L^{2}-DETERMINANT CLASS AND APPROXIMATION OF L^{2}-BETTI NUMBERS

THOMAS SCHICK

Abstract. A standing conjecture in L^{2}-cohomology says that every finite CW -complex X is of L^{2}-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class G of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free products of these. If, in addition, X is L^{2}-acyclic, we also show that the L^{2}-determinant is a homotopy invariant — giving a short and easy proof independent of and encompassing all known cases. Under suitable conditions we give new approximation formulas for L^{2}- Betti numbers.

Keywords: L^{2}-determinant, L^{2}-Betti numbers, approximation, L^{2}-torsion, homotopy invariance MSC: 58G26 (Primary), 55N25, 55P29 (Secondary)

1. Introduction

For a finite CW -complex X with fundamental group π, L^{2}-invariants of the universal covering X̃ are defined in terms of the combinatorial Laplacians Δ_{∗} on C_{(2)}^{∗}(X̃) = C_{cell}^{∗}(X̃) ⊗_{ℤπ} l^{2}(π), which, after the choice of a cellular base, is a finite direct sum of copies of l^{2}(π). Δ_{p} = (c_{p} ⊗ id)^{∗}(c_{p} ⊗ id) + (c_{p−1} ⊗ id)(c_{p−1} ⊗ id)^{∗} becomes a matrix over ℤπ ⊂ N π in this way, which acts on l^{2}(π)^{d} via multiplication from the left. Here N π is the group von Neumann algebra with its natural trace tr_{π}, defined as follows:

1.1. Definition. For Δ = (a_{ij}) ∈ M(d × d, N π) set

tr_{π}(Δ) := ∑_{i} tr_{π}(a_{ii})

where tr_{π}(a) := a_{1} = (a, 1)_{l^{2}(π)} is the coefficient of the trivial group element if a = ∑_{g∈π} λ_{g}g ∈ N π ⊂ l^{2}(π).

Particularly important are the spectral density functions

F_{p}(λ) := F_{Δ_{p}}(λ) := tr_{π} χ_{[0,λ]}(Δ_{p}). (1.2)

The L^{2}-Betti numbers are defined as

b_{p}^{(2)}(X) := b_{p}^{(2)}(Δ_{p}) := F_{Δ_{p}}(0) = dim_{π}(ker(Δ_{p})).

These are invariants of the homotopy type of X.

Another important invariant is the regularized determinant:

Date: last compiled: October 12, 2000, last edited after: Oct 11, 2000. 1991 Mathematics Subject Classification. Primary: 58G26, Secondary: 55N25, 55P29. Key words and phrases. L^{2}-determinant, L^{2}-Betti numbers, approximation, L^{2}-torsion, ho-

motopy invariance.

1.3. Definition. For a positive and self-adjoint operator Δ ∈ M(d × d, N) over a finite von Neumann algebra N with spectral density function F_{Δ} define

ln det_{N} (Δ) := \begin{cases}∫_{0^{+}}^{∞} ln(λ)dF_{Δ}(λ); & if the integral converges \\ −∞; & otherwise \end{cases}

Sometimes, this regularized determinant is called Fuglede-Kadison determinant.

This gives rise to the definition:

1.4. Definition. A self-adjoint operator Δ as above is said to be of π-determinant class if and only if

∫_{0^{+}}^{1} ln λ dF_{Δ}(λ) > −∞.

The space X is said to be of determinant class if the Laplacian Δ_{p} is of πdeterminant class for every p.

1.5. Conjecture. Every finite CW -complex is of determinant class.

If X is of π-determinant class and all L^{2}-Betti numbers are zero, then we can define its (additive) L^{2}-Reidemeister torsion

T^{(2)}(X) := ∑_{p}(−1)^{p}p ln det_{π} Δ_{p}.

Burghelea et.al. [2] show that L^{2}-Reidemeister torsion is equal to L^{2}-analytical torsion (for closed manifolds) and therefore is a generalization of the volume of a hyperbolic manifold in the following sense: for odd dimensional compact manifolds whose interior admits a complete hyperbolic metric of finite volume the analytic torsion is proportional to the volume of the interior [15] with non-zero constant of proportionality [9].

Lück and Rothenberg [14, 3.12] show that this torsion is an invariant of the simple homotopy type of X. They conjecture

1.6. Conjecture. L^{2}-Reidemeister torsion is a homotopy invariant,

and prove [14, 3.11]:

1.7. Theorem. L^{2}-Reidemeister torsion is a homotopy invariant of L^{2}-acyclic finite CW -complexes with fundamental group π if and only if for every A ∈ Gl(d × d, ℤπ) the regularized determinant is zero: ln det_{π}(A^{∗}A) = 0.

1.8. Remark. In fact, ln det_{π}(·^{∗}·) factors through the Whitehead group of π. The corresponding homomorphism is denoted Φ by Lück-Rothenberg, but we will write ln det_{π} for the map on W h(π) as well.

1.9. Remark. Mathai and Rothenberg [16, 2.5] extend the study of L^{2}-determinants from the L^{2}-acyclic to the general case, dealing with determinant lines instead of complex numbers. Without any difficulty, this could be done in our more general situation as well. Because this would only complicate the notation and the generalization is transparent, we will not carry this out.

Suppose that π is residual, i.e. it contains a nested sequence of normal subgroups π = π_{1} ⊃ π_{2} ⊃ … such that ⋂_{i} π_{i} = {1}. Then we construct the corresponding coverings X_{i} of X with fundamental group π_{i}. L^{2}-invariants can be defined for arbitrary normal coverings (the relevant von Neumann algebra is the one of the group of deck transformations). We conjecture 1.10. Conjecture. In the situation just described, for every p, the L^{2}-Betti numbers b_{p}^{(2)}(X_{i}) converge as i → ∞ and

lim_{i→∞} b_{p}^{(2)}(X_{i}) = b_{p}^{(2)}(X).

The projections π → π/π_{i} induce maps p_{i} : M(d × d, ℤπ) → M(d × d, ℤπ/π_{i}), and the Laplacian on X_{i} (considered as such a matrix) is just the image of the Laplacian on X̃. Therefore, Conjecture 1.10 follows from the following conjecture about such matrices (and, in fact is equivalent as follows from the proof of [13, 2.2]).

1.11. Conjecture. For A ∈ M(d × d, ℤπ) set A_{i} := p_{i}(A). Then

lim_{i→∞} dim_{π/π_{i}}(ker A_{i}) = dim_{π}(ker A).

The Conjectures 1.5, 1.6 and 1.10 are proven for residually finite groups by Lück [11]. Using the ideas of Lück in a different context, Dodziuk and Mathai prove Conjecture 1.5 for amenable π [6, 0.2]. They also establish an approximation theorem for L^{2}-Betti numbers of a slightly different type in this case.

Clair proves in [3] Conjectures 1.5, 1.6 and 1.10 for residually amenable fundamental groups. To prove Conjecture 1.6 he uses [16, Proposition 2.5], where Mathai and Rothenberg give a proof of 1.6 for amenable π. Unfortunately, this proof is very complicated and not complete (some steps even seem to be wrong). In this paper we will obtain in particular an independent (and much easier) proof of this result (and of the results of Clair).

One can interpret the cohomology of the covering spaces as cohomology with coefficients in l^{2}(π/π_{i}). Michael Farber [7] generalizes Lück’s results to certain other sequences of finite dimensional coefficients which converge to l^{2}(π).

The aim of this paper is to extend the above results (except Farber’s) to the following larger class of groups.

1.12. Definition. Let G be the smallest class of groups which contains the trivial group and is closed under the following processes:

• If U < π is any subgroup such that U ∈ G and the discrete homogeneous space π/U admits a π-invariant metric which makes it to an amenable discrete homogeneous space, then π ∈ G.

• If π = dirlim_{i∈I} π_{i} is the direct limit of a directed system of groups π_{i} ∈ G, then π ∈ G, too.

• If π = invlim_{i∈I} π_{i} is the inverse limit of a directed system of groups π_{i} ∈ G, then π ∈ G, too.

• The class G is closed under taking subgroups.

Here we use the following definition (compare 4.1):

1.13. Definition. A discrete homogeneous space π/U is called amenable if on π/U we find a π-invariant metric d : π/U × π/U → ℕ such that sets of finite diameter are finite and such that ∀K > 0, ϵ > 0 there is a finite subset ∅ ≠ X ⊂ π/U with

|{x ∈ π/U; d(x, X) ≤ K and d(x, π/U − X) ≤ K}| ≤ ϵ |X| .

This is in particular fulfilled if U is a normal subgroup and π/U is an amenable group.

It follows immediately from the definition that G contains all amenable groups, is closed under directed unions and is residually closed. Section 2 contains more results on the structure of G.

The main theorem of the paper is the following:

1.14. Theorem. Suppose π belongs to the class G. Then for every CW-complex with fundamental group π which has finitely many cells in each dimension, Conjectures 1.5 and 1.6 are true. The approximation results 1.10 and 1.11 (and generalizations thereof, compare Section 6) are valid under the condition that all the occuring groups belong to G.

The more general form of the approximation result can be applied to get more cases and new proofs of the following conjecture which is called the Atiyah conjecture [1]:

1.15. Conjecture. Suppose π is torsion free. Then dim_{π}(ker A) ∈ ℤ whenever A ∈ M(d×d, ℤπ). Equivalently, whenever X is a finite CW -complex with π_{1}(X) = π then b_{p}^{(2)}(X) ∈ ℤ ∀p ∈ ℤ.

We will address this in a forthcoming paper. Linnell [10] proves that the Atiyah conjecture is true for abelian groups, free groups, and for extensions with elementary amenable quotient.

1.16. Remark. So far, no example of a countable group which does not belong to the class G has been constructed. Good candidates for such examples are finitely generated simple groups which are not amenable, e.g. groups containing a free group with two generators.

On the other hand, we can not give an example of a non-residually amenable group which belongs to G, either. In any case, our description of G leads easily to many properties like being closed under direct sums and free products (Proposition 2.7), which (if true at all) are probably much harder to establish for the class of residually amenable groups.

In fact, we prove a little bit more than Theorem 1.14. Namely, we show that the relevant properties are stable under the operations characterizing the class G. We use the following definitions:

1.17. Definition. Let C be any property of discrete groups. It is said to be

• stable under direct/inverse limits if C is true for π whenever π is a direct/inverse limit of a directed system of groups which have property C.

• subgroup stable if every subgroup U < π of a group with property C shares this property, too.

• stable under amenable extensions if π has property C whenever it contains a subgroup U with property C such that the homogeneous space π/U is amenable in the sense of Definition 1.13.

The properties we have in mind are listed in the following definition:

1.18. Definition. Let π be a discrete group. We say

• π is of determinant class if Δ is of π-determinant class for every Δ ∈ M(d × d, ℤπ) which is positive and self-adjoint;

• π has semi-integral determinant if ln det_{π}(Δ) ≥ 0 for every Δ ∈ M(d × d, ℤπ) which is positive self-adjoint. In particular every such Δ is of π-determinant class, i.e. π itself is of determinant class;

• π has Whitehead-trivial determinant if ln det_{π}(A^{∗}A) = 0 ∀A ∈ W h(π).

1.19. Theorem. Whitehead-trivial determinant is stable under direct and inverse limits and is subgroup stable.

1.20. Remark. In the light of this theorem, we can use the fact that the Fuglede-Kadison determinant must be trivial on trivial Whitehead groups, e.g. for every torsion free discrete and cocompact subgroup of a Lie group with finitely many components [8, 2.1]. Waldhausen shows that the Whitehead group is trivial for another class of groups, including torsion free one-relator groups and many fundamental groups of 3-manifolds [18, 17.5].

The validity of the isomorphism conjecture of Farrell and Jones would imply that the Whitehead group is trivial if π is torsion free.

We can extend G to the class G′ which is the smallest class of groups containing G and in addition any class of groups whose Whitehead group is (known to be) trivial, and which is closed under taking subgroups, direct and inverse limits. Then every group in G′ has Whitehead trivial determinant, i.e. L^{2}-torsion is a homotopy invariant for L^{2}-acyclic finite CW -complexes with such a fundamental group.

1.21. Theorem. The property “semi-integral determinant” is stable under direct and inverse limits, subgroup stable and stable under amenable extensions.

We now show that the complicated proofs of homotopy invariance of L^{2}-torsion in [11, 16, 3] can be replaced by a very short and easy argument.

1.22. Theorem. If for a group π and ∀A ∈ W h(π) we have ln det_{π}(A^{∗}A) ≥ 0, then the Fuglede-Kadison determinant is trivial on W h(π).

In particular, semi-integral determinant implies Whitehead-trivial determinant.

Proof. A ∈ W h(π) implies A has an inverse B ∈ W h(π). Now by [12, 4.2]

0 = ln det_{π}(id) = ln det_{π}((AB)^{∗}AB) = \underbrace{ln det_{π}(A^{∗}A)}_{≥0} + \underbrace{ln det_{π}(B^{∗}B)}_{≥0}

and the statement follows.

It follows from the induction principle 2.2 and the fact that the trivial group has semi-integral determinant (Lemma 6.8) that 1.21 and 1.22 imply the first part of our main theorem 1.14.

For the readers convenience, the first section after this introduction could give a more detailed account to L^{2}-invariants, group von Neumann algebras and their dimension theory. Instead she is referred to older sources, e.g. [11] and [6]. One of the reasons is that many results of this paper are obtained by an elaboration of the basic methods of Lück’s paper and to some extend the paper of Dodziuk and Mathai.

The actual plan of the paper is as follows: In Section 2 we give a closer description of our class G. Sections 3–6 contain the proofs the the theorems stated in this introduction. In Section 7 we discuss the question how one can generalize the approximation result 1.11 to matrices over the complex group ring. Section 8 gives a slight generalization of the determinant class conjecture 1.5.

2. Properties of the class G

We proceed with another definition of the class G, similar to the description Linnell gives of his class C in [10, p. 570].

2.1. Definition. For each ordinal α, we define a class of groups G_{α} inductively. • G_{0} consists of the trivial groups.

• If α is a limit ordinal, then G_{α} is the union of G_{β} with β < α.

• If α has a predecessor α − 1, then G_{α} consists of all groups which – are subgroups of groups of G_{α−1}, or

– contain a subgroup U which belongs to G_{α−1} such that the quotient space is an amenable homogeneous space as defined in 1.13, or

– are direct or inverse limits of directed systems of groups in G_{α−1}. By definition, a group is in G if it belongs to G_{α} for some ordinal α.

The class G is defined by (transfinite) induction. Therefore, properties of the groups in G can be proven by induction, too. More precisely, the following induction principle is valid:

2.2. Proposition. Suppose a property C of groups is shared by the trivial group, and the following is true:

• whenever K has property C and K < π with π/K an amenable homogeneous space, then π has property C as well;

• whenever π is a direct or inverse limit of a directed system of groups with the property C, then π has property C

• property C is inherited by subgroups. Then property C is shared by all groups in the class G.

Proof. The proof of the induction principle is done by transfinite induction.

By assumption, C holds for G_{0}. We have to establish C for every group in G_{α}, granted its validity for groups in G_{β} for all β < α. If α is a limit ordinal, this is trivial. If α has a predecessor α − 1, the assumptions just match the definition of G_{α}, so the statement follows.

Now, we study properties of the class G.

2.3. Proposition. The class G is closed under directed unions.

Proof. A directed union is a special case of a directed direct limit.

2.4. Proposition. The class G is residually closed. This means that if π contains a nested sequence of normal subgroups π_{1} ⊃ π_{2} ⊃ … with trivial intersection and if π/π_{i} ∈ G ∀i then also π ∈ G.

Proof. The inverse system of groups π/π_{i} has some inverse limit G. The system of maps π → π/π_{i} induces a homomorphism π → G. If g ∈ π is mapped to 1 ∈ G, then g has to be mapped to 1 ∈ π/π_{i} ∀i, i.e. g ∈ ⋂_{i} π_{i} = {1}. As a directed limit, G ∈ G, and as a subgroup of G, also π ∈ G.

2.5. Theorem. If U belongs to G and ϕ : U → U is any group homomorphism, then the “mapping torus”-extension of U with respect to ϕ

π = ⟨u ∈ U, t|t^{−1}ut = ϕ(u), u · v = (uv); ∀u, v ∈ U⟩

also belongs to G (if ϕ is injective, this is a special example of an HNN-extension).

Proof. There is a canonical projection π → ℤ sending u ∈ U to 0 and t to 1. Denote its kernel by K. We will show that K belongs to G, then so does π because it is an extension of K with amenable quotient ℤ.

Now K is the direct limit of the sequence

U ⟶^{ϕ} U ⟶^{ϕ} U ⟶^{ϕ} U …

and belongs to G, which is closed under taking direct limits.

2.6. Remark. Although such a mapping-torus extension of a finitely presented group is finitely presented again, the kernel K we used in the proof may very well not even admit a finite set of generators. This is one instance where it is useful to allow arbitrary groups, even if one is only interested in fundamental groups of finite CW -complexes.

For the next property, we use the induction principle.

2.7. Proposition. G is closed under forming 1. direct sums and direct products;

2. arbitrary inverse limits;

3. free products.

Proof. We have to check the conditions for the induction principle 2.2. Fix π ∈ G.

1. If U < G then U × π < G × π. If G × π ∈ G, the same is true for U × π. If G is the (direct or inverse) limit of the directed system of groups G_{i}, then G × π is the limit of the system G_{i} × π (compare Lemma 2.9 or 2.10). If (by assumption) G_{i} × π ∈ G, then G × π ∈ G. Finally, if U < G, and G/U is amenable, then U × π < G × π with the same amenable quotient. Therefore, U × π ∈ G ⟹ G × π ∈ G.

The induction principle (and induction on the number of factors) now implies that G is closed under direct sums. Since a direct product is the inverse limit of finite direct sums (indexed by the directed system of finite subsets of the indexing set), G is closed under direct products, too.

2. An arbitrary inverse limit is by construction a subgroup of a direct product. Therefore the first assertion implies the second one.

3. The case of free products is a little bit more complicated. As first step we prove: ∗_{i∈I}ℤ/4 ∈ G for every index set I. This is the direct limit of finite free products of copies of ℤ/4, therefore we have to prove the statement for finite I. Now ∗_{i=1}^{n}ℤ/4 is a subgroup of ℤ/4 ∗ ℤ/4 (contained in the kernel of the projection onto one factor by 2.8), and ℤ/4 ∗ ℤ/4 is virtually free by [5, IV.1.9] and [17, I Theorem 7], i.e. an extension of a free and therefore by [4, page 57] residually finite (the free group) with an amenable (finite) group. Therefore ℤ/4 ∗ ℤ/4 belongs to G, and this implies the first step.

Next we show that π ∗ (∗_{j∈J}ℤ/4) ∈ G for every set J. We prove this using the induction principle. For π = 1 this is the conclusion of the first step. If π is a limit of (π_{i})_{∈I}, or a subgroup of G, then π ∗ (∗_{j∈J}ℤ/4) is a subgroup of the limit of π_{i} ∗ (∗_{j∈J}ℤ/4) (compare Lemma 2.9 and Lemma 2.10) or a subgroup of G ∗ (∗_{j∈J}ℤ/4), and we can apply that G is subgroup-closed. If U < π and π/U is amenable, π ∗ (∗_{j∈J}ℤ/4) acts on π/U. We get a new point stabilizer, which is isomorphic to the free product of U with ∗_{G/U}(∗_{i∈I}ℤ/4) by Lemma 2.8. Fortunately, the induction hypothesis applies with the free product of an arbitrary number of copies of ℤ/4.

As the next step we show that ∗_{i∈I}π ∈ G. This follows (as ∗_{i∈I}π is a direct limit) from the corresponding statement for I finite, and these are subgroups of π ∗ ℤ/4, contained in the kernel of the projection onto ℤ/4 by Lemma 2.8.

π_{1} ∗ π_{2} is contained in (π_{1} × π_{2}) ∗ (π_{1} × π_{2}), and the general statement follows by induction and taking limits.

In the proof of Proposition 2.7 we have used the following well known lemmas.

2.8. Lemma. Let K be the kernel of the group homomorphism q : G → Q and H a group. Then the kernel of q ∗ 1 : G ∗ H → Q is the free product

(∗_{i∈Q}H^{g_{i}}) ∗ K

where {g_{i}}_{i∈Q} is a system of representatives for G/K ≅ Q.

Proof. The subgroup generated by the conjugates H^{g_{i}} and K is contained in the kernel of q∗1. For the converse, take an element of the kernel and write it in normal for a_{1}h_{1} … a_{n}h_{n} with a_{k} ∈ G and h_{k} ∈ H. Check that it is possible in a unique way to rewrite it as a product of factors in H^{g_{i}} and K.

2.9. Lemma. If π is the direct limit of a system of groups π_{i} and G is any group, then π ∗ G is the direct limit of π_{i} ∗ G, and π × G is the direct limit of π_{i} × G.

Proof. There are obvious maps from π_{i} ∗ G to π ∗ G and from π_{i} × G to π × G.

Suppose one has consistent maps from π_{i} ∗G (or π_{i} ×G) to some group X. Since π_{i} and G both are subgroups of π_{i} ∗ G (or of π_{i} × G) , this means that we have a consistent family of maps on π_{i} multiplied with a fixed map on G. Therefore (from the properties of products) there exists exactly one map from π ∗G (or π ×G) to X making all the diagrams commutative (for the abelian product note that the union of the images of the π_{i} in X commutes with the image of G).

2.10. Lemma. If π is an inverse limit of a system of groups π_{i} and G is any group, then π ∗ G is contained in the inverse limit X of π_{i} ∗ G.

The inverse limit of π_{i} × G is π × G.

Proof. First, we look at the free products:

We have a consistent family of homomorphisms from π ∗ G to π_{i} ∗ G, therefore a homomorphism from π ∗ G to X. An element x = p_{1}g_{1} … p_{n}g_{n} ∈ π ∗ G with p_{i} ∈ π and g_{i} ∈ G is in the kernel of this homomorphism if and only if it is mapped to 1 ∈ π_{i} ∗G for every i ∈ I. This can not happen if 1 ≠ x ∈ π. It remains to check the case 1 ≠ g_{1} ∈ G. We may assume that g_{2} ≠ 1 if and only if p_{2} ≠ 1. If ϕ_{i} : π → π_{i} is the natural homomorphism, then x is mapped to ϕ_{i}(p_{1})g_{1}ϕ_{i}(p_{2})g_{2} … g_{n} ∈ π_{i} ∗ G. If this is trivial, but g_{1} ≠ 1, necessarily ϕ_{i}(p_{2}) = 1 ∀i. This implies p_{2} = 1, i.e. x = p_{1}g_{1}, since we wrote x in normal form. But then (ϕ_{i} ∗ id)(x) ≠ 1 ∀i, and the kernel of the map to X is trivial, as required.

For the abelian product, let X be a group together with a consistent family of morphisms to π_{i} × G. These have the form x ↦ (ϕ_{i}(x), f_{i}(x)). Composition with the projections to π_{i} or to G shows that ϕ_{i} is a consistent family of morphisms to π_{i}, and f = f_{i} : X → G all coincide. Let ϕ : X → π be the limit. Then ϕ × f : X → π × G is a unique homomorphism which makes all relevant diagrams commutative. By the universal property of inverse limits, the statement follows.

3. Passage to subgroups

Suppose U ⊂ π is a subgroup of a discrete group. A positive self-adjoint matrix A ∈ M(d×d, ℤU) can also be considered as a matrix over ℤπ. Denote the operators with A_{U} and A_{π}, respectively. Recall the following well known fact:

3.1. Proposition. The spectral density functions of A_{U} and A_{π} coincide.

Proof. Choose a set of representatives {g_{i}}_{i∈I} with 0 ∈ I and g_{0} = 1, to write π = ⨿_{i∈I}Ug_{i}. Then

l^{2}(π)^{d} = ⨁_{i∈I} l^{2}(U)^{d}g_{i}.

With respect to this splitting, the action of A_{π} on l^{2}(π) is diagonal and, restricted to each of the summands l^{2}(U)^{d}g_{i}, is multiplication by A_{U} from the left. It follows that every spectral projection χ_{[0,λ]}(A_{π}) is diagonal with χ_{[0,λ]}(A_{U}) on the diagonal. Then (where e_{k} = (0, …, 0, 1, 0, …, 0))

\begin{matrix}F_{A_{π}}(λ) = & ∑_{k=1}^{d}⟨χ_{[0,λ]}(A_{π})e_{k}^{π}, e_{k}^{π}⟩ = ∑_{k=1}^{d}⟨χ_{[0,λ]}(A_{π})e_{k}^{U} · 1, e_{k}^{U} · 1⟩ \\ = & ∑_{k=1}^{d}⟨χ_{[0,λ]}(A_{U})e_{k}^{U}, e_{k}^{U}⟩ = F_{A_{U}} (λ). \end{matrix}

3.2. Corollary. The properties of Definition 1.18 are inherited by subgroups.

In particular, we have proven the subgroup part of Theorems 1.19 and 1.21.

4. Amenable extensions

4.1. Definition. A discrete homogeneous space π/U is called amenable if on π/U we find a π-invariant integer-valued metric d : π/U × π/U → ℕ such that

• sets of finite diameter are finite

• for every K > 0, ϵ > 0 there is a non-empty finite subset ∅ ≠ X ⊂ π/U with

|N_{K}(X)| ≤ ϵ |X|

where N_{K}(X) := U_{K}(X)∩U_{K}(π/U −X) is the K-neighborhood of the boundary of X (with U_{K}(X) := {x ∈ π/U; d(x, X) ≤ K}).

A nested sequence of finite subsets K_{1} ⊂ K_{2} ⊂ … is called an amenable exhaustion of π/U if ⋃ K_{n} = π/U and if ∀K > 0 and ϵ > 0 we find N ∈ ℕ so that |N_{K}(K_{i})| ≤ ϵ |K_{i}| ∀i ≥ N.

4.2. Lemma. Every amenable homogeneous space π/U admits an amenable exhaustion.

Proof. For n, K ∈ ℕ we find X_{n,K} with |N_{K}(X_{n,K})| ≤ \frac{1}{n} |X_{n,K}|. Fix some base point o ∈ π/U. Since π acts transitively on π/U and the metric is π-invariant, we may assume after translation that the base point is contained in each of the X_{n,K}. Now we construct the exhaustion E_{i} inductively. Set E_{1} := X_{1,1}. For the induction suppose E_{1}, …, E_{n} are constructed with |N_{k}(E_{k})| ≤ \frac{1}{k} |E_{k}| for k = 1, …, n. Suppose E_{n} ⊂ U_{d}(o) with d ∈ ℕ and d ≥ n + 1. Set E_{n+1} := E_{n} ∪ U_{n}(o) ∪ X_{n+1,2d}. Then U_{d}(E_{n+1}) ⊂ U_{2d}(X_{n+1,2d}) by the triangle inequality and because o ∈ X_{n+1,2d}. Moreover U_{d}(π/U − E_{n+1}) ⊂ U_{2d}(π/U − X_{n+1,2d}). Consequently N_{n+1}(E_{n+1}) ⊂ N_{d}(E_{n+1}) ⊂ N_{2d}(X_{n+1,2d}). It follows

|N_{n+1}(E_{n+1})| ≤ |N_{2d}(X_{n+1,2d})| ≤ \frac{1}{n + 1} |X_{n+1,2d}| ≤ \frac{1}{n + 1} |E_{n+1}| .

Since U_{n}(o) ⊂ E_{n+1} we also have ⋃_{n∈ℕ} E_{n} = π/U. The claim follows.

4.3. Example. If U is a normal subgroup and π/U is an amenable group, it is an amenable homogeneous space.

4.4. Definition. Suppose π is a group with subgroup U and amenable quotient π/U. Choose an amenable cover X_{1} ⊂ X_{2} ⊂ ⋯ ⊂ π/U. For B ∈ M(d × d, N U) set

tr_{m}(B) := \frac{1}{|X_{m}|} tr_{U}(B).

For Δ ∈ M(d × d, N π) positive and self-adjoint set Δ_{m} := P_{m}ΔP_{m} where P_{m} = diag(p_{m}) with p_{m} ∈ B(l^{2}(π)) given by projection onto the closed subspace generated by the inverse image of X_{m}. Then Δ_{m} no longer belongs to N π but still to N U and we define (by slight abuse of notation)

F_{Δ_{m}}(λ) := tr_{m}(χ_{[0,λ]}(Δ_{m})),
ln det_{U}(Δ_{m}) := ∫_{0^{+}}^{∞} ln(λ) dF_{Δ_{m}}(λ)  using the new F_{Δ_{m}}.

Here Δ_{m} is considered as operator on the image of P_{m}. This subspace is N Uisomorphic to l^{2}(U)^{d|X_{m}|}.

Note that there are two meanings of F_{Δ_{m}}(λ) and ln det_{U}(Δ_{m}) (using either tr_{U} or tr_{m}), but in the amenable case we will always use the variant where we divide by the volume of the sets X_{m}.

The following is one of the key lemmas which make our method work:

4.5. Lemma. In the situation above, we find K ∈ ℝ independent of m, so that

∥Δ∥ ≤ K  and  ∥Δ_{m}∥ ≤ K  ∀m ∈ ℕ.

Proof. This is an immediate consequence of the fact that ∥P ∥ ≤ 1 for every projection P and Δ_{m} = P_{m}ΔP_{m} with projections P_{m}.

We now establish the second key lemma. It generalizes a corresponding result of Dodziuk/Mathai [6, 2.3] where U is trivial. We need the result only for matrices over ℂπ, but for possible other applications we prove a more general statement here.

4.6. Lemma. Let p(x) ∈ ℂ[x] be a polynomial. Suppose Δ ∈ M(d × d, N π). Then

tr_{π} p(Δ) = lim_{m→∞} tr_{m} p(Δ_{m}).

Proof. By linearity it suffices to prove the statement for the monomials x^{N}, N ∈ ℕ.

Pull the metric on π/U back to π to get some semi-metric on π. Denote the inverse image of X_{k} in π with X′_{k}.

We have to compare (Δ^{N}ge_{k}, ge_{k}) and (Δ_{m}^{N}ge_{k}, ge_{k}) for g = g_{i} ∈ X′_{m}, in particular for those g_{i} with B_{a}(g_{i}) ⊂ X′_{m}. Of course, we don’t find a ∈ ℝ such that the difference is zero. However, we will show that (for fixed N) we can find a such that the difference is sufficiently small.

First observe that P_{m}ge_{k} = ge_{k} if g ∈ X′_{m}, and (since P_{m} is self-adjoint)

((P_{m}ΔP_{m})^{N}ge_{k}, ge_{k}) = (ΔP_{m}Δ … P_{m}Δge_{k}, ge_{k}).

Now the following sum is a telescope and therefore

(4.7) ΔP_{m}Δ … P_{m}Δ =

Δ^{N} − Δ(1 − P_{m})Δ^{N−1} − ΔP_{m}Δ(1 − P_{m})Δ^{N−2} − ⋯ − ΔP_{m} … Δ(1 − P_{m})Δ.

It follows for g ∈ X′_{m}

\begin{aligned}|(Δ^{N}ge_{k}, ge_{k}) − (Δ_{m}^{N}ge_{k}, ge_{k})| & ≤ ∑_{i=1}^{N−1} |((1 − P_{m})Δ^{i}ge_{k}, (Δ^{∗}P_{m})^{N−i}ge_{k})| \\ & ≤ ∑_{i=1}^{N−1} |(1 − P_{m})Δ^{i}ge_{k}| · ∥Δ^{∗}∥^{N−i} . \end{aligned}

Here we used the fact that the norm of a nontrivial projector is 1 and |ge_{k}| = 1.

Fix ϵ > 0. For i = 1, …, N − 1 and k = 1, …, d we have Δ^{i}ge_{k} ∈ l^{2}(π)^{d}. It follows that we find R > 0 so that

|(1 − P_{B_{R}(g)})Δ^{i}ge_{k}| ≤ ϵ (4.8)

where P_{B_{R}(g)} is the projector onto the closed subspace spanned by the elements in ⋃_{k=1}^{d} B_{R}(g)e_{k}. Since Δ and the semi-metric are π-invariant, this holds for every g ∈ π with R independent of g. If the range of P_{B_{R}(g)} is contained in the range of P_{m}, i.e. if B_{R}(g) ∈ X′_{m} then (4.8) implies

|(1 − P_{m})Δ^{i}ge_{k}| ≤ ϵ

(since we replace by zero even more Fourier coefficients (in the standard orthonormal base of l^{2}(π)^{d} coming from π)). Taken together, we get

|tr_{π} Δ^{N} − tr_{m} Δ_{m}^{N}| ≤ \frac{1}{|X_{m}|} ∑_{k=1}^{d} ∑_{i∈X_{m}} |(Δ^{N}g_{i}e_{k}, g_{i}e_{k}) − ((Δ_{m})^{N}g_{i}e_{k}, g_{i}e_{k})|

≤ \frac{1}{|X_{m}|} ∑_{k=1}^{d} ∑_{i∈X_{m}} ∑_{j=1}^{N−1} |(1 − P_{m})Δ^{j}g_{i}e_{k}| · ∥Δ^{∗}∥^{N−j}

≤ \frac{1}{|X_{m}|} ∑_{k=1}^{d} ∑_{i∈X_{m}−N_{R}(X_{m})} ∑_{j=1}^{N−1} \underbrace{|(1 − P_{m})Δ^{j}g_{i}e_{k}|}_{<ϵ since B_{R}(g_{i}) ⊂ X′_{m} by def. of N_{R}(X_{m})} ∥Δ^{∗}∥^{N−j}

+ \frac{1}{|X_{m}|} ∑_{i∈N_{R}(X_{m})} ∑_{k=1}^{d} ∑_{j=1}^{N−1} |(1 − P_{m})Δ^{j}g_{i}e_{k}| · ∥Δ^{∗}∥^{N−j}

≤ ϵ \underbrace{dN max_{j=1,...,N−1}\{∥Δ^{∗}∥^{j}\}}_{=:C_{N}} +\frac{|N_{R}(X_{m})|}{|X_{m}|} \underbrace{dN \underbrace{∥1 − P_{m}∥}_{≤2} max_{j=1...N}\{∥Δ∥^{j} · ∥Δ^{∗}∥^{N−j}\}}_{=:C′_{N}} .

Note that C_{N} and C′_{N} are independent of m and ϵ. Since X_{m} is an amenable exhaustion, for every R we find m_{R} so that \frac{|N_{R}(X_{m})|}{|X_{m}|} is smaller than ϵ for every m ≥ m_{R}. Since ϵ was arbitrary, the assertion of the lemma follows.

5. Direct and inverse limits

5.1. Remark. In this section, we study the properties of Definition 1.18 for direct and inverse limits. However, we will only deal with the apparently weaker statements that each of the conditions holds for every Δ of the form Δ = A^{∗}A. The general case is a consequence of this since we can easily compare the self-adjoint Δ with Δ^{2} = Δ^{∗}Δ, because F_{Δ}(λ) = F_{Δ^{2}}(λ^{2}).

We describe now the situation we are dealing with in this section:

5.2. Definition. Suppose the group π is the direct or inverse limit of a directed system of groups π_{i}, i ∈ I. The latter means that we have a partial ordering < on I, and ∀i, j ∈ I we find k ∈ I with i < k and j < k. In the case of a direct limit, let p_{i} : π_{i} → π the natural maps, in the case of an inverse limit, p_{i} : π → π_{i}.

Suppose A ∈ M(d × d, ℂπ) is given.

If π is an inverse limit, let A_{i} = p_{i}(A) be the image of A under the projection M(d, π) → M(d, π_{i}). Set Δ := A^{∗}A. Then Δ_{i} = (A_{i})^{∗}A_{i} (this follows from the algebraic description of the adjoint [11, p. 465]). In particular, all of the operators Δ_{i} are positive. Define

tr_{i}(Δ_{i}) := tr_{π_{i}}(Δ_{i}).

F_{Δ_{i}}(λ) is defined using the trace on the von Neumann algebra π_{i} (i ∈ I).

If we want to give a similar definition in the case where π is a direct limit, we have to make additional choices. Namely, let A = (a_{kl}) with a_{kl} = ∑_{g∈π} λ_{kl}^{g}g. Then, only finitely many of the λ_{kl}^{g} are nonzero. Let V be the corresponding finite collection of g ∈ π. Since π is the direct limit of π_{i} we find j_{0} ∈ I such that V ⊂ p_{j_{0}}(π_{j_{0}}). Choose an inverse image for each g in π_{j_{0}}. This gives a matrix A_{j_{0}} ∈ M(d × d, π_{j_{0}}), which is mapped to A_{i} ∈ M(d × d, π_{i}) for i > j_{0}. Now we apply the above constructions to this net (A_{i})_{i>j_{0}}. Note that this definitely depends on the choices.

For notational convenience, we choose some j_{0} ∈ I also when we deal with an inverse limit.

Now, we will establish in this situation the two key lemmas corresponding to Lemma 4.5 and Lemma 4.6.

For the first lemma, instead of working with the norm of operators, we will use another invariant which gives an upper bound for the norm but is much easier to read off:

5.3. Definition. Let π be a discrete group, Δ ∈ M(d × d, ℤπ). Set

K(Δ) := d^{2} max_{i,j}{|a_{i,j}|_{1}} where |·|_{1} is the L^{1}-norm on ℂπ ⊂ l^{1}(π).

5.4. Lemma. Adopt the situation of Definition 5.2. One can find K ∈ ℝ, independent of i, such that

∥A_{i}∥ ≤ K ∀i > j_{0}  and  ∥A∥ ≤ K.

Proof. Lück [11, 2.5] shows that ∥A_{i}∥ ≤ K(A_{i}). It follows from the construction of A_{i} that K(A_{i}) ≤ K(A) in the case of an inverse limit, and K(A_{i}) ≤ K(A_{j_{0}}) in the case of a direct limit, with j_{0} as above. In both cases, we obtain a uniform bound for ∥A_{i}∥.

5.5. Lemma. Adopt the situation of Definition 5.2. Let p(x) ∈ ℂ[x] be a polynomial. There exists i_{0} ∈ I depending on the matrix A and on p such that

tr_{π}(p(A)) = tr_{i}(p(A_{i})) ∀i > i_{0}.

Proof. Suppose π is the inverse limit of the π_{i}. We follow [11, 2.6]. Let p(A) = (∑_{g∈π} λ_{g}^{kl}g)_{k,l=1,...,d}. Then

tr_{π}(p(A)) = ∑_{k} λ_{1}^{kk}  and  tr_{π_{i}}(p(A_{i})) = ∑_{k} ∑_{g∈ker p_{i}} λ_{g}^{kk}.

Since only finitely many of λ_{g}^{ij} ≠ 0 and π is the inverse limit of the π_{i} we find i_{0} ∈ I such that λ_{g}^{kk} ≠ 0 and g ∈ ker p_{i_{0}} implies g = 1. For i > i_{0} the assertion is true.

If π is the direct limit we have chosen A_{j_{0}} with p_{j_{0}}(A_{j_{0}}) = A. Then p_{j_{0}}(p(A_{j_{0}})) = p(A). However, there may be a g ∈ π_{j_{0}} with nontrivial coefficient in p(A_{j_{0}}) with p_{j_{0}}(g) = 1, and this means that the relevant traces may differ. But still there are only finitely many g with nontrivial coefficient in p(A_{j_{0}}), and since π is the direct limit of (π_{i})_{i>j_{0}} we find i_{0} such that p_{i_{0}}(g) = 1 for g ∈ π_{i_{0}} with nontrivial coefficient in p(A_{i_{0}}) implies g = 1. Then the above reasoning shows tr_{π_{i}}(p(A_{i})) = tr_{π}(p(A)) ∀i > i_{0}.

6. Approximation properties and proofs of stability statements

In this section, we use the information gathered so far to proof the statements of the introduction, in particular Theorem 1.21. This is done by studying limits of operators, therefore the same treatment yields approximation results for L^{2}-Betti numbers. The precise statements and conditions are given below.

We have already dealt with the passage to subgroups.

For the rest of this section, assume the following situation:

6.1. Situation. The group π is the direct or inverse limit of a directed system of groups π_{i}, or an amenable extension U → π → π/U (write π_{i} = U also in this case).

As described in 4.4 or 5.2, any matrix Δ over ℂπ then gives rise to matrices Δ_{i} over π_{i} (after the choice of an inverse image in the case of a direct limit, and after the choice of an amenable exhaustion for amenable extensions). Without loss of generality we assume that Δ = A^{∗}A for another matrix A over ℂπ (this is explained in Remark 5.1). We also get spectral density functions F_{A_{i}}(λ) defined using the group π_{i} (remember that in the amenable case there is an additional normalization).

The problem now is to obtain information about F_{Δ}(λ) from the family F_{Δ_{i}}(λ). In particular, we want to show that F_{A_{i}}(0) converges to F_{A}(0). (Translated to

geometry this means that certain L^{2}-Betti numbers converge.)

In short: we have

• a group π and a matrix Δ ∈ M(d × d, ℂπ)

• a family (Δ_{i})_{i∈I} of matrices over ℂπ_{i} which approximate Δ (I is a directed system)

• positive and normal trace functionals tr_{i} (on a von Neumann algebra which contains Δ_{i}) which are normalized in the following sense: If Δ = id ∈ M(d × d, ℤπ) then tr_{i}(Δ_{i}) = d ∀i.

• If Δ lives over ℤπ, then Δ_{i} is a matrix over ℤπ_{i}.

6.2. Definition. Define

\overline{F_{Δ}}(λ) := lim sup_{i∈I} F_{Δ_{i}}(λ),
\underline{F_{Δ}}(λ) := lim inf_{i∈I} F_{Δ_{i}}(λ).

Remember lim sup_{i∈I}{x_{i}} = inf_{i∈I}{sup_{j>i}{x_{j}}}; lim inf_{i∈I}{x_{i}} = sup_{i∈I}{inf_{j>i}{x_{j}}}.

6.3. Definition. Suppose F : [0, ∞) → ℝ is monotone increasing (e.g. a spectral density function). Then set

F^{+}(λ) := lim_{ϵ→0^{+}} F (λ + ϵ)

i.e. F^{+} is the right continuous approximation of F. In particular, we have defined \overline{F_{Δ}}^{+} and \underline{F_{Δ}}^{+}.

6.4. Remark. Note that by our definition a spectral densitiy function is right continuous, i.e. unchanged if we perform this construction.

To establish the first step in our program we have to establish the following functional analytical lemma (compare [11] or [3]):

6.5. Lemma. Let N be a finite von Neumann algebra with positive normal and normalized trace tr_{N}. Choose Δ ∈ M(d × d, N) positive and self-adjoint. If for a function p_{n} : ℝ → ℝ

χ_{[0,λ]}(x) ≤ p_{n}(x) ≤ \frac{1}{n}χ_{[0,K]}(x) + χ_{[0,λ+1/n]}(x) ∀0 ≤ x ≤ K (6.6)

and if ∥Δ∥ ≤ K then

F_{Δ}(λ) ≤ tr_{N} p_{n}(Δ) ≤ \frac{1}{n}d + F_{Δ}(λ + 1/n).

Here χ_{S}(x) is the characteristic function of the subset S ⊂ ℝ.

Proof. This is a direct consequence of positivity of the trace, of the definition of spectral density functions and of the fact that tr_{N} (1 ∈ M(d × d, N)) = d by the definition of a normalized trace.

Now one of the key technical results follows. Corresponding special cases are [11, 2.3(1)], [6, 2.1(1)], [3, 2.1], and the proof is essentially always the same.

6.7. Proposition. For every λ ∈ ℝ we have

\begin{matrix}\overline{F_{Δ}}(λ) ≤ F_{Δ}(λ) = \underline{F_{Δ}}^{+}(λ) ≤ \underline{F_{Δ}}^{+}(λ), \\ F_{Δ}(λ) = \underline{F_{Δ}}^{+}(λ) = \overline{F_{Δ}}^{+}(λ). \end{matrix}

Proof. The proof only depends on our key lemmas 4.5, 4.6, 5.4, 5.5. These say • ∃K ∈ ℝ such that ∥Δ_{i}∥ ≤ K ∀i ∈ I

• For every polynomial p ∈ ℂ[x] we have tr_{π}(p(Δ)) = lim_{i} tr_{i}(p(Δ_{i})).

For each λ ∈ ℝ choose polynomials p_{n} ∈ ℝ[x] such that inequality (6.6) is fulfilled. Note that by the first key lemma we find a uniform upper bound K for the spectrum of all of the Δ_{i}. Then by Lemma 6.6

F_{Δ_{i}}(λ) ≤ tr_{i}(p_{n}(Δ_{i})) ≤ F_{Δ_{i}}(λ + \frac{1}{n}) + \frac{d}{n}

We can take the limes inferior and superior and use the second key lemma to get

\overline{F_{Δ}}(λ) ≤ tr_{π}(p_{n}(Δ)) ≤ \underline{F_{Δ}}(λ + \frac{1}{n}) + \frac{d}{n}.

Now we take the limes for n → ∞. We use the fact that tr_{π} is normal and p_{n}(Δ) converges strongly inside a norm bounded set to χ_{[0,λ]}(Δ). Therefore the convergence even is in the ultra strong topology.

This implies

\overline{F_{Δ}}(λ) ≤ F_{Δ}(λ) ≤ \underline{F_{Δ}}^{+}(λ).

For ϵ > 0 we can now conclude since \underline{F_{Δ}} and \overline{F_{Δ}} are monotone

F_{Δ}(λ) ≤ \underline{F_{Δ}}(λ + ϵ) ≤ \overline{F_{Δ}}(λ + ϵ) ≤ F_{Δ}(λ + ϵ).

Taking the limit ϵ → 0^{+} gives (since F_{Δ} is right continuous)

F_{Δ}(λ) = \overline{F_{Δ}}^{+}(λ) = \underline{F_{Δ}}^{+}(λ).

Therefore both of the inequalities are established.

The next step is to prove convergence results without taking right continuous approximations (at least for λ = 0). We are able to do this only under additional assumptions:

• From now on, Δ and therefore also Δ_{i} ∀i ∈ I are matrices over the integral group ring.

The following well known statement is used as start for the induction (observed e.g. in [11]).

6.8. Lemma. The trivial group has semi-integral determinant.

Proof. Take Δ ∈ M(d×d, ℤ) positive and self-adjoint. Then det_{1}(Δ) is the product of all nonzero eigenvalues and therefore the lowest nonzero coefficient in the characteristic polynomial. In particular, it is an integer ≠ 0 and ln det_{1}(Δ) ≥ 0.

Now we give a proof of Theorem 1.21 and prove the corresponding approximation result.

6.9. Theorem. Suppose π_{i} has semi-integral determinant ∀i ∈ I, then the same is true for π, and dim_{π}(ker Δ) = F_{Δ}(0) = lim_{i} F_{Δ_{i}}(0).

Proof. Choose K ∈ ℝ such that K > ∥Δ∥ and K > ∥Δ_{i}∥ ∀i. This is possible because of Lemma 4.5 or 5.4. Then

ln det_{π_{i}}(Δ_{i}) = ln(K)(F_{Δ_{i}}(K) − F_{Δ_{i}}(0)) − ∫_{0^{+}}^{K} \frac{F_{Δ_{i}}(λ) − F_{Δ_{i}}(0)}{λ} dλ.

If this is (by assumption) ≥ 0, then since F_{Δ_{i}}(K) = tr_{i}(1_{d}) = d by our normalization

∫_{0^{+}}^{K} \frac{F_{Δ_{i}}(λ) − F_{Δ_{i}}(0)}{λ} dλ ≤ ln(K)(d − F_{Δ_{i}}(0)) ≤ ln(K)d.

We want to establish the same estimate for Δ. If ϵ > 0 then

∫_{ϵ}^{K} \frac{F_{Δ}(λ) − F_{Δ}(0)}{λ} dλ = ∫_{ϵ}^{K} \frac{\underline{F_{Δ}}^{+}(λ) − F_{Δ}(0)}{λ} dλ = ∫_{ϵ}^{K} \frac{\underline{F_{Δ}}(λ) − F_{Δ}(0)}{λ}

(since the integrand is bounded, the integral over the left continuous approximation is equal to the integral over the original function)

≤ ∫_{ϵ}^{K} \frac{\underline{F_{Δ}}(λ) − \overline{F_{Δ}}(0)}{λ}
= ∫_{ϵ}^{K} \frac{lim inf_{i} F_{Δ_{i}}(λ) − lim sup_{i} F_{Δ_{i}}(0)}{λ}
≤ ∫_{ϵ}^{K} \frac{lim inf_{i}(F_{Δ_{i}}(λ) − F_{Δ_{i}}(0))}{λ}
≤ lim inf_{i} ∫_{ϵ}^{K} \frac{F_{Δ_{i}}(λ) − F_{Δ_{i}}(0)}{λ} ≤ d ln(K).

Since this holds for every ϵ > 0, we even have

∫_{0^{+}}^{K} \frac{F_{Δ}(λ) − F_{Δ}(0)}{λ} ≤ ∫_{0^{+}}^{K} \frac{\underline{F_{Δ}}(λ) − \overline{F_{Δ}}(0)}{λ} dλ
≤ sup_{ϵ>0} lim inf_{i} ∫_{ϵ}^{K} \frac{F_{Δ_{i}}(λ) − F_{Δ_{i}}(0)}{λ} dλ
≤d ln(K).

The second integral would be infinite if lim_{δ→0} \underline{F_{Δ}}(δ) ≠ \overline{F_{Δ}(0)}. It follows from 6.7 that lim sup_{i} F_{Δ_{i}}(0) = F_{Δ}(0). Since we can play the same game for every subnet of I, also lim inf_{i} F_{Δ_{i}}(0) = F_{Δ}(0) i.e. the approximation property is true.

To estimate the determinant note that in the above inequality

sup_{ϵ>0} lim inf_{i} ∫_{ϵ}^{K} \frac{F_{Δ_{i}}(λ) − F_{Δ_{i}}(0)}{λ} dλ
≤ lim inf_{i} sup_{ϵ>0} ∫_{ϵ}^{K} \frac{F_{Δ_{i}}(λ) − F_{Δ_{i}}(0)}{λ} dλ = lim inf_{i} ∫_{0^{+}}^{K} \frac{F_{Δ_{i}}(λ) − F_{Δ_{i}}(0)}{λ} dλ
≤ ln(K)(d − F_{Δ_{i}}(0)).

Therefore

ln det_{π}(Δ) = ln(K)(d − F_{Δ}(0)) − ∫_{0^{+}}^{K} \frac{F_{Δ}(λ) − F_{Δ}(0)}{λ} dλ

≥ ln(K)(d − lim_{i} F_{Δ_{i}}(0)) − lim inf_{i} ∫_{0^{+}}^{K} \frac{F_{Δ_{i}}(λ) − F_{Δ_{i}}(0)}{λ} dλ

= lim sup_{i} (ln(K)(d − F_{Δ_{i}}(0)) − ∫_{0^{+}}^{K} \frac{F_{Δ_{i}}(λ) − F_{Δ_{i}}(0)}{λ} dλ)

= lim sup_{i} ln det_{π_{i}}(Δ_{i}) ≥ 0.

6.10. Remark. We can establish the approximation results only under the assumption that the groups π_{i} have good properties, e.g. belong to the class G. It is interesting to note that for amenable groups every quotient group is amenable and belongs to G. It follows that Conjecture 1.10 holds if π_{1}(X) is amenable without additional assumptions.

6.11. Remark. It would be possible to give geometrical interpretations of the more general approximation results similar to 1.10. However, this seems to be rather artificial and therefore is omitted here.

Proof of Theorem 1.19. It remains to show that Whitehead trivial determinant is stable under direct and inverse limits.

Proof of Theorem 1.19. We are still in the situation described at the beginning of this section, and, in addition, we assume that Δ is invertible in M(d × d, ℤπ) with inverse B ∈ M(d × d, ℤπ). If π is an inverse limit then Δ_{i} and B_{i} are images of projections of Δ and B, and therefore remain inverse to each other.

In case π is a direct limit, we first lifted Δ to some Δ_{j_{0}}. We may assume that we also can lift B to B_{j_{0}}. Then Δ_{j_{0}}B_{j_{0}} is mapped to the identity over π. Since it has only finitely many nonzero coefficients, there is j_{1} such that the image of Δ_{j_{0}}B_{j_{0}} over π_{j_{1}} already is the identity, and similar for B_{j_{0}}Δ_{j_{0}}. Therefore, we may assume that the lifts Δ_{j_{0}} and B_{j_{0}} are inverse to each other. The same is then true for Δ_{i} and B_{i} for i > j_{0}, i.e. Δ_{i} represents an element in W h(π_{i}).

By assumption ln det_{π_{i}}(Δ_{i}^{∗}Δ_{i}) = 0. Note that the proof of Theorem 6.9 applies to our situation and we conclude ln det_{π}(Δ^{∗}Δ) ≥ 0. Since Δ ∈ W h(π) was arbitrary, Theorem 1.22 implies the result.

6.12. Remark. It is not possible to proceed along similar lines in the case that U has an amenable quotient (even if π is amenable itself). The problem is that we approximate the matrix Δ (over ℤπ) by matrices over ℤU of larger and larger dimension. One can show that these matrices are invertible over N U if Δ itself was invertible. However, even if the inverse of Δ is a matrix over ℤπ, in general this is not true for the approximating matrices over ℤU.

This finishes the proof of Theorems 1.19, 1.21 and 1.14. We proceed with some side remarks.

7. Complex approximation

In this section, we will address the question whether the approximation results we have obtained in section 6 are valid not only for matrices over the integral group ring, but also over the complex group ring. In particular, we adopt Situation 6.1: A group π is approximated by groups π_{i}, and a matrix Δ = A^{∗}A by matrices Δ_{i}.

Essentially, we will give a positive answer to our question only for free abelian groups. We start with a general observation.

7.1. Lemma. Suppose in the situation 6.1 that ker Δ = 0. Then the approximation result holds without integrality assumptions: lim_{i} F_{Δ_{i}}(0) = 0 = dim_{π}(ker Δ).

More generally, if λ is not an eigenvalue of Δ then lim_{i} F_{Δ_{i}}(λ) = F_{Δ}(λ).

Proof. We know that \underline{F} _{Δ}^{+}(x) = F_{Δ}(x) for every x ∈ ℝ. If \underline{F}_{Δ}(λ) ≤ F_{Δ}(λ) − ϵ then \underline{F} _{Δ}^{+}(x) ≤ F_{Δ}(λ)−ϵ for every x < λ, i.e. F_{Δ}(x) ≤ F_{Δ}(λ)−ϵ ∀x < λ. By assumption, the eigenspace of Δ to λ is trivial, therefore F_{Δ} is continuous at λ and ϵ can only be zero.

7.2. Proposition. If π = ℤ^{n} and contained in the inverse limit of quotient groups {π_{i}}_{i∈ℕ} then the approximation result holds ∀Δ ∈ ℂπ.

Proof. Embed ℂ[ℤ^{n}] into its ring of fractions. Let A ∈ M(d × d, ℂ[ℤ^{n}]). Linear algebra gives X, Y ∈ Gl(d × d, ℂ(ℤ^{n})) with A = X diag(1, …, 1, 0, …, 0)Y. Collecting the denominators in X and Y we find 0 ≠ c ∈ ℂ[ℤ^{n}] ∩ Gl(1, ℂ(ℤ^{n})) and X′, Y′ ∈ M(d × d, ℂ[ℤ^{n}]) ∩ Gl(d, ℂ(ℤ^{n})) such that cA = X′DY′ with D = diag(1, …, 1, \underbrace{0, …, 0}_{r}). The corresponding equation holds after passage to matri-

ces c_{i}, A_{i}, X′_{i}, Y′_{i} over π_{i}. Observe that for V = l^{2}(π_{(i)})^{d} and π- or π_{i}-morphisms u, v : V → V the exact sequence

0 → ker(v) ↪ ker(uv) →^{v} ker(u),

the inclusions im(uv) ⊂ im(u) and the exact sequences

0 → ker(u) → V →^{u} im(u) → 0 0 → ker(uv) → V →^{uv} im(uv) → 0

together with additivity of the von Neumann dimension dim imply

max{dim(ker(u)), dim(ker(v))} ≤ dim(ker(uv)) ≤ dim(ker(u)) + dim(ker(v)).

Because they are invertible over the field of fractions ℂ(ℤ^{n}) the operators c, X′ and Y′ have trivial kernel. The above inequalities applied to c_{i}, A_{i}, X′_{i}, Y′_{i} then imply by Lemma 7.1

r = lim_{i} dim_{π_{i}}(ker(X′_{i}DY′_{i})) = lim_{i} dim_{π_{i}} ker(c_{i}A_{i}) = lim_{i} dim_{π_{i}} ker(A_{i}).

But we also have r = dim_{ℤ^{n}}(ker(A)), therefore the desired result.

8. Quotients

To enlarge the class G, it is important to find other operations under which our main properties —determinant class and semi-integrality— are inherited.

We indicate just one partial result:

8.1. Proposition. Suppose 1 → F → π →^{p} Q → 1 is an extension of groups and |F | < ∞, and π is of determinant class. Then also Q is of determinant class.

Proof. We only indicate the proof, which was discussed with M. Farber during a conference in Oberwolfach, and which uses the theory of virtual characters of Farber [7]. l^{2}(π) corresponds to the Dirac character δ_{1}. The representations V_{k} of the finite group F give rise to characters χ_{k} of π with support contained in F. Since l^{2}(F) = ⨁ μ_{k}V_{k}, δ_{1} = \frac{1}{|F |} ∑ μ_{k}χ_{k} with μ_{k} > 0. Since the operator Δ we are interested in arises from c^{∗} ⊗ id on C^{∗} ⊗ l^{2}(π), F_{Δ}(λ) = ∑ \frac{μ_{k}}{|F |}F_{Δ}^{χ_{k}}(λ) (compare [7, 7.2]). Now the trivial representation V_{1} of F corresponds to the quotient representation l^{2}(Q), and F_{Δ}^{χ_{1}} is just F_{p(Δ)}^{Q}. By assumption ∫_{0+}^{∞} ln(λ) dF_{Δ}(λ) > −∞. Since ∫ ln(λ) dF_{Δ}^{χ_{k}}(λ) < ∞ ∀k, it follows in particular

∫_{0^{+}}^{∞} ln(λ) dF_{p(Δ)}^{Q}(λ) > −∞.

Since p is surjective, this is true for every matrix over ℤπ we have to consider. This concludes the proof.

8.2. Remark. If we have an extension 1 → ℤ^{n} → π →^{p} Q → 1, we can not write the character of l^{2}(π) as a direct sum, but as a direct integral (over the dual space ℤ̂^{n} = T^{n}). Then F_{Δ}(λ) = ∫_{T^{n}} F_{Δ}^{χ_{η}}(λ) dη, and F_{Δ}^{χ_{1}}(λ) = F_{p(Δ)}^{Q}(λ). After establishing an appropriate continuity property, we can conclude as above that if π is of determinant class then

∫_{0^{+}}^{∞} ln(λ) dF_{p(Δ)}^{Q}(λ) > −∞,

i.e. also Q is of determinant class.

References

[1] Atiyah, M.: “Elliptic operators, discrete groups and von Neumann algebras”, Astérisque 32, 43–72 (1976)

[2] Burghelea, D., et al.: “Analytic and Reidemeister torsion for representations in finite type Hilbert modules”, Geometric and Functional Analysis 6, 751–859 (1996)

[3] Clair, B.: “Residual amenability and the approximation of L^{2}-invariants”, Michigan Math. J. 46, 331–346 (1999)

[4] Cohen, M.M.: “Combinatorial group theory: a topological approach”, vol. 14 of LMS student texts, Cambridge University Press (1989)

[5] Dicks, W. and Dunwoody, M.J.: “Groups acting on graphs”, No. 17 in Cambridge studies in Advanced Mathematics, Cambridge University Press (1989)

[6] Dodziuk, J. and Mathai, V.: “Approximating L^{2}-invariants of amenable covering spaces: A combinatorial approach”, J. of Functional Analysis 154, 359–378 (1998)

[7] Farber, M.: “Geometry of growth: Approximation theorems for L^{2}-invariants”, Math. Annalen 311, 335–376 (1998)

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

[9] Hess, E. and Schick, T.: “Non-vanishing of L^{2}-torsion of hyperbolic manifolds”, manuscr. mathem. 97, 329–334 (1998)

[10] Linnell, P.: “Division rings and group von Neumann algebras”, Forum Math. 5, 561–576 (1993)

[11] Lück, W.: “Approximating L^{2}-invariants by their finite-dimensional analogues”, Geometric and Functional Analysis 4, 455–481 (1994)

[12] Lück, W.: “L^{2}-torsion and 3-manifolds”, in: Johannson, Klaus (ed.), Conference Proceedings and Lecture Notes in Geometry and Topology, Volume III, Knoxville 1992: Lowdimensional topology, 75–107, International Press (1994)

[13] Lück, W.: “L^{2}-invariants of regular coverings of compact manifolds and CW -complexes”, to appear in “Handbook of Geometry”, Elsevier (1998)

[14] Lück, W. and Rothenberg, M.: “Reidemeister torsion and the K-theory of von Neumann algebras”, K-theory 5, 213–264 (1991)

[15] Lück, W. and Schick, T.: “L^{2}-torsion of hyperbolic manifolds of finite volume”, Geom. Funct. Anal. 9, 518–567 (1999)

[16] Mathai, V. and Rothenberg, M.: “On the homotopy invariance of L^{2} torsion for covering spaces”, Proc. of the AMS 126, 887–897 (1998)

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

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

Fachbereich Mathematik, Universität Münster, Einsteinstr. 62, 48149 Münster, Fax: ++49 251/83 38370

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