 1.4]). — Let S be a ﬁnite subset of a group G (containing the identity element e ∈ G). Let X be a space. (i) A homotopy S-action (ϕ, H) on X consists of continuous maps ϕ : X → X for g ∈ S and homotopies H : X×[0, 1]→ X for g, h ∈ S with gh ∈ S such that H (−, 0) = g,h g,h ϕ ◦ ϕ and H (−, 1) = ϕ holds for g, h ∈ S with gh ∈ S. Moreover, it is required g h g,h gh that H (−, t) = ϕ = id for all t ∈[0, 1]; e,e e X (ii) For g ∈ S let F (ϕ, H) be the set of all maps X → X of the form x → H (x, t) where g r,s t ∈[0, 1] and r, s ∈ S with rs = g; K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 117 (iii) Given a subset A ⊂ G × X let S (A) ⊂ G × X denote the set ϕ,H −1 ga b, y |∃x ∈ X, a, b ∈ S, f ∈ F (ϕ, H), f ∈ F (ϕ, H) a b satisfying (g, x) ∈ A, f (x) = f (y) . n−1 n 1 Then deﬁne inductively S (A) := S (S (A)); ϕ,H ϕ,H ϕ,H (iv) Let (ϕ, H) be a homotopy S-action on X andU be an open cover of G × X.Wesay thatU is S-long with respect to (ϕ, H) if for every (g, x) ∈ G × X there is U ∈U |S| containing S (g, x) where |S| is the cardinality of S. ϕ,H We will use the following variant of [4, Deﬁnition 1.8]. Deﬁnition 5.3 (Almost transfer reducible). — Let G be a group andF be a family of subgroups. We will say that G is almost transfer reducible overF if there is a number N such that for any ﬁnite subset S of G we can ﬁnd (i) a contractible, compact, controlled N-dominated, metric space X ([3, Deﬁnition 0.2]), equipped with a homotopy S-action (ϕ, H) and (ii) a G-invariant coverU of G × X of dimension at most N that is S-long. Moreover we require that for all U ∈U the subgroup G := {g ∈ G | gU = U} belongs toF.(Here we use the G-action on G×X given by g · (h, x) = (gh, x).) The original deﬁnition for transfer reducibility requires in addition that gUand U are disjoint if U ∈U and g ∈ / G , in other words each U is required to be anF - subset. One can also drop this condition from the notion of “strongly transfer reducible” introduced in [24, Deﬁnition 3.1]. A group satisfying this weaker version will be called almost strongly transfer reducible. The result corresponding to [4, Theorem 1.1] (respectively [24, Theorem 1.1]) is as follows. Proposition 5.4. —LetF be a family of subgroups of a group G and letF be the family of subgroups of G that contain a member ofF as a ﬁnite index subgroup. (i) If G is almost transfer reducible overF , it satisﬁes the K-theoretic FJC up to dimension 1 with respect toF and the L-theoretic FJC with respect toF . (ii) If G is almost strongly transfer reducible overF,itsatisﬁesthe K-theoretic FJC with respect toF and the L-theoretic FJC with respect toF . Proof. — (i) The proof can be copied almost word for word from the proof of [4, Theorem 1.1]. The only difference is that we no longer know that the isotropy groups of the action of G on the geometric realization of the nerve ofU belong toF,and this action may no longer be cell preserving. But it is still simplicial and therefore we can 118 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING just replace by its barycentric subdivision by the following Lemma 5.5 provided we replaceF byF . (The precise place where this makes a difference is in the proof of [4, Proposition 3.9].) (ii) The L-theory part follows from part (i) since almost strongly transfer reducible implies almost transfer reducible. For the proof of the K-theory part we have to adapt Wegner’s proof of [24,The- orem 1.1]. The necessary changes concern [24, Proposition 3.6] and are similar to the changes discussed above, but as Wegner’s argument is somewhat differently organized we have to be a little more careful. First we observe that for any ﬁxed M > 0 the second assertion in [24, Proposition 3.6] can be strengthened to (5.1)Mk · d f (g, x), f (h, y) ≤ d (g, x), (h, y) for all(g, x), (h, y) ∈ G×X. ,S,k, To do so we only need to set n := M · 4Nk instead of 4Nk in the second line of Wegner’s proof; then k can be replaced by M · k in the denominator of the ﬁnal expression on p.786. (Of course X, , , and f depend now also on M.) Then we can replace by its barycentric subdivision . Using Lemma 5.5 we conclude from (5.1) with sufﬁciently large M > 0that • d (f (g, x), f (h, y)) ≤ for all (g, x), (h, y) ∈ G×X satisfying the inequality d ((g, x), (h, y)) ≤ k. ,S,k, This assertion still guarantees that the maps (f ) induce a functor as needed on the right hand side of the diagram on p. 789 of [24]. With this change Wegner’s argument proves the K-theory part of (ii). (Alternatively one can strengthen Lemma 5.5 below and check that the l -metric under barycentric subdivision changes only up to Lipschitz equivalence. Thus [24,Propo- sition 3.6] remains in fact true for .) F be the family of subgroups of G.LetF be the Lemma 5.5. —Let G be a group, and let family of subgroups of G that contain a member ofF as a ﬁnite index subgroup. Let be a simplicial complex with a simplicial G-action such that the isotropy group of each vertex is contained inF.Let 1 1 1 1 be the barycentric subdivision. Denote by d the l -metric on and by d the l -metric on (i) The group G acts cell preserving on . All isotropy groups of lie inF . In particular is a G-CW-complex whose isotropy groups belong toF ; (ii) Given a number > 0 and a natural number N, there exists a number > 0 depending only on and N such that the following holds: If dim( ) ≤ N and x, y ∈ satisfy 1 1 d (x, y)<,then d (x, y)< . Proof. — The elementary proof of assertion (i) is left to the reader. The proof of assertion (ii) is an obvious variation of the proof of [4, Lemma 9.4 (ii)]. Namely, any simplicial complex can be equipped with the l -metric. With those metrics the inclusion of K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 119 a subcomplex is an isometric embedding. Let be the simplicial complex with the same vertices as , such that any ﬁnite subset of the vertices spans a simplex. The inclusions i : and i : are isometric embeddings. The images i(x), i(y) of any two points x, y ∈ are contained in a closed simplex of dimension at most 2N + 1. Thus it sufﬁces to consider the case where is replaced by the standard (2N + 1)-simplex. A compactness argument gives the result in that case. Remark 5.6. —Often thefamilyF is closed under ﬁnite overgroups. In this case F =F and it is really easier to work with the weaker notion of almost transfer reducible instead of transfer reducible. However, there are situations in which a family that is not closed under ﬁnite overgroups is important. For example, in [10] the family of virtually cyclic groups of type I is considered. Let G be a group and F be ﬁnite group. We will think of elements of the F-fold F F F product G as functions g : F → G. For a ∈ F, g ∈ G we write l (g) ∈ G for the function b → g(ba); this deﬁnes a left action of F on G and the corresponding semi-direct product is the wreath product G F with multiplication gag a = gl (g )aa for g, g ∈ G and a, a ∈ F. If G acts on a set X, then we obtain an action of G FonX . In formulas this action is given by (g · x)(b) := g(b) · x(b); (a · x)(b) := x(ba), and hence (ga · x)(b) = g(b) · x(ba) F F for g ∈ G , x ∈ X and a, b ∈ F. We will sometimes also write l (x) for a · x.Let now S ⊂ G and (ϕ, H) be a homotopy S-action on a space X. Set S F := {sa | s ∈ S , a ∈ F}⊂ G F. Then we obtain a homotopy S F-action (ϕ, ˆ H) on X . In formulas this is given by ϕ ˆ (x) (b) := ϕ x(ba) ; sa s(b) H (x, t) (b) := H x baa , t , sa,s a s(b),s (ba) for t ∈[0, 1], s, s ∈ S , a, a , b ∈ Fwith sas a = sl (s )aa ∈ S F. Hence (sl (s )(b) = a a s(b)s (ba) ∈ S and the right hand side is deﬁned. It is easy to check that if X is a con- tractible, compact, controlled N-dominated, metric space, then the same is true for X provided we replace N by N ·|F|. Proof of Theorem 5.1. — Let us postpone the “strong”-case until the end of the proof. Because of Proposition 5.4 it sufﬁces to show that G F is almost transfer reducible over F .Let S be a ﬁnite subset of G F. By enlarging it we can assume that it has the form S F 120 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING for some ﬁnite subset S ⊂ G. Pick a ﬁnite subset S ⊂ Gsuch thatS ⊂ S and |S |≥|S F|. (If G is ﬁnite, then G F ∈F and there is nothing to prove.) As G is transfer reducible and hence in particular almost transfer reducible, there is a number N, (depending only on G, not on S or S) a compact, contractible, controlled N-dominated, metric space X, a homotopy S -action (ϕ, H) on X, and a G-invariant S -long open coverU of G×Xof dimension at most N such that for all U ∈U we have G ∈F . As pointed out before, X is a compact, contractible, controlled N ·|F|-dominated, metric space. For u : F →U , u F F let V := {(g, x) ∈ G ×X | (g(b), x(b)) ∈ u(b) for all b ∈ F}.Weobtainanopencover u F F |F| V := {V | u : F →U } of G ×X of dimension at most (N + 1) − 1. This cover is invariant for the G F-action deﬁned by g · (h, x) (b) := g(b)h(b), x(b) ; a · (h, x) (b) := h(ba), x(ba) , F F F for g, h ∈ G , x ∈ X and a, b ∈ F. As we have (G ) u = G , it follows that (G V u(b) b∈F −1 F F) ∈F for all V ∈V . Now we pull backV to a coverU := {p (V) | V ∈V } of G F×X F F F along the G F-equivariant map p : G F×X → G ×X , (ga, x) → (g, a · x).Here G F F F F operates on G F × X via left multiplication on the ﬁrst factor and on G × X via the operation deﬁned above. The deﬁnition of the homotopy S F-action on X gives H (−, t) = H (−, t) ◦ l = l ◦ H (−, t) sa,s a s,l (s ) aa aa l (s),l (s ) a −1 −1 −1 a a a for s, s ∈ S , a, a ∈ F. For ¯ s := sl (s ), a ¯ := aa we have ¯ sa ¯ = sas a and consequently F F F F (5.2)F (ϕ, ˆ H) = F (ϕ ˆ| , H| ) ◦ l = l ◦ F (ϕ ˆ| , H| ). ¯ sa ¯ ¯ s G G a ¯ a ¯ l (¯ s) G G −1 a ¯ 1 F F Let us insert this into the deﬁnition of (S F) (hb, x) with h ∈ G , b ∈ F, x ∈ X . ϕ, ˆ H 1 F Pick any (h b , x ) ∈ (S F) (hb, x) with h ∈ G , b ∈ F. Then there are elements ϕ, ˆ H s, s ∈ S , a, a ∈ Fand f ∈ F (ϕ, ˆ H), f ∈ F (ϕ, ˆ H) such that sa s a −1 f (x) = f x , and h b = hb(sa) s a . ¯ ¯ F F Using the ﬁrst equality in (5.2)weﬁnd f ∈ F (ϕ ˆ| , H| ) such that f = f ◦ l . Using s G G a F F −1 −1 the second equality in (5.2)weﬁnd f ∈ F (ϕ ˆ| , H| ) such that f ◦ l = l ◦ f . l (s) G G ab ab −1 ba −1 −1 −1 Then f ◦ l = l ◦ f ; equivalently l ◦ f = f ◦ l . Similarly, using both equations b ab ba ˜ ˜ ˜ F F −1 −1 in (5.2) again, we ﬁnd f ∈ F (ϕ ˆ| , H| ) such that l ◦ f = f ◦ l . l (s ) G G ba ba a −1 ba F 1 We claim that p(h b , x ) belongs to (S ) (p(hb, x)).Wehave p(h b , x ) = ϕ ˆ| ,H| F F G G −1 −1 (h , b · x ) and p(hb, x) = (h, b · x).From h b = hb(sa) s a we conclude h = hl −1 (s s ) ba −1 and b = ba a . Now the equations −1 ˜ ˜ ˜ f (b · x) = l −1 f (x) = l −1 f x = f ba a · x = f b x ba ba K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 121 −1 −1 h = hl −1 s s = h l −1 (s) l −1 s ba ba ba prove our claim. Summarizing we have shown that for any A ⊂ G F × X we have 1 F p (S F) (A) ⊂ S p(A) . ϕ, ˆ H ϕ ˆ| ,H| F F G G By induction then for all n n F p (S F) (A) ⊂ S p(A) . ϕ, ˆ H ϕ ˆ| ,H| F F G G F F Since the S -homotopy action on X is deﬁned componentwise we have F n S (g, x) ⊂ S g(a), x(a) . ϕ,H ϕ ˆ| ,H| K K G G a∈F Recall the deﬁnition of S from the beginning of the proof. Since the coverU of G × Xis |S | S -long, thereisfor each a ∈ Fa u(a) ∈U with (S ) (h(a), x(ab)) ⊂ u(a).Thusweobtain ϕ,H |SF| |SF| |SF| p (S F) (hb, x) ⊂ S (h, b · x) ⊂ S h(a), x(ab) ϕ,H ϕ, ˆ H ϕ ˆ| ,H| F F G G a∈F |S | ⊂ S h(a), x(ab) ⊂ u(a) = V . ϕ,H a∈F a∈F |SF| −1 u So (S F) (hb, x) ⊂ p (V ). Hence the coverU is S F-long. ϕ, ˆ H If X is equipped with a strong homotopy action (see [24, Section 2]), we obtain a strong homotopy action on X .Informulasitisgiven by (g a , t ,..., t , g a , x)(b) n n n 1 0 0 := g (b), t , g (ba ), t , g (ba a ), ..., n n n−1 n n−1 n−2 n n−1 g (ba ... a ), x(ba ... a ) 0 n 1 n 0 F F for g ,..., g ∈ G , a ,..., a , b ∈ F, t ,..., t ∈[0, 1], x ∈ X .Wealsohavehere 0 n 0 n 1 n (g a , t ,..., t , g a , −) n n n 1 0 0 = g , t , l (g ), t , l (g ), t ,..., l (g ), − ◦ l n n a n−1 n−1 a a n−2 n−2 a ...a 0 a ...a n n n−1 n 1 n 0 = l ◦ l −1 g , t , l −1 (g ), t ,..., l −1 (g ), − . a ...a (a ...a ) n n (a ...a ) n−1 n−1 0 n 0 n 0 n−1 0 a For g¯ := g l (g )l (g )... l (g ) ∈ G , a ¯ := a ... a ∈ F, n ∈ N we have g¯a ¯ = n a n−1 a a n−2 a ...a 0 n 0 n n n−1 n 1 g a g a ... g a and consequently we have analogously to (5.2) n n n−1 n−1 0 0 F F F F F (, S F, n) = F | , S , n ◦ l = l ◦ F | , S , n . g¯a ¯ g¯ a ¯ a ¯ l (g¯) G G −1 a ¯ With this observation the proof can be carried out in exactly the same way as in the case of a homotopy action. 122 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING Remark 5.7. — The proof of Theorem 5.1 given above only uses that G is almost (strongly) transfer reducible overF , not that G is (strongly) transfer reducible. Conse- quently, Theorem 5.1 remains true if we replace the assumption “(strongly) transfer re- ducible” by the weaker assumption “almost (strongly) transfer reducible”. 6. The Farrell-Jones conjecture with wreath products Deﬁnition 6.1. —Agroup G is said to satisfy the L-theoretic Farrell-Jones Conjecture with wreath products with respect to the familyF if for any ﬁnite group F the wreath product G F satisﬁes the L-theoretic Farrell-Jones Conjecture with respect to the familyF (in the sense of Deﬁnition 0.2). If the familyF is not mentioned, it is by default the family VCyc of virtually cyclic subgroups. There are similar versions with wreath products of the K-theoretic Farrell-Jones Conjecture and the K-theoretic Farrell-Jones Conjecture up to dimension 1. The FJC with wreath products has ﬁrst been used in [15] to deal with ﬁnite exten- sions, see also [23, Deﬁnition 2.1]. Remark 6.2. — The inheritance properties of the FJC for direct products, sub- groups, exact sequences and directed colimits hold also for the FJC with wreath products and can be deduced from the corresponding properties of the FJC itself. See for exam- ple [19, Lemma 3.2, 3.15, 3.16, Satz 3.5]. Foragroup G andtwo ﬁnitegroups F and F we have (H F ) F ⊂ H (F F ) 1 2 1 2 1 2 and F F is ﬁnite. In particular, if G satisﬁes the FJC with wreath products, then the 1 2 same is true for any wreath product G F with F ﬁnite. The main advantage of the FJC with wreath products is that in addition it passes to overgroups of ﬁnite index. Let G be an overgroup of G of ﬁnite index, i.e., G ⊂ G , [G : G] < ∞. Let S denote a system of representatives of the cosets G /G. Then −1 N := sGs is a ﬁnite index, normal subgroup of G .Now G can be embedded in s∈S N G /N (see [11, Section 2.6], [15, Section 2]). This implies that G satisﬁes the FJC with wreath products, because N G /N does by the inheritance properties discussed before. Theorem 6.3. —The L-theoretic FJC with wreath products holds for GL (Z). Proof. — We proceed by induction over n.As GL (Z) is ﬁnite, the induction begin- ning is trivial. Let F be a ﬁnite group. Since GL (Z) is transfer reducible overF by Theorem 3.1 n n it follows from Theorem 5.1 (i) that GL (Z) F satisﬁes the L-theoretic FJC with respect to (F ) . It remains to replace (F ) by the family VCyc. By the Transitivity Principle 0.5 n n it sufﬁces to prove the L-theoretic FJC (with respect to VCyc) for all groups H ∈ (F ) . Because the FJC with wreath products passes to products and ﬁnite index over- groups, see Remark 6.2 it sufﬁces to consider H ∈F . n K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 123 Combining the induction assumption for GL (Z), k < n with the inheritance prop- erties for direct products, exact sequences of groups and subgroups (see Remark 6.2)itis easy to reduce the FJC with wreath products for members ofF to the class of virtually poly-cyclic groups. Wreath products of virtually poly-cyclic groups with ﬁnite groups are again virtually poly-cyclic. Thus the result follows in this case from [1]. Because for ﬁnite F the wreath product GL (Z) F can be embedded into GL (Z) n m for some m > n, there is really no difference between the FJC and the FJC with wreath products for the collection of groups GL (Z), n ∈ N. Nevertheless, as discussed in the introduction, for L-theory the induction only works for the FJC with wreath products. Remark 6.4. — We also conclude that a hyperbolic group G satisﬁes the K- and L-theoretic FJC with wreath products. For K-theory (without ﬁnite wreath products) this has already been proved in [5]. A hyperbolic group is strongly transfer reducible over VC yc by [24, Example 3.2] and in particular transfer reducible over VC yc. Hence it satisﬁes the K-theoretic FJC in all dimensions and the L-theoretic FJC with respect to the family VC yc by Theorem 5.1 (ii). By the transitivity principle 0.5 it sufﬁces to show the FJC for all groups from VC yc . Since those groups are virtually polycyclic the FJC holds for them by [1, Theorem 0.1]. Notice that a group is hyperbolic if a subgroup of ﬁnite index is hyperbolic. Never- theless, it is desirable to have the wreath product version for hyperbolic groups also since it inherits to colimits of hyperbolic groups and many constructions of groups with exotic properties occur as colimits of hyperbolic groups. 7. Proof of the general theorem Lemma 7.1. —Let R be a ring whose underlying abelian group is ﬁnitely generated. Then both the K-theoretic and the L-theoretic FJC hold for GL (R) and SL (R). n n Proof. — Since the FJC passes to subgroups (by [7]) we only need to treat GL (R). n k Choose an isomorphism of abelian groups h : R − → Z × T for some natural number k and a ﬁnite abelian group T. We obtain an injection of groups f h n n k GL (R) aut R − → aut R − → aut Z × T , n R Z Z where f is the forgetful map and h comes by conjugation with h . Since the FJC passes to subgroups, it sufﬁces to prove the FJC for aut (Z × T). There is an obvious exact sequence of groups k k 1 → hom Z , T → aut Z × T → GL (Z) × aut (T) → 1. Z Z k Z Since hom (Z , T) and aut (T) are ﬁnite and GL (Z) satisﬁes the FJC, Lemma 7.1 fol- Z Z k lows from [1, Corollary 1.12]. 124 ARTHUR BARTELS, WOLFGANG LÜCK, HOLGER REICH, AND HENRIK RÜPING Proof of General Theorem. — Let G be a group which is commensurable to a subgroup H ⊆ GL (R) for some natural number n. We have to show that G satisﬁes the FJC with wreath products. We have explained in Remark 6.2 that the FJC with wreath products passes to overgroups of ﬁnite index and all subgroups. Therefore it sufﬁces to show that the FJC with wreath products holds for GL (R). Consider a ﬁnite group H. Let R H be the twisted group ring of H with coefﬁcients in R where the H-action on R is given by permuting the factors. Since R is ﬁnitely H H generated as abelian group by assumption, the same is true for R H. Hence GL (R H) satisﬁes the FJC by Lemma 7.1. There is an obvious injective group homomorphism GL (R) H = GL (R) n n H H H → GL (R H). It extends the obvious group monomorphism GL (R) = GL (R ) → n n n GL (R H) via H → GL (R H), h → h · I . Since the FJC passes to subgroups it holds n n n for GL (R) H. Acknowledgements We are grateful to Dan Grayson for fruitful discussions about his paper [16]. We also thank Enrico Leuzinger for answering related questions. The work was ﬁnancially supported by SFB 878 Groups, Geometry and Actions in Mün- ster, the HCM (Hausdorff Center for Mathematics) in Bonn, and the Leibniz-Preis of the second author. Parts of the paper were developed during the Trimester Program Rigidity at the HIM (Hausdorff Research Institute for Mathematics) in Bonn in the fall of 2009. REFERENCES 1. A. BARTELS,T.FARRELL,and W. LÜCK, The Farrell-Jones conjecture for cocompact lattices in virtually connected Lie groups. arXiv:1101.0469v1 [math.GT], 2011. 2. A. BARTELS and W. LÜCK, On twisted group rings with twisted involutions, their module categories and L-theory, in Cohomology of Groups and Algebraic K-Theory. Advanced Lectures in Mathematics, vol. 12, pp. 1–55, International Press, Somerville, 2009. 3. A. BARTELS and W. LÜCK, Geodesic ﬂow for CAT(0)-groups, Geom. Topol., 16 (2012), 1345–1391. 4. A. BARTELS and W. LÜCK, The Borel conjecture for hyperbolic and CAT(0)-groups, Ann. Math. (2), 175 (2012), 631– 5. A. BARTELS,W.LÜCK,and H. REICH, The K-theoretic Farrell-Jones conjecture for hyperbolic groups, Invent. Math., 172 (2008), 29–70. 6. A. BARTELS,W.LÜCK,and H. REICH, On the Farrell-Jones conjecture and its applications, Topolog y, 1 (2008), 57–86. 7. A. BARTELS and H. REICH, Coefﬁcients for the Farrell-Jones conjecture, Adv. Math., 209 (2007), 337–362. 8. M. R. BRIDSON and A. HAEFLIGER, Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999. Die Grundlehren der mathematischen Wissenschaften, Band 319. 9. K. S. BROWN, Cohomology of Groups. Graduate Texts in Mathematics, vol. 87. Springer, New York, 1982. 10. J. F. DAVIS,F.QUINN,and H. REICH, Algebraic K-theory over the inﬁnite dihedral group: a controlled topology ap- proach, Topolog y, 4 (2011), 505–528. 11. J. D. DIXON and B. MORTIMER, Permutation Groups. Graduate Texts in Mathematics, vol. 163. Springer, New York, 12. P. B. EBERLEIN, Geometry of Nonpositively Curved Manifolds, University of Chicago Press, Chicago, 1996. 13. F. T. FARRELL andL.E.JONES, Isomorphism conjectures in algebraic K-theory, J. Am. Math. Soc., 6 (1993), 249–297. K- AND L-THEORY OF GROUP RINGS OVER GL (Z) 125 14. F. T. FARRELL andL.E.JONES, Rigidity for aspherical manifolds with π ⊂ GL (R), Asian J. Math., 2 (1998), 215–262. 1 m 15. F. T. FARRELL andS.K.ROUSHON, The Whitehead groups of braid groups vanish, Int. Math. Res. Not., 10 (2000), 515–526. 16. D. R. GRAYSON, Reduction theory using semistability, Comment. Math. Helv., 59 (1984), 600–634. 17. P. GRIFFITHS and J. HARRIS, Principles of Algebraic Geometry, Wiley-Interscience/Wiley, New York, 1978 Pure and Applied Mathematics. 18. S. HELGASON, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, vol. 80, Academic Press/Harcourt Brace Jovanovich, New York, 1978. 19. P. KÜHL, Isomorphismusvermutungen und 3-Mannigfaltigkeiten. Preprint, arXiv:0907.0729v1 [math.KT], 2009. 20. W. LÜCK, Survey on classifying spaces for families of subgroups, in Inﬁnite Groups: Geometric, Combinatorial and Dynamical Aspects. Progress in Mathematics, vol. 248, pp. 269–322, Birkhäuser, Basel, 2005. 21. W. LÜCK and H. REICH, The Baum-Connes and the Farrell-Jones conjectures in K- and L-theory, in Handbook of K-Theory, vols.1,2, pp. 703–842, Springer, Berlin, 2005. 22. J. NEUKIRCH, Algebraic Number Theory, Springer, Berlin, 1999 Translated from the 1992 German original and with a note by N. Schappacher, With a foreword by G. Harder. 23. S. K. ROUSHON, The Farrell-Jones isomorphism conjecture for 3-manifold groups, K-Theory, 1 (2008), 49–82. 24. C. WEGNER, The K-theoretic Farrell-Jones conjecture for CAT(0)-groups, Proc.Am. Math.Soc., 140 (2012), 779–793. A. B. Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 60, 48149 Münster, Germany bartelsa@math.uni-muenster.de W. L.,H.R. Mathematisches Institut, Rheinische Wilhelms-Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany wolfgang.lueck@him.uni-bonn.de H. R. henrik.rueping@hcm.uni-bonn.de H. R. Institut für Mathematik, Freie Universität Berlin, Arnimallee 7, 14195 Berlin, Germany holger.reich@fu-berlin.de Manuscrit reçu le 11 avril 2012 Manuscrit accepté le 6 mai 2013 publié en ligne le 25 mai 2013.

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K- and L-theory of group rings over GL n (Z)

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