Computations of K-and L-Theory of Cocompact Planar Groups

by Wolfgang Lück and Roland Stamm ^{∗}

Abstract

The verification of the isomorphism conjectures of Baum & Connes and Farrell & Jones for certain classes of groups is used to compute the algebraic K- and L-theory and the topological K-theory of cocompact planar groups (= cocompact N.E.C-groups) and of groups G appearing in an extension 1 → ℤ^{n} → G → π → 1 where π is a finite group and the conjugation π-action on ℤ^{n} is free outside 0 ∈ ℤ^{n}. These computations apply for instance to two-dimensional crystallographic groups and cocompact Fuchsian groups. Key words: K-theory, L-theory, planar groups 1991 mathematics subject classification: 19B28, 19D50, 19G24, 19K99

Introduction

The goal of this paper is to compute the algebraic K-groups K_{p}(ℤG) for p ≤ 1, the algebraic L-groups L_{p}(ℤG) for p ∈ ℤ (mostly after inverting 2) of the integral group ring ℤG and the topological K-groups K_{p}(C_{r}^{∗}(G)) for p ∈ ℤ of the reduced C^{∗}-algebra C_{r}^{∗}(G) for certain infinite (discrete) groups G. Namely, we assume that G is either a cocompact planar group or that there is an exact sequence 1 → ℤ^{n} → G → π → 1, where π is a finite group and the conjugation action of π on ℤ^{n} is free outside 0 ∈ ℤ^{n}. A cocompact planar group is a discontinuous group of isometries of S^{2}, ℝ^{2} or ℍ^{2} with compact quotient. More information about these groups and the result of the explicit computations will be given in Section 4 (see Theorem 4.5, Theorem 4.31).

For our techniques to work it is crucial to have very good information on the structure of the finite subgroups – as well as their normalizers – and the infinite virtually cyclic subgroups of cocompact planar groups. More explicitly, we use that all maximal finite subgroups are either cyclic or dihedral, and that a common subgroup of any two different maximal finite subgroups has at most 2 elements. Furthermore, the normalizer N_{G}M of a maximal finite subgroup M satisfies N_{G}M = M, except if M is generated by a single reflection, in which case N_{G}M ≅ ℤ×ℤ/2. These exceptions are responsible for the summands indexed by B and B′′ in Theorem 4.5, wheras the other maximal finite subgroups correspond to the summands indexed by A. Finally, we have a complete list of the infinite virtually cyclic subgroups of a cocompact planar group; they are given as subgroups of very simple amalgams. This will allow us to reduce the computations in algebraic K- and L-theory from the family of virtually cyclic subgroups to the family of finite subgroups. All the facts about cocompact planar groups mentioned above will be recollected in Theorem 4.3 and Lemma 4.6 and 4.7.

∗Roland Stamm is supported by the Graduiertenkolleg “Algebraische Geometrie und Zahlentheorie” in Münster. Parts of this paper grew out of his Dissertation.

Examples of cocompact planar groups are cocompact Fuchsian groups. Next we give the result in this comparatively easy case as an illustration.

Theorem 1 Let F be a cocompact Fuchsian group with presentation

F = ⟨a_{1}, b_{1}, …, a_{g}, b_{g}, c_{1}, …, c_{t} | c_{1}^{γ_{1}} = … = c_{t}^{γ_{t}} = c_{1}^{−1} ⋯ c_{t}^{−1}[a_{1}, b_{1}] ⋯ [a_{g}, b_{g}] = 1⟩

for integers g, t ≥ 0 and γ_{i} > 1. Then

(a) The inclusions of the maximal subgroups ℤ/γ_{i} = ⟨c_{i}⟩ induce an isomorphism

⨁_{i=1}^{t} Wh_{q}(ℤ/γ_{i}) ⟶^{≅} Wh_{q}(F)

for q ≤ 1. If the isomorphism conjecture for algebraic K-theory (see 1.9 and Theorem 1.10) holds for F also in dimensions q ≥ 2, then this is an isomorphism for all q ∈ ℤ. (Information about Wh_{q}(ℤ/γ_{i}) is given in Theorem 3.2 (d);

(b) There are isomorphisms

L_{q}(ℤF)[1/2] ≅ \{ \begin{matrix}(1 + ∑_{i=1}^{t} [\frac{γ_{i}}{2} ]) · ℤ[1/2] & q ≡ 0 (4); \\ (2g) · ℤ[1/2] & q ≡ 1 (4); \\ (1 + ∑_{i=1}^{t} [\frac{γ_{i}−1}{2}]) · ℤ[1/2] & q ≡ 2 (4); \\ 0 & q ≡ 3 (4), \end{matrix}

where [r] for r ∈ ℝ denotes the largest integer less than or equal to r.

(c) There are isomorphisms

K_{q}(C_{r}^{∗}(F)) ≅ \{ \begin{matrix}(2 + ∑_{i=1}^{t}(γ_{i} − 1)) · ℤ & q = 0; \\ (2g) · ℤ & q = 1. \end{matrix}

We will give more information about the algebraic L-theory of cocompact Fuchsian groups without inverting 2 in Remark 4.32. The algebraic K-theory in dimensions ≤ 1 of cocompact Fuchsian groups has been computed in [7].

Other examples of cocompact planar groups are two-dimensional crystallographic groups. Their K- and L-theory is explicitly computed in Section 5. For a two-dimensional crystallographic group G the algebraic K-theory of ℤG in dimensions ≤ 1 has already been determined in [32], and the topological K-theory of C_{r}^{∗}(G) in [51].

In Section 6 we will prove a result which is similar to that of Theorem 4.5 but applies to certain virtually abelian groups groups whose classifying space for the family of finite subgroups \underline{E}G = E(G, FIN) is of higher dimension. (For an infinite cocompact planar group G, a model for \underline{E}G is ℝ^{2} or ℍ^{2} with the obvious G-action, and for the groups G appearing below, ℝ^{n} with a certain G-action is a model for \underline{E}G.)

Theorem 2 Let 1 → ℤ^{n} → G → π → 1 be a group extension for a finite group π such that the conjugation action of π on ℤ^{n} is free, i.e. the only element in π with a fixed point in ℤ^{n} different from zero is the identity element in π. Let {M_{α} | α ∈ A} be a complete system of representatives of conjugacy classes of maximal finite subgroups of G. Then

(a) The natural map induced by the inclusions of subgroups

⨁_{α∈A} Wh_{q}(M_{α}) → Wh_{q}(G)

is an isomorphism for q ≤ 1, and K_{q}(ℤG) is trivial for q ≤ −2.

If the isomorphism conjecture in algebraic K-theory (see 1.9 and Theorem 1.10) holds also for q ≥ 2, then the map above is an isomorphism for all q ∈ ℤ;

(b) There are short exact sequences

0 → ⨁_{α∈A} L̃_{q}(ℤM_{α})[1/2] → L_{q}(ℤG)[1/2] → H_{q}(G\\underline{E}G; L(ℤ))[1/2] → 0,

where L(ℤ) is the L-theory spectrum associated to the ring ℤ, H_{∗}(−; L(ℤ)) is the associated homology theory and the first map is induced by the various inclusions M_{α} → G.

If we invert 2|π|, this sequence splits and we obtain isomorphisms

(⨁_{α∈A} L̃_{q}(ℤM_{α}) [\frac{1}{2|π|}]) ⊕ H_{q}(π\T^{n}; L(ℤ)) [\frac{1}{2|π|}] ⟶^{≅} L_{q}(ℤG) [\frac{1}{2|π|}] ,

where the π-action on T^{n} is induced by the conjugation action of π on ℤ^{n}.

(c) There are short exact sequences

0 → ⨁_{α∈A} K̃_{q}(C_{r}^{∗}(M_{α})) → K_{q}(C_{r}^{∗}(G)) → K_{q}(G\\underline{E}G) → 0,

where K_{q}(G\\underline{E}G) is the topological complex K-homology of G\\underline{E}G and the first map is induced by the various inclusions M_{α} → G.

If we invert |π|, this sequence splits and we obtain isomorphisms

(⨁_{α∈A} K̃_{q}(C_{r}^{∗}(M_{α})) [\frac{1}{|π|}]) ⊕ K_{q}(π\T^{n}) [\frac{1}{|π|}] ⟶^{≅} K_{q}(C_{r}^{∗}(G)) [\frac{1}{|π|}] .

We will present more detailed information on the L-theory (without inverting 2) of groups as in Theorem 2 in Remark 6.5. Furthermore, we can generalize the methods of Theorem 1 and 2 to yield similar results for groups which are given as extensions of the form 1 → ℤ^{n} → G → π → 1, where π ≅ D_{2m} is a dihedral group of order 2m such that the subgroup ℤ/m acts freely on ℤ^{n}. See Remark 6.6 for more information on this.

Our computations use the isomorphism conjectures in K-and L-theory due to Farrell and Jones and to Baum and Connes which are known to be true for the groups we consider here. We exploit the unified treatment of these conjectures of [15]. Thus the computation is reduced to the investigation of the homology of certain spaces over the orbit category with coefficients in K- and L-spectra over the orbit category which will be carried out by homological methods, mainly Mayer-Vietoris sequences. There are various spectral sequences to compute these homology groups but they turn out to be too complicated even for the relatively elementary groups we consider here. It seems to be very hard to compute these Kand L-groups integrally (or after inverting 2) for more general groups even if one assumes in the K-theory case that one does know the K-theory of integral group rings of finite subgroups. Rationally these computations can be done via Chern characters and lead to rather general and explicit formulas, since the existence of the Chern characters guarantees that the relevant spectral sequences collapses [27]. The integral computations of K- and Lgroups presented here exploit the explicit knowledge and special properties of the virtually cyclic and finite subgroups and their normalizers of the groups under consideration (see Lemma 4.6 and Lemma 6.3).

The paper is organized as follows:

1. Review of the Isomorphism Conjectures in K- and L-Theory
2. Preliminaries about Spaces over the Orbit Category
3. Preliminary Computations of K- and L-Groups of Finite Groups
4. Cocompact Planar Groups
5. Two-Dimensional Crystallographic Groups
6. Extensions of Finite Groups with a Free Abelian Group

References We thank the referee for his detailed and very helpful report.

1 Review of the Isomorphism Conjectures in K- and L-Theory

We want to review the isomorphism conjectures in K- and L-theory as far as we will need here. Since we want to do this in the language of spaces and spectra over a category we give some basic facts about these notions. More information can be found for instance in [15], [18].

Given a (discrete) group G, a family F of subgroups is a set of subgroups of G closed under taking subgroups and conjugates. Our main examples for families will be the families TR, FIN, VC and ALL respectively consisting of the trivial subgroup, finite subgroups, virtually cyclic subgroups and all subgroups respectively. Recall that G is virtually cyclic if G is finite or contains ℤ as subgroup of finite index. The orbit category Or(G, F) of G with respect to F has as objects homogeneous spaces G/H with H ∈ F and as morphisms G-maps. If F is the family ALL of all subgroups, we abbreviate Or(G, ALL) by Or(G).

A contravariant (pointed) Or(G)-space is a contravariant functor from Or(G) to the category of (pointed) spaces. A morphism between contravariant (pointed) Or(G)-spaces is a natural transformation. A G-space X defines a contravariant Or(G)-space by assigning to G/H its H-fixed point set X^{H} = map_{G}(G/H, X). A covariant Or(G)-spectrum is a covariant functor from the category Or(G) into the category Spectra of spectra. An object E in Spectra is a sequence of spaces (E_{n})_{n∈ℤ} together with structure maps σ_{n} : ΣE_{n} → E_{n+1} for each n ∈ ℤ. A map f : E → F of spectra is a sequence (f_{n} : E_{n} → F_{n})_{n∈ℤ} of maps satisfying f_{n+1}∘σ_{n}^{E} = σ_{n}^{F}∘Σf_{n} for all n ∈ ℤ. The q-th homotopy group π_{q}(E) of a spectrum E for q ∈ ℤ is the colimit colim_{n→∞} π_{q+n}(E_{n}) with respect to the obvious maps π_{q+n}(E_{n}) → π_{q+n+1}(E_{n+1}) induced by the structure maps σ_{n} and the suspension homomorphisms. Next we review our main examples of covariant Or(G)-spectra.

Let Groupoids be the category of groupoids. Let Groupoids^{inj} be the subcategory of Groupoids which has the same objects as Groupoids and as morphisms covariant functors F : G_{0} → G_{1} which are faithful, i.e. for any two objects x, y in G_{0} the induced map mor_{G_{0}}(x, y) ⟶ mor_{G_{1}}(F (x), F (y)) is injective. A left G-set S defines a groupoid G^{G}(S), where Ob(G^{G}(S)) = S and mor(s, t) = {g ∈ G | gs = t} for s, t ∈ S. The composition law is given by group multiplication. Obviously a map of left G-sets defines a functor of the associated groupoids. The category G(G/H) is equivalent to the groupoid associated with H which has one object and H as set of morphisms, and hence G(G/H) can serve as a substitute for the subgroup H. Thus we obtain a covariant functor

G^{G} : Or(G) → Groupoids^{inj} .  (1.1)

In [15, section 2] covariant functors

K^{alg} : Groupoids ⟶ Spectra ;
L : Groupoids ⟶ Spectra ;
K^{top} : Groupoids^{inj} ⟶ Spectra

are constructed using [34] and [40] in the algebraic context. (Unfortunately there is a problem in the actual construction of K^{top} concerning the pairing μ in [15, page 217] which will be corrected elsewhere. This does not affect the results of this paper.) We denote their composition with G^{G} by the same letters or by the following abbreviations and obtain covariant functors

K = K_{G}^{alg} : Or(G) ⟶ Spectra ;  (1.2)
L = L_{G} : Or(G) ⟶ Spectra ;  (1.3)
K = K_{G}^{top} : Or(G) ⟶ Spectra .  (1.4)

Notice that π_{q}(L(G/H)) ≅ L_{q}(ℤH), π_{q}(K^{alg}(G/H)) ≅ K_{q}(ℤH) and π_{q}(K^{top}(G/H)) ≅ K_{q}(C_{r}^{∗}(H)), where C_{r}^{∗}(H) is the reduced C^{∗}-algebra of the group H and K_{q}(C_{r}^{∗}(H)) denotes its topological K-theory. Functoriality for a G-map G/H → G/K, g′H ↦ g′g^{−1}K corresponds under this isomorphism to induction with respect to the injective homomorphism H → K given by h ↦ ghg^{−1}.

If ∗ denotes the trivial groupoid consisting of one morphism, there is for any groupoid G the canonical projection pr : G → ∗. Denote by L̃(G) the homotopy fiber of the map of spectra L(pr) : L(G) → L(∗). Thus we obtain covariant functors

K̃ = K̃_{G}^{alg} : Or(G) ⟶ Spectra ;  (1.5)
L̃ = L̃_{G} : Or(G) ⟶ Spectra ;  (1.6)
K̃ = K̃_{G}^{top} : Or(G, FIN) ⟶ Spectra .  (1.7)

Notice that we have defined K̃_{G}^{top} only for Or(G, FIN). The problem is that pr : G → ∗ is not a morphism in Groupoids^{inj} . However, if we replace the reduced C^{∗}-algebra by the maximal C^{∗}-algebra, then K^{top} is indeed a functor on Groupoids , and for amenable groups such as all finite groups and virtually abelian groups the natural map from the maximal to the reduced C^{∗}-algebra is an isomorphism [33, Theorem 7.3.9 on page 243]. Notice that we only need K̃ = K̃_{G}^{top} : Or(G, FIN) ⟶ Spectra since it will be only applied in context with Or(G)-spaces X for which X(G/H) is empty for infinite H.

The homology H_{q}^{G}(X; E) of a contravariant Or(G)-space X with coefficients in the covariant Or(G)-spectrum E is defined for q ∈ ℤ in [15, section 4] using Or(G)-CW - approximations. The above homology groups are functorial in X and E. We get from [15, Lemma 4.4]

Lemma 1.8 H_{p}^{G}(X, A; E) is an unreduced homology theory on pairs of contravariant Or(G)- spaces which satisfy the WHE-axiom and the disjoint union axiom.

To be more precise, homology theory means that homotopic maps of pairs of contravariant Or(G)-spaces induce the same maps on the homology groups, that there are long exact sequences of pairs (X, A), and that for any commutative diagram of contravariant Or(G)- spaces

[figure: X_{0}; i_{1}; X_{1}; X_{2}; i_{2}; j_{2}; X; j_{1}]

the map (j_{2}, i_{1}) : (X_{2}, X_{0}) ⟶ (X, X_{1}) induces an isomorphism on homology, provided that the evaluation of the diagram at any object G/H yields a pushout of spaces with a cofibration of spaces as upper horizontal arrow. We will frequently use the associated long exact Mayer-Vietoris sequence

… ⟶^{δ}H_{p}^{G}(X_{0}; E) → H_{p}^{G}(X_{1}; E) ⊕ H_{p}^{G}(X_{2}; E) → H_{p}^{G}(X; E) ⟶^{∂} H_{p−1}^{G}(X_{0}; E) → ….

The disjoint union axiom says that for an arbitrary disjoint union the obvious map from the direct sum of the homology groups of the various summands to the homology of the disjoint union is an isomorphism. The WHE-axiom requires that a weak homotopy equivalence of contravariant Or(G)-spaces induces an isomorphism on homology, where a map f : X → Y of Or(G)-spaces or spectra is called n-connected resp. a weak homotopy equivalence, if the map f(G/H) : X(G/H) → Y (G/H) is n-connected resp. a weak homotopy equivalence for every object G/H. In order to guarantee the WHE-axiom, Or(G)-CW -approximations are used in the definitions. Notice, however, that we will almost everywhere do calculations with the Or(G)-space X itself and not with its Or(G)-CW -approximations, which will be quite convenient since the Or(G)-spaces X we will deal with will very often be very simple. Namely, we will consider the Or(G)-spaces ⋆_{G,F} associated to a family F, which assigns to an object G/H the space ∗ consisting of one point if H belongs to F and the empty set ∅ otherwise. For these spaces it will be comparatively easy to check whether the necessary conditions are satisfied for the square above to get a Mayer-Vietoris sequence.

Notice in the sequel that for any covariant Or(G)-spectrum E there is a canonical isomorphism

H_{q}^{G}(⋆_{G,ALL}; E) ⟶^{≅} π_{q}(E(G/G))

which comes from the fact that ⋆_{G,ALL} is an Or(G)-CW -complex. The isomorphism conjecture for a group G, a family of subgroups F and an Or(G)-spectrum E says that the map induced by the inclusion ⋆_{G,F} → ⋆_{G,ALL}

H_{q}^{G}(⋆_{G,F}; E) → H_{q}^{G}(⋆_{G,ALL}; E) = π_{q}(E(G/G))  (1.9)

is an isomorphism for all q ∈ ℤ. The philosophy is to compute the groups of interest π_{q}(E(G/G)) by the values of E(G/H) on the groups in H ∈ F. The isomorphism conjectures of Farrell and Jones for algebraic K-theory and L-theory are the special cases where E is given by the Or(G)-spectra K and L of (1.2) and (1.3) and F is the family VC of virtually cyclic subgroups of G. The Baum-Connes conjecture is the special case where E is given by the Or(G)-spectra K^{top} of (1.4) and F is the family FIN of finite subgroups of G. The Farrell and Jones isomorphism conjecture and the Baum-Connes conjecture provide tools to compute π_{q}(K^{alg}(G/G)) = K_{q}(ℤG), π_{q}(L(G/G)) = L_{q}(ℤG) and π_{q}(K^{top}(G/G)) = K_{q}(C_{r}^{∗}(G)) in terms of data given by the virtually cyclic subgroups or the finite subgroups of G. If G is torsion free, these conjectures predict that for q ∈ ℤ

Wh_{q}(G) ≅ 0;
L_{q}(ℤG) ≅ H_{q}(BG; L(ℤ));
K_{q}(C_{r}^{∗}(G)) ≅ K_{q}(BG),

where H_{q}(BG; L(ℤ)) is the homology of the classifying space BG with respect to the Ltheory spectrum associated to the ring ℤ, K_{q}(BG) is the topological complex K-homology of BG and Wh_{q}(G) denotes the reduced negative or zero-th K-group K̃_{q}(ℤG) for q ≤ 0, the ordinary Whitehead group Wh(G) for q = 1 and Waldhausen’s definition for q ≥ 1 in terms of the fiber of the assembly map BG_{+} ∧ K(ℤ) → K(ℤG).

We mention that the assembly map appearing in the original conjectures of Farrell and Jones and of Baum and Connes are different from the one presented here. Their identification is discussed in [15, page 239, page 247-248] and based on [11] and [39, Proposition 8.4 on page 421].

Theorem 1.10 Let G be a planar group with compact orbit space or a virtually finitely generated abelian group. Then the assembly maps

H_{q}^{G}(⋆_{G,VC}; K^{alg}) → H_{q}^{G}(⋆_{G,ALL}; K^{alg}) = K_{q}(ℤG)  for q ≤ 1;
H_{q}^{G}(⋆_{G,FIN}; L)[1/2] → H_{q}^{G}(⋆_{G,ALL}; L)[1/2] = L_{q}(ℤG)[1/2]  for q ∈ ℤ;
H_{q}^{G}(⋆_{G,FIN}; K^{top}) → H_{q}^{G}(⋆_{G,ALL}; K^{top}) = K_{q}(C_{r}^{∗}(G))  for q ∈ ℤ

are isomorphisms. The first map is surjective for q = 2.

Here and in the sequel A[1/m] for an integer m ≥ 1 means A⊗_{ℤ}ℤ[1/m] for ℤ[1/m] = {a·m^{b} ∈ ℚ | a, b ∈ ℤ}. Thus A[1/m] is obtained from A by inverting m. The claim for algebraic K- and L-theory is a consequence of the results of [18, Proposition 2.3, Proposition 2.4 and Remark 2.1.3], [50] and Theorem 2.4. Since we will invert 2 in (almost) all our L-theory calculations we do not have to distinguish between the various decorations of L-groups such as L^{h}, L^{s} or L^{−∞} as they differ by 2-torsion because of the Rothenberg sequences. The Baum-Connes-conjecture has been proven for a very large class of groups including virtually finitely generated abelian groups and cocompact planar groups [21, Theorem 1.12].

2 Preliminaries about Spaces over the Orbit Category

In this section we prove some facts about spaces over the orbit category and their homology which will be needed later.

Given a homomorphism of groups i : H → G, there is an induced functor I = I(i) : Or(H) → Or(G) sending H/K to G ×_{i} H/K = G/i(K). Given a (covariant or contravariant) Or(G)-space Y, we obtain a (covariant or contravariant) Or(H)-space I^{∗}Y called the restriction of Y with I by the composition Y ∘I. Given a covariant Or(H)-space X, its induction I_{∗}X is the covariant Or(G)-space defined in [15, Definition 1.8]. Its value at the object G/K of Or(G) is the quotient space

∐_{H/L∈Or(H)} X(H/L) × map_{G}(G/K, G/i(L))/ ∼

where ∼ is the equivalence relation generated by (X(ϕ)(x), ψ) ∼ (x, I(ϕ)∘ψ) for a H-map ϕ : H/L → H/L′, a G-map ψ : G/K → G(i(L)) and x ∈ X(H/L′). There is an adjunction homeomorphism for an Or(G)-space X and an Or(G)-space Y (see [15, Lemma 1.9])

hom_{Or(G)}(I_{∗}X, Y) ⟶^{≅} hom_{Or(H)}(X, I^{∗}Y).  (2.1)

In the sequel we use the identification of the Weyl group W_{G}H = N_{G}H/H with the automorphism group aut_{G}(G/H, G/H) which sends gH ∈ N_{G}H/H to the G-map R_{g^{−1}} : G/H → G/H g′H ↦ g′g^{−1}H. Notice that aut_{G}(G/H, G/H) = map_{G}(G/H, G/H) holds for finite H but not in general (see [26, Example 1.32 on page 22]).

Lemma 2.2 Let G be a group with subgroups H, L ⊂ G. Let {H_{α} | α ∈ A} be a complete system of representatives of L-conjugacy classes of subgroups of L which are G-conjugated to H. To every index α ∈ A choose an isomorphism μ_{α} : G/H → G/H_{α}. Let I = I(L) be the functor induced by the inclusion L ↪ G. Then we have for K ⊂ L:

(a) The following map is a natural equivalence of functors Or(L) → Sets

T (L/K) : ∐_{α∈A} map_{L}(L/H_{α}, L/K) ×_{W_{L}H_{α}} W_{G}H_{α} ⟶ map_{G}(G/H, G/K)
[ϕ, ψ] ↦ (Id_{G} ×_{L}ϕ)∘ψ∘μ_{α};

(b) There is a homeomorphism, natural in X,

∐_{α∈A} X(L/H_{α}) ×_{W_{L}H_{α}} W_{G}H_{α} ⟶ I_{∗}X(G/H)

for every contravariant Or(L)-space X;

(c) There is a natural isomorphism

H_{q}^{G}(I_{∗}X; E) ⟶^{≅} H_{q}^{L}(X; E)

for E one of the spectra associated to G and L in (1.2), (1.3), (1.4), (1.5), (1.6) and (1.7);

(d) The map from {g ∈ L\G/N_{G}H | gHg^{−1} ⊂ L} to the set of L-conjugacy classes of subgroups of L being G-conjugated to H which sends LgN_{G}H to (gHg^{−1})_{L} is bijective.

\underline{\underline{Proof}} : (a), (b) and (d) are elementary consequences of the definitions. (c) follows from the facts that induction with I : Or(L) → Or(G) sends an Or(L)-CW -approximation to an Or(G)-CW -approximation and that the canonical map E_{L} → I^{∗}E_{G} is an equivalence of Or(L)-spectra together with the adjunction (2.1) and [15, Lemma 4.6].

Sometimes we can use smaller families than VC such as the family FIN of finite subgroups as explained in the next result. Notice that for n = ∞ and m = 1 it is just [18, Theorem A.10].

Definition 2.3 Let G be a group, and let F be a family of subgroups of G. For a subgroup H of G, we define H ∩ F to be the family of subgroups of H given as {H ∩ K | K ∈ F}.

Theorem 2.4 Let F ⊂ G be families of subgroups of the group G. Let m ≥ 1 and n be integers. Suppose for every H ∈ G that the assembly map

H_{q}^{H}(⋆_{H,H∩F}; I(H)^{∗}E)[1/m] → H_{q}^{H}(⋆_{H,ALL}; I(H)^{∗}E)[1/m]

is an isomorphism for q ≤ n. Then the relative assembly map

H_{q}^{G}(⋆_{G,F}; E)[1/m] → H_{q}^{G}(⋆_{G,G}; E)[1/m]

is an isomorphism for q ≤ n.

\underline{\underline{Proof}} : In the sequel we use the identification

H_{p}^{G}(⋆_{G,F}; E) = π_{q}(hocolim_{Or(G,F)} E)

as explained in [15, section 3]. By assumption the map

π_{q} ( hocolim_{Or(H,H∩F)} I(H)^{∗}E) [1/m] → π_{q}(E(G/H))[1/m]

is an isomorphism for q ≤ n and H ∈ G. A standard spectral sequence argument applied to the Atiyah-Hirzebruch spectral sequence [15, Theorem 4.7] or the Bousfield-Kan spectral sequence [9, XXII 5.7 on page 339] shows that the induced map

π_{q} (hocolim_{Or(G,G)} hocolim_{Or(H,H∩F)} I(H)^{∗}E) [1/m] → π_{q}(hocolim_{Or(G,G)} E)[1/m]

is an isomorphism for q ≤ n. There is an equivalence of categories

Or(G, F) ↓ (G/H) ⟶^{≅} Or(H, H ∩ F) (G/F → G/H) ↦ H/F,

where Or(G, F) ↓ (G/H) denotes the category of objects over G/H. For E′(G/F → G/H) := E(G/F) we get an isomorphism

π_{q} ( hocolim_{G/H∈Or(G,G)} hocolim_{Or(H,H∩F)} I(H)^{∗}E) ≅ π_{q} ( hocolim_{G/H∈Or(G,G)} hocolim_{Or(G,F)↓G/H} E′) .

By [24, Theorem 2.4] the homotopy colimits commute, hence there is an isomorphism for q ≤ n

π_{q} (hocolim_{Or(G,G)} E) [1/m] ≅ π_{q} ( hocolim_{G/H∈Or(G,G)} hocolim_{Or(H,H∩F)} I(H)^{∗}E) [1/m]
≅ π_{q} ( hocolim_{G/H∈Or(G,G)} hocolim_{Or(G,F)↓G/H} E′) [1/m]
≅ π_{q} ( hocolim_{G/H∈Or(G,F)} hocolim_{Or(G,G)↓G/H} E′) [1/m]
≅ π_{q} (hocolim_{Or(G,F)} E) [1/m].

Recall that a classifying space E(G, F) for a family F of subgroups of G is a G-CW - complex whose H-fixed point set is contractible if H ∈ F and empty otherwise. Such a G-space is unique up to G-homotopy. In particular E(G, TR) is a model for EG. We abbreviate \underline{E}G = E(G, FIN) for the family FIN of finite subgroups.

For the reader’s convenience we briefly sketch a different more geometric proof of Theorem 2.4. Namely, given a model for E(G, G), one can construct a model for E(G, F) by replacing each cell G/H × D^{n} in E(G, G) by G ×_{H} E(H, H ∩ F) × D^{n} and then use Mayer-Vietoris sequences. Notice that the assumption in Theorem 2.4 implies that for each H ∈ G the projection induces an isomorphism

H_{q}^{H}(⋆_{H,H∩F}; I(H)^{∗}E)[1/m] = H_{p}^{G}(G ×_{H} E(H, H ∩ F); E)[1/m]

⟶^{≅} H_{p}^{G}(G/H; E)[1/m] = H_{q}^{H}(⋆_{H,ALL}; I(H)^{∗}E)[1/m],

where we interpret a G-space as a Or(G)-space by assigning to G/L its L-fixed point set.

The next result follows from the definitions of Wh_{q}(G) in [47, Definition 15.6 on page 228 and Proposition 15.7 on page 229], from the definition of H_{∗}^{G}(X, A; E) in [15, Section 4] and from [15, Lemma 7.6].

Lemma 2.5 For a group G there is an isomorphism

H_{q}^{G}(⋆_{G,ALL}, ⋆_{G,TR}; K^{alg}) = \{ \begin{matrix}Wh_{q}(G) & q ≥ 2; \\ Wh(G) = Wh_{1}(G) & q = 1; \\ K̃_{0}(ℤG) = Wh_{0}(G) & q = 0; \\ K_{q}(ℤG) = Wh_{q}(G) & q ≤ −1, \end{matrix}

which is natural in G.

Lemma 2.6 The assembly map

H_{q}^{Or(G)}(⋆_{G,FIN}; K) ⟶^{≅} H_{q}^{Or(G)}(⋆_{G,ALL}; K)

is an isomorphism for any q ∈ ℤ if G = ℤ or ℤ/2 ∗ ℤ/2, and an isomorphism for q ≤ 2 if G = ℤ/2 × ℤ, ℤ/2 × (ℤ/2 ∗ ℤ/2), ℤ/4 ∗_{ℤ/2} ℤ/4, or ℤ/4 ∗_{ℤ/2} (ℤ/2)^{2}.

\underline{\underline{Proof}} : We begin with the case G = ℤ. A model for \underline{E}G = E(G, FIN) is the universal covering of S^{1}. Therefore the source of the assembly map above reduces to K_{q}(ℤ) ⊕ K_{q−1}(ℤ) and the assembly map itself is the restriction of the Bass-Heller-Swan isomorphism (see [4, Chapter XII], [5] and [37, Corollary to Theorem 8 on p. 114])

K_{q}(ℤ) ⊕ K_{q−1}(ℤ) ⊕ Nil_{q}(ℤ) ⊕ Nil_{q}(ℤ) → K_{q}(ℤ[ℤ])

to the first two summands. Since the ring ℤ is regular and hence all its Nil-groups are trivial, the assembly map is an isomorphism for G = ℤ.

For G = ℤ/2 ∗ ℤ/2 = ℤ ⋊ ℤ/2 there is a model for \underline{E}G with ℝ as underlying space such that G\\underline{E}G is the unit intervall. Then the assembly map in question can be identified with the obvious map Wh_{q}(ℤ/2) ⊕ Wh_{q}(ℤ/2) → Wh_{q}(ℤ/2 ∗ ℤ/2) which is bijective by a result of Waldhausen [47, Corollary 11.5 and the following Remark].

By Theorem 3.2 (e), the Nil-groups of ℤ[ℤ/2 × ℤ^{l}] are trivial for q ≤ 2 and l ≥ 0. So the claim follows for G = ℤ × ℤ/2 from the Bass-Heller-Swan theorem [4, Chapter XII], [5] and [37, Corollary to Theorem 8 on p. 114].

In the remaining cases G is of the shape G = G_{1} ∗_{ℤ/2} G_{2} for finite groups G_{i} such that Wh_{q}(G_{i}) = 0 for q ≤ 1 (see Theorem 3.2 (div)). The claim for q ≤ −1 follows from [19, Theorem 2.1]. We obtain from [47, Theorem 1 on page 137] the exact sequence

Wh_{2}(ℤ/2) → Wh_{2}(G_{1}) ⊕ Wh_{2}(G_{2}) → Wh_{2}(G) → Wh(ℤ/2) → Wh(G_{1}) ⊕ Wh(G_{2})

→ Wh(G) → K̃_{0}(ℤ[ℤ/2]) → K̃_{0}(ℤG_{1}) ⊕ K̃_{0}(ℤG_{2}) → K̃_{0}(ℤG) → 0

where it is not clear a priori why the last map is surjective. This follows from the exact sequence, obtained by applying the same argument to G × ℤ = (G_{1} × ℤ) ∗_{ℤ/2×ℤ} (G_{2} × ℤ),

Wh(G_{1} × ℤ) ⊕ Wh(G_{2} × ℤ) → Wh(G × ℤ) → K̃_{0}(ℤ[ℤ/2 × ℤ]),

the computation K̃_{0}(ℤ/2 × ℤ) = 0 from Theorem 3.2 (diii) and (e) and the fact that K̃_{0}(ℤG_{1}) ⊕ K̃_{0}(ℤG_{2}) → K̃_{0}(ℤG) is a natural direct summand in Wh(G_{1} × ℤ) ⊕ Wh(G_{2} × ℤ) → Wh(G × ℤ) by the Bass-Heller-Swan decomposition. In order to get the claim for q = 0, 1, 2, one constructs an exact sequence as above but now with H_{q}^{G}(⋆_{G,FIN}; K^{alg}) instead of Wh_{q}(G), together with a map between these exact sequences, and uses the five-lemma. The last exact sequence is the Mayer-Vietoris sequence associated to the G-CW -model for \underline{E}G which has one 1-cell with isotropy group ℤ/2 and two 0-cells with isotropy groups G_{1} and G_{2} [43, Theorem 7 on page 32 and Corollary on page 36].

Lemma 2.7 Let G be a group and let L ⊂ G be a normal subgroup such that there is an epimorphism f : G → L inducing the identity on L. Denote by SUB(L) the family of subgroups of G which consists of the subgroups of L. Then there is an isomorphism

H_{q}^{G}(⋆_{G,SUB(L)}; L) → H_{q}(B(G/L); L(ℤL)),

and analogously for both versions of K-theory and all the reduced versions of K- and Ltheory.

\underline{\underline{Proof}} : We only treat the case of the L-theory spectrum L, the others are completely analogous. We get from the definitions and [15, section 7] an isomorphism

H_{q}^{G}(⋆_{G,SUB(L)}; L) ≅ π_{q}(E(G/L)_{+} ∧_{G/L} L(G/L)),

since EG/L regarded as a G-space via the projection G → G/L is a model for the classifying space E(G, SUB(L)) of G for the family SUB(L). From f we obtain a morphism in Groupoids^{inj}

f_{∗} : G^{G}(G/L) → G^{L}(L/L).

Notice that G/L = aut_{G}(G/L, G/L) acts on G^{G}(G/L) in the obvious way and that this action induces a non-trivial action on L(G^{G}(G/L)) although it induces a trivial action on the homotopy groups. If we equip G^{L}(L/L) with the trivial G/L-action, the map f_{∗} is Lequivariant. Since f_{∗} is an equivalence of categories, the induced map π_{q}(L(G^{G}(G/L))) → π_{q}(L(G^{L}(L/L))) is an isomorphism. Hence the induced map

π_{q}(E(G/L)_{+} ∧_{G/L} L(G/L)) ⟶^{≅} π_{q}(E(G/L)_{+} ∧_{G/L} L(G^{L}(L/L))) = H_{q}(B(G/L); L(ℤL))

is an isomorphism [15, Lemma 4.6].

Definition 2.8 If E is a spectrum, we denote the generalized homology of a space X which is associated to E by H_{∗}(X; E). If E is the topological K-theory spectrum K = K(ℂ), then we also write K_{∗}(X) = H_{∗}(X; K).

Lemma 2.9 Let G be a discrete group. Let A be a ring with ℤ ⊂ A ⊂ ℚ such that the order of any finite subgroup of G is invertible in A.

(a) Let H_{∗} be any generalized homology theory. Then we obtain a natural isomorphism

H_{∗}(BG) ⊗_{ℤ} A ⟶^{≅} H_{∗}(G\\underline{E}G) ⊗_{ℤ} A.

(b) There is a long exact sequence

… → H_{q+1}(G\\underline{E}G; L(ℤ)) → H_{q}^{G}(⋆_{G,FIN}; L̃) → H_{q}^{G}(⋆_{G,FIN}; L)
→ H_{q}(G\\underline{E}G; L(ℤ)) → …,

where L(ℤ) is the L-theory spectrum associated to the ring ℤ. This sequence splits after tensoring with A, yielding isomorphisms

H_{q}(G\\underline{E}G; L(ℤ)) ⊗_{ℤ} A ⊕ H_{q}^{G}(⋆_{G,FIN}; L̃) ⊗_{ℤ} A ⟶^{≅} H_{q}^{G}(⋆_{G,FIN}; L) ⊗_{ℤ} A.

(c) For topological K-theory we obtain the long exact sequence

… → K_{q+1}(G\\underline{E}G) → H_{q}^{G}(⋆_{G,FIN}; K̃) → H_{q}^{G}(⋆_{G,FIN}; K)
→ K_{q}(G\\underline{E}G) → ….

This sequence splits after tensoring with A yielding isomorphisms

K_{q}(G\\underline{E}G) ⊗_{ℤ} A ⊕ H_{q}^{G}(⋆_{G,FIN}; K̃) ⊗_{ℤ} A ⟶^{≅} H_{q}^{G}(⋆_{G,FIN}; K) ⊗_{ℤ} A.

\underline{\underline{Proof}} : (a) By the Atiyah-Hirzebruch spectral sequence it suffices to check the claimed isomorphism in the special case where H_{∗} is the cellular homology H_{∗}. The claim follows from the fact that the projection induces a homology equivalence of projective AG-chain complexes C_{∗}(EG) ⊗_{ℤ} A → C_{∗}(\underline{E}G) ⊗_{ℤ} A which is then an AG-chain homotopy equivalence and hence induces a chain homotopy equivalence C_{∗}(EG) ⊗_{ℤG} A → C_{∗}(\underline{E}G) ⊗_{ℤG} A.

(b) There are natural maps of Or(G)-spectra L̃ → L → \underline{L(ℤ)}, where \underline{L(ℤ)} denotes the constant Or(G)-spectrum with value L(ℤ) = L(∗). Since its evaluation at an object G/H is a fibration of spectra, it induces a long exact sequence

… → H_{q+1}^{G}(X; \underline{L(ℤ)}) ⟶^{δ_{q+1}} H_{q}^{G}(X; L̃) → H_{q}^{G}(X; L) → H_{q}^{G}(X; \underline{L(ℤ)}) → …

for any contravariant Or(G)-space X. Notice that it suffices to check exactness for any Or(G)-CW -complex X and hence for any Or(G)-space of the form map_{G}(G/?, G/K) for any fixed object G/K in Or(G), where the claim reduces to the exactness of the long homotopy sequence associated to a fibration. We get from [15, Lemma 7.6] an identification H_{q}^{G}(⋆_{G,FIN}; \underline{L(ℤ)}) with H_{q}(G\\underline{E}G; L(ℤ)) and thus the desired long exact sequence by taking X = ⋆_{G,FIN}.

The composition

H_{q}(BG; L(ℤ)) = H_{q}^{G}(⋆_{G,TR}; L) → H_{q}(⋆_{G,FIN}; L) → H_{q}(G\\underline{E}G; L(ℤ))

becomes an isomorphism after tensoring with A by assertion (a) and thus induces the splitting of the long exact sequence after tensoring with A.

The proof of (c) is analogous to that of assertion (b) taking into account that K_{q}(Y) = H_{q}(Y; K^{top}(ℂ)) holds by definition.

3 Preliminary Computations of K- and L-Groups of Finite Groups

In this section we state some computations about K-and L-groups for finite groups which we will use later in the computations for infinite groups.

Definition 3.1 Let π be a finite group. By q(π), r(π) and c(π), we denote the number of irreducible rational, real respectively complex representations of π. Let r_{ℂ}(π) be the number of isomorphisms classes of irreducible real π-representations V which are of complex type, i.e. aut_{ℝπ}(V) ≅ ℂ. Let RO(π) resp. R(π) be the real resp. complex representation ring. For a positive integer m, we let ℤ/m denote the cyclic group of order m, and D_{2m} is the dihedral group of order 2m.

Theorem 3.2 For a finite group π, we have the following.

(a) There are isomorphisms RO(π) ≅ ℤ^{r(π)} and R(π) ≅ ℤ^{c(π)}. The number q(π) is the number of conjugacy classes of cyclic subgroups in π, the number c(π) is the number of conjugacy classes of elements in π and the number r(π) is the number of ℝ-conjugacy classes of elements in π, where g_{1} and g_{2} in π are ℝ-conjugated if g_{1} and g_{2} or g_{1}^{−1} and g_{2} are conjugated;

(b)

K_{q}(C_{r}^{∗}(π)) ≅ \{ \begin{matrix}R(π) ≅ ℤ^{c(π)} & q = 0; \\ 0 & q = 1; \end{matrix}

In particular

K_{0}(C_{r}^{∗}(ℤ/m)) ≅ ℤ^{m}
K_{0}(C_{r}^{∗}(D_{2m})) ≅ \{ \begin{matrix}ℤ^{m/2+3} & m ≡ 0 (2); \\ ℤ^{(m−1)/2+2} & m ≡ 1 (2); \end{matrix}

(c)

L_{q}(ℤπ)[1/2] ≅ L_{q}(ℝπ)[1/2] ≅ \{ \begin{matrix}ℤ[1/2]^{r(π)} & q ≡ 0 (4); \\ ℤ[1/2]^{r_{ℂ}(π)} & q ≡ 2 (4); \\ 0 & q ≡ 1, 3 (4); \end{matrix}

In particular,

L_{0}(ℤ[ℤ/m])[1/2] ≅ ℤ[1/2]^{[(m+2)/2]};
L_{2}(ℤ[ℤ/m])[1/2] ≅ ℤ[1/2]^{[(m−1)/2]};
L_{0}(ℤ[D_{2m}])[1/2] ≅ \{ \begin{matrix}ℤ[1/2]^{m/2+3} & m ≡ 0 (2); \\ ℤ[1/2]^{(m−1)/2+2} & m ≡ 1 (2); \end{matrix}
L_{2}(ℤ[D_{2m}])[1/2] ≅ 0;

(d) (i) K_{q}(ℤπ) = 0 for q ≤ −2; (ii) The rank of Wh(π) as an abelian group is r(π) − q(π). We have

Wh(ℤ/m) ≅ ℤ^{[m/2]+1−δ(m)},

where δ(m) is the number of divisors of m and [m/2] is the largest integer less or equal to m/2;

(iii) We have Wh_{q}(π) = 0 for q ≤ 1 for the following finite groups π = {1}, ℤ/2, ℤ/3, ℤ/4, ℤ/2×ℤ/2, D_{6}, D_{8}. If l is a prime, then K_{−1}(ℤ[ℤ/l]) = K_{−1}(ℤ[ℤ/l×ℤ/l]) = 0. We have

Wh_{q}(ℤ/6) ≅ \{ \begin{matrix}0 & q = 0, 1; \\ ℤ & q = −1; \end{matrix}
Wh_{q}(D_{12}) ≅ \{ \begin{matrix}0 & q = 0, 1; \\ ℤ & q = −1; \end{matrix}

(iv) We have

Wh_{2}(π) = 0, for π = {1}, ℤ/2, ℤ/3, ℤ/4;
| Wh_{2}(ℤ/6)| ≤ 2;
Wh_{2}(D_{6}) = ℤ/2;
Wh_{2}((ℤ/2)^{2}) ≥ (ℤ/2)^{2}.

The assembly map H_{2}(Bℤ/2; K(ℤ)) → K_{2}(ℤ[ℤ/2]) is an isomorphism;

(e) There are isomorphisms for an integer n ≥ 0 and a prime number l and q ∈ ℤ

L_{q}(ℤ[ℤ^{n} × π])[1/2] ≅ ⨁_{i=0}^{n} \binom{n}{i} · L_{q−i}(ℤ[π])[1/2];

K_{q}(C_{r}^{∗}(ℤ^{n} × π)) ≅ ⨁_{i=0}^{n} \binom{n}{i} · K_{q−i}(C_{r}^{∗}(π));

K_{q}(ℤ[ℤ^{n} × ℤ/l]) ≅ K_{q}(ℤ[ℤ/l]) ⊕ n · K_{q−1}(ℤ[ℤ/l]) ⊕ \binom{n}{2} · K_{q−2}(ℤ[ℤ/l]) q ≤ 2;
Wh_{q}(ℤ^{n} × ℤ/l) ≅ Wh_{q}(ℤ/l) ⊕ n · Wh_{q−1}(ℤ/l) ⊕ \binom{n}{2} · Wh_{q−2}(ℤ/l)  q ≤ 2;
Nil_{q}(ℤ[ℤ^{n} × ℤ/l]) = 0  q ≤ 2;

\underline{\underline{Proof}} : (a) is proven in [42, Theorem 7 on page 19, Corollary 2 on page 96, page 102, page 106].

(b) follows from Morita equivalence applied to ℂ[π] ≅ ∏_{i=1}^{c(π)} M(n_{i}, n_{i}, ℂ) and the computation K_{0}(ℂ) = ℤ and K_{1}(ℂ) = 0.

(c) follows from [40, Proposition 22.34 on page 253].

(d) The computations of K_{q}(ℤπ) for q ≤ −1 follow from [13]; that K_{−1}(ℤπ) = 0 for π = ℤ/l or ℤ/l × ℤ/l can also be found in [4, Theorem 10.6, p. 695]. For information about Wh(π) we refer to [31]. The vanishing of K̃_{0}(ℤπ) is proven for π = D_{6} in [41, Theorem 8.2] and for π = D_{8} in [41, Theorem 6.4]. The cases π = ℤ/2, ℤ/3, ℤ/4, ℤ/6, and (ℤ/2)^{2} are in [14, Corollary 5.17]. Finally, K̃_{0}(ℤD_{12}) = 0 follows from [14, Theorem 50.29 on page 266] and the fact that ℚD_{12} as a ℚ-algebra splits into copies of ℚ and matrix algebras over ℚ, so its maximal order has vanishing class group by Morita equivalence.

The claims about Wh_{2}(ℤ/n) for n = 2, 3, 4, 6 and for Wh_{2}((ℤ/2)^{2}) are taken from [16, Proposition 5], [17, p.482] and [46, p. 218 and 221]. We get K_{2}(ℤD_{6}) ≅ 3 · ℤ/2 from [46, Theorem 3.1]. The assembly map H_{2}(Bℤ/2; K(ℤ)) → K_{2}(ℤ[ℤ/2]) is an isomorphism by [17, Theorem on p. 482]. Now construct a commutative diagram

[figure: H_{2}(Bℤ/2; K(ℤ)); ≅; H_{2}(BD_{6}; K(ℤ)); K_{2}(ℤ[ℤ/2]); ≅; K_{2}(ℤD_{6})]

whose lower horitontal arrow is split injective and whose upper horizontal arrow is an isomorphism by the Atiyah-Hirzebruch spectral sequence. Hence the right vertical arrow is split injective and Wh_{2}(D_{6}) = ℤ/2.

(e) The claim for L-groups follows from the Shaneson splitting [44, Theorem 5.1] and for topological K-groups for instance from the more general Voiculescu-Pimsner sequence [8, Theorem 10.2.1 on page 83]. The claim for the algebraic K-groups follows for q ≤ 0 from [4, Theorem 10.6 on page 695]. To prove the vanishing of the Nil_{1}- and Nil_{2}-terms, consider the following cartesian square of rings:

[figure: ℤ[ℤ/l × ℤ^{k}]; t ↦ exp(2πi/l); ℤ[exp(2πi/l)][ℤ^{k}]; ℤ[ℤ^{k}]; t ↦ 1; ℤ/l[ℤ^{k}],]

where t is a generator of ℤ/l. Let ξ := exp(2πi/l). By [30, Theorem 3.3 and 6.2] and the methods of [46, Section 1], this diagram yields a long exact Mayer Vietoris sequence

K_{3}(ℤ/l[ℤ^{k}]) → K_{2}(ℤ[ℤ/l × ℤ^{k}]) → K_{2}(ℤ[ℤ^{k}]) ⊕ K_{2}(ℤ[ξ][ℤ^{k}]) → K_{2}(ℤ/l[ℤ^{k}])
→ K_{1}(ℤ[ℤ/l × ℤ^{k}]) → K_{1}(ℤ[ℤ^{k}]) ⊕ K_{1}(ℤ[ξ][ℤ^{k}]).

The rings ℤ/l, ℤ[ξ] and ℤ are regular, so they have trivial Nil-terms in any dimension [37, Corollary to Theorem 8 on p. 122]. Furthermore, since ℤ/l is a field, its higher algebraic K-groups are finite [36, Theorem 8 on p. 583], and the K-groups of a Dedekind ring (such as ℤ[ξ]) are finitely generated [38, Theorem 1 on p. 179]. Therefore, the groups K_{i}(ℤ/l[ℤ^{k}]), K_{i}(ℤ[ℤ^{k}]) and K_{i}(ℤ[ξ][ℤ^{k}]) are finitely generated for i = 1, 2, 3, and hence so is K_{i}(ℤ[ℤ/l × ℤ^{k}]) for i = 1, 2. Using the fact that the Nil-groups are either trivial or infinitely generated [35], we conclude that Nil_{i}(ℤ[ℤ/l × ℤ^{k−1}]) = 0 for i = 1, 2.

4 Cocompact Planar Groups

In this section we calculate the K- and L-theory of planar groups, giving explicit formulas in terms of their signature (Theorem 4.5 and Theorem 4.31). We begin with reviewing some facts about planar groups.

Definition 4.1 A planar group (sometimes also called N.E.C. group = Non-Euclidean crystallographic group) is a discontinuous group G of isometries of the two-sphere S^{2}, the Euclidean plane ℝ^{2} or the hyperbolic plane ℍ^{2}. It is cocompact if the quotient manifold S which is the quotient of S^{2}, ℝ^{2} or ℍ^{2} under the G-action is compact.

It is known (see, e.g. [48, Theorem 3 and 4], [52, Theorem 4.5.6 on page 119]) that a cocompact planar group has the following presentation. For fixed integers r_{i} ≥ 1, s ≥ 0, t ≥ 0, g ≥ 0 and h ≥ 0 such that h = 0 or g = h, generators are

\begin{matrix}(1) & x_{i,j}, & 1 ≤ i ≤ s, 1 ≤ j ≤ r_{i}, \\ (2) & e_{i}, & 1 ≤ i ≤ s, \\ (3) & c_{k}, & 1 ≤ k ≤ t, \\ (4) & a_{p}, & 1 ≤ p ≤ g, \\ (5) & b_{p}, & 1 ≤ p ≤ h, \end{matrix}

and relations are given by

(6)  x_{i,j}^{2} = 1,
(7) (x_{i,j}x_{i,j+1})^{n_{i,j}} = 1, 1 ≤ j < r_{i}, n_{i,j} ≥ 2,
(8)  x_{i,r_{i}}e_{i}x_{i,1}e_{i}^{−1} = 1,
(9) c_{k}^{γ_{k}} = 1, γ_{k} ≥ 2,
(10) e_{1}^{−1} ⋯ e_{s}^{−1}c_{1}^{−1} ⋯ c_{t}^{−1}y = 1,

where y = a_{1}^{2} ⋯ a_{g}^{2} if h ≠ g and y = [a_{1}, b_{1}] ⋯ [a_{g}, b_{g}] if h = g. On the other hand, it is also known [52, Theorem 4.7.1 on page 122] that every group with a presentation as above is a cocompact planar group.

Associated to such a presentation is the so-called signature which encodes the presentation as follows

(g, ±, [γ_{1}, γ_{2}, … γ_{t}], {(n_{1,1}, n_{1,2}, …, n_{1,r_{1}−1}), … (n_{s,1}, n_{s,2}, … n_{s,r_{s}−1})}),  (4.2)

where g ≥ 0 is an integer, ± a sign, [γ_{1}, γ_{2}, … γ_{t}] is an ordered set of integers γ_{i} ≥ 0 and (n_{i,1}, n_{i,2}, … n_{i,r_{i}−1}) is an ordered set of integers n_{i,j} ≥ 2 for i = 1, 2, … s. It is allowed that [γ_{1}, γ_{2}, … γ_{t}] consists of the empty symbol [ ], i.e. t = 0. Similiarly (n_{i,1}, n_{i,2}, … n_{i,r_{i}−1}) may consist of the empty symbol ( ), i.e. r_{i} = 1. It is also possible that s = 0, then the fourth entry in the signature consists of the empty set { }. It is clear how a presentation as above defines a signature and vice versa, where of course the sign is positive if S is orientable, or equivalently, h = g in the presentation, and the sign is negative if S is non-orientable, or equivalently, h ≠ g in the presentation. In [28, Section 9], we find necessary and sufficient conditions for two such presentations or signatures respectively to describe isomorphic groups and the proof that two planar groups are algebraically isomorphic if and only if they are geometrically isomorphic.

As mentioned above, the quotient manifold S of a cocompact planar group is orientable if and only if h = g. The number of boundary components is s, and the genus is g. If G acts on ℝ^{2} or the hyperbolic plane ℍ^{2}, then this G-space is a model for \underline{E}G = E(G, FIN) and in particular S is a model for G\\underline{E}G. This can be found in [52, Section 4.2] or follows from the more general result [3, Corollary 4.14]. A cocompact planar group G is finite if and only if it acts on S^{2}. One easily computes using the Atiyah-Hirzebruch spectral sequence

H_{0}(S; ℤ) = ℤ;

H_{1}(S; ℤ) = \{ \begin{matrix}ℤ^{2g} & , if S is orientable and s = 0; \\ ℤ^{2g+s−1} & , if S is orientable and s > 0; \\ ℤ/2 ⊕ ℤ^{g−1} & , if S is non-orientable and s = 0; \\ ℤ^{g+s−1} & , if S is non-orientable and s > 0; \end{matrix}

H_{2}(S; ℤ) = \{ \begin{matrix}ℤ & , if S is orientable and s = 0; \\ 0 & , else; \end{matrix}
K_{0}(S) = H_{0}(S, ℤ) ⊕ H_{2}(S; ℤ);
K_{1}(S) = H_{1}(S; ℤ);
H_{p}(S; L(ℤ))[1/2] = H_{p}(S; ℤ)[1/2] for 0 ≤ p ≤ 3.

As was mentioned in the introduction, the following result yields complete control over the structure of the finite subgroups of a cocompact planar group, thus allowing us to compute its K- and L-theory in terms of the maximal finite subgroups.

Theorem 4.3 Let G be an infinite cocompact planar group.

(a) The system {⟨c_{k}⟩} ∐{⟨x_{i,j}, x_{i,j+1}⟩} is a complete system of representatives of the conjugacy classes of maximal finite subgroups with (M)_{G} ≠ (⟨x_{i,j}⟩)_{G}. (Note that if r_{i} = 1 in the presentation of G, then ⟨x_{i,1}⟩ is a maximal finite subgroup isomorphic to ℤ/2, but we want to exclude these from the above system). The group ⟨c_{k}⟩ is cyclic of order γ_{k} and the group ⟨x_{i,j}, x_{i,j+1}⟩ is the dihedral group D_{2n_{i,j}} of order 2n_{i,j}. In particular any finite subgroup of G is cyclic or dihedral;

(b) Every element of finite order is conjugate to an element x_{i,j}, c_{k}^{q} or (x_{i,j}x_{i,j+1})^{q} for some q ∈ ℤ. The subgroups ⟨c_{k}^{q}⟩ and ⟨(x_{i,j}x_{i,j+1})^{q}⟩ are either trivial or have finite normalizers;

(c) Among the non-trivial powers of the c_{k} and the x_{i,j}x_{i,j+1} only (x_{i,j}x_{i,j+1})^{q} and (x_{i,j}x_{i,j+1})^{−q} are conjugate, and none of these powers is conjugate to one of the x_{i,j};

(d) If g ∈ G has infinite order, then C_{G}(g) is isomorphic to one of the following groups:

ℤ, ℤ/2 × ℤ, ℤ × ℤ, ℤ × (ℤ/2 ∗ ℤ/2), ℤ ⋊ ℤ, or ℤ ∗_{2ℤ} (2ℤ × ℤ/2).

If s = 0 in the presentation of G, then an element g ∈ G of infinite order has a centralizer isomorphic to ℤ, ℤ^{2}, or ℤ ⋊ ℤ. If G is a cocompact Fuchsian group, i.e. s = 0 and F acts orientation preserving on ℍ^{2}, then an element g ∈ G of infinite order has a centralizer isomorphic to ℤ;

(e) The centralizer of a reflection generator x_{i,j} is either ℤ/2 × ℤ or ℤ/2 × (ℤ/2 ∗ ℤ/2), where the first factor ℤ/2 is ⟨x_{i,j}⟩.

\underline{\underline{Proof}} : (b) follows from [23, Corollary 2 on p. 742] and (c) from [52, Theorem 4.8.1 on page 126]. Assertion (a) is a consequence of (c) and [23, Corollary 2 on p. 742]. We conclude (d) from [22, Corollary 4 on p. 67], [23, Theorem 4 on p. 743] and the fact that ℤ^{2} cannot be a subgroup of a cocompact Fuchsian group. Assertion (e) is proven in [23, Theorem 4 on p. 743].

Definition 4.4 Let G be a cocompact planar group with a presentation as above. Let {M_{α}|α ∈ A} be a full system of representatives of the conjugacy classes of maximal finite subgroups of G with (M_{α})_{G} ≠ (⟨x_{i,j}⟩)_{G}, and let {H_{β} | β ∈ B} be a full system of representatives of conjugacy classes of subgroups generated by a single reflection generator x_{i,j}. For α ∈ A for which M_{α} is not cyclic, denote by C_{α} ⊂ M_{α} the cyclic subgroup which is conjugated in G to a cyclic subgroup generated by an element of the form x_{i,j}x_{i,j+1} (see Theorem 4.3 (a) and (c)). For α ∈ A with cyclic M_{α}, put C_{α} = M_{α}. Let B′ = {β ∈ B | N_{G}H_{β} ≅ ℤ/2 × (ℤ/2 ∗ ℤ/2)} and B′′ = {β ∈ B | N_{G}H_{β} ≅ ℤ/2 × ℤ)}. Then B = B′ ∐ B′′ by Theorem 4.3 (e).

Define for α ∈ A a direct summand SL̃_{q}(ℤM_{α}) of L̃_{q}(ℤM_{α}) as the kernel of the splitepimorphism

\begin{aligned}0 : & L̃_{q}(ℤM_{α}) & → & 0 & M_{α} ≅ ℤ/k; \\ u_{∗} : & L̃_{q}(ℤM_{α}) & → & L̃_{q}(ℤ[ℤ/2]) & M_{α} ≅ D_{2k} k odd; \\ u_{∗} ⊕ v_{∗} : & L̃_{q}(ℤM_{α}) & → & L̃_{q}(ℤ[ℤ/2]) ⊕ L̃_{q}(ℤ[ℤ/2]) & M_{α} ≅ D_{2k} k even, \end{aligned}

where u : M_{α} → ℤ/2 is the epimorphism with C_{α} as kernel and v : M_{α} → ℤ/2 is some homomorphism for which v(C_{α}) = ℤ/2. Obviously u has a section s : ℤ/2 → M_{α} so that u_{∗} is split surjective. Since there are homomorphisms s_{1}, s_{2} : ℤ/2 → M_{α} such that u∘s_{1} and v∘s_{2} are the identity and v∘s_{1} is trivial, (s_{1})_{∗} + (s_{2})_{∗} is a section of u_{∗} ⊕ v_{∗}. Explicitly we get from Theorem 3.2

SL̃_{q}(ℤM_{α})[1/2] ≅ \{ \begin{matrix}ℤ[1/2]^{[k/2]} & q ≡ 0 (4) & M_{α} & ≅ & ℤ/k; \\ ℤ[1/2]^{[k/2]} & q ≡ 0 (4) & M_{α} & ≅ & D_{2k}, k ≥ 2; \\ ℤ[1/2]^{[(k−1)/2]} & q ≡ 2 (4) & M_{α} & ≅ & ℤ/k; \\ 0 & & & & otherwise. \end{matrix}

Define the direct summand SK̃_{q}(C_{r}^{∗}(M_{α})) of K̃_{q}(C_{r}^{∗}(M_{α})) analogously to SL̃_{q}(ℤ[M_{α}]). We get explicitly from Theorem 3.2

SK̃_{q}(C_{r}^{∗}(M_{α})) ≅ \{ \begin{matrix}ℤ^{k−1} & q ≡ 0 (2) & M_{α} & ≅ & ℤ/k; \\ ℤ^{[k/2]} & q ≡ 0 (2) & M_{α} & ≅ & D_{2k}, k ≥ 2; \\ 0 & & & & otherwise. \end{matrix}

Theorem 4.5 Let G be an infinite cocompact planar group with a presentation as above and quotient manifold S, and let M_{α} and H_{β} be as in Definition 4.4. We have

(a) The map induced by the inclusions

⨁_{α∈A} Wh_{q}(M_{α}) ⟶^{≅} Wh_{q}(G)

is an isomorphism for q ≤ 1 and surjective for q = 2; (b) We have split exact sequences

0 → (⨁_{β∈B} H_{q}(BW_{G}H_{β}; L(ℤ)) ⊕ ⨁_{α∈A} SL̃_{q}(ℤM_{α})) [1/2]

→ L_{q}(ℤG)[1/2] → H_{q}(S; L(ℤ))[1/2] → 0,

and

H_{q}(BW_{G}H_{β}; L(ℤ))[1/2] ≅ \{ \begin{matrix}ℤ[1/2] & , q ≡ 0 (4); \\ ℤ[1/2] & , β ∈ B′′, q ≡ 1 (4); \\ 0 & , else; \end{matrix}

(c) We have split exact sequences

0 → ⨁_{β∈B} K_{0}(BW_{G}H) ⊕ ⨁_{α∈A} SK̃_{0}(C_{r}^{∗}(M_{α})) → K_{0}(C_{r}^{∗}(G)) → K_{0}(S) → 0;

0 → ⨁_{β∈B′′} K_{1}(Bℤ) → K_{1}(C_{r}^{∗}(G)) → K_{1}(S) → 0,

and

K_{0}(BW_{G}H_{β}) ≅ ℤ for β ∈ B.

If the isomorphism conjecture in algebraic K-theory (see 1.9 and Theorem 1.10) holds also in dimension 2, the homomorphism appearing in Theorem 4.5 (a) is an isomorphism also for q = 2. Furthermore, if s = 0 and the isomorphism conjecture is true in any dimension, then it is an isomorphism for any q ∈ ℤ. Notice that in the case s = 0 we have B = ∅, so that the families E and E_{α} appearing in the proof of Theorem 4.5 are just the families T R.

The proof of Theorem 4.5 needs some preparations. The following lemma will ensure that the application of Lemma 2.2 yields tractable results.

Lemma 4.6 Let G be an infinite cocompact planar group. Define for a maximal finite subgroup M the cyclic subgroup C_{M} as in Definition 4.4. Then

(a) If H ⊂ G is finite and M ⊂ G is maximal finite, (M)_{G} ≠ (⟨x_{i,j}⟩)_{G}, with H ∩ C_{M} ≠ 1, then N_{G}(H ∩ C_{M}) and N_{G}H are finite;

(b) Let M ⊂ G be maximal finite, (M)_{G} ≠ (⟨x_{i,j}⟩)_{G}, and H ⊂ M be a subgroup with

H ∩ C_{M} ≠ 1. Then

M = N_{G}M = N_{G}(H ∩ C_{M}).

If M and N are maximal finite subgroups, (M)_{G}, (N)_{G} ≠ (⟨x_{i,j}⟩)_{G}, with C_{M} ∩ C_{N} ≠ 1, then M = N;

(c) Let M ⊂ G be maximal finite, (M)_{G} ≠ (⟨x_{i,j}⟩)_{G}. If K_{1}, K_{2} ⊂ M are subgroups with K_{i} ∩ C_{M} ≠ 1 and (K_{1})_{G} = (K_{2})_{G}, then (K_{1})_{M} = (K_{2})_{M} and N_{G}K_{1} = N_{M}K_{1} and N_{G}K_{2} = N_{M}K_{2}; (d) Let D_{2m} = ⟨s, t | s^{m} = t^{2} = 1, tst = s^{−1}⟩ be the dihedral group of order 2m. If m

is odd, D_{2m} contains up to conjugacy precisely one subgroup of order 2, namely ⟨t⟩ with trivial Weyl group. If m is even, then D_{2m} has up to conjugacy three subgroups of order 2, namely, ⟨s^{m/2}⟩ with normalizer D_{2m}, ⟨st⟩ with normalizer ⟨s^{m/2}, st⟩ and ⟨t⟩ with normalizer ⟨s^{m/2}, t⟩. The subgroup ⟨s⟩ is a characteristic subgroup of D_{2m} if m ≥ 3.

\underline{\underline{Proof}} : (a) By Theorem 4.3 (b), H ∩C_{M} is conjugated in G to a non-trivial cyclic subgroup generated by some power of an element x_{i,j}x_{i,j+1} or c_{k}, and N_{G}(H ∩ C_{M}) is finite. Hence also N_{G}H is finite since the centralizer C_{G}H has finite index in N_{G}H and is contained in N_{G}(H ∩ C_{M}).

(b) We get M ⊂ N_{G}(H ∩ C_{M}) since H ∩ C_{M} is normal in M. Obviously M ⊂ N_{G}M. As both N_{G}(H ∩ C_{M}) and N_{G}M are finite by assertion (a), we get M = N_{G}M = N_{G}(H ∩ C_{M}) from the maximality of M. Applying this to H = C_{M} ∩ C_{N} we conclude M = N.

(c) Choose g ∈ G with gK_{1}g^{−1} = K_{2}. Then K_{2} ⊂ gMg^{−1} ∩ M and hence gMg^{−1} = M by assertion (b). Again from assertion (b) we conclude g ∈ M. If we apply this to the special case K_{1} = K_{2}, we conclude N_{G}K_{1} = N_{M}K_{1} and N_{G}K_{2} = N_{M}K_{2}.

(d) is a direct calculation. This finishes the proof of Lemma 4.6.

Lemma 4.7 Let G be an infinite cocompact planar group. Let Γ ≤ G be infinite virtually cyclic. Then Γ is isomorphic to one of the following groups, where D denotes the infinite dihedral group ℤ/2 ∗ ℤ/2:

ℤ, ℤ/2 × ℤ, D, ℤ/2 × D, ℤ/4 ∗_{ℤ/2} ℤ/4 ≅ ℤ ⋊ ℤ/4, ℤ/4 ∗_{ℤ/2} (ℤ/2)^{2}.

If s = 0 in the presentation of G, then G only contains infinite virtually cyclic subgroups isomorphic to ℤ or D.

\underline{\underline{Proof}} : It is known that for an infinite virtually cyclic group Γ there is an exact sequence 1 → π → Γ → Q → 1 for finite π such that Q is ℤ or D [19, Lemma 2.5]. First, suppose that s = 0. We have to show that π is trivial. Since D contains a normal subgroup Q′ isomorphic to ℤ, it suffices to treat the case Q ≅ ℤ, otherwise substitute Γ by p^{−1}(Q′). The subgroup p^{−1}(|π|! · ℤ) of Γ is isomorphic to π × ℤ and hence contained in C_{G}(g) for some element g of infinite order. Hence π is a subgroup of a torsion free group by Theorem 4.3 (d) and hence trivial.

Now assume that s ≠ 0. If Q = ℤ, i.e. Γ = π⋊ℤ with π finite, then Γ contains π×|π|!·ℤ, so π is by Theorem 4.3 (d) a subgroup of a group whose finite subgroups all have order ≤ 2 [43, Corollary on page 36]. Hence π is trivial or ℤ/2. This means that Γ is either ℤ or ℤ/2 × ℤ.

It remains to treat the case Q = D, i.e., where Γ surjects onto D with finite kernel π. Then π is either cyclic or dihedral by Theorem 4.3 (a). Suppose π = (ℤ/2)^{2}. Then D acts on π, and Γ contains an element g of infinite order which centralizes π, i.e. (ℤ/2)^{2}×⟨g⟩ ≤ C_{G}(g), which is a contradiction to Theorem 4.3 (d). If π is nonabelian dihedral, then π contains a unique cyclic subgroup of order ≥ 3 by Lemma 4.6 (d) on which D acts. Again, this implies that Γ contains an element of infinite order which is centralized by a cyclic group of order ≥ 3, which contradicts Theorem 4.3 (d). By the same argument, we see that π cannot be cyclic of order ≥ 3. If π is trivial, Γ is D; if π is ℤ/2, Γ has to be one of the following groups: ℤ/2 × D, ℤ/4 ∗_{ℤ/2} ℤ/4, or ℤ/4 ∗_{ℤ/2} (ℤ/2)^{2} by [10, Theorem IV.3.12 on page 93].

Lemma 4.8 Let G be as in Theorem 4.5. There is a bijection of sets

f : ∐_{α∈A,|M_{α}|≡0 (4)}{(K)_{M_{α}} | K ⊂ M_{α}, K ∩ C_{α} = 1, K ≠ 1} ⟶^{≅} ∐_{β∈B′} {(U)_{W_{G}H_{β}} | |U| = 2}.

\underline{\underline{Proof}} : Given an element (K)_{M_{α}} in \{(K)_{M_{α}} | K ⊂ M_{α}, K∩C_{α} = 1, K ≠ 1\} for |M_{α}| ≡ 0 (4), we can choose β ∈ B′ with (K)_{G} = (H_{β})_{G} and g ∈ G with gKg^{−1} = H_{β}, and M_{α} is dihedral because of Theorem 4.3 (a) and (c). Since |M_{α}| ≡ 0 (4), we conclude β ∈ B′ from Theorem 4.3 (e) and Lemma 4.6 (d). There is a unique element u_{α} of order 2 in C_{α}, and u_{α} is central in M_{α}. We have u_{α}Ku_{α}^{−1} = K and hence gu_{α}g^{−1} ∈ N_{G}H_{β}. Notice that gu_{α}g^{−1} ∉ H_{β} because of Theorem 4.3 (c). If pr : N_{G}H_{β} → W_{G}H_{β} is the canonical projection, put U = ⟨pr(gu_{α}g^{−1})⟩ and define f((K)_{M_{α}}) = (U)_{W_{G}H_{β}}. This is independent of the choice of g since any other choice is of the shape ug for some u ∈ N_{G}H_{β}.

Next we define the inverse f^{−1}. Given β ∈ B′ and (U)_{W_{G}H_{β}} with |U| = 2, pr^{−1}(U)) contains H_{β} and has order 4. Choose α ∈ A and g ∈ G with gpr^{−1}(U)g^{−1} ⊂ M_{α}. Then define f^{−1}((U)_{W_{G}H_{β}}) = (gH_{β}g^{−1})_{M_{α}}. This is independent of the choice of g, since any other choice is of the form ug for u ∈ G such that gpr^{−1}(U)g^{−1} ⊂ M_{α} ∩u^{−1}M_{α}u and hence u ∈ M_{α} by Lemma 4.6 (b).

One easily checks that f^{−1} is indeed an inverse of f.

Now we are ready to prove Theorem 4.5.

\underline{\underline{Proof}} : (a) Recall that \{M_{α}|α ∈ A\} is a full system of representatives of the conjugacy classes of maximal finite subgroups of G which are not conjugate to some ⟨x_{i,j}⟩. Denote by E_{α} the family of subgroups K ≤ M_{α} with (K)_{G} ≤ (⟨x_{i,j}⟩)_{G} and by E the family of finite subgroups of G with (K)_{G} ≤ (⟨x_{i,j}⟩)_{G}. Consider the following diagram of Or(G)-spaces

[figure: ∐_{α∈A}(I_{α})_{∗}(⋆_{M_{α},E_{α}}); ∐_{α∈A}(I_{α})_{∗}(⋆_{M_{α},ALL}); ⋆_{G,E}; ⋆_{G,FIN}]  (4.9)

where the horizontal maps are given by inclusions and the vertical maps are the adjoints under the homeomorphism (2.1) of the obvious maps of Or(M_{α})-spaces for the obvious functor I_{α} : Or(M_{α}) → Or(G). Its evaluation at an object G/H gives pushouts of spaces with a cofibration as upper horizontal map since we get for this evaluation from Lemma 2.2 (b) and Lemma 4.6 (c) for appropriate Y

\begin{matrix}Y & ⟶^{Id} & Y \\ ↓ & & ↓ \\ ∗ & ⟶^{Id} & ∗ \end{matrix}  \begin{matrix}∅ & ⟶ & ∗ \\ ^{Id}↓ & ^{Id}↓ \\ ∅ & ⟶ & ∗, \end{matrix}  ∅ ⟶^{Id} ∅ ; ^{Id}↓  ^{Id}↓ ; ∅ ⟶^{Id} ∅,

if H ∈ E,  if H ∈ FIN , H ∉ E,  else.

Hence we obtain from Lemma 1.8, Lemma 2.2 (c) and (4.9) the isomorphism for q ∈ ℤ

⨁_{α∈A} H_{q}^{M_{α}} (⋆_{M_{α},ALL}, ⋆_{M_{α},E_{α}}; K) ⟶^{≅} H_{q}^{G} (⋆_{G,FIN} , ⋆_{G,E}; K)  (4.10)

and the long exact Mayer-Vietoris sequence

… → ⨁_{α∈A} H_{q}^{M_{α}} (⋆_{M_{α},E_{α}}; K) → H_{q}^{G}(⋆_{G,E}; K) ⨁_{α∈A} K_{q}(C_{r}^{∗}(M_{α})) → H_{q}^{G}(⋆_{G,FIN}; K) → …(4.11)

If L is ℤ/2, then the assembly map H_{q}^{L}(⋆_{L,TR}; K) → H_{q}^{L}(⋆_{L,ALL}; K) is an isomorphism for q ≤ 2 by Theorem 3.2 (div) and Lemma 2.5. Hence the following maps are isomorphisms by Lemma 1.8 and Theorem 2.4 for q ≤ 2

H_{q}^{M_{α}} (⋆_{M_{α},ALL}, ⋆_{M_{α},TR}; K) ⟶^{≅} H_{q}^{M_{α}} (⋆_{M_{α},ALL}, ⋆_{M_{α},E_{α}}; K) ;  (4.12)
H_{q}^{G} (⋆_{G,FIN} , ⋆_{G,TR}; K) ⟶^{≅} H_{q}^{G} (⋆_{G,FIN} , ⋆_{G,E}; K) .  (4.13)

Combining (4.10), (4.12) and (4.13) yields an isomorphism for q ≤ 2

⨁_{α∈A} H_{q}^{M_{α}} (⋆_{M_{α},ALL}, ⋆_{M_{α},TR}; K) ⟶^{≅} H_{q}^{G} (⋆_{G,FIN} , ⋆_{G,TR}; K) .  (4.14)

By Lemma 2.6 and Lemma 4.7 the assumptions in Theorem 2.4 are satisfied. Thus Theorem 2.4 and Lemma 1.8 imply that the inclusion ⋆_{G,FIN} , ⋆_{G,TR} → ⋆_{G,VC}, ⋆_{G,TR} induces an isomorphism

H_{q}^{G} (⋆_{G,FIN} , ⋆_{G,TR}; K) ⟶^{≅} H_{q}^{G} (⋆_{G,VC}, ⋆_{G,TR}; K) .  (4.15)

Now the assertion (a) follows from the Isomorphism Conjectures (see Theorem 1.10), Lemma 2.5 and the isomorphisms (4.14) and (4.15).

(b) is proven analogously to assertion (c), only easier because we invert 2.

(c) We get from Theorem 4.3 (e)

W_{G}H_{β} ≅ \{ \begin{matrix}ℤ ⋊ ℤ/2 & , if β ∈ B′; \\ ℤ & , if β ∈ B′′, \end{matrix}  (4.16)

where ℤ/2 acts on ℤ by − Id. The induced action of ℤ/2 on K_{q}(S^{1}) is by Id if q = 0 and by − Id if q = 1. Hence we obtain the following isomorphisms from the Atiyah-Hirzebruch spectral sequence, applied to the obvious fibration S^{1} → BW_{G}H_{β} → Bℤ/2:

K_{0}(BW_{G}H_{β}) ≅ K_{0}(S^{1}) ≅ ℤ;  (4.17)
K_{1}(BW_{G}H_{β})[1/2] ≅ K_{0}(S^{1})[1/2] ≅ ℤ[1/2].  (4.18)

We obtain a long exact sequence

… → K_{q+1}(S) → H_{q}^{G}(⋆_{G,FIN}; K̃) → K_{q}(C_{r}^{∗}(G)) → K_{q}(S) → …  (4.19)

and its rational splitting into short split-exact sequences from Theorem 1.10 and Lemma 2.9 (c). Define a diagram of Or(G)-spaces

[figure: ∐_{β∈B} I(H_{β})_{∗}⋆_{N_{G}H_{β},TR}; ∐_{β∈B} I(H_{β})_{∗}⋆_{N_{G}H_{β},SUB(H_{β})}; ⋆_{G,TR}; ⋆_{G,E}]

where I(H_{β}) : Or(N_{G}H_{β}) → Or(G) is the functor induced by the inclusion, SUB(H_{β}) is the family of subgroups of N_{G}H_{β} consisting of the subgroups of H_{β}, the horizontal maps are induced by inclusions and the vertical maps by the obvious maps of Or(N_{G}H_{β})-spaces and the adjunction (2.1). One easily checks using Lemma 2.2 (b) that evaluation of this diagram at any object in Or(G) is a push-out of spaces with a cofibration as upper horizontal map. Since K̃(G/1) has trivial homotopy groups, we get from this diagram, Lemma 1.8 and Lemma 2.2 (c) an isomorphism

⨁_{β∈B} H_{q}^{N_{G}H_{β}}(⋆_{N_{G}H_{β},SUB(H_{β}}); K̃) ⟶^{≅} H_{q}^{G}(⋆_{G,E}; K̃).

By Theorem 4.3 (e), H_{β} is a direct factor of N_{G}H_{β}. Hence we get from Lemma 2.7

H_{q}^{N_{G}H_{β}}(⋆_{N_{G}H_{β},SUB(H_{β}}); K̃) ⟶^{≅} H_{q}(BW_{G}H_{β}; K̃(C_{r}^{∗}(ℤ/2))).  (4.20)

The ring homomorphism ℂ[ℤ/2] → ℂ sending a + bt to a − b induces a map of spectra K̃(C_{r}^{∗}(ℤ/2)) → K(ℂ) which induces an isomorphism on homotopy groups. Hence it yields an isomorphism [15, Lemma 4.6]

H_{q}(BW_{G}H_{β}; K̃(C_{r}^{∗}(ℤ/2))) ⟶^{≅} K_{q}(BW_{G}H_{β}).  (4.21)

Thus we obtain isomorphisms

⨁_{β∈B} K_{q}(BW_{G}H_{β}) ≅ H_{q}^{G}(⋆_{G,E}; K̃);  (4.22)
⨁_{(K)_{Mα}, K∈E\{1}} K_{q}(BW_{M_{α}}K) ≅ H_{q}^{M_{α}}(⋆_{M_{α},E_{α}}; K̃),  (4.23)

where the construction of the second isomorphism is analogous to the one for G, just replace G by M_{α} and E by E_{α}. The long exact Mayer-Vietoris sequence (4.11) becomes under the isomorphisms (4.22) and (4.23)

… → ⨁_{α∈A} ⨁_{(K)_{Mα}, K∈E\{1}} K_{q}(BW_{M_{α}}K) → ⨁_{β∈B} K_{q}(BW_{G}H_{β}) ⊕ ⨁_{α∈A} K̃_{q}(C_{r}^{∗}(M_{α}))

→ H_{q}^{G}(⋆_{G,FIN}; K̃) → ⨁_{α∈A} ⨁_{(K)_{Mα}, K∈E} K_{q−1}(BW_{M_{α}K}) → …. (4.24)

The composition of the homomorphism induced by the inclusions

⨁_{(K)_{Mα}, K∈E\{1}} K̃_{q}(C_{r}^{∗}(K)) → K̃_{q}(C_{r}^{∗}(M_{α}))

with the homomorphism appearing in the definition of SK̃_{q}(ℤM_{α}) is an isomorphism since it

can be identified with Id : 0 → 0, Id : K̃_{q}(ℤ[ℤ/2]) → K̃_{q}(ℤ[ℤ/2]) or ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) : K̃_{q}(ℤ[ℤ/2])⊕

K̃_{q}(ℤ[ℤ/2]) → K̃_{q}(ℤ[ℤ/2]) ⊕ K̃_{q}(ℤ[ℤ/2]) respectively. We obtain a commutative diagram with split exact columns and an isomorphism as lower horizontal map:

0  0

[figure: ⨁_{(K)_{Mα}, K∈E\{1}} K̃_{q}(BW_{M_{α}}K); SK̃_{q}(C_{r}^{∗}(M_{α}))]

[figure: ⨁_{(K)_{Mα}, K∈E\{1}} K_{q}(BW_{M_{α}}K); K̃_{q}(C_{r}^{∗}(M_{α}))]

[figure: ⨁_{(K)_{Mα}, K∈E\{1}} K_{q}(∗); ≅; K̃_{q}(C_{r}^{∗}(M_{α}))/SK̃_{q}(C_{r}^{∗}(M_{α}))]

[figure: 0; 0]

Hence the sequence (4.24) reduces to the long exact sequence

… → ⨁_{α∈A} ⨁_{(K)_{Mα}, K∈E\{1}} K̃_{q}(BW_{M_{α}}K) ⟶^{ϵ_{q}} ⨁_{β∈B} K_{q}(BW_{G}H_{β}) ⊕ ⨁_{α} SK̃_{q}(C_{r}^{∗}(M_{α}))

→ H_{q}^{G}(⋆_{G,FIN}; K̃) → ⨁_{α∈A} ⨁_{(K)_{Mα}, K∈E\{1}} K̃_{q−1}(BW_{M_{α}}K) ⟶^{ϵ_{q−1}} ….  (4.25)

Since W_{M_{α}}K is trivial or ℤ/2 for K ∈ E \ {1} by Theorem 4.3 (e), K̃_{q}(BW_{M_{α}}K)[1/2] is trivial for all q ∈ ℤ by the Atiyah-Hirzebruch spectral sequence. From (4.19) and (4.25), we therefore get the long exact sequence

K_{q+1}(S)[1/2] ⟶^{δ_{q+1}} ⨁_{β∈B} K_{q}(BW_{G}H_{β})[1/2] ⊕ ⨁_{α∈A} SK̃_{q}(C_{r}^{∗}(M_{α}))[1/2]

→ K_{q}(C_{r}^{∗}(G))[1/2] → K_{q}(S)[1/2] ⟶^{δ_{q}} …,  (4.26)

which splits into short split-exact sequences rationally. Since the boundary map δ_{q+1} in (4.26) is rationally zero and its target does not contain torsion (see (4.17) and (4.18)), δ_{q+1} itself is zero. Hence the long exact sequence (4.26) reduces to short exact sequences

0 → ⨁_{β∈B} K_{q}(BW_{G}H_{β})[1/2] ⊕ ⨁_{α∈A} SK̃_{q}(C_{r}^{∗}(M_{α})[1/2] → K_{q}(C_{r}^{∗}(G))[1/2] → K_{q}(S)[1/2] → 0.

In the case of L-theory, we are done at this point because H_{q}(S; L(ℤ))[1/2] is free. It remains to show the claim for topological K-theory without inverting 2.

To do this, we further investigate the homomorphism ϵ_{q} appearing in the long exact sequence (4.25). For β ∈ B′′, K_{q}(BW_{G}H) = K_{q}(S^{1}) is torsion free, and so is SK̃_{q}(C_{r}^{∗}(M_{α})). Since K̃_{q}(BW_{M_{α}}H) is 2-torsion, it suffices to determine the kernel and cokernel of the part denoted in the same way:

ϵ_{q} : ⨁_{α∈A} ⨁_{(K)_{Mα}, K∈E\{1}} K̃_{q}(BW_{M_{α}}K) → ⨁_{β∈B′} K_{q}(BW_{G}H_{β}).  (4.27)

Fix α ∈ A, (K)_{M_{α}}, K ∈ E \ {1} and β ∈ B′. The part of ϵ_{q} going from the summand in the source belonging to (K)_{M_{α}} to the summand in the target belonging to β is denoted by

ϵ_{q}(α, (K), β) : K̃_{q}(BW_{M_{α}}K) → K_{q}(BW_{G}H_{β}).  (4.28)

If (H_{β})_{G} ≠ (K)_{G}, then ϵ_{q}(α, (K), β) is trivial. If H_{β} = K, then ϵ_{q}(α, (K), β) is induced by the obvious inclusion W_{M_{α}}K → W_{G}K.

Using the bijection f in Lemma 4.8 and the facts that for K ⊂ M_{α}, K ≠ 1, |M_{α}| ≡ 0 (4) the condition K ∩ C_{α} = 1 is equivalent to (K)_{G} = (x_{i,j})_{G} for appropriate i, j and that W_{M_{α}}K = 1 and hence K̃_{q}(BW_{M_{α}}K) = 0 if |M_{α}| ≡ 2 (4), one easily constructs a commutative diagram with isomorphisms as vertical maps

[figure: ⨁_{α∈A} ⨁_{(K)_{Mα}, K∈E\{1}} K̃_{q}(BW_{M_{α}}K); ϵ_{q}; ⨁_{β∈B′} K_{q}(BW_{G}H_{β}); ⨁_{β∈B′} ⨁_{(U)_{W_{G}H_{β}} ,|U|=2} K̃_{q}(BU); ≅; ⨁_{β∈B′} μ(H_{β})_{q}; ⨁_{β∈B′} K_{q}(BW_{G}H_{β}); Id]

where μ(H_{β})_{q} : ⨁_{(U)_{W_{G}H_{β}} ,|U|=2} K̃_{q}(BU) → K_{q}(BW_{G}H_{β}) is induced by the various inclusions

U → W_{G}H_{β}. Since we get K̃_{0}(Bℤ/2) = 0 and K̃_{1}(Bℤ/2) = ℤ[1/2]/ℤ = ℤ/2^{∞} from the Atiyah-Hirzebruch spectral sequence and [25, Proposition 2.11], the long exact sequence (4.25) becomes under this identification

0 → ⨁_{β∈B} K_{0}(BW_{G}H_{β}) ⊕ ⨁_{α∈A} SK̃_{0}(C_{r}^{∗}(M_{α})) → H_{0}^{G}(⋆_{G,FIN}; K̃)

→ ⨁_{β∈B′} ⨁_{(U)_{W_{G}H_{β}} ,|U|=2} K̃_{1}(BU) ⟶^{⨁_{β∈B′} μ(H_{β})_{1}⊕0} ⨁_{β∈B′} K_{1}(BW_{G}H_{β}) ⊕ ⨁_{β∈B′′} K_{1}(BW_{G}H_{β})

→ H_{1}^{G}(⋆_{G,FIN}; K̃) → 0  (4.29)

Now the map K̃_{1}(Bℤ/2)^{2} → K_{1}(B(ℤ/2∗ℤ/2) induced by the inclusion of two non-conjugated ℤ/2 is an isomorphism by the Mayer-Vietoris sequence, so the exact sequence (4.29) reduces to the isomorphisms

⨁_{β∈B} K_{0}(BW_{G}H_{β}) ⊕ ⨁_{α∈A} SK̃_{0}(C_{r}^{∗}(M_{α})) ⟶^{≅} H_{0}^{G}(⋆_{G,FIN}; K̃),
⨁_{β∈B′′} K_{1}(S^{1}) ⟶^{≅} H_{1}^{G}(⋆_{G,FIN}; K̃).

This shows that the abelian groups H_{q}^{G}(⋆_{G,FIN}; K̃) are free, so from the rationally split exact sequence (4.19), we get exact sequences

0 → ⨁_{β∈B′′} K_{1}(S^{1}) → K_{1}(C_{r}^{∗}(G)) → K_{1}(S) → 0

0 → ⨁_{β∈B} K_{0}(BW_{G}H_{β}) ⊕ ⨁_{α∈A} SK̃_{0}(C_{r}^{∗}(M_{α})) → K_{0}(C_{r}^{∗}(G)) → K_{0}(S) → 0.

It remains to show that the exact sequences above split. The group K_{0}(S) is always free abelian. If s = 0 and h ≠ g, then the quotient space is a closed non-orientable surface, so K_{1}(S) is not free but contains a direct summand ℤ/2. On the other hand, in this case B and hence B′′ is empty, so K_{1}(C_{r}^{∗}(G)) ≅ K_{1}(S). If h = g or s > 0, then K_{1}(S)) is free, so the claim follows.

Next we state the K-groups and L-theory explicitly in terms of the signature.

Definition 4.30 Let G be an infinite cocompact planar group with signature

(g, ±, [γ_{1}, γ_{2}, … γ_{t}], {(n_{1,1}, n_{1,2}, … n_{1,r_{1}−1}), … (n_{s,1}, n_{s,2}, … n_{s,r_{s}−1})})

Define b′ to be the number of elements n_{i,j} which are even and b′′ to be the number of integers i ∈ {1, 2, … s} for which r_{i} = 1 or each n_{i,j} for j = 1, 2, …, r_{i} − 1 is odd. (With this definition, b′ = b′′ = 0 if s = 0.)

Theorem 4.31 Let G, b′ and b′′ be given as in Definition 4.30. Recall that for a real number r we denote by [r] the largest integer which is less than or equal to r. Then

L_{0}(ℤG)[1/2] = (b′ + b′′ + 1 + ∑_{k=1}^{t}[γ_{k}/2] + ∑_{i=1}^{s} ∑_{j=1}^{r_{i}−1}[n_{i,j}/2]) · ℤ[1/2];

L_{1}(ℤG)[1/2] = \{ \begin{matrix}(b′′ + 2g) · ℤ[1/2] & for +, s = 0; \\ (b′′ + 2g + s − 1) · ℤ[1/2] & for +, s > 0; \\ (b′′ + g + s − 1) · ℤ[1/2] & for −; \end{matrix}

L_{2}(ℤG)[1/2] = \{ \begin{matrix}(1 + ∑_{k=1}^{t}[(γ_{k} − 1)/2]) · ℤ[1/2] & for +, s = 0; \\ (∑_{k=1}^{t}[(γ_{k} − 1)/2]) · ℤ[1/2] & otherwise; \end{matrix}
L_{3}(ℤG)[1/2] = 0;

K_{0}(C_{r}^{∗}(G)) = \{ \begin{matrix}(b′ + b′′ + 2 + ∑_{k=1}^{t}(γ_{k} − 1) + ∑_{i=1}^{s} ∑_{j=1}^{r_{s}−1}[n_{i,j}/2]) · ℤ & for +, s = 0; \\ (b′ + b′′ + 1 + ∑_{k=1}^{t}(γ_{k} − 1) + ∑_{i=1}^{s} ∑_{j=1}^{r_{s}−1}[n_{i,j}/2]) · ℤ & otherwise; \end{matrix}

K_{1}(C_{r}^{∗}(G)) = \{ \begin{matrix}(b′′ + 2g) · ℤ & for +, s = 0; \\ (b′′ + 2g + s − 1) · ℤ & for +, s > 0; \\ ℤ/2 ⊕ (b′′ + g − 1) · ℤ & for −, s = 0; \\ (b′′ + g + s − 1) · ℤ & for −, s > 0. \end{matrix}

\underline{\underline{Proof}} : This follows from Theorem 4.5 and the computation of K_{q}(S) and H_{p}(S; L(ℤ))[1/2] stated above as soon as we know that b′ and b′′ respectively is the cardinality of B′ and B′′ respectively. (The sets B′ and B′′ have been introduced in Definition 4.4). This follows for b′ from Theorem 4.3 (e) and Lemma 4.8 since the sets {(K)_{M_{α}} | K ∩ C_{α} = 1} for α ∈ A with |M_{α}| ≡ 0 (4) and the sets {(U)_{W_{G}H_{β}} | |U| = 2} for β ∈ B′ appearing in Lemma 4.8 all have cardinality 2.

Notice that x_{i,j} is conjugated to x_{i,j+1} if n_{i,j} is odd by Lemma 4.6 (d), x_{i,1} is always conjugated to x_{i,r_{i}} and N_{G}⟨x_{i,j}⟩ cannot be isomorphic to ℤ/2 × ℤ for even n_{i,j} by Lemma 4.6 (d). This shows for any β ∈ B′′ that there is an index i ∈ {1, 2, … s} such that ⟨x_{i,j}⟩ is conjugated to H_{β} for each j ∈ {1, 2, …, r_{i}}. Hence it remains to show that for two elements x_{i,j} and x_{i′,j′} which are conjugated, i = i′ holds. This follows from the geometric descriptions in terms of fundamental polygons (see e.g. [28], [48], [52]). Namely, a conjugating element g ∈ G maps the fixed point set of the reflections x_{i,j} and x_{i′,j′} to one another. Hence the images of their fixed point set under the quotient map onto S agree. This shows that x_{i,j} and x_{i′,j′} belong to the same boundary component of S and thus i = i′.

Remark 4.32 A cocompact Fuchsian group is a planar groups acting isometrically on the hyperbolic plane with s = 0, h = g, i.e. the quotient space is a closed orientable surface of genus g. Hence Theorem 4.5 implies Theorem 1 mentioned in the introduction.

We want briefly sketch the computation of L_{n}^{ϵ}(ℤG) without inverting 2 for a Fuchsian group F as in Theorem 1. For this purpose we will need for L^{ϵ} for ϵ = −∞, p, h or s that the isomorphism conjecture is true for G with respect to the family of virtually cyclic subgroups. Notice that Farrell and Jones [18] formulate their Isomorphism Conjecture only for L^{−∞} and that they have shown that it cannot be true simultaneously for ϵ = h and ϵ = s in the case G = ℤ^{2} × ℤ/5 [20]. However, using the various Rothenberg sequences together with the explicit computations of the lower and middle K-theory and the five lemma one can show in this particular case that the isomorphism conjecture with respect to VC and without inverting 2 is true for all of the decorated L-groups L^{−∞}, L^{p}, L^{h}, L^{s} if it holds for one of them. Since it is known for L^{−∞} for F [18, Theorem 2.1 and Remark 2.1.3], the computations below are true without any assumptions.

Let V be any subgroup of F isomorphic to ℤ/2∗ℤ/2. Its commutator [V, V] is an infinite cyclic subgroup which is characteristic in V. The centralizer C_{G}[V, V] is again an infinite cyclic subgroup by Theorem 4.3 (d). This implies that V_{max} := N_{G}[V, V] contains V and is isomorphic to ℤ/2 ∗ ℤ/2. Since [V, V] is a characteristic subgroup of V, we conclude N_{G}V ⊂ V_{max}. Let V ⊂ W ⊂ G be subgroups such that both V and W are isomorphic to ℤ/2 ∗ ℤ/2. Then [V, V] ⊂ [W, W] ⊂ C_{G}[W, W] ⊂ C_{G}[V, V]. Since C_{G}[V, V] is abelian, we get C_{G}[V, V] ⊂ C_{G}[W, W]. This implies C_{G}[V, V] = C_{G}[W, W] and hence V_{max} = W_{max}.

Let {V_{δ} | δ ∈ D} be a full system of representatives of the conjugacy classes of subgroups V ⊂ G which are maximal among the subgroups of G isomorphic to ℤ/2 ∗ ℤ/2. Hence each subgroup V ⊂ G with V ≅ ℤ/2 ∗ ℤ/2 is subconjugated to precisely one V_{δ}, namely the one with (V_{δ}) = (V_{max}), and we have N_{G}V = N_{V_{max}}V.

Using these facts one easily verifies that the following diagram is a pushout of contravariant Or(G)-spaces with a cofibration as upper horizontal map:

[figure: ∐_{δ∈D}(I_{δ})_{∗}(⋆_{V_{δ},FIN}); ∐_{δ∈D}(I_{δ})_{∗}(⋆_{V_{δ},ALL}); ⋆_{G,FIN}; ⋆_{G,VC}.]

Hence this diagram yields a long exact sequence

… → ⨁_{δ∈D} UNil_{q+1} → H_{q}^{F} (⋆_{F,FIN}; L^{ϵ}) → L_{q}^{ϵ}(ℤF) → ⨁_{δ∈D} UNil_{q} → …,

where UNil_{q} ≅ H_{q}^{G}(⋆_{V_{δ},ALL}, ⋆_{V_{δ},FIN}; L) is the UNil-term appearing in the splitting

L̃_{q}^{ϵ}(ℤ[ℤ/2]) ⊕ L̃_{q}^{ϵ}(ℤ[ℤ/2]) ⊕ UNil_{q} → L̃_{q}^{ϵ}(ℤ[ℤ/2 ∗ ℤ/2])

due to [12, Theorem 10] and

… → H_{q+1}(S; L(ℤ)) → ⨁_{i=1}^{t} L̃_{q}^{ϵ}(ℤ[ℤ/γ_{i}]) → H_{q}^{F} (⋆_{F,FIN}; L^{ϵ}) → H_{q}(S; L(ℤ)) → …,

where the last sequence splits after inverting the least common multiple m of the γ_{i}’s. The first exact sequence splits, too; a splitting is given by the obvious map

⨁_{δ∈D} UNil_{q} → ⨁_{δ∈D} L_{q}^{ϵ}(ℤV_{δ}) → L_{q}^{ϵ}(ℤF).

Hence we obtain an exact sequence which splits after inverting m

… → H_{q+1}(S; L(ℤ)) → ⨁_{i=1}^{t} L̃_{q}^{ϵ}(ℤ[ℤ/γ_{i}]) ⊕ ⨁_{δ∈D} UNil_{q} → L_{q}^{ϵ}(ℤF) → H_{q}(S; L(ℤ)) → ….

From now on suppose that each γ_{i} is odd. Then the number m above is odd. Since H_{q}(S; L(ℤ)) and ⨁_{i=1}^{t} L̃_{q}^{ϵ}(ℤ[ℤ/γ_{i}]) (see Theorem 3.2 (c)) contain no odd torsion, we obtain isomorphisms for q ∈ ℤ

⨁_{i=1}^{t} L̃_{q}^{ϵ}(ℤ[ℤ/γ_{i}]) ⊕ H_{q}(S; L(ℤ)) ≅ L_{q}^{ϵ}(ℤF).

Explicitly, we get from the computations of L̃_{q}^{ϵ}(ℤ/γ_{i}) in [1, Theorem 1, 3 and 5] and [2, Corollary 4.3 on page 58] for ϵ = p and s

L_{q}^{ϵ}(ℤF) ≅ \{ \begin{matrix}ℤ/2 ⨁ (1 + ∑_{i=1}^{t} \frac{γ_{i}−1}{2}) · ℤ & q ≡ 0 (4); \\ (2g) · ℤ & q ≡ 1 (4); \\ ℤ/2 ⨁ (1 + ∑_{i=1}^{t} \frac{γ_{i}−1}{2}) · ℤ & q ≡ 2 (4); \\ (2g) · ℤ/2 & q ≡ 3 (4). \end{matrix}

For ϵ = h, there is no general formula known for the 2-torsion contained in L̃_{2q}^{h}(ℤ[ℤ/m]), m odd, since it is given by the term Ĥ^{2}(ℤ/2; K̃_{0}(ℤ[ℤ/m])), see [1, Theorem 2].

5 Two-Dimensional Crystallographic Groups

In this section we give a complete description of the algebraic K- and L-groups of the integral group ring and the topological K-theory of the reduced C^{∗}-algebra of all 2-dimensional crystallographic groups. The algebraic K-theory in dimension ≤ 1 has been determined in [32], and the C^{∗}-K-theory has been computed in [51]. Note the difference between the group K_{0}(C_{r}^{∗}(Cmm)) given here and in [51, page 102]. We believe that this difference comes from a fixed point which has been overlooked in [51]. If we use the methods of [51], we get ℤ^{6}, as in the following table. As far as we know, this is the first computation of the L-groups of 2-dimensional crystallographic groups.

A two-dimensional crystallographic group is the same as a cocompact planar group G acting on ℝ^{2}. The signatures of crystallographic groups have been listed in [28, page 1204]. Hence the results for the L-theory and the topological K-theory below follow directly from Theorem 4.31. In the computation of the L-theory of the groups P 3 and P g we do not have to invert 2 because these two groups only contain infinite virtually cyclic groups which are isomorphic to ℤ. Hence the isomorphism conjecture for the family of finite subgroups without inverting 2 is true in this case, and a careful analysis as in the proof of Theorem 4.5 (c) shows our results. In the case of P 3, note that K̃_{0}(ℤ[ℤ/3]) = 0 by Theorem 3.2 (diii), so there is no 2-torsion coming from the maximal finite subgroups for any decoration ϵ.

The computations of Wh_{q}(ℤG) for q ≤ 1 follow directly from Theorem 3.2 (d), Theorem 4.3 (a) and Theorem 4.5 (a). For q = 2 we assume that the the isomorphism conjecture in algebraic K-theory (see 1.9 and Theorem 1.10) holds also in dimension 2. In some cases we can drop this assumption for the computation of Wh_{2}(ℤG). Since the assembly map is surjective in dimension 2, we can at least conclude Wh_{2}(G) = 0 if the second Whitehead groups of all finite subgroups vanish. Furthermore, in some cases (like P 6 or P 31m), there is only one finite subgroup with (possibly) non-trivial Wh_{2} which splits off from G, allowing us to compute Wh_{2}(G).

Our notation for the two-dimensional crystallographic groups follows that of [28].

Group  Signature  Wh_{q} ≠ 0, q ≤ 2  L_{q}(ℤG)  K_{q}(C_{r}^{∗}(G))
P 1  (1, +, [ ], { })  L_{0}  = ℤ ⊕ ℤ/2 K_{0} = ℤ^{2}
L_{1}  = ℤ^{2}  K_{1} = ℤ^{2}
L_{2}  = ℤ ⊕ ℤ/2
L_{3}  = (ℤ/2)^{2}
P 2 (0, +, [2, 2, 2, 2], \{ \})  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{5} K_{0} = ℤ^{6}
L_{1} [\frac{1}{2}] = 0  K_{1} = 0
L_{2} [\frac{1}{2}] = ℤ [\frac{1}{2}]
L_{3} [\frac{1}{2}] = 0
P 3 (0, +, [3, 3, 3], { })  L_{0}  = ℤ^{4} ⊕ ℤ/2 K_{0} = ℤ^{8}
L_{1}  = 0  K_{1} = 0
L_{2}  = ℤ^{4} ⊕ ℤ/2
L_{3}  = 0
P 4 (0, +, [2, 4, 4], \{ \})  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{6} K_{0} = ℤ^{9}
L_{1} [\frac{1}{2}] = 0  K_{1} = 0
L_{2} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{3}
L_{3} [\frac{1}{2}] = 0
P 6 (0, +, [2, 3, 6], \{ \}) K_{−1} = ℤ  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{6} K_{0} = ℤ^{10}
Wh_{2} = Wh_{2}(ℤ/6) L_{1} [\frac{1}{2}] = 0  K_{1} = 0
L_{2} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{4}
L_{3} [\frac{1}{2}] = 0
Cm (1, −, [ ], \{( )\}) L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{2} K_{0} = ℤ^{2}
L_{1} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{2} K_{1} = ℤ^{2}
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P m (0, +, [ ], \{( ), ( )\})  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{3} K_{0} = ℤ^{3}
L_{1} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{3} K_{1} = ℤ^{3}
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P g (2, −, [ ], { }) L_{0} = ℤ ⊕ ℤ/2 K_{0} = ℤ
L_{1}  = ℤ ⊕ ℤ/2 K_{1} = ℤ ⊕ ℤ/2
L_{2}  = ℤ/2
L_{3}  = (ℤ/2)^{2}
Cmm (0, +, [2], \{(2, 2)\}) Wh_{2} = Wh_{2}(D_{4})^{2} L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{6} K_{0} = ℤ^{6}
L_{1} [\frac{1}{2}] = 0  K_{1} = 0
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0

Group  Signature  Wh_{q} ≠ 0, q ≤ 2  L_{q}(ℤG)  K_{q}(C_{r}^{∗}(G))
P mm (0, +, [ ], \{(2, 2, 2, 2)\}) Wh_{2} = Wh_{2}(D_{4})^{4}  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{9} K_{0} = ℤ^{9}
L_{1} [\frac{1}{2}] = 0  K_{1} = 0
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P mg (0, +, [2, 2], \{( )\})  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{4} K_{0} = ℤ^{4}
L_{1} [\frac{1}{2}] = ℤ [\frac{1}{2}] K_{1} = ℤ
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P gg (1, −, [2, 2], \{ \}) L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{3} K_{0} = ℤ^{3}
L_{1} [\frac{1}{2}] = 0  K_{1} = ℤ/2
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P 3m1 (0, +, [3], \{(3)\}) Wh_{2} = ℤ/2  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{4} K_{0} = ℤ^{5}
L_{1} [\frac{1}{2}] = ℤ [\frac{1}{2}] K_{1} = ℤ
L_{2} [\frac{1}{2}] = ℤ [\frac{1}{2}]
L_{3} [\frac{1}{2}] = 0
P 31m (0, +, [ ], \{(3, 3, 3)\}) Wh_{2} = (ℤ/2)^{3}  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{5} K_{0} = ℤ^{5}
L_{1} [\frac{1}{2}] = ℤ [\frac{1}{2}] K_{1} = ℤ
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P 6m (0, +, [ ], \{(2, 3, 6)\}) Wh_{2} = Wh_{2}(D_{4}) ⊕ ℤ/2  L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{8} K_{0} = ℤ^{8}
⊕ Wh_{2}(D_{12}) L_{1} [\frac{1}{2}] = 0 K_{1} = 0
K_{−1} = ℤ  L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P 4m (1, +, [ ], \{(2, 4, 4)\}) Wh_{2} = Wh_{2}(D_{4}) ⊕ Wh_{2}(D_{8})^{2} L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{9} K_{0} = ℤ^{9}
L_{1} [\frac{1}{2}] = 0  K_{1} = 0
L_{2} [\frac{1}{2}] = 0
L_{3} [\frac{1}{2}] = 0
P 4g (0, +, [4], \{(2)\}) Wh_{2} = Wh_{2}(D_{4}) L_{0} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{5} K_{0} = ℤ^{6}
L_{1} [\frac{1}{2}] = 0  K_{1} = 0
L_{2} [\frac{1}{2}] = ℤ [\frac{1}{2}]^{1}
L_{3} [\frac{1}{2}] = 0

6 Extensions of Finite Groups with a Free Abelian Group

In this section we give the proof of Theorem 2. Some preparations are needed.

Lemma 6.1 Let G be as in Theorem 2. Then

(a) Every finite subgroup of G is isomorphic to a subgroup of π. For a non-trivial finite subgroup H ⊂ G its normalizer N_{G}H := {g ∈ G | gHg^{−1} = H} is finite;

(b) Every infinite virtually cyclic subgroup of G is either cyclic and contained in ℤ^{n} or isomorphic to ℤ/2 ∗ ℤ/2.

\underline{\underline{Proof}} : (a) Let H ⊂ G be finite. Then p|_{H} has trivial kernel, since it is a finite subgroup of ℤ^{n}. Hence H embeds into π.

Consider g ∈ N_{G}H. The map c_{g} on H given by conjugation with g is an automorphism of the finite group H and therefore of finite order itself, so there is a natural number k with (c_{g})^{k} = c_{g^{k}} = Id. Thus we have g^{k}hg^{−k} = h for all h ∈ H. Hence we get g^{km} ∈ ℤ^{n} and hg^{km}h^{−1} = g^{km} for all h ∈ H and m = |π|. Since H is non-trivial and π acts freely on ℤ^{n}, we get g^{km} = 1. Thus, every element of N_{G}H has finite order and cannot lie in the kernel of p. Hence N_{G}H is isomorphic to a subgroup of π, which is finite.

(b) Let C ⊂ G be infinite virtually cyclic. There is an extension 1 → ℤ → C ⟶^{q} F → 1 with a finite group F. Restricting p to C, we get an extension 1 → Z → C → p(C) → 1 where Z is a nontrivial subgroup of ℤ^{n}. Its image under q is a subgroup of the finite group F and hence Z ∩ ker(q) has finite index in Z. Therefore, Z must be infinite cyclic. Denote a generator of Z by z. As aut(ℤ) ≅ ℤ/2 and the action of p(C) on Z is free, we must have an injection p(C) → ℤ/2. If p(C) = 1, C ≅ Z. If p(C) = ℤ/2, choose t ∈ C whose image generates p(C). Then we must have t^{2} = z^{k} for some k ∈ ℤ since p(t)^{2} = 1. Thus,

z^{2k} = t^{4} = tz^{k}t = tt(t^{−1}z^{k}t) = z^{k}z^{−k} = 1 ⇒ k = 0,

and C is a nontrivial semi-direct product ℤ ⋊ ℤ/2 ≅ ℤ/2 ∗ ℤ/2.

Lemma 6.2 Let π be a finite group acting linearly on ℝ^{n} such that the action is free outside the origin. If |π| is odd, there exists to every prime divisor p of |π| a unique subgroup of π of order p. If |π| is even, there is a unique (and therefore central) element of order 2 in π.

\underline{\underline{Proof}} : If |π| is even, π contains exactly one non-trivial element of order two [29, Remark on page 624]. Hence it remains to treat the case where |π| is odd.

By [49, Theorems 5.3.1, 5.3.2 and 5.4.1], π is generated by two elements A, B such that for m and n the order of A and B, respectively, one has (m, n) = 1, BAB^{−1} = A^{r} with r^{n} ≡ 1 (m), and if d is the multiplicative order of r in ℤ/m, then for every prime p dividing d one has p|\frac{n}{d} and p^{2}|n. In particular there is a split extension 1 → ⟨A⟩ → π → ⟨B⟩ → 1. Any element x ∈ π can be written uniquely as A^{a}B^{b} for a ∈ ℤ/m and b ∈ ℤ/n.

Suppose that x^{p} = 1. Since x^{p} can be written as x = A^{c}B^{pb} for some integer c ∈ ℤ/m, we conclude pb = 0 (n). Suppose that p does not divide n. Then this implies B^{b} = 1 and hence x = A^{a} with ap = 0 (m). Hence x lies in the subgroup ⟨A^{m/p}⟩. Therefore ⟨A^{m/p}⟩ is the unique subgroup of π of order p if p does not divide n. Suppose that p divides n. Then p does not divide m and pb = 0 (n), and hence we can write b = k \frac{n}{p}. We have p|\frac{n}{d}, and so

B^{\frac{n}{p}} AB^{− \frac{n}{p}} = A^{r^{\frac{n}{p}}} = A^{r^{d \frac{n}{pd}}} = A.

Therefore x^{p} = A^{ap}B^{pb} = 1 which implies x ∈ ⟨B^{\frac{n}{p}} ⟩. Hence ⟨B^{\frac{n}{p}} ⟩ is the unique subgroup of π of order p if p divides n.

Lemma 6.3 Let G be as in Theorem 2. Then

(a) The intersection of any two different maximal finite subgroups of G is trivial;

(b) For H ⊂ G a non-trivial finite subgroup there is precisely one G-conjugacy class of subgroups (M) with M ⊂ G maximal finite such that (H) ≤ (M);

(c) Let M ⊂ G be a maximal finite subgroup and K, L ⊂ M. Then N_{M}K = N_{G}K and (K)_{M} = (L)_{M} ⇔ (K)_{G} = (L)_{G}.

\underline{\underline{Proof}} : We give only the proof of assertion (a), the other assertions are direct consequences using the conclusion from Lemma 6.1 (a) that N_{G}M = M holds for a maximal finite subgroup M ⊂ G.

Let M, N ⊂ G be maximal finite groups with non-trivial intersection H := M ∩ N. If |M| and |N| are odd, H contains a unique (normal) subgroup U of order p for any p dividing |H| by Lemma 6.2. Since M and N contain exactly one subgroup of order p, they both normalize U, so we must have N_{G}U = M = N, since M and N are maximal and N_{G}U is finite by Lemma 6.1 (a).

If |M| and |N| are even, M and N contain exactly one element of order two, say t_{M} and t_{N} respectively. These are central in M resp. N, so they are both contained in the finite group N_{G}H. Since N_{G}H is finite and acts freely on ℤ^{n} as well by Lemma 6.1 (a), it contains a unique element of order 2, so t_{M} = t_{N} =: t. Again, we have N_{G}⟨t⟩ = M = N because of the maximality of M and N and the finiteness of N_{G}⟨t⟩.

Suppose |M| is even, |N| is odd. Then H contains a subgroup U of prime order with N_{G}U = N. On the other hand, H ⊂ M is centralized by the unique element of order 2 in M, so t ∈ N_{G}U = N which is a contradiction.

Now we can give the proof of Theorem 2. \underline{\underline{Proof}} : (a) Consider the commutative diagram of contravariant Or(G)-spaces

[figure: ∐_{α∈A}(I_{α})_{∗}(⋆_{M_{α},TR}); ∐_{α∈A}(I_{α})_{∗}(⋆_{M_{α},ALL}); ⋆_{G,TR}; ⋆_{G,FIN}]  (6.4)

where the horizontal maps are given by inclusions and the vertical maps are the adjoints under the adjunction homeomorphism (2.1) of the inclusions of Or(M_{α})-spaces for the obvious functor I_{α} : Or(M_{α}) → Or(G). The evaluation of the diagram of Or(G)-spaces above at an object G/H gives pushouts of spaces with a cofibration as upper horizontal map. We get for this evaluation from Lemma 2.2 (b) and Lemma 6.3 (b) and (c)

\begin{matrix}∐_{α∈A} ∗ ×_{M_{α}} G & ⟶^{Id} & ∐_{α∈A} ∗ ×_{M_{α}} G \\ ↓ & & ↓ \\ ∗ & ⟶^{Id} & ∗ \end{matrix}  \begin{matrix}∅ & ⟶ & ∗ \\ ^{Id}↓ & ^{Id}↓ \\ ∅ & ⟶ & ∗ \end{matrix}  ∅ ⟶^{Id} ∅ ; ^{Id}↓  ^{Id}↓ ; ∅ ⟶^{Id} ∅

H = 1,  H ∈ FIN , H ≠ 1,  else.

Lemma 1.8, Theorem 1.10, Lemma 2.2 (c), Theorem 2.4, Lemma 2.6 and Lemma 6.1 (b) show that each map in the following composition is an isomorphism

⨁_{α∈A} H_{q}^{G}(⋆_{M_{α},ALL}, ⋆_{M_{α},TR}; K) ⟶^{≅} H_{q}^{G}(⋆_{G,FIN} , ⋆_{G,TR}; K)

⟶^{≅} H_{q}^{G}(⋆_{G,VC}, ⋆_{G,TR}; K) ⟶^{≅} H_{q}^{G}(⋆_{G,ALL}, ⋆_{G,TR}; K).

The assertions now follow from Lemma 2.5 and the fact that K_{q}(ℤK) = 0 for a finite group K and q ≤ −2 [13].

(b) Lemma 1.8 and Lemma 2.2 (c) applied to (6.4) and L̃ yield an isomorphism

⨁_{α∈A} H_{q}^{M_{α}}(⋆_{M_{α},ALL}; L̃) ⟶^{≅} H_{q}^{G}(⋆_{G,FIN}; L̃).

Together with Theorem 1.10 and Lemma 2.9 (b) we obtain an exact sequence which splits into short exact sequences after tensoring with ℤ [\frac{1}{2|π|}]

… → H_{q+1}(G\\underline{E}G; L(ℤ))[1/2] ⟶^{δ_{q+1}} ⨁_{α∈A} L̃_{q}(ℤM_{α})[1/2] → L_{q}(ℤG)[1/2]

→ H_{q}(G\\underline{E}G; L(ℤ))[1/2] ⟶^{δ_{q}} …,

Since δ_{q} ⊗ ℚ is trivial and its target ⨁_{α∈A} L̃_{q−1}(ℤM_{α})[1/2] is free as a ℤ[1/2]-module by Theorem 3.2 (c), δ_{q} is trivial for all q ∈ ℤ. Hence assertion (b) follows if we can find an isomorphism

H_{q}(G\\underline{E}G; L(ℤ)) [\frac{1}{2|π|}] ≅ H_{q}(π\T^{n}; L(ℤ)) [\frac{1}{2|π|}] .

Since ℤ^{n} is torsion free and therefore acts freely on \underline{E}G, ℤ^{n}\\underline{E}G is a model for Bℤ^{n}. Choose a homotopy equivalence f : ℤ^{n}\\underline{E}G → T^{n} which induces on π_{1} = H_{1} the obvious map H_{1}(ℤ^{n}\E(G, FIN)) ≅ ℤ^{n} ≅ H_{1}(ℝ^{n}/ℤ^{n}) = H_{1}(T^{n}). This map f respects the obvious π = ℤ^{n}\G-operation on ℤ^{n}\\underline{E}G and the π-action on T^{n} by conjugation up to homotopy since the isomorphism H_{1}(f) is π-equivariant. Thus f induces a ℤπ-isomorphism H_{p}(f) :

H_{p}(ℤ^{n}\\underline{E}G) ⟶^{≅} H_{p}(T^{n}) for p ≥ 0. By the Atiyah-Hirzebruch spectral sequence f induces

an isomorphism H_{p}(ℤ^{n}\\underline{E}G; L(ℤ))⊗_{ℤπ}ℤ [\frac{1}{2|π|}] ⟶^{≅} H_{p}(T^{n}; L(ℤ))⊗_{ℤπ}ℤ [\frac{1}{2|π|}] . Now the claim

follows since the projections induce isomorphisms

H_{p}(ℤ^{n}\\underline{E}G; L(ℤ)) ⊗_{ℤπ} ℤ [\frac{1}{2|π|}] ⟶^{≅} H_{p}(G\\underline{E}G; L(ℤ)) [\frac{1}{2|π|}];
H_{p}(T^{n}; L(ℤ)) ⊗_{ℤπ} ℤ [\frac{1}{2|π|}] ⟶^{≅} H_{p}(π\T^{n}; L(ℤ)) [\frac{1}{2|π|}] .

(c) is proven analogously to assertion (b). This finishes the proof of Theorem 2.

Remark 6.5 Let G be as in Theorem 2. We want briefly sketch the computation of L_{n}^{ϵ}(ℤG) without inverting 2. For this purpose we will need for L^{ϵ} for ϵ = −∞, p, h or s that the isomorphism conjecture is true for G with respect to the family of virtually cyclic subgroups. Notice that Farrell and Jones [18] formulate their Isomorphism Conjecture only for L^{−∞} and that they have shown that it cannot be true simultaneously for ϵ = h and ϵ = s in the case G = ℤ^{2} × ℤ/5 [20]. However, using the various Rothenberg sequences together with the explicit computations of the lower and middle K-theory and the five lemma one can show in this particular case that the isomorphism conjecture with respect to VC and without inverting 2 is true for all of the decorated L-groups L^{−∞}, L^{p}, L^{h}, L^{s} if it holds for one of them. Since it is known for L^{−∞} for G as in Theorem 2 [18, Theorem 2.1 and Remark 2.1.3], the computations below are true without any assumptions. Notice that ℤ^{2} × ℤ/5 does not fall under the groups appearing in Theorem 2.

Let {V_{δ} | δ ∈ D} be a full system of representatives of the conjugacy classes of subgroups V ⊂ G which are maximal among the subgroups of G isomorphic to ℤ/2 ∗ ℤ/2. Notice for any virtually cyclic subgroup V ⊂ G with V ≅ ℤ/2 ∗ ℤ/2 that N_{G}V is subconjugated to precisely one V_{δ}. One can show by the same methods as before that one obtains long exact sequences

… → ⨁_{δ∈D} UNil_{q+1} → H_{q}^{G}(⋆_{G,FIN}; L^{ϵ}) → L_{q}^{ϵ}(ℤG) → ⨁_{δ∈D} UNil_{q} → …,

where UNil_{q} is the UNil-term appearing in [12, Theorem 10] and

… → H_{q+1}(G\\underline{E}G; L(ℤ)) → ⨁_{α} L̃_{q}^{ϵ}(ℤM_{α}) → H_{q}^{G}(⋆_{G,FIN}; L^{ϵ}) → H_{q}(G\\underline{E}G; L(ℤ)) → …,

where the last sequence splits after inverting |π|. The first exact sequence splits, too; a splitting is given by the obvious map

⨁_{δ∈D} UNil_{q} → ⨁_{δ∈D} L̃_{q}^{ϵ}(ℤV_{δ}) → L_{q}^{ϵ}(ℤG).

Hence we obtain an exact sequence which splits after inverting |π|

… → H_{q+1}(G\\underline{E}G; L(ℤ)) → ⨁_{α} L̃_{q}^{ϵ}(ℤM_{α})⊕⨁_{δ∈D} UNil_{q} → L_{q}^{ϵ}(ℤG) → H_{q}(G\\underline{E}G; L(ℤ)) → ….

From now on suppose that |π| is odd. Since ⨁_{α} L̃_{q}^{ϵ}(ℤM_{α}) contains no odd torsion (see Theorem 3.2 (c)) and G contains no subgroup of order 2, we obtain a short exact sequence which splits after inverting |π|

0 → ⨁_{α} L̃_{q}^{ϵ}(ℤM_{α}) → L_{q}^{ϵ}(ℤG) → H_{q}(G\\underline{E}G; L(ℤ)) → 0.

Remark 6.6 A variation of the proofs of Theorems 4.5 and 2 yields a similar result in the case of an extension 1 → ℤ^{n} → G ⟶^{p} π → 1, where π is now a dihedral group D_{2m} = ⟨s, t | s^{m} = t^{2} = (st)^{2} = 1⟩ (note that by Lemma 6.2, π cannot act freely on ℤ^{n} if m > 1) such that the cyclic subgroup ⟨s⟩ of order m acts freely on ℤ^{n}. We only have to replace the system {H_{β}} in Theorem 4.5 by a full system of representatives of conjugacy classes of subgroups H of order 2 with p(H) ∩ ⟨s⟩ = 1 and to replace the system {M_{α}} by a full system of conjugacy classes of maximal finite subgroups M of G with p(M) ∩ ⟨s⟩ ≠ 1. Put B′ = {β ∈ B | p(N_{G}H_{β}) ≠ p(H_{β})} and B′′ = B \ B′. Then the claims and proofs for algebraic K- and L-theory are the same as in Theorem 4.5. Let m(t) and m(β) respectively be the non-negative integer satisfying ℤ^{m(t)} ≅ (ℤ^{n})^{⟨t⟩} and ℤ^{m(β)} ≅ (ℤ^{n})^{p(H_{β})} respectively. For instance for the topological K-theory, we get

(a) Suppose that m is odd. Then B = B′′ and we obtain short exact sequences

0 → |B| · K_{q}(T^{m(t)}) ⊕ ⨁_{α} SK̃_{q}(C_{r}^{∗}(M_{α})) → K_{q}(C_{r}^{∗}(G)) → K_{q}(G\\underline{E}G) → 0;

(b) Suppose m is even. Then we obtain the long exact sequence

… → K_{q+1}(G\\underline{E}G) → H_{q}^{G}(⋆_{G,FIN}; K̃) → K_{q}(C_{r}^{∗}(G)) → K_{q}(G\\underline{E}G) → …,

the short exact sequence

0 → ⨁_{β∈B} K_{0}(T^{m(β)}) ⊕ ⨁_{α} SK̃_{0}(C_{r}^{∗}(M_{α})) → H_{0}^{G}(⋆_{G,FIN}; K̃) →

⨁_{β∈B′} coker (K_{0}(T^{m(β)}) → K_{0}(ℤ/2\T^{m(β)})) → 0

and the isomorphism

⨁_{β∈B′′} K_{1}(T^{m(β)}) ⟶^{≅} H_{1}^{G}(⋆_{G,FIN}; K̃),

where ℤ/2 acts on ℤ^{m(β)} by − Id and thus on T^{m(β)}.

If we invert 2, these exact sequences reduce to short exact sequences

0 → ⨁_{β∈B} K_{q}(BW_{G}H_{β})[1/2] ⊕ ⨁_{α∈A} SK̃_{q}(C_{r}^{∗}(M_{α}))[1/2] → K_{q}(C_{r}^{∗}(G))[1/2]

→ K_{q}(G\\underline{E}G)[1/2] → 0

(c) If we invert m, the short exact sequences above split into short exact sequences and isomorphisms respectively and we obtain isomorphisms

K_{q}(D_{2m}\T^{n})[1/m] ⊕ ⨁_{β∈B} K_{q}(BW_{G}H)[1/m] ⊕ ⨁_{α∈A} SK̃_{q}(C_{r}^{∗}(M_{α})[1/m]

⟶^{≅} K_{q}(C_{r}^{∗}(G))[1/m].

For m(β) > 1, the group coker (K_{0}(T^{m(β)}) → K_{0}(ℤ/2\T^{m(β)})) is non-trivial.

For the proof of the above results cf. [45].

References

[1] Bak, A.: “The Computation of Surgery Groups of Finite Groups with Abelian 2-Hyperelementary Subgroups”, Lecture Notes in Mathematics 551, 384 – 409, Springer (1976).

[2] Bak, A. and Kolster, A.: “The computation of odd dimensional projective surgery groups of finite groups”, Topology 21, 35 – 63 (1982).

[3] H. Abels: “A universal proper G-space”, Math. Z. 159, 143 – 158 (1978).

[4] Bass, H.: “Algebraic K-theory”, Benjamin (1968).

[5] Bass, H., Heller, A., and Swan, R.: “The Whitehead group of a polynomial extension”, Publ. Math. IHES 22, 61 – 80 (1964).

[6] Baum, P., Connes, A., and Higson, N.: “Classifying space for proper actions and K-theory of group C^{∗}-algebras”, in: Doran, R.S. (ed.), C^{∗}-algebras,Contemporary Mathematics 167, 241 – 291 (1994).

[7] Berkhove, E., Juan-Pineda, D. and Pearson, K.: “The lower algebraic K-theory of Fuchsian groups”, preprint (1998).

[8] Blackadar, B.: “K-theory for operator algebras”, M.S.R.I. Monographs 5, Springer (1986).

[9] Bousfield, A.K. and Kahn, D.M.: “Homotopy limits, completions and localizations”, Lecture notes in mathematics 304, Springer (1972).

[10] Brown, K.S.: “Cohomology of groups”, Graduate texts in Mathematics 87, Springer (1982).

[11] Carlsson, G. and Pedersen, E.K.: “Controlled algebra and the Baum-Connes conjecture”, in preparation (1998).

[12] Cappell, S.: “Unitary nilpotnent groups and Hermitian K-theory I”, Bull. of the AMS 80, 1117 – 1122 (1974).

[13] Carter, D.: “Lower K-theory of finite groups”, Comm. Alg. 8, 1927 – 1937 (1980).

[14] Curtis, C.W. and Reiner, I.: “Methods of representation theory volume II ”, Wiley (1987).

[15] J. Davis and W. Lück, Spaces over a Category, Assembly Maps in Isomorphism Conjecture in K-and L-Theory, K-theory 15, 201 – 252 (1998).

[16] Dennis, R.K., Keating, M. and Stein, M.: “Lower bounds for the order of K_{2}(ℤG) and Wh_{2}(G)”, Math. Ann. 223, 97 – 103 (1976).

[17] Dunwoody, M.J.: “K_{2}(ℤπ) for π a group of order 2 or 3”, J. London Math. Soc. (2) 11, 481 – 490 (1975).

[18] Farrell, F.T. and Jones, L.E.: “Isomorphism conjectures in algebraic K-theory”, J. of the AMS 6, 249 – 298 (1993).

[19] Farrell, F.T. and Jones, L.E.: “The lower algebraic K-theory for virtually infinite cyclic groups”, K-theory 9, 13 – 30 (1995).

[20] Farrell, F.T. and Jones, L.E., private communication (2000).

[21] Higson, N. and Kasparov, G.: “Operator K-theory for groups which act properly and isometrically on Hilbert Space”, preprint (1997).

[22] Hoare, A.H.M., Karrass, A. and Solitar, D.: “Subgroups of infinite index in Fuchsian groups”, Math. Z. 125, 59 – 69, (1972).

[23] Hoare, A.H.M., Karrass, A. and Solitar, D.: “Subgroups of NEC Groups”, Comm. Pure Appl. Math. 26, 731 – 744, (1973).

[24] Hollender, J. and Vogt, R.M.: “Modules of topological spaces, applications to homotopy limits and E_{∞}-structures”, Arch. Math. 59, 115 – 129 (1992).

[25] Knapp, K.: “On the K-homology of classifying spaces”, Math. Ann. 233, 103 – 124 (1978).

[26] W. Lück: “Transformation groups and algebraic K-theory”, Lecture Notes in Mathematics 1408, Springer (1989).

[27] W. Lück: “Chern characters for proper equivariant homology theories and applications to K- and L-theory”, Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Münster (2000).

[28] Macbeath, A.M.: “The classification of non-Eucledian plane crystallographic groups.”, Can. J. Math 19, 1192 – 1205 (1967).

[29] Milnor, J.: “Groups which act on S^{n} without fixed points”, Amer. J Math 79, 623 - 630 (1957).

[30] Milnor, J.: “Introduction to algebraic K-theory”, Annals of mathematics studies 72, Princeton University Press (1971).

[31] Oliver, R.: “Whitehead groups of finite groups”, LMS Lecture Notes Series 132, Cambridge University Press (1989).

[32] Pearson, K.: “Algebraic K-theory of two-dimensional crystallographic groups”, Ktheory 14, 265 – 280 (1998).

[33] Pedersen, G.K.: “C^{∗}-algebras and their automorphisms groups”, LMS monograph 14, Academic Press (1989).

[34] Pedersen, E.K. and Weibel, C.A.: “A non-connective delooping of algebraic Ktheory”, in “Algebraic and Geometric Topology”, proc. conf. Rutgers Univ., New Brunswick 1983, Lecture Notes in Mathematics 1126, Springer, 166 – 181 (1985).

[35] Prasolov, A.V.: “Infiniteness of the Groups Nil”, Math. Notes 32, pp. 484 – 485 (1983).

[36] Quillen, D.: “On the Cohomology and K-Theory of the General Linear Group over a Finite Field”, Ann. Math 96, 552 – 586 (1972).

[37] Quillen, D.: “Higher algebraic K-theory. I.”, in: Proceedings of Battelle Seattle Algebraic K-theory Conference 1972, vol. I , Lecture notes in mathematics 341, 85 – 147, Springer (1973).

[38] Quillen, D.: “Finite Generation of the Groups K_{i} of Rings of Algebraic Integers”, Lecture notes in mathematics 341, 179 – 198, Springer (1973).

[39] Quinn, F.: “Ends of maps II”, Inventiones Math. 68, 353 - 424 (1982).

[40] Ranicki, A.: “Algebraic L-theory and topological manifolds”, Cambridge Tracts in Mathematics 102, Cambridge University Press (1992).

[41] Reiner, I.: “Class groups and Picard groups of group rings and orders”, Conference Board of the Mathematical sciences 26 (1976).

[42] Serre, J.P.: “Linear representations of finite groups”, Graduate texts in Mathematics 42 , Springer (1977).

[43] Serre, J-P.: “Trees”, Springer (1980).

[44] Shaneson, J.: “Wall’s surgery obstruction group for G × Z”, Ann. of Math. 90, 296 –334 (1969).

[45] Stamm, R.: “The K- and L-Theory of Certain Discrete Groups”, Ph. D. thesis, Münster (1999).

[46] Stein, M. R.: “Excision and K_{2} of group rings”, J. of Pure and Appl. Alg. 18, 213 – 224 (1980).

[47] Waldhausen, F.: “Algebraic K-theory of generalized free products I,II”, Ann. of Math. 108, 135 – 256 (1978).

[48] Wilkie, H. C.: “On non-Euclidean crystallographic groups”, Math. Z. 91, 87 – 102 (1966).

[49] Wolf, J.A.: “Spaces of constant curvature”, Publish or Perish, 5th edn. (1984).

[50] Yamasaki, M.: “L-groups of crystallographic groups”, Invent. Math. 88, 571 – 602 (1987).

[51] Yang, M.: “Crossed Products by Finite Groups Acting on Low Dimensional Complexes and Applications”, Ph.D. thesis, University of Saskatchewan, Saskatoon (1997).

[52] Zieschang, H., Vogt, E., Coldewey, H.-D.: “Surfaces and planar discontinuous groups”, Lecture notes in mathematics 835, Springer (1980).

Addresses:

Wolfgang Lück and Roland Stamm Institut für Mathematik und Informatik Westfälische Wilhelms-Universtität Einsteinstr. 62, 48149 Münster, Germany lueck@math.uni-muenster.de, stammr@math.uni-muenster.de http://wwwmath.uni-muenster.de/math/u/lueck