Contemporary Mathematics

Novikov-Shubin invariants for arbitrary group actions and their positivity

Wolfgang Lück, Holger Reich, and Thomas Schick

Dedicated to Mel Rothenberg on the occasion of his 65th birthday

Abstract. We extend the notion of Novikov-Shubin invariant for free Γ-CW - complexes of finite type to spaces with arbitrary Γ-actions and prove some statements about their positivity. In particular we apply this to classifying spaces of discrete groups.

1. Introduction

In [8] the first author extended the notion of L^{2}-Betti number for free Γ-CW - complexes of finite type to topological spaces with arbitrary Γ-actions. The key ingredient there is the notion of a dimension function for arbitrary modules over a finite von Neumann algebra A, extending the classical notion for finitely generated projective A-modules defined in terms of the von Neumann trace of an associated projection. These notions turned out to be useful in particular in the case where Γ is amenable (see [8], [9]).

In this paper we carry out an analogous program for Novikov-Shubin invariants. So we will introduce for arbitrary A-modules in Section 2 the equivalent notion of capacity (which is essentially the inverse of the Novikov-Shubin invariant and was introduced for finitely presented A-modules in [2] and [7]) and study its main properties. This enables us to define the p-th capacity for a space X with an arbitrary action of a discrete group. Originally Novikov-Shubin invariants were defined in terms of the heat kernel of the universal covering of a compact Riemannian manifold in [10], [11].

We will use the extension to study Novikov-Shubin invariants of groups in Section 3. The key observation is that a group Γ which may have a model of finite type for BΓ may contain interesting normal subgroups for which the classical definition does not apply because its classifying space is not even of finite type. We are in particular interested in the conjecture that for a regular covering of a CW -complex of finite type the Novikov-Shubin invariants are always positive, or equivalently, the capacities are always finite [6, Conjecture 7.1]. Our main results

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

1991 Mathematics Subject Classification. 55R40 (primary), 58G25,55T10 (secondary).

Key words and phrases. Novikov-Shubin invariants, elementary amenable groups.

in this direction are Theorems 3.7 and 3.9 which say among other things that α_{p}(Γ) ≥ 1 for all p ≥ 1 if Γ contains ℤ^{n} as normal subgroup for some n ≥ 1 and that α_{p}(Γ) ≥ 1 for p = 1, 2 if Γ is elementary amenable and contains no infinite locally finite subgroup. In particular the first and second Novikov-Shubin invariants of the universal covering of a CW -complex X of finite type are greater or equal to 1 if π_{1}(X) satisfies the condition above since they agree with the ones for π_{1}(X).

We mention that α_{p} of a space or group is the Novikov-Shubin invariant associated to the p-th differential. Sometimes Novikov-Shubin invariants are also defined in terms of the Laplacian. If we denote the latter ones by α̃_{p}, the connection between these two invariants is α̃_{p} = min{α_{p+1}, α_{p}}. Our normalization is such that α_{1} = α̃_{0} of the universal covering of S^{1} is 1. The p-th capacity will essentially be the inverse of the (p + 1)-th Novikov-Shubin invariant.

2. Capacity of modules

There is the notion of the Novikov-Shubin invariant α(M) ∈ [0, ∞] ⨿ {∞^{+}} of a finitely presented A-module M as defined in [7, Definition 3.1]. It is the Novikov-Shubin invariant of the spectral density function of a morphism of Hilbert A-modules f : l^{2}(A)^{m} → l^{2}(A)^{n} which corresponds to a presentation matrix A ∈ M(m, n, A) for M = coker(A : A^{m} → A^{n}). We want to extend it to arbitrary Amodules. For convenience we will use the notion of capacity (see [2, 4.8]) which is essentially the reciprocal of the Novikov-Shubin invariant and extend it to arbitrary A-modules. In the sequel we will use the notion and properties of the dimension dim(M) of an A-module introduced in [7, Theorem 0.6].

Definition 2.1. We call an A-module M measurable if it is the quotient of a finitely presented A-module L with dim(L) = 0. We call an A-module M cofinalmeasurable if each finitely generated submodule is measurable. In particular this implies dim M = 0.

We will see later that the following definition of capacity is particularly well behaved on the the class of cofinal-measurable modules.

Definition 2.2. Let L be a finitely presented A-module with dim(L) = 0. Define its capacity

c(L) ∈ {0^{−}} ∪ [0, ∞]

by

c(L) := \frac{1}{α(L)},

where 0^{−} is a new formal symbol different from 0, and 0^{−1} = ∞, ∞^{−1} = 0 and (∞^{+})^{−1} = 0^{−}. Let M be a measurable A-module. Define

c′(M) := inf{c(L) | L finitely presented, dim(L) = 0, M quotient of L}.

Let N be an arbitrary A-module. Define (see also Notation 2.5)

c′′(N) := sup{c′(M) | M measurable, M ⊂ N}.

Note that dim(N) is not necessarily zero. In fact c′′(N) measures the size of the largest zero-dimensional submodule of N (compare [8, 2.15]).

The invariants take value in {0^{−}} ⨿ [0, ∞]. We define an order < on this set by the usual one on [0, ∞) and the rule 0^{−} < r < ∞ for r ∈ [0, ∞). For two elements r, s ∈ [0^{−}, ∞] we define another element r + s = s + r in this set by the ordinary addition in [0, ∞) and by

0^{−} + r = r and ∞ + r = ∞ for r ∈ [0^{−}, ∞]

We have introduced c(M) as the inverse of α(M) because then c(M) becomes bigger if M becomes bigger and some of the formulas become nicer. Notice that for a finitely presented A-module M with dim(M) = 0 we have c(M) = 0^{−} if and only if M is trivial. Hence a measurable A-module M satisfies c′(M) = 0^{−} if and only if it is trivial. A measurable A-module is finitely generated but not necessarily finitely presented. We have for an arbitrary A-module N that c′′(N) = 0^{−} if and only if any A-map f : M ⟶ N from a finitely presented A-module M with dim(M) = 0 to N is trivial. A cofinal-measurable A-module M is trivial if and only if c′′(M) = 0^{−} (see also Example 2.10).

We will show that our different definitions of capacity coincide and prove their basic properties. For this we need

Lemma 2.3. Let 0 → K → P → Q → 0 be an exact sequence of finitely presented A-modules of dimension zero. Then

\begin{aligned}c(K) & ≤ c(P), c(Q) ≤ c(P) \\ c(P) & ≤ c(K) + c(Q). \end{aligned}

Proof. By [7, 3.4] we find resolutions 0 → F_{K} → F_{K} → K → 0 and 0 → F_{Q} → F_{Q} → Q → 0 with F_{K} and F_{Q} finitely generated free. Now we can construct a resolution 0 → F → F → P → 0 with F = F_{K} ⊕ F_{Q} which fits into a short exact sequence of resolutions. Application of [6, Lemma 1.12] to this situation together with the equivalence of finitely generated Hilbert A-modules and finitely generated projective algebraic A-modules as in [7] gives the result.

Proposition 2.4. 1. If M is a finitely presented A-module which satisfies dim(M) = 0, then M is measurable and

c′(M) = c(M).

2. If M is a measurable A-module, then M is cofinal-measurable and

c′′(M) = c′(M).

Notation 2.5. In view of Proposition 2.4 we sometimes will not distinguish between c, c′ and c′′ in the sequel.

Proof of Proposition 2.4. If M and N are finitely presented with dimension zero and p : N → M is surjective, then c(M) ≤ c(N) by Lemma 2.3. We use that ker p is finitely presented by [7, Theorem 0.2]. The first statement follows.

Suppose M is measurable and M_{0} ⊂ M is finitely generated. We have to show that M_{0} is measurable and c′(M_{0}) ≤ c′(M). Suppose L is finitely presented with dim(L) = 0 and there is an epimorphism f : L ⟶ M. As M_{0} is finitely generated, we can find a finitely generated module K ⊂ L with f(K) = M_{0}. As K is finitely generated and L is finitely presented, L/K is finitely presented. Since the category of finitely presented A-modules is abelian [7, Theorem 0.2], K is finitely presented. Hence M_{0} is measurable. Moreover c(K) ≤ c(L) by Lemma 2.3. Therefore

c′(M_{0}) ≤ inf_{K as above} c(K) ≤ inf_{L as above} c(L) = c′(M).

Before we give a list of the basic properties of capacity, we state a simple lemma which will be used repeatedly during the proof.

Lemma 2.6. Suppose M is a measurable A-module and p : F → M a projection of a finitely generated free module onto M. Then

c(M) = inf{c(F/K)| K ⊂ ker p finitely generated with dim(F/K) = 0}.

The set on the right hand side is nonempty if and only if M is measurable.

Proof. Let d denote the number on the right. Every F/K as above is finitely presented and projects onto M. From Definition 2.2 we get d ≥ c(M). On the other hand let q : L → M be an epimorphism with L finitely presented and dim L = 0. We can lift p to a map f with q ∘ f = p. Since the category of finitely presented modules is abelian [7, Theorem 0.2] the kernel of f is finitely generated. Moreover dim(F/ ker f) = dim im f ≤ dim L = 0, ker f ⊂ ker p and Lemma 2.3 implies c(F/ ker f) ≤ c(L). So for every L we found an F/K with c(F/K) ≤ c(L). This implies d ≤ c(M).

Theorem 2.7. 1. Let 0 ⟶ M_{0} ⟶^{i} M_{1} ⟶^{p} M_{2} ⟶ 0 be an exact sequence

of A-modules. Then (a) c(M_{0}) ≤ c(M_{1}).

(b) c(M_{2}) ≤ c(M_{1}), provided that M_{1} is cofinal-measurable.

(c) c(M_{1}) ≤ c(M_{0}) + c(M_{2}) if dim M_{1} = 0.

2. Let M = ⋃_{i∈I} M_{i} be a directed union of submodules. Then

c(M) = sup{c(M_{i}) | i ∈ I}.

3. Let M be the colimit colim_{i∈I} M_{i} of a directed system of A-modules with structure maps ϕ_{ij} : M_{i} → M_{j}. Then

c(M) ≤ lim inf_{i∈I} c(M_{i}) (:= sup_{i∈I}{inf_{j≥i} c(M_{j})}) .

If every M_{i} is measurable and ϕ_{ij} is surjective for every j ≥ i, then

c(M) = inf_{i∈I} c(M_{i}).

4. Let {M_{i} | i ∈ I} be a family of A-modules. Then

c (⊕_{i∈I}M_{i}) = sup{c(M_{i}) | i ∈ I}.

Remark 2.8. Because zero-dimensional modules will be most important for us, we give a reminder of basic properties of the dimension, which are stated in or follow from [8]:

1. If 0 → M_{0} → M_{1} → M_{2} → 0 is an exact sequence of A-modules, then

dim(M_{1}) = 0 ⟺ dim(M_{0}) = dim(M_{2}) = 0.

2. dim(⊕_{i∈I}(M_{i})) = 0 ⟺ dim(M_{i}) = 0 for all i ∈ I.

3. If M = ⋃_{i∈I} M_{i}, then dim M = 0 ⟺ dim M_{i} = 0 for all i ∈ I.

4. If M = colim_{i∈I} M_{i} is the colimit of a directed system, then

dim M ≤ lim inf_{i∈I} dim M_{i}.

Proof of Theorem 2.7. 1a) Every measurable submodule of M_{0} is a measurable submodule of M_{1}, therefore c′′(M_{0}) ≤ c′′(M_{1}).

1b) For every measurable submodule M of M_{2} we find a finitely generated submodule N_{M} ⊂ M_{1} which projects onto M. If M_{1} is cofinal-measurable, then N_{M} is measurable and c′(N_{M}) ≥ c′(M) since on the left we take the infimum over a smaller set of numbers. Therefore

c′′(M_{1}) ≥ sup_{M⊂M_{2} measurable} c′(N_{M}) ≥ sup_{M⊂M_{2} measurable} c′(M) = c′′(M_{2}).

1c) Step 1: We prove c(M_{1}) ≤ c(M_{0}) + c(M_{2}) if M_{0} is measurable and M_{2} is finitely presented. We will also see, that this implies, that M_{1} is measurable:

By [7, Lemma 3.4] there is a finitely generated free A-module F_{2} and an exact sequence 0 ⟶ F_{2} ⟶^{i_{2}} F_{2} ⟶ M_{2} ⟶ 0. Let q : F_{0} → M_{0} be a projection of a finitely generated free A-module onto M_{0}.

We get the following commutative diagram with exact rows and columns where the F_{i} are finitely generated free:

0  0  0
↑  ↑  ↑
0 ⟶ M_{0} ⟶ M_{1} ⟶ M_{2} ⟶ 0
↑  ↑  ↑
0 ⟶ F_{0} ⟶ F_{1} ⟶ F_{2} ⟶ 0
↑  ↑  ↑
0 ⟶ K_{0} ⟶ K_{1} ⟶ i_{2}(F_{2}) ⟶ 0
↑  ↑  ↑
0  0  0

Let K′_{0} ⊂ K_{0} be a finitely generated submodule with dim F_{0}/K′_{0} = 0 (Lemma 2.6). We can consider K′_{0} also as submodule of K_{1}. Let s : i_{2}(F_{2}) ⟶ K_{1} be a section of the epimorphism K_{1} ⟶ i_{2}(F_{2}). Let K′_{1} be the (finitely generated!) submodule of K_{1} generated by K′_{0} and the image of s. We obtain the exact sequence

0 → K′_{0} → K′_{1} → i_{2}(F_{2}) → 0.

Going to the quotients, we obtain a commutative diagram with epimorphisms as vertical maps whose lower row is an exact sequence of finitely presented A-modules

0 ⟶ M_{0} ⟶ M_{1} ⟶ M_{2} ⟶ 0
↑  ↑  ↑^{id}
0 ⟶ F_{0}/K′_{0} ⟶ F_{1}/K′_{1} ⟶ M_{2} ⟶ 0.

Note that this implies M_{1} to be measurable. By Lemma 2.3

c(M_{1}) ≤^{1b)} c(F_{1}/K′_{1}) ≤ c(F_{0}/K′_{0}) + c(M_{2}).

This holds for every K′_{0} as above, therefore also for the infimum c(M_{0}) (by Lemma 2.6) in place of c(F_{0}/K′_{0}).

Step 2: We prove the inequality if M_{0} and M_{1} are arbitrary and M_{2} is finitely presented: Choose N_{1} ⊂ M_{1} measurable. Let N_{0} := i^{−1}(N_{1}) and N_{2} := p(N_{1}). Let L_{1} be a finitely presented module with dim(L_{1}) = 0 projecting onto N_{1}. We get a commutative diagram with exact rows and surjective columns

0 ⟶ N_{0} ⟶ N_{1} ⟶ N_{2} ⟶ 0
↑  ↑  ↑^{id}
0 ⟶ K_{0} ⟶ L_{1} ⟶ N_{2} ⟶ 0.

N_{2} is the image of the composition L_{1} → M_{2} and K_{0} is the kernel of this map. Since L_{1} and M_{2} are finitely presented, by [7, Theorem 0.2] the same is true for N_{2} and K_{0}. In particular N_{0} is measurable. Therefore by Step 1

c(M_{1}) = sup_{N_{1}⊂M_{1} measurable} c(N_{1}) ≤ sup(c(N_{0}) + c(N_{2})) ≤^{1a)} c(M_{0}) + c(M_{2}).

Step 3: We prove the inequality if M_{1} is measurable: Obviously M_{2} is also measurable. Let f : L_{2} → M_{2} be a projection with L_{2} finitely presented and dim(L_{2}) = 0. By the pull back construction we obtain a commutative diagram with exact rows and epimorphisms as vertical arrows

0 ⟶ M_{0} ⟶ M_{1} ⟶ M_{2} ⟶ 0
id↑  ↑  ^{f}↑
0 ⟶ M_{0} ⟶ X_{1} ⟶ L_{2} ⟶ 0.

Note that X_{1} as a submodule of L_{2} ⊕ M_{1} is cofinal-measurable by the proof of Proposition 2.4.2. Then by the last step and 1b)

c(M_{1}) ≤^{1b)} c(X_{1}) ≤ c(M_{0}) + c(L_{2}).

This holds for every L_{2} as above, therefore also for the infimum c(M_{2}).

Step 4: Finally M_{0}, M_{1} and M_{2} are arbitrary A-modules. Suppose N_{1} ⊂ M_{1} is measurable. We get the exact sequence 0 → N_{0} → N_{1} → N_{2} → 0 as above with N_{i} ⊂ M_{i}. Then by the previous step and 1a)

c(N_{1}) ≤ c(N_{0}) + c(N_{2}) ≤^{1a)} c(M_{0}) + c(M_{2}).

This holds for all N_{1} as above and passing to the supremum yields the desired inequality.

2.) This follows from 1a) and the fact, that each finitely generated and in particular each measurable submodule L ⊂ M is contained in some M_{i}.

3.) Let N ⊂ M be a measurable submodule. It suffices to show

c(N) ≤ lim inf_{i∈I} c(M_{i}).

Since N is finitely generated and M is the colimit of a directed system, we find i_{0} and a finitely generated N_{i_{0}} ⊂ M_{i_{0}} which projects onto N. For i ≥ i_{0} set N_{i} := ϕ_{i_{0}i}(N_{i_{0}}) ⊂ M_{i}. Then c(N_{i}) ≤ c(M_{i}) by 1a) and (since I is directed) lim inf_{i≥i_{0}} c(N_{i}) ≤ lim inf_{i∈I} c(M_{i}). Observe that N_{i} projects onto N for all i ≥ i_{0}. We will show that there is i_{1} ∈ I such that N_{i} is measurable for i ≥ i_{1}. Then by 1b) c(N) ≤ c(N_{i}) for i ≥ i_{1} and therefore also c(N) ≤ lim inf_{i≥i_{1}} c(N_{i}) which implies the assertion.

Let p_{i_{0}} : F → N_{i_{0}} be a projection of a finitely generated free module onto N_{i_{0}} and let p : F → N be the composed projection. By Lemma 2.6, since N is measurable, we find a finitely generated submodule K ⊂ ker(p) with dim(F/K) = 0. Because K is finitely generated and I is a directed system, we find i_{1} ≥ i_{0} such that ϕ_{i_{0}i_{1}}p_{i_{0}}(K) = 0. Then ϕ_{i_{0}i}p_{i_{0}} induces a projection of the finitely presented zero-dimensional module F/K onto N_{i} for every i ≥ i_{1}. Therefore these N_{i} are measurable and the inequality follows.

For the second part suppose that every M_{i} is measurable and every ϕ_{ij} is surjective. 1b) implies that lim inf c(M_{i}) = inf c(M_{i}). It remains to prove c(M) ≥ inf c(M_{i}). Choose some a ∈ I and a projection p_{a} : F → M_{a} with F finitely generated free. By composition we get a projection p : F → M. Let K ⊂ ker(p) be finitely generated with dim(F/K) = 0. Since I is a directed system, we find b ≥ a so that with p_{b} := ϕ_{ab}p_{a} already p_{b}(K) = 0. Since ϕ_{ab} is surjective, p_{b} : F → M_{b} is onto. Therefore by 1b)

c(F/K) ≥ c(M_{b}) ≥ inf c(M_{i}).

This holds for any finitely generated K as above, therefore also for the infimum in place of c(F/K) which by Lemma 2.6 is c(M).

4.) Since ⊕_{i∈I}M_{i} = ⋃\begin{matrix}J⊂I \\ J finite \end{matrix} ⊕_{i∈J}M_{i} and because of 2.) we may assume I is

finite. By induction we restrict to the case ⊕_{i∈I}M_{i} = M_{0} ⊕ M_{1}. Because of 1a) it remains to prove

(2.9)  c (M_{0} ⊕ M_{1}) ≤ sup{c(M_{0}), c(M_{1})}.

If M_{0} and M_{1} are finitely presented (2.9) follows from [6, Lemma 1.12] in the same way as Lemma 2.3 does. If M_{0} and M_{1} are measurable, we can choose epimorphisms L_{0} → M_{0} and L_{1} → M_{1} with L_{0} and L_{1} finitely presented and of dimension zero. Note that L_{0} ⊕ L_{1} is finitely presented and therefore the result for the finitely presented case implies c(M_{0} ⊕ M_{1}) ≤ c(L_{0} ⊕ L_{1}) ≤ sup{c(L_{0}), c(L_{1})}. Since this holds for every choice of L_{0} and L_{1} we can pass to the infimum and get (2.9). Now let M_{0} and M_{1} be arbitrary modules. Then every measurable submodule N ⊂ M_{0} ⊕ M_{1} is contained in N_{0} ⊕ N_{1} where N_{0} and N_{1} are the images of N under projection to M_{0} and M_{1}. In particular they are measurable as quotients of a measurable module. Applying the result in the measurable case we get c(N) = c(N_{0} ⊕ N_{1}) ≤ sup{c(N_{0}), c(N_{1})} ≤ sup{c(M_{0}), c(M_{1})}. Passing to the supremum on the left yields (2.9) for arbitrary modules M_{0} and M_{1}. This finishes the proof of 2.7.

There are examples showing that the inequality 1c) in Theorem 2.7 is sharp. Moreover the assumption on cofinality in 1b) is necessary by the following example.

Example 2.10. We construct a non-trivial finitely generated A-module M with dim(M) = 0 which contains no non-trivial measurable A-submodule. In particular M is not cofinal-measurable and c(M) = 0^{−}. Moreover, we construct a quotient module N of M with c(N) > c(M).

Take A = L^{∞}(S^{1}) which can be identified with the group von Neumann algebra N (ℤ). Let χ_{n} be the characteristic function of the subset {exp(2πit) | t ∈ [1/n, 1 − 1/n]} of S^{1}. Let P_{n} be the submodule in P = L^{∞}(S^{1}) generated by χ_{n}. It is a direct summand. Hence the quotient P/P_{n} is a finitely generated projective A-module of dimension 2/n. Projectivity implies c(P/P_{n}) = 0^{−}. Define I := ⋃_{n∈ℕ} P_{n} ⊂ L^{∞}(S^{1}) and put M = L^{∞}(S^{1})/I. Then M = colim_{n∈ℕ} P/P_{n} is a finitely generated A-module with dim(M) = 0 and c(M) = 0^{−} by Theorem 2.7.3 and Remark 2.8

Observe that the same argument applies to any von Neumann algebra A with a directed system P_{i} ⊂ P of direct summands of a projective A-module P such that dim P_{i} < dim P but dim P = sup_{i∈I} dim P_{i}.

For the quotient example, put N_{n} = L^{∞}(S^{1})/((z − 1)^{n}) for positive integers n ≥ 0, where ((z − 1)^{n}) is the ideal generated by the function S^{1} ⟶ ℂ sending z to (z −1)^{n}. Obviously I ⊂ ((z − 1)^{n}) so that N_{n} is a quotient of M. The A-module N_{n} is finitely presented with α(N_{n}) = 1/n [7, Example 4.3]. Hence we get

c(M) = 0^{−}; but c(N_{n}) = n.

Lemma 2.11. 1. A finitely generated submodule of a measurable A-module

is again measurable.

2. A quotient module of a measurable A-module is again measurable.

3. An A-module M is cofinal-measurable if and only if it is the union of its measurable submodules.

4. Submodules and quotient modules of cofinal-measurable A-modules are again cofinal-measurable.

5. Let 0 ⟶ M_{0} ⟶^{i} M_{1} ⟶^{p} M_{2} ⟶ 0 be an exact sequence of A-modules. If M_{0} and M_{2} are cofinal-measurable, then M_{1} is cofinal-measurable.

6. The full subcategory of the abelian category of all A-modules consisting of cofinal-measurable modules is abelian and closed under colimits over directed systems. Given r ∈ {0^{−}} ⨿ [0, ∞], this is also true for the full subcategory of cofinal-measurable A-modules M with c(M) ≤ r.

7. If C_{∗} is an A-chain complex of cofinal-measurable A-modules, then its homology H_{p}(C_{∗}) is cofinal-measurable for all p. Moreover c(H_{p}(C)) ≤ c(C_{p}).

Proof. 1.) This follows from the proof of Proposition 2.4. 2.) This is obvious. 3.) Since M is the union of its finitely generated submodules, it is the union of its measurable submodules provided that M is cofinal-measurable. Suppose that M is the union of its measurable submodules and L ⊂ M is finitely generated. There are finitely many measurable submodules K_{1}, K_{2}, …, K_{r} such that L is contained in the submodule K generated by K_{1}, K_{2}, …, K_{r}. Obviously K and therefore L is measurable by assertions 1.). 4.) follows from assertionss 1.) and 2.). 5.) We have to show for a finitely generated submodule M′_{1} ⊂ M_{1} that it is measurable. Let M′_{2} ⊂ M_{2} be the finitely generated submodule p(M′_{1}) and M′_{0} ⊂ M_{0} be i^{−1}(M′_{1}). Since M_{2} is cofinal-measurable, M′_{2} is measurable. Choose a finitely presented A-module M′′_{2} with dim(M′′_{2}) = 0 together with an epimorphism f : M′′_{2} ⟶ M′_{2}. The pull back construction yields a commutative square with exact rows and epimorphisms as vertical arrows

0 ⟶ M′_{0} ⟶ M′_{1} ⟶ M′_{2} ⟶ 0
id↑  \overline{f}↑  ^{f}↑
0 ⟶ M′_{0} ⟶^{i′′} M′′_{1} ⟶^{p′′} M′′_{2} ⟶ 0.

Since M′_{1} is finitely generated, there is a finitely generated submodule M′′′_{1} ⊂ M′′_{1} such that \overline{f}(M′′′_{1}) = M′_{1}. Let M′′′_{2} ⊂ M′′_{2} be the finitely generated submodule p′′(M′′′_{1}) and M′′′_{0} ⊂ M′_{0} be (i′′)^{−1}(M′′′_{1}). We obtain an exact sequence

0 ⟶ M′′′_{0} ⟶ M′′′_{1} ⟶ M′′′_{2} ⟶ 0.

Since M′′′_{2} is a finitely generated submodule of the finitely presented A-module M′′_{2}, the quotient M′′_{2}/M′′′_{2} is finitely presented. Since the category of finitely presented A-modules is abelian [7, Theorem 0.2], the A-module M′′′_{2} is finitely presented. Since M′′′_{0} is the kernel of an epimorphism of the finitely generated Amodule M′′′_{1} onto the finitely presented A-module M′′′_{2}, M′′′_{0} is finitely generated. As M′′′_{0} is a submodule of the cofinal-measurable A-module M_{0}, the A-module M′′′_{0} is measurable. Since M′′′_{0} is measurable and M′′′_{2} finitely presented, M′′′_{1} is measurable as follows from the first step of the proof of Theorem 2.7.1c). Hence M′_{1} is measurable by assertions 2.), since it is a quotient of M′′′_{1}. 6.) If M = colim_{i∈I} M_{i} is the colimit of a directed system with ψ_{i} : M_{i} → M, then M is the directed union of ψ_{i}(M_{i}). If all M_{i} are cofinal-measurable, then their quotients ψ_{i}(M_{i}) are cofinal-measurable by assertions 4.) and the same is true for M by assertions 3.). The assertions now follows from 4.), 5.) and Theorem 2.7.1. 7.) follows from 4.) and Theorem 2.7.1.

Finally we discuss the behaviour of these notions under induction and restriction for subgroups i : Δ → Γ. The functor i_{∗} was already investigated in [8, Theorem 3.3] where it is shown that i_{∗} is exact and dim_{N (Δ)}(M) = dim_{N (Γ)}(i_{∗}M).

Lemma 2.12. Let i : Δ ⟶ Γ be an inclusion of groups, then i induces an inclusion i : N (Δ) → N (Γ).

1. If M is a measurable or cofinal-measurable respectively N (Δ)-module, then the N (Γ)-module i_{∗}M := N (Γ)⊗_{N (Δ)}M is measurable or cofinal-measurable respectively and

c_{N (Δ)}(M) = c_{N (Γ)}(i_{∗}M).

For an arbitrary N (Δ)-module N we have c(N) ≤ c(i_{∗}N).

2. If the index of Δ in Γ is finite and M is an arbitrary N (Δ)-module, then

c_{N (Δ)}(M) = c_{N (Γ)}(i_{∗}M).

3. If the index of Δ in Γ is finite, N is an N (Γ)-module and i^{∗}N the N (Δ)- module obtained by restriction, then

c_{N (Δ)}(i^{∗}N) = c_{N (Γ)}(N)

and i^{∗}N is measurable or cofinal-measurable if and only if N has the same property.

Proof. 1.) First suppose M is a finitely presented zero-dimensional N (Δ)-module. Choose a resolution 0 → F →^{f} F → M → 0 with a finitely generated free module F as in [7, Lemma 3.4]. Apply the proof of [6, Lemma 3.6] to f, taking into account the equivalence of free Hilbert N Γ-modules and free algebraic N Γ-modules of [7]. It follows that i_{∗}M is a finitely presented N (Γ)-module with dim_{N (Γ)}(i_{∗}M) = 0 and c_{N (Δ)}(M) = c_{N (Γ)}(i_{∗}M).

Next let M be a measurable N (Δ)-module. Let p : F → M be a projection of a finitely generated free N (Δ)-module onto M. Set K := ker(p). Then i_{∗}p : i_{∗}F → i_{∗}M is surjective with kernel i_{∗}K (since i_{∗} is exact). If K_{1} ⊂ K is finitely generated with dim F/K_{1} = 0 (such a module exists by Lemma 2.6), then i_{∗}K_{1} ⊂ i_{∗}K is also finitely generated with

dim(i_{∗}F/i_{∗}K_{1}) =^{exactness} dim i_{∗}(F/K_{1}) = dim F/K_{1} = 0.

Since i_{∗}F/i_{∗}K_{1} is finitely presented and projects onto i_{∗}M the latter module is measurable. Moreover by Lemma 2.6 and the first step applied to F/K_{1}

c(i_{∗}M) ≤ inf_{K_{1}} c(i_{∗}F/i_{∗}K_{1}) = inf_{K_{1}} c(F/K_{1}) = c(M).

Choose on the other hand a finitely generated N (Γ)-submodule L ⊂ i_{∗}K with dim(i_{∗}F)/L = 0. For L we find finitely many generators ∑ u_{i} ⊗ k_{i} ∈ i_{∗}K = N (Γ) ⊗_{N (Δ)} K. Let L′ be the submodule of K generated by all the k_{i}. Then L ⊂ i_{∗}L′, therefore

0 ≤ dim F/L′ = dim i_{∗}F/i_{∗}L′ ≤ dim i_{∗}F/L = 0

and (since F/L′ is finitely presented and by 2.7.1b))

c(M) ≤ c(F/L′) = c(i_{∗}F/i_{∗}L′) ≤ c(i_{∗}F/L).

Since this holds for arbitrary L as above, Lemma 2.6 implies c(M) ≤ c(i_{∗}M).

If M is cofinal-measurable, then it is the union ∪_{i∈I}M_{i} over the directed system of its measurable N (Δ)-submodules. Since i_{∗} is exact, the N (Γ)-module i_{∗}M is the union ⋃_{i∈I} i_{∗}M_{i} over the directed system of measurable N (Γ)-submodules i_{∗}M_{i}. We conclude from Theorem 2.7.2 and the previous step that i_{∗}M is cofinalmeasurable and

c_{N (Δ)}(M) = sup_{i∈I}{c_{N (Δ)}(M_{i})} = sup_{i∈I}{c_{N (Γ)}(i_{∗}M_{i}))} = c_{N (Γ)}(i_{∗}M).

Last let M be an arbitrary N (Δ)-module. Since every measurable N (Δ)- submodule of M induces an N (Γ)-submodule of i_{∗}M of the same capacity, c(M) ≤ c(i_{∗}M) by Definition 2.2. 3.) We begin with studying the restriction. Here i^{∗}N (Γ) = ⊕_{i=1}^{[Γ:Δ]}N (Δ) since the same holds for ℂΓ as a ℂΔ-module and N (Γ) = N (Δ) ⊗_{ℂΔ} ℂΓ. This observation and the proof of [6, Lemma 3.6] imply that if N is a finitely presented N (Γ)-module, then i^{∗}N is finitely presented as N (Δ)-module and

dim_{N (Δ)}(i^{∗}N) = [Γ : Δ] dim_{N (Γ)}(N); c_{N (Δ)}(i^{∗}N) = c_{N (Γ)}(N).

If N is arbitrary and L → N a projection of a finitely presented zero-dimensional N (Γ)-module, then i^{∗}L → i^{∗}N is a corresponding projection and c(L) = c(i^{∗}L). If on the other hand L′ is a zero-dimensional N (Δ)-module projecting onto i^{∗}N, then i_{∗}L′ = N (Γ)⊗_{N (Δ)}L′ is a finitely presented N (Γ)-module naturally projecting onto N with the same dimension and capacity. In particular N is measurable if and only if i^{∗}N is measurable and by Definition 2.2 the capacities coincide in this case.

Any measurable submodule of an N (Γ)-module N restricts to a measurable N (Δ)-submodule of i^{∗}N with the same capacity. On the other hand, if U ⊂ i^{∗}N is a measurable N (Δ)-submodule and V is the N (Γ)-module generated by U, then V is a quotient of i_{∗}U and U a submodule of V, therefore by assertions 1.) and Theorem 2.7.1 V is measurable and

c(U) = c(i_{∗}U) ≥ c(V) = c(i^{∗}V) ≥ c(U).

Definition 2.2 implies c(N) = c(i^{∗}N). The cofinal-measurable case is proven as above. 2.) Let M be an N (Δ)-module. Then i^{∗}i_{∗}M ≅ ⊕_{i=1}^{[Γ:Δ]}M. By 1.) and 3.)

c(M) ≤ c(i_{∗}M) = c(i^{∗}i_{∗}M) = c(⊕_{i=1}^{[Γ:Δ]}M) = c(M).

3. Capacity of groups

In this section we apply the concepts developed so far to define and study Novikov-Shubin invariants respectively capacities for arbitrary spaces with an arbitrary action of a discrete group. In particular we will apply them to arbitrary groups G via the classifying space EG.

Definition 3.1. Let X be a topological space with an action of the discrete group Γ. Let H_{p}^{Γ}(X; N (Γ)) be the N (Γ)-module given by the p-th homology of the chain complex N (Γ) ⊗_{ℤΓ} C_{∗}^{sing}(X), where C_{∗}^{sing}(X) is the singular chain complex of X. Define the p-th capacity of X

c_{p}(X; N (Γ)) := c(H_{p}^{Γ}(X; N (Γ))).

Define the p-th capacity of the group Γ by

c_{p}(Γ) := c_{p}(EΓ; N (Γ)),

where EΓ is any universal free Γ-space.

A group is locally finite, nilpotent, abelian, … respectively, if any finitely generated subgroup is finite, nilpotent, abelian, … respectively. A group is virtually nilpotent, abelian, … respectively, if it contains a subgroup of finite index which is nilpotent, abelian, … respectively.

If S is a finite set of generators for the group G, let b_{S}(k) be the number of elements in G which can be written as a word in k letters of S ∪ S^{−1} ∪ {1}. The group G has polynomial growth of degree not greater than n if there is C with b_{S}(k) ≤ C · k^{d} for all k ≥ 1. This property is a property of G and independent of the choice of the finite set S of generators. We say that G has polynomial growth if it has polynomial growth of degree not greater than n for some n > 0. A finitely generated group Γ is nilpotent if Γ possesses a finite lower central series

Γ = Γ_{1} ⊃ Γ_{2} ⊃ … ⊃ Γ_{s} = {1} Γ_{k+1} = [Γ, Γ_{k}].

Let n_{i} be the rank of the finitely generated abelian group Γ_{i}/Γ_{i+1} and let n be the integer ∑_{i≥1} i · n_{i}. Suppose that \overline{Γ} contains Γ as subgroup of finite index. Then for any finite set S of generators of \overline{Γ} there is a constant C > 0 such that C^{−1} · k^{n} ≤ b_{S}(k) ≤ C · k^{n} holds for any k ≥ 1 and in particular \overline{Γ} has polynomial growth precisely of degree n [1, page 607 and Theorem 2 on page 608].

A famous result of Gromov [3] says that a finitely generated group has polynomial growth if and only if it is virtually nilpotent.

Proposition 3.2. 1. If Γ is not locally finite, then H_{0}^{Γ}(EΓ; N (Γ)) is measurable and

c_{0}(Γ) = inf{c_{0}(Γ′)| Γ′ < Γ infinite finitely generated}.

If Γ is locally finite, then c_{0}(Γ) = 0^{−} but H_{0}^{Γ}(EΓ; N (Γ)) is non-trivial and is in particular not cofinal-measurable.

2. Suppose that Γ is finitely generated. Then

c_{0}(Γ) = \begin{cases}0^{−}; & if Γ is finite or non-amenable, \\ 1/n; & if Γ has polynomial growth of degree n, \\ 0; & if Γ is infinite and amenable, but not virtually nilpotent. \end{cases}

This computes c_{0} for every finitely generated group.

3. Let Γ be an arbitrary group. Then

Γ is locally finite or non-amenable ⟺ c_{0}(Γ) = 0^{−}.

If Γ is locally virtually nilpotent but not locally finite, then

c_{0}(Γ) = inf{c_{0}(Γ′)| Γ′ < Γ infinite finitely generated nilpotent}.

If Γ is amenable and contains a subgroup which is finitely generated but not virtually nilpotent, then

c_{0}(Γ) = 0.

Note that every group belongs to one of the categories and that c_{0}(Γ) > 0 implies that Γ is locally virtually nilpotent but not locally finite.

The above also applies to c_{0} of arbitrary path-connected Γ-spaces by [8, 4.10]. Observe that H_{0}^{Γ}(EΓ; N (Γ)) is not cofinal-measurable if Γ is finite or locally finite. This is responsible for the clumsiness of some of the statements below because Theorem 2.7 1b) becomes false without the condition cofinal-measurable as shown in Example 2.10.

Proof. Remember that H_{0}^{Γ}(EΓ; N (Γ)) = N (Γ) ⊗_{ℂΓ} ℂ which has dimension zero if and only if Γ is infinite and is trivial if and only if Γ is nonamenable by [8, 4.10]. Moreover N (Γ) ⊗_{ℂΓ} ℂ is finitely presented if Γ is finitely generated. 1.) If Γ′ is finitely generated infinite N (Γ′) ⊗_{ℂΓ′} ℂ is measurable and by Lemma 2.12.1 N (Γ)⊗_{N (Γ′)} N (Γ′)⊗_{ℂΓ′} ℂ is measurable and c(N (Γ)⊗_{N (Γ′)} N (Γ′)⊗_{ℂΓ′} ℂ) = c(N (Γ′) ⊗_{ℂΓ′} ℂ). If Γ is not locally finite the system of infinite finitely generated subgroups is cofinal and therefore

N (Γ) ⊗_{ℂΓ} ℂ = colim N (Γ) ⊗_{ℂΓ′} ℂ,

where the colimit is taken over the directed system of infinite finitely generated subgroups. Now the second part of Theorem 2.7.3 yields the claim if Γ is not locally finite.

The proof of Theorem 3.7.3 shows for locally finite Γ

H_{0}^{Γ}(EΓ; N (Γ)) = colim_{Δ⊂Γ} H_{0}^{Δ}(EΔ; N (Γ)) = colim_{Δ⊂Γ} H_{0}(EΔ) ⊗_{ℂΓ} N (Γ)),

where Δ ⊂ Γ runs though the finite subgroups. For finite Δ, the ℂΔ-module H_{0}(EΔ) is projective and hence the N (Γ)-module H_{0}^{Δ}(EΔ; N (Γ)) is projective which implies c(H_{0}^{Δ}(EΔ; N (Γ))) = 0^{−}. By Theorem 2.7.3

c(Γ) = c(H_{0}^{Γ}(EΓ; N (Γ))) = 0^{−}.

Since Γ is non-amenable, H_{0}^{Γ}(EΓ; N (Γ)) is non-trivial and hence cannot be cofinalmeasurable.

2.) If Γ is finite or non-amenable, then N (Γ) ⊗_{ℂΓ} ℂ is finitely generated projective or trivial and therefore c_{0}(Γ) = 0^{−}. If Γ is infinite and amenable, then H_{0}^{Γ}(EΓ; N (Γ)) is finitely presented and zero-dimensional but non-trivial, therefore c_{0}(Γ) ≥ 0. The rest is the content of Lemma 3.3.

3.) This follows by combining the above results.

Lemma 3.3. Suppose Γ is a finitely generated infinite group. Then c_{0}(Γ) = 1/n if Γ has polynomial growth precisely of rate n. If Γ is not virtually nilpotent, then c_{0}(Γ) ≤ 0.

Proof. First observe that Γ has polynomial growth of precisely degree n if and only if the recurrency probablity p(k) of the natural random walk on Γ decreases polynomially with exponent n/2, i.e. there is a constant C > 0 with C^{−1} · k^{−n/2} ≤ p(n) ≤ C · k^{−n/2} for k ≥ 1 and it is not virtually nilpotent if and only if for any n > 0 there is C(n) > 0 satisfying p(k) ≤ C · k^{−n} for k ≥ 1. These are results of Varopoulos [12], compare [13, 6.6 and 6.7].

Now we have to translate this statement to information about the spectrum of the Laplacian. We will prove the following result from which our lemma follows: the finitely generated group Γ has polynomially decreasing recurrence probability with exponent n/2 if and only if c_{0}(Γ) = 1/n.

This can be deduced from [12]. We will give a self-contained and simple proof. If S is a finite set of generators of Γ, then

H_{0}^{Γ}(EΓ; N (Γ)) = coker(⊕_{s∈S}N (Γ) ⟶^{d_{0}=⊕(s−1)} N (Γ)).

Therefore c_{0}(Γ) is the inverse of the Novikov-Shubin invariant α_{0}(Γ) of d_{0}, which we compute from the combinatorial Laplacian Δ_{0} = 1−P. Here P is the transition operator P (g) = (1/|S|) · ∑_{s∈S} sg. The recurrence probability is given by

p(k) := (P^{k}(e), e) = tr(P^{k}),

and the spectral density function of Δ_{0} by

F (λ) = tr(χ_{[1−λ,1]}(P)).

All of the operators in question are positive and therefore for k ∈ ℕ and 0 < λ < 1

(1 − λ)^{k}χ_{[1−λ,1]}(P) ≤ P^{k} ≤ (1 − λ)^{k}χ_{[0,1−λ]}(P) + χ_{[1−λ,1]}(P) ≤ 1. (3.4)

Application of the trace to these inequalities gives

(1 − λ)^{k}F (λ) ≤ p(k) ≤ (1 − λ)^{k} + F (λ) for all 0 < λ < 1. (3.5)

The first inequality implies if 0 < λ < 1

\frac{ln F (λ)}{ln λ} ≥ \frac{ln p(k)}{ln λ} − k \frac{ln(1 − λ)}{ln λ}.

If p(k) ≤ Ck^{−a} for C > 0, then, putting k = [λ^{−1}] (the largest integer not larger than λ^{−1}) we see

α_{1}(Γ) = 2 lim inf_{λ→0^{+}} \frac{ln F (λ)}{ln λ} ≥ 2 lim_{λ→0} (a\frac{ln λ}{ln λ} + \frac{ln C}{ln λ} − \frac{ln(1 − λ)}{λ ln λ}) = 2a.

Suppose now that p(k) ≥ Ck^{−a}. Choose ϵ > 0. Putting k := [λ^{−(1+ϵ)}] + 1 the second part of (3.5) implies

F (λ) ≥ Cλ^{a(1+ϵ)} (\frac{[λ^{−1−ϵ}] + 1}{λ^{−1−ϵ}})^{−a} − (1 − λ)^{[λ^{−(1+ϵ)}]+1}.

Using lemma 3.6 below with δ = C/2 and a(1 + ϵ) instead of a this implies

(1 − λ)^{[λ^{−1−ϵ}]+1} ≤^{1−λ<1} (1 − λ)^{−1−ϵ} ≤ \frac{C}{2} λ^{a(1+ϵ)}

⟹ \frac{ln F (λ)}{ln λ} ≤ \underbrace{\frac{ln(C (([λ^{−1−ϵ}] + 1)/λ^{−1−ϵ})^{−a} − C/2)}{ln λ}}_{→ 0 if λ → 0} +a(1 + ϵ)\frac{ln λ}{ln λ}.

Since the inequality is true for arbitrary ϵ > 0 we conclude

α_{0}(Γ) = 2 lim inf_{λ→0^{+}} \frac{ln F (λ)}{ln λ} ≤ 2a.

Now Lemma 3.3 follows.

Lemma 3.6. For arbitrary ϵ, δ, a > 0 one finds λ_{0} > 0 so that

(1 − λ)^{λ^{−1−ϵ}} ≤ δλ^{a} for all 0 < λ < λ_{0}.

Proof. Note that for 0 < λ < 1 the stated inequality is equivalent to

λ^{−1−ϵ} ln(1 − λ) ≤ ln δ + a ln λ
⟺ 1 ≥ \frac{(ln δ + a ln λ)λ^{1+ϵ}}{ln(1 − λ)} .

For λ → 0 the right hand side tends to 0 which can be seen using l’Hospital’s rule.

Theorem 3.7. 1. Let Δ ⊂ Γ be a subgroup of finite index. Then

c_{n}(Γ) = c_{n}(Δ).

Moreover H_{n}^{Δ}(EΔ; N (Δ)) is cofinal-measurable if and only if H_{n}^{Γ}(EΓ; N (Γ)) is cofinal-measurable.

2. Let Δ ⊂ Γ be a normal subgroup. Suppose that H_{q}^{Δ}(EΔ; N (Δ)) is cofinalmeasurable for q ≤ n. Then we get for p = 0, 1, …, n that H_{p}^{Γ}(EΓ; N (Γ)) is cofinal-measurable and

c_{p}(Γ) ≤ ∑_{q=0}^{p} c_{q}(Δ).

3. If there is a cofinal system of subgroups Δ ⊂ Γ with H_{p}^{Δ}(EΔ; N (Δ)) cofinalmeasurable, then H_{p}^{Γ}(EΓ; N (Γ)) is cofinal-measurableand

c_{n}(Γ) ≤ lim inf{c_{n}(Δ)},

where Δ runs over the cofinal system.

4. If n ≥ 1 and Γ = ℤ^{n} or ℤ^{∞} = ⊕_{i=1}^{∞}ℤ, then H_{p}^{Γ}(EΓ; N (Γ)) is cofinalmeasurable for every p and

\begin{aligned}c_{p}(ℤ^{n}) & = \begin{cases}1/n & if 0 ≤ p ≤ n − 1; \\ 0^{−} & if p ≥ n; \end{cases} \\ c_{p}(ℤ^{∞}) & ≤ 0 if p ≥ 0. \end{aligned}

(Remark: It is possible to show c_{p}(ℤ^{∞}) = 0 for all p ≥ 0.)

5. Suppose that Γ is a finitely generated virtually nilpotent group but not finite. Then H_{p}^{Γ}(EΓ; N (Γ)) is cofinal-measurable for p ≥ 0 and

c_{0}(Γ) + c_{1}(Γ) ≤ 1;
c_{p}(Γ) ≤ 1 for p ≥ 1.

The same holds for Γ locally virtually nilpotent but not locally finite.

Proof. 1.) Let i : N (Δ) ⟶ N (Γ) be the ring homomorphism induced by the inclusion. Since N (Δ) ⊗_{ℤΔ} ℤΓ is isomorphic to N (Γ) as N (Δ)-ℤΓ-bimodule and since EΓ viewed as Δ-space is a model for EΔ, we get i^{∗}H_{n}^{Γ}(EΓ; N (Γ)) = H_{n}^{Δ}(EΔ; N (Δ)). The statement now follows from Lemma 2.12. 2.) There is a spectral sequence converging to H_{p+q}^{Γ}(EΓ; N (Γ)) whose E_{1}-term is given by

E_{p,q}^{1} = H_{q}^{Δ}(EΔ; N (Γ)) ⊗_{ℤπ} C_{p}(Eπ) = ⊕_{I_{p}}i_{∗}H_{q}^{Δ}(EΔ; N (Δ)),

where i : Δ ⟶ Γ is the inclusion and I_{p} the set of p-cells of Bπ. We conclude from Theorem 2.7.4 and Lemma 2.12.1 that E_{p,q}^{1} is cofinal-measurable for q ≤ n and c(E_{p,q}^{1}) = c_{q}(Δ). We conclude from Lemma 2.11.7 that E_{p,q}^{∞} is cofinalmeasurable for q ≤ n and c(E_{p,q}^{∞}) ≤ c_{q}(Δ). Theorem 2.7.1 and 2.11.5 implies that H_{q}^{Γ}(EΓ; N (Γ)) is cofinal-measurable for q ≤ n and c_{q}(Γ) ≤ ∑_{p=0}^{q} c_{p}(Δ) for 0 ≤ p ≤ q. 3.) Since Γ is the union of the subgroups Δ, one can choose a model for EΓ such that for each subgroup Δ the model EΔ is a subcomplex of EΓ and EΓ is the union of all the EΔ’s. For instance take as model for EΓ the infinite join Γ∗Γ∗. . .. Hence

EΓ = colim_{Δ⊂Γ} Γ ×_{Δ} EΔ;
H_{p}^{Γ}(EΓ; N (Γ)) = colim_{Δ⊂Γ} H_{p}^{Δ}(EΔ; N (Γ)),

where Δ ⊂ Γ runs through the finitely generated subgroups. Since H_{p}^{Δ}(EΔ; N (Γ)) is N (Γ)-isomorphic to N (Γ)⊗_{N (Δ)}H_{p}^{Δ}(EΔ; N (Δ)) the claim follows from Theorem 2.7.3 and Lemma 2.12.1. 4.) A direct computation shows the result for Γ = ℤ^{n}. For ℤ^{∞} apply assertions 3. 5.) By 3.) we can assume that Γ is finitely generated infinite and virtually nilpotent. We claim that such a group Γ is either virtually abelian or contains a normal subgroup Δ such that there exists a central extension 1 ⟶ ℤ ⟶ Δ ⟶ ℤ^{2} ⟶ 1. This is proven as follows.

Recall that subgroups and quotient groups of nilpotent groups are nilpotent again, any nilpotent group contains a normal torsionfree group of finite index and the center of a non-trivial nilpotent group is non-trivial. Now choose a normal torsionfree subgroup Γ_{0} of Γ of finite index and inspect the exact sequence 1 ⟶ cent(Γ_{0}) ⟶ Γ_{0} ⟶ Γ_{0}/ cent(Γ_{0}) ⟶ 1. If Γ_{0}/ cent(Γ_{0}) is finite, Γ is virtually abelian. Suppose that Γ_{0}/ cent(Γ_{0}) is infinite. By inspecting the analogous sequence for a normal torsionfree subgroup Γ_{1} ⊂ Γ_{0}/ cent(Γ_{0}) of finite index and using the fact that Γ_{1}/ cent(Γ_{1}) is either finite or contains ℤ as normal subgroup, one sees that Γ_{0}/ cent(Γ_{0}) contains ℤ as subgroup of finite index or contains both ℤ and ℤ^{2} as normal subgroups. Since cent(Γ_{0}) has at least rank 1, the claim follows for Γ_{0} and hence for Γ.

If Γ is virtually abelian the assertion 5.) follows from 1.) and 4.). Suppose that Δ is a normal subgroup of Γ and that there is a central extension 1 ⟶ ℤ ⟶ Δ ⟶ ℤ^{2} ⟶ 1. Because of 2.) c_{0}(Γ) + c_{1}(Γ) ≤ 2 · c_{0}(Δ) + c_{1}(Δ). Hence it remains to show

2 · c_{0}(Δ) + c_{1}(Δ) ≤ 1.

One can realize Δ as the fundamental group of a closed 3-manifold M which is a principal S^{1}-bundle over T^{2}. Hence M is a Seifert manifold whose base orbifold has Euler characteristic χ = 0 and the computation in [6, Theorem 4.1] shows c_{0}(Δ) = c_{1}(Δ) = 1/3 if the Euler class e(M) = 0 and c_{0}(Δ) = 1/4 and c_{1}(Δ) = 1/2 if e(M) ≠ 0. This finishes the proof of Theorem 3.7.

Definition 3.8. Given a class of groups X, let LX be the class of groups Γ for which any finitely generated subgroup Δ belongs to X. Given classes of groups X and Y, let X Y be the class of groups Γ which contain a normal subgroup Δ ⊂ Γ with Δ ∈ X and Γ/Δ ∈ Y. The class E of elementary amenable groups is defined as the smallest class of groups which contains all abelian and all finite groups, is closed under extensions, taking subgroups, forming quotient groups and under directed unions. The class of finite groups is denoted by F.

Theorem 3.9. Let C be the class of groups Γ for which H_{p}^{Γ}(EΓ; N (Γ)) is cofinal-measurable for all p ≥ 0 and c_{0}(Γ) + c_{1}(Γ) ≤ 1 holds.

1. Finite and locally finite groups do not belong to C.

2. If the infinite elementary amenable group Γ contains no infinite locally finite subgroup, then Γ belongs to C.

3. If Γ contains a normal subgroup Δ which belongs to C, then Γ belongs to C.

4. Let Γ be the amalgamated product Γ_{0} ∗_{Δ} Γ_{1} for a common subgroup Δ of Γ_{0} and Γ_{1}. Suppose that Δ, Γ_{0} and Γ_{1} belong to C and that c_{0}(Δ) ≤ 0, then Γ belongs to C.

5. L(C ∪ F) = C ∪ LF.

6. Suppose for the group Γ that c_{0}(Γ) > 0 or that it contains ℤ^{n} for some n ≥ 1 as a normal subgroup. Then Γ belongs to C and moreover

c_{p}(Γ) ≤ 1 also holds for p ≥ 1.

Proof. 1.) H_{0}^{Γ}(EΓ; N (Γ)) is not cofinal-measurable if Γ is (locally) finite. 5.) We have to show for a group Γ ∈ L(C ∪ F) which is not locally finite that Γ belongs to C. Since any infinite finitely generated subgroup Γ′ ⊂ Γ belongs to C by assumption and these groups form a cofinal system of subgroups, we get from Theorem 3.7.3 that H_{p}^{Γ}(EΓ; N (Γ)) is cofinal-measurable. and

c_{1}(Γ) = lim inf{c_{1}(Γ′)} ≤ 1.

If c_{0}(Γ) ≤ 0 we are done, otherwise we know from 3.2.3 that Γ is locally virtually nilpotent but not locally finite and the result follows from 3.7.5. 3.) We get from Theorem 3.7.2 that H_{p}^{Γ}(EΓ; N (Γ)) is cofinal-measurable for p ≥ 0 and

(3.10)  c_{1}(Γ) ≤ c_{0}(Δ) + c_{1}(Δ) ≤ 1.

It remains to show c_{0}(Γ) + c_{1}(Γ) ≤ 1. If c_{0}(Γ) ≤ 0, this follows from (3.10). If c_{0}(Γ) > 0 we are again in the case, where Γ is locally virtually nilpotent but not locally finite and can apply 3.7.5. 2.) We use the following description of the class of elementary-amenable groups [4, Lemma 3.1]. Let B be the class of all groups which are finitely generated and virtually free abelian. Define for each ordinal α

E_{0} =  {1}  ;
E_{α} = (LE_{α−1})B , if α is a successor ordinal
E_{α} = ∪_{β<α}E_{β} , if α is a limit ordinal.

Then E = ∪_{α≥0}E_{α} is the class of elementary amenable groups. For any elementary amenable group Γ there is a least ordinal α with Γ ∈ E_{α} and we use transfinite induction to show that Γ belongs to C, provided that Γ is infinite and contains no infinite locally finite subgroup.

The induction begin α = 0 is obvious. If α is a limit ordinal, the induction step is clear. Suppose that α is a successor ordinal. Then there is an extension 1 ⟶ Δ ⟶ Γ ⟶ π ⟶ 1 such that Δ ∈ LE_{α−1} and π ∈ B. Every finitely generated subgroup Δ′ ⊂ Δ belongs to E_{α−1}. By assumption Γ and therefore Δ and Δ′ contain no infinite locally finite subgroup. The induction hypothesis implies that Δ′ is finite or belongs to C. From assertions 5.) we get Δ ∈ L(C ∪F) = C ∪LF. Since Δ is not infinite locally finite either Δ is finite or Δ ∈ C. In the second case apply 3.). If Δ is finite π must be infinite. Since FB = B we have Γ infinite virtually free abelian. In this case the statement follows from Theorem 3.7.1 and 3.7.5. 4.) This follows from the long Mayer-Vietoris sequence, Theorem 2.7.1, Lemma 2.11 and Lemma 2.12. One has to argue as above for the case c_{0}(Γ) > 0. 6.) Suppose c_{0}(Γ) > 0. Then Γ is locally virtually nilpotent but not locally finite and the statement is 3.7.5. Now suppose Γ contains ℤ^{n} as a normal subgroup for n ≥ 1. Then the result follows from 3.7.4 and 3.7.2.

4. Final Remarks

Remark 4.1. We mention that the proof for Theorem 3.9.2 goes also through if one enlarges the class E as defined by transfinite induction in the proof by substituting the class B of virtually abelian groups by any bigger class B′ with the properties that B′ ⊂ C, and FB′ = B′.

Remark 4.2. Let 1 ⟶ Δ ⟶^{i} Γ ⟶^{i} ℤ ⟶ 1 be an extension of groups. Suppose that Δ is locally finite. Then H_{p}(EΓ; N (Γ)) is trivial for p ≥ 2 and for p = 1 equal to the kernel K of the N (Γ)-map

N (Γ) ⊗_{ℂΔ} ℂ ⟶ N (Γ) ⊗_{ℂΔ} ℂ : u ⊗ n ↦ u(t − 1) ⊗ n

for some t ∈ N (Γ) which maps to a generator of ℤ under p. If we would know that K is trivial, then Γ would belong to C and it would suffice in Theorem 3.9.2 to assume that Γ itself is not locally finite instead of assuming that Γ contains no infinite locally finite subgroup.

Remark 4.3. So far we know no counterexample to the following statement: If Γ is elementary amenable and not locally-finite, then H_{p}(EΓ; N (Γ)) is cofinalmeasurable for all p ≥ 0 and

(4.4)  c_{p}(Γ) ≤ 1;  for p ≥ 0.

To prove this, it suffices to show inequality 4.4 for any group Γ such that there is an extension 1 ⟶ Δ ⟶ Γ ⟶ ℤ ⟶ 1 with a group Δ which already satisfies inequality 4.4. Then the proof of Theorem 3.9.2 would go through.

On the other hand one can construct a fiber bundle F ⟶ E ⟶ S^{1} of closed manifolds with simply-connected fiber F such that c_{p}(Ẽ) is arbitrary large. This follows from the observation that in this case one can read off c_{p}(Ẽ) from the automorphism of H_{p}(F) induced by the monodromy map F ⟶ F and one can realize any element in GL(n, ℤ) as the automorphism induced on H_{3}(∏_{i=1}^{n} S^{3}) by an automorphism of ∏_{i=1}^{n} S^{3}.

References

[1] Bass, H.: “The degree of polynomial growth of finitely generated nilpotent groups”, Proc. of LMS 25, 603–614 (1972)

[2] Farber, M.S.: “Homological algebra of Novikov-Shubin invariants and Morse inequalities”, Geometric Analysis and Functional Analysis 6, 628 - 665 (1996)

[3] Gromov, M. : “Groups of polynomial growth and expanding maps”, Publ. Math. IHES 53, 53-73 (1981)

[4] Kropholler, P.H., Linnell, P.A., and Moody, J.A.: “Applications of a new K-theoretic theorem to soluble group rings”, Proc.of the AMS 104, 675–684 (1988)

[5] Lott, J.: “Heat kernels on covering spaces and topological invariants”, Journal of Differential Geometry 35, 471–510 (1992)

[6] Lott, J. and Lück, W.: “L^{2}-topological invariants of 3-manifolds”, Inventiones Math. 120, 15 -60 (1995)

[7] Lück, W.: “Hilbert modules and modules over finite von Neumann algebras and applications to L^{2}-invariants”, Math. Annalen 309, 247 - 285 (1997)

[8] Lück, W.: “Dimension theory of arbitrary modules over finite von Neumann algebras and L^{2}-Betti numbers I: Foundations”, Journal für reine und angewandte Mathematik 495, 135 - 162 (1998)

[9] Lück, W.: “Dimension theory of arbitrary modules over finite von Neumann algebras and L^{2}-Betti numbers II: Applications to Grothendieck groups”, J. für Reine und Angewandte Mathematik 496, 213 – 236 (1998)

[10] Novikov, S. and Shubin, M.: “Morse inequalities and von Neumann invariants of nonsimply connected manifolds”, Sov. Math. Dok. 34, 79-82 (1987)

[11] Novikov, S. and Shubin, M.: “Morse inequalities and von Neumann II_{1}-factors”, Dokl. Akad. Nauk. 34 no. 1, 289 - 292 (1986), Soviet. Math. Dokl. 34 no. 1, 79 - 82 (1987)

[12] Varopoulos, N. : “Random walks and Brownian motion on manifolds”, Symposia Mathematica XXIX, 97 - 109 (1988)

[13] Woess, W.: “Random walks on infinite graphs and groups — A survey on selected topics”, Bull. of the London Math. Soc. 26, 1–60 (1994)

Fachbereich Mathematik — Universität Münster, Einsteinstr. 62 — 48149 Münster

E-mail address: wolfgang.lueck@math.uni-muenster.de
E-mail address: holger.reich@math.uni-muenster.de
E-mail address: thomas.schick@math.uni-muenster.de
URL: http://www.uni-muenster.de/math/u/lueck/