RESIDUAL AMENABILITY AND THE APPROXIMATION OF L^{2}-INVARIANTS

BRYAN CLAIR

Abstract. We generalize Lück’s Theorem to show that the L^{2}-Betti numbers of a residually amenable covering space are the limit of the L^{2}-Betti numbers of a sequence of amenable covering spaces. We show that any residually amenable covering space of a finite simplicial complex is of determinant class, and that the L^{2}-torsion is a homotopy invariant for such spaces. We give examples of residually amenable groups, including the Baumslag-Solitar groups.

Introduction

In 1994, Wolfgang Lück [15] proved the beautiful theorem that if X is a finite simplicial complex with residually finite fundamental group, the L^{2}-Betti numbers of the universal covering of X can be approximated by the ordinary Betti numbers of a sequence of finite coverings of X. In fact, the question of approximation dates back to Kazhdan [12] (see also [10, Pg. 231]) but only an inequality was known. Dodziuk and Mathai [8] have shown a result analogous to Lück’s Theorem in the situation where the covering transformation group is amenable. Specifically, they show that the L^{2}-Betti numbers of an amenable covering X̃ of X can be approximated by the ordinary Betti numbers of a sequence of Følner subsets of X̃. This paper generalizes Lück’s Theorem to the case where the cover of X has residually amenable transformation group, a large class of groups that includes the residually finite groups of Lück’s Theorem and the amenable groups of Dodziuk and Mathai.

In this paper, we also consider L^{2}-torsion. At first, L^{2}-torsion was defined for L^{2}- acyclic covering spaces. The L^{2}-analytic torsion was first studied in [18] and [14], and L^{2}-Reidemeister-Franz torsion was first studied in [6], see also [16]. Equality of the combinatorial and analytic L^{2}-torsions was proven in 1996 [4].

In order to define these L^{2}-torsions, one needs to establish decay near zero of the spectral density function for the L^{2}-Laplacian. In the case of a residually finite covering, Lück [15] derives an elegant estimate on the spectral density functions for the finite covers, which in the limit gives the necessary decay for the combinatorial L^{2}-Laplacian. Lück also proves the homotopy invariance of L^{2}-combinatorial torsion in this case.

In [5], the combinatorial and analytic torsion invariants were defined more generally as volume forms on L^{2}-cohomology, the decay condition on the spectrum now replaced by a similar condition known as determinant class. The results of [4] extend to show the equality of these more general combinatorial and analytic L^{2}- torsions.

Date: January 17,1999.

Dodziuk and Mathai [8] show that coverings with amenable covering group are of determinant class. Mathai and Rothenberg [19] have recently extended Lück’s results to prove the homotopy invariance of L^{2}-torsion in that case. Although an error in their proof has been pointed out, the error is easily fixable and is corrected in the Appendix of this paper.

A main result of this paper is that coverings with residually amenable covering group are of determinant class, and that L^{2}-torsion of such spaces is a homotopy invariant.

On a different note, Farber [9] has generalized Lück’s Theorem in a new direction, viewing it as a statement about flat bundles rather than finite coverings. In particular, he gives precise conditions for the convergence of L^{2}-Betti numbers of finite non-regular covers. A reasonable direction for future work would be to try and extend the results of this paper using his techniques.

We now formulate the main results of this paper. Let Y be a connected simplicial complex. Suppose that a finitely generated group π acts freely and simplicially on Y so that X = Y/π is a finite simplicial complex.

Suppose there is a nested sequence of normal subgroups π = Γ_{1} ⊃ Γ_{2} ⊃ ⋯ such that ⋂_{n=1}^{∞} Γ_{n} = {e}. Form Y_{n} = Y/Γ_{n}, so that Y_{1}, Y_{2}, … are a tower of covering spaces of X.

Say that π is residually finite if there exist Γ_{n}’s as above, so that the quotients π/Γ_{n} are all finite. Then each Y_{n} is a finite complex, and Lück’s Theorem [15] states that

b_{j}^{(2)}(Y : π) = lim_{n→∞} \frac{1}{|Γ_{n}|}b_{j}(Y_{n})

where b_{j}^{(2)}(Y : π) is the jth L^{2}-Betti number of Y, and b_{j}(Y_{n}) is the ordinary jth Betti number of Y_{n}.

We generalize Lück’s Theorem to the situation where π is residually amenable, meaning there exist Γ_{n}’s as above so that the quotients π/Γ_{n} are all amenable. The first main result of this paper is

Theorem 0.1 (Approximation Theorem). Suppose Y is a simplicial complex, π acts freely and simplicially on Y, and X = Y/π is a finite simplicial complex. If π is residually amenable, then

b_{j}^{(2)}(Y : π) = lim_{n→∞} b_{j}^{(2)}(Y_{n} : π/Γ_{n}).

The next result gives more evidence for the determinant class conjecture, which states that any regular covering space of a finite simplicial complex is of determinant class. For π residually finite this follows from [15], and it was shown for π amenable in [8].

Theorem 0.2 (Determinant Class Theorem). Suppose Y is a simplicial complex, π acts freely and simplicially on Y, and X = Y/π is a finite simplicial complex. If π is residually amenable, then Y is of determinant class.

Now we turn to the problem of homotopy invariance of L^{2}-torsion. Let M and N be compact cell complexes, M̃ and Ñ regular π-covering spaces. As in [19], a homotopy equivalence f : M → N induces a canonical isomorphism f̃_{∗}^{∗} : det \overline{H}_{(2)}^{∗}(Ñ) → det \overline{H}_{(2)}^{∗}(M̃) of determinant lines of L^{2}-cohomology.

Let ϕM̃ ∈ det \overline{H}_{(2)}^{∗}(M̃) denote the combinatorial L^{2}-torsion of M.

Theorem 0.3 (Homotopy Invariance of L^{2}-Torsion). Suppose f : M → N is a homotopy equivalence of compact cell complexes, and M̃ and Ñ are regular covering spaces with residually amenable covering group. Then via the above identification of determinant lines of L^{2}-cohomology,

ϕM̃ = ϕÑ ∈ det \overline{H}_{(2)}^{∗}(M̃).

This provides more evidence for the conjecture in [19] (see also [15]) that L^{2}- torsion is always a homotopy invariant when the covering spaces in question are of determinant class.

This paper is organized as follows. The first section covers preliminaries on residually amenable groups, and exhibits interesting examples. The second section proves the main technical theorem. It essentially states that the L^{2}-spectra of Laplacians on the Y_{n} approximate the L^{2}-spectrum of Laplacian on Y. Finally, the third section proves the three main theorems.

The results of this paper first appeared in the author’s thesis [7]. T. Schick has independently arrived at similar results. Thanks to Mel Rothenberg for suggesting this direction of research, and also to Kevin Whyte and Shmuel Weinberger for helpful discussions.

1. Preliminaries

1.1. Residual Properties.

Definition 1.1. Let C be a nonempty class of groups (though possibly containing only one group). A group π is residually C if for any element g ∈ π, g ≠ e, there exists a quotient group π′(g) belonging to C such that g ↦ g′ ∈ π′(g) with g′ ≠ e.

This paper will use a condition equivalent to residuality which holds for certain classes, for example any C which is closed under products and subgroups. For such a class, a countable group π is residually C if and only if there is a nested sequence of normal subgroups π = Γ_{1} ⊃ Γ_{2} ⊃ ⋯ such that π/Γ_{n} belongs to C and ⋂_{n=1}^{∞} Γ_{n} = {e}. For other basic theorems concerning residual properties of groups, we refer to [17].

When C = {finite groups} we say that π is a residually finite group.

1.2. Amenability. Let π be a finitely generated discrete group, with word metric d. We use the following characterization of amenability, due to Følner.

Definition 1.2. π is amenable if there is a sequence of finite subsets {Λ_{k}}_{k=1}^{∞} such that for any fixed δ > 0

lim_{k→∞} \frac{#\{∂_{δ}Λ_{k}\}}{#\{Λ_{k}\}} = 0

where ∂_{δ}Λ_{k} = {γ ∈ π : d(γ, Λ_{k}) < δ and d(γ, π − Λ_{k}) < δ} is a δ-neighborhood of the boundary of Λ_{k}.

Examples of amenable groups include finite groups, abelian groups, nilpotent groups and solvable groups, and groups of subexponential growth. Amenability for discrete groups is preserved by the following five processes [20, Prop. 0.16]:

1. Taking subgroups;

2. Forming quotient groups;

3. Forming group extensions by amenable groups; 4. Forming upward directed unions of amenable groups;

5. Forming a direct limit of amenable groups.

Free groups with two or more generators, and fundamental groups of closed negatively curved manifolds are not amenable.

1.3. Residual Amenability.

Definition 1.3. If π is residually C, where C = {amenable groups}, we say that π is residually amenable.

Recall that the derived subgroups π^{(i)} of a group π are defined by π^{(0)} = π and π^{(i+1)} = [π^{(i)}, π^{(i)}]. Say that π is solvable if π^{(i)} = {e} for some i. The rank of π is then defined to be the smallest i for which π^{(i)} = {e}.

Contained in the class of residually amenable groups is the class of residually solvable groups. Residually solvable groups are naturally characterized in Prop 1.1. Free products of residually solvable groups are themselves residually solvable. This follows from the fact that solvability is a root property as discussed in [17].

Proposition 1.1. π is residually solvable if and only if ⋂_{i=1}^{∞} π^{(i)} = {e}.

Proof. Since π/π^{(i)} is solvable, if ⋂_{i=1}^{∞} π^{(i)} = {e} then π is residually solvable. Now suppose π is residually solvable, and let g ∈ ⋂_{i=1}^{∞} π^{(i)}. For any map f : π → S with S solvable of rank k, we have f(g) ∈ f(π^{(i)}) ⊆ S^{(i)} = {e} for i > k. Since the image of g is trivial in any solvable quotient of π we must have g = e.

Proposition 1.2. If Γ is residually solvable, S is solvable, and π is an extension

1 → Γ →^{ι} π →^{κ} S → 1

then π is residually solvable.

Proof. Suppose S has rank k. Then κ(π^{(k)}) ⊆ S^{(k)} = {e}, so π^{(k)} ⊆ Γ and therefore π^{(k+i)} ⊆ Γ^{(i)} for all i ≥ 0. Then ⋂_{i=1}^{∞} π^{(i)} ⊆ ⋂_{i=1}^{∞} Γ^{(i)} = {e}.

Example 1.1. For nonzero integers p and q, define the Baumslag-Solitar group BS(p, q) by

BS(p, q) = ⟨a, b | a^{−1}b^{p}a = b^{q}⟩ .

A group π is Hopfian if π/Γ ≅ π implies Γ = {e}. The family of groups BS(p, q) were first defined in [1], where it was shown that BS(p, q) is Hopfian if and only if p and q are meshed, which means p|q, q|p, or p and q have exactly the same set of prime divisors. As any finitely generated residually finite group is Hopfian, the groups BS(p, q) are not residually finite when p and q are not meshed.

Kropholler shows in [13] that the second derived subgroup π^{(2)} is free when π = BS(p, q), for any p, q. When p and q are not both ±1, π^{(2)} is free on two or more generators and therefore π is not amenable. However, π is residually solvable since it is an extension

1 → π^{(2)} → π → π/π^{(2)} → 1

with π^{(2)} residually solvable and π/π^{(2)} solvable (of rank 2).

More generally, let π be any non-cyclic group which is the fundamental group of a graph of infinite cyclic groups. Then from [13], π^{(2)} is free and the above argument shows π is residually solvable.

Example 1.2. We show that the amalgamated free product of two abelian groups is residually solvable. Slightly more generally, suppose we have two solvable groups A and B, and two monomorphisms from an abelian group H into the centers of A and B given by α : H → Z(A) and β : H → Z(B). Then the free product with amalgamation A ∗_{H} B is residually solvable.

To see this, let N = {(α(h), β(h)^{−1})|h ∈ H} ⊂ A × B. Since H is abelian, N is a subgroup of A × B. N is normal because H includes into the centers of both A and B. Note that if H is only known to be normal in A and B, N is unlikely to be normal in A × B.

Let K be the kernel of the natural map A ∗_{H} B → (A × B)/N. In A ∗_{H} B, K has trivial intersection with all conjugates of A and B, hence K is free by a well known theorem of group actions on trees (see for example [2, pg 54]). Then A ∗_{H} B is an extension of the solvable group (A × B)/N by the residually solvable group K, and so A ∗_{H} B is residually solvable.

Example 1.3. It is shown in [21] that any HNN-extension of a finitely generated abelian group is residually solvable.

Example 1.4. R.J. Thompson, G. Higman, K. Brown, and E.A. Scott have produced various classes of finitely presented infinite simple groups (see [22]). The example of Scott contains a free subgroup on two generators and is therefore not amenable and not residually amenable.

2. Main Technical Theorems

As in the introduction, suppose Y is a simplicial complex, π acts freely and simplicially on Y, and X = Y/π is a finite simplicial complex. Suppose we have a nested sequence of normal subgroups π = Γ_{1} ⊃ Γ_{2} ⊃ ⋯ such that ⋂_{n=1}^{∞} Γ_{n} = {e}. Define Y_{n} = Y/Γ_{n}.

Suppose X has a_{j} cells in dimension j, and choose a lift to Y of each j-cell of X. These choices give a basis over l^{2}(π) of the space C_{(2)}^{j}(Y) of j dimensional l^{2}-cochains on Y. The lifts also descend to give bases of C_{(2)}^{j}(Y_{n}) over l^{2}(π/Γ_{n}).

Denote by Δ and Δ_{n} the Laplacian on C_{(2)}^{j}(Y) and C_{(2)}^{j}(Y_{n}) respectively. All arguments to follow will apply to a specific value of j, but this dependence will not be indicated.

Let {E(λ) : λ ∈ [0, ∞)} and {E_{n}(λ) : λ ∈ [0, ∞)} denote the (right continuous) family of spectral projections of Δ and Δ_{n}. Since Δ is π-equivariant, so are E(λ) = χ_{[0,λ]}(Δ) for λ ∈ [0, ∞). Similarly, E_{n}(λ) are π/Γ_{n}-equivariant. Let F, F_{n} : [0, ∞) → [0, ∞) denote the spectral density functions

F (λ) = Tr_{π} E(λ)
F_{n}(λ) = Tr_{π/Γ_{n}} E_{n}(λ).

We now set

\overline{F} (λ) = lim sup_{n→∞} F_{n}(λ); \underline{F} (λ) = lim inf_{n→∞} F_{n}(λ)
\overline{F}^{+}(λ) = lim_{δ→0^{+}} \overline{F} (λ + δ); \underline{F}^{+}(λ) = lim_{δ→0^{+}} \underline{F} (λ + δ).

With the above notation, we state the main technical results of this paper.

Theorem 2.1. For all λ ∈ [0, ∞),

F (λ) = \overline{F}^{+}(λ) = \underline{F}^{+}(λ)

Theorem 2.2. Suppose there is some right continuous function s : [0, ε) → [0, ∞), with s(0) = 0 and so that for all n and for all λ ∈ [0, ε) we have

F_{n}(λ) − F_{n}(0) ≤ s(λ)

then

1. \overline{F} (λ) and \underline{F} (λ) are right continuous at zero and one has the equalities

\overline{F} (0) = \overline{F}^{+}(0) = F (0) = \underline{F} (0) = \underline{F}^{+}(0).

2. For all λ ∈ [0, ε),

F (λ) − F (0) ≤ s(λ).

These theorems and their proofs are similar to Lück [15, Theorem 2.3], but here they are stated so as to require no conditions on the quotient groups π/Γ_{n}. The residual finiteness condition in Lück’s Theorem or the residual amenability condition of this paper are required to provide s, the uniform decay at zero of the spectral density functions for the covers Y_{n}.

To show the two technical theorems, we first prove a number of preliminary lemmas.

Lemma 2.1. There exists a number K > 1 such that the operator norms of Δ and Δ_{n} are smaller than K for all n.

Proof. Choosing lifts of cells of X, we have identified the space of l^{2}-cochains on Y with ⨁_{i=1}^{a} l^{2}(π). The combinatorial Laplacian Δ is then described by an a × a matrix B with entries in ℤ[π], acting by right multiplication. The Laplacian Δ_{n} is described by the same matrix B, now acting by right multiplication on ⨁_{i=1}^{a} l^{2}(π/Γ_{n})

For u = ∑_{g∈π} λ_{g} · g ∈ ℂ[π] define |u|_{1} = ∑_{g∈π}|λ_{g}|. Choose K > 1 so that

K ≥ a · ∑_{j=1}^{b} max {|B_{ij}|_{1}; i = 1, …, a}.

The proof then proceeds exactly as in [15, Lemma 2.5].

Lemma 2.2. Let p(μ) be a polynomial. There is a number n_{0}, depending only on the system of groups π = Γ_{1} ⊃ Γ_{2} ⊃ … such that for all n ≥ n_{0}

Tr_{π} p(Δ) = Tr_{π/Γ_{n}} p(Δ_{n})

Proof. We identify Δ with an a×a matrix B with entries in ℤ[π], as in the previous Lemma.

Fix elements g_{0}, g_{1}, …, g_{r} ∈ π and λ_{0}, λ_{1}, …, λ_{r} ∈ ℝ such that g_{0} = e, g_{i} ≠ e, and λ_{i} ≠ 0 for 1 ≤ i ≤ r so that

∑_{j=1}^{a}(p(B))_{j,j} = ∑_{i=0}^{r} λ_{i}g_{i}.

Then

Tr_{π} p(Δ) = λ_{0}.

The Laplacian Δ_{n} on Y_{n} is also described by the matrix B, now acting on ⨁_{i=1}^{a} l^{2}(π/Γ_{n}) by right multiplication. Then

Tr_{π/Γ_{n}} p(Δ_{n}) = ∑_{i=1}^{r} λ_{i} Tr_{π/Γ_{n}} R(g_{i})

where R(g_{i}) : l^{2}(π/Γ_{n}) → l^{2}(π/Γ_{n}) is right multiplication with g_{i}.

Since the intersection of the Γ_{i}’s is trivial, there is a number n_{0} such that for n ≥ n_{0} none of the elements g_{i} for 1 ≤ i ≤ r lies in Γ_{n}. Since Γ_{n} is normal, we conclude for n ≥ n_{0} and i ≠ 0

Tr_{π/Γ_{n}} R(g_{i}) = 0.

Then for n ≥ n_{0}

Tr_{π} p(Δ) = λ(0) = Tr_{π/Γ_{n}} p(Δ_{n}).

Lemma 2.3. Let K be as in Lemma 2.1. Let {p_{k}(μ)}_{k=1}^{∞} be a sequence of polynomials, uniformly bounded on [0, K], such that for the characteristic function χ_{[0,λ]}(μ) of the interval [0, λ],

lim_{k→∞} p_{k}(μ) = χ_{[0,λ]}(μ)

holds for each μ ∈ [0, K]. Then

lim_{k→∞} Tr_{π} p_{k}(Δ) = F (λ).

Proof. This lemma and its proof are identical to [15, Lemma 2.7]

We now prove Theorem 2.1. Fix λ ≥ 0. Define for k ≥ 1 a continuous function f_{k} : ℝ → ℝ by

f_{k}(μ) = \begin{cases}1 + \frac{1}{k} & μ ≤ λ \\ 1 + \frac{1}{k} − k(μ − λ) & λ ≤ μ ≤ λ + \frac{1}{k} \\ \frac{1}{k} & λ + \frac{1}{k} ≤ μ \end{cases}

Clearly χ_{[0,λ]}(μ) < f_{k+1}(μ) < f_{k}(μ) and f_{k}(μ) converges to χ_{[0,λ]}(μ) for all μ ∈ [0, ∞). For each k choose a polynomial p_{k} such that χ_{[0,λ]}(μ) < p_{k}(μ) < f_{k}(μ) holds for all μ ∈ [0, K], where K is as in Lemma 2.1. The polynomials p_{k} satisfy the conditions of Lemma 2.3.

Because χ_{[0,λ]}(μ) ≤ p_{k}(μ) for all μ ∈ [0, ∥Δ_{n}∥], we have

F_{n}(λ) = Tr_{π/Γ_{n}}(χ_{[0,λ]}(Δ_{n}))

≤ Tr_{π/Γ_{n}}(p_{k}(Δ_{n})). (2.1)

On the other hand, we have p_{k}(μ) ≤ 1 + \frac{1}{k} for μ ∈ [0, λ + \frac{1}{k}] and p_{k}(μ) ≤ \frac{1}{k} for μ ∈ [λ + \frac{1}{k} , K]. So

Tr_{π/Γ_{n}}(p_{k}(Δ_{n})) ≤ Tr_{π/Γ_{n}}((1 + \frac{1}{k})χ_{[0,λ+ \frac{1}{k} ]}(Δ_{n}))
+ Tr_{π/Γ_{n}}((\frac{1}{k})χ[λ+ \frac{1}{k} , K](Δ_{n}))
= (1 + \frac{1}{k})F_{n}(λ + \frac{1}{k})
+ \frac{1}{k} (F_{n}(K) − F_{n}(λ + \frac{1}{k}))
= F_{n}(λ + \frac{1}{k}) + \frac{1}{k} F_{n}(K)

(2.2)

Now notice F_{n}(K) = Tr_{π/Γ_{n}}(χ_{[0,K]}(Δ_{n})). But χ_{[0,K]}(Δ_{n}) is the identity on the space C_{(2)}^{j}(Y_{n}), which is identified with ⨁_{i=1}^{a_{j}} l^{2}(π/Γ_{n}). Thus

F_{n}(K) = a_{j}. (2.3)

By Lemma 2.2, there is a number n_{0}(k) for each polynomial p_{k} such that for n ≥ n_{0}(k)

Tr_{π} p_{k}(Δ) = Tr_{π/Γ_{n}}(p_{k}(Δ_{n})).

Then for n ≥ n_{0}(k), the equations (2.1), (2.2), and (2.3) give

F_{n}(λ) ≤ Tr_{π} p_{k}(Δ) ≤ F_{n}(λ + \frac{1}{k}) + \frac{1}{k} a_{j}

Taking limits as n → ∞,

\overline{F} (λ) ≤ Tr_{π} p_{k}(Δ) ≤ \underline{F} (λ + \frac{1}{k}) + \frac{1}{k} a_{j}

Taking limits as k → ∞, and using Lemma 2.3,

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

We have for all ϵ > 0

F (λ) ≤ \underline{F}^{+}(λ) ≤ \underline{F} (λ + ϵ) ≤ \overline{F} (λ + ϵ) ≤ F (λ + ϵ)

and since lim_{ϵ→0+} F (λ + ϵ) = F (λ) we get

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

This finishes the proof of Theorem 2.1.

Next we show Theorem 2.2. We suppose there is some right continuous function s : [0, ε) → [0, ∞), with s(0) = 0 and so that for all n and for all λ ∈ [0, ε) we have

F_{n}(λ) − F_{n}(0) ≤ s(λ).

Taking the limit inferior and limit superior for n → ∞ gives:

\underline{F} (λ) ≤ \underline{F} (0) + s(λ) and \overline{F} (λ) ≤ \overline{F} (0) + s(λ).

Taking the limit for λ → 0 gives

\underline{F}^{+}(0) ≤ \underline{F} (0) and \overline{F}^{+}(0) ≤ \overline{F} (0).

And finally, since \underline{F} and \overline{F} are increasing,

\underline{F}^{+}(0) = \underline{F} (0) and \overline{F}^{+}(0) = \overline{F} (0).

We already know \overline{F}^{+}(0) = F (0) = \underline{F}^{+}(0) from Theorem 2.1, and this proves Theorem 2.2.1. Since s is right continuous, we conclude:

\overline{F}^{+}(λ) ≤ F (0) + s(λ)

and Theorem 2.2.2 follows from Theorem 2.1. This finishes the proof of Theorem 2.2.

The following pair of Lemmas are proved in [15, Lemma 3.3]. We will need them in the proof of Theorem 0.2 in the last section.

Lemma 2.4. Suppose lim_{n→∞} F_{n}(0) = F (0) (which follows from Theorem 2.2 when the decay hypothesis is satisfied). Then

∫_{0^{+}}^{K} \frac{F (λ) − F (0)}{λ}dλ ≤ lim inf_{n→∞} ∫_{0^{+}}^{K} \frac{F_{n}(λ) − F_{n}(0)}{λ}dλ

Lemma 2.5. Suppose the F_{n} have a uniform spectral gap, that is there is some ϵ > 0 so that F_{n}(λ) = F_{n}(0) for all n and for all λ ≤ ϵ. Then

∫_{0^{+}}^{K} \frac{F (λ) − F (0)}{λ}dλ ≥ lim sup_{n→∞} ∫_{0^{+}}^{K} \frac{F_{n}(λ) − F_{n}(0)}{λ}dλ

and therefore using the previous Lemma,

∫_{0^{+}}^{K} \frac{F (λ) − F (0)}{λ}dλ = lim_{n→∞} ∫_{0^{+}}^{K} \frac{F_{n}(λ) − F_{n}(0)}{λ}dλ.

Remark. In Lemma 2.5, Lück gives a sharper condition than simply demanding a spectral gap, but it will not be needed here.

3. Proofs Of The Main Theorems

In this chapter, we prove the approximation theorem, determinant class theorem, and homotopy invariance of L^{2}-torsion for residually amenable groups.

3.1. The Approximation Theorem. Proof of Theorem 0.1 (Approximation Theorem).

Observe that the jth L^{2}-Betti numbers of Y and Y_{n} are given by

b_{(2)}^{j}(Y : π) = F (0); b_{(2)}^{j}(Y_{n} : π/Γ_{n}) = F_{n}(0).

Therefore Theorem 0.1 will follow directly from Theorem 2.2.1, if we can establish a uniform decay of F_{n} near zero.

Since π/Γ_{n} is amenable, [8, Theorem 2.1.3] applies, and we have a constant K > 1 so that

F_{n}(λ) − F_{n}(0) ≤ a_{j} \frac{log K}{− log λ} = s(λ) (3.1)

for all 0 < λ < 1. The constant K can be any number larger than max(∥Δ_{n}∥, 1), and therefore can be chosen independently of n, by Lemma 2.1.

Remark. In the case of Lück’s Theorem, the groups π/Γ_{n} are finite, so the Laplacian on Y_{n} is a finite, self-adjoint matrix. Lück then proves, for any self-adjoint matrix, an elegant estimate on the number of eigenvalues which are less than a fixed λ. The estimate is weakened if the product of the nonzero eigenvalues is small, but this product must be at least one as the Laplacian has integer entries.

Dodziuk and Mathai make use of the same fundamental estimate when proving the approximation theorem for amenable groups.

3.2. Results For Manifolds. Suppose M is a compact Riemannian manifold, and M̃ is a regular covering space for M with residually amenable transformation group π. Let Δ̃ = d^{∗}d + dd^{∗} denote the Laplacian on L^{2} j-forms on M̃. It is π-equivariant and essentially self-adjoint, so has a spectral family of π-equivariant projections {Ẽ_{j}(λ)|λ ∈ [0, ∞)}. Each E_{j}(λ) has Schwartz kernel, and one can then form the analytic spectral density function F̃_{j}(λ) = Tr_{π} Ẽ_{j}(λ).

Following arguments in [8], one can investigate the analytic spectral density function F̃(λ) of Δ̃. To relate this analytic Laplacian to the combinatorial situation in the previous sections, let X be a triangulation of M and lift to get a triangulation Y of M̃.

The spectrum of Δ̃ is said to have a gap at zero if the spectral projection Ẽ(λ) = Ẽ(0) for some λ > 0. Because the von Neumann dimension is faithful, Δ̃ has a spectral gap at zero if and only if F̃(λ) = F̃(0) for some λ > 0.

Theorem 3.1. (Gap Criterion) The spectrum of Δ̃ has a gap at zero if and only if there is a λ > 0 such that

lim_{n→∞} F_{n}(λ) − F_{n}(0) = 0.

Proof. Since F̃ and F are dilation equivalent [11], Δ̃ will have a spectral gap if and only if Δ does. By Theorem 2.1, F (λ) = F (0) if and only if

lim_{n→∞} F_{n}(λ) − F_{n}(0) = 0.

Theorem 3.2. (Spectral Density Estimate) There are constants C > 0 and ε > 0 independent of λ, such that for all λ ∈ (0, ε)

F̃(λ) − F̃(0) ≤ \frac{C}{− log λ}.

Proof. This follows directly from dilation equivalence and the estimate 3.1.

The spectral density estimate is interesting as it provides evidence supporting the well known conjecture that the Novikov-Shubin invariants of a closed manifold are positive.

3.3. The Determinant Class Theorem. Recall that a covering space Y of a finite simplicial complex X is of determinant class if for each j,

−∞ < ∫_{0^{+}}^{1} log λdF_{j}(λ),

where F_{j}(λ) denotes the von Neumann spectral density function of the combinatorial Laplacian Δ_{j} on L^{2} j-cochains.

We will prove that every residually amenable covering of a finite simplicial complex is of determinant class. The appendix of [3] contains a proof that every residually finite covering of a compact manifold is of determinant class. Their proof is based on Lück’s approximation of von Neumann spectral density functions. Since an analogous approximation holds in the setting of this paper, we can apply the argument of [3] to prove Theorem 0.2. The fact that our coverings are infinite necessitates some modifications.

Proof of Theorem 0.2 (Determinant Class Theorem).

As with the rest of this paper, this proof will proceed for a fixed j which will be suppressed in the notation.

Denote by Det_{π/Γ_{n}} Δ′_{n} the Fuglede-Kadison determinant of Δ_{n} restricted to the orthogonal complement of its kernel. It is given by the following Lebesgue-Stieltjes integral,

log Det_{π/Γ_{n}} Δ′_{n} = ∫_{0^{+}}^{∞} log λdF_{n}(λ) = ∫_{0^{+}}^{K} log λdF_{n}(λ)

with K as in Lemma 2.1. That is, ∥Δ_{n}∥ ≤ K.

As Y_{n} is an amenable cover of X, we are in the situation of Dodziuk-Mathai [8]. Their proof of the Determinant Class Theorem for amenable coverings [8, Thm 0.2] shows that

log Det_{π/Γ_{n}} Δ′_{n} ≥ 0,

which is stronger than simply determinant class - the key point being that the bound is uniform in n. It is worth remarking that their proof and the corresponding proof in [3] require a similar statement for the Laplacians on finite approximations. In the finite case, the Laplacian is an integer matrix, so the product of its nonzero eigenvalues is a positive integer and therefore at least 1.

By Lemma 3.1 at the end of this section,

0 ≤ log Det_{π/Γ_{n}} Δ′_{n} = (log K)(F_{n}(K) − F_{n}(0)) − ∫_{0^{+}}^{K} \frac{F_{n}(λ) − F_{n}(0)}{λ}dλ

which gives

∫_{0^{+}}^{K} \frac{F_{n}(λ) − F_{n}(0)}{λ}dλ ≤ (log K)(F_{n}(K) − F_{n}(0)).

Now from Lemma 2.4,

∫_{0^{+}}^{K} \frac{F (λ) − F (0)}{λ}dλ ≤ lim inf_{n→∞} ∫_{0^{+}}^{K} \frac{F_{n}(λ) − F_{n}(0)}{λ}dλ
≤ lim inf_{n→∞} (log K)(F_{n}(K) − F_{n}(0)).

(3.2)

Since we have uniform decay (3.1) of the F_{n} near 0, Theorem 2.2.1 applies and lim_{n→∞} F_{n}(0) = F (0). Since K ≥ ∥Δ_{n}∥ for all n and K ≥ ∥Δ∥,

F_{n}(K) = F (K) = a_{j}

for all n, so (3.2) becomes

∫_{0^{+}}^{K} \frac{F (λ) − F (0)}{λ}dλ ≤ (log K)(F(K) − F(0)). (3.3)

This shows in particular that the left hand integral exists, and again applying Lemma 3.1,

log Det_{π} Δ′ = (log K)(F(K) − F(0)) − ∫_{0^{+}}^{K} \frac{F (λ) − F (0)}{λ}dλ ≥ 0.

Since this is true for all j, Y is of determinant class.

Lemma 3.1. Let F : [0, ∞) → [0, ∞) be any non-decreasing function satisfying lim_{λ→0} F (λ) = 0. Then for any K > 0,

∫_{0^{+}}^{K} log λdF (λ) = (log K)(F (K)) − ∫_{0^{+}}^{K} \frac{F (λ)}{λ}dλ

where the left hand integral exists if and only if the right hand integral exists.

Proof. Fixing 1 > ϵ > 0, integration by parts yields

(3.4) ∫_{ϵ}^{K} log λdF (λ) = (log K)(F (K)) − (log ϵ)(F (ϵ)) − ∫_{ϵ}^{K} \frac{F (λ)}{λ}dλ.

Now −(log ϵ)(F (ϵ)) is non-negative, and ∫_{ϵ}^{K} log λdF (λ) is decreasing as ϵ → 0. Thus, if ∫_{0^{+}}^{K} \frac{F (λ)}{λ}dλ exists then so does ∫_{0^{+}}^{K} log λdF (λ).

On the other hand, if ∫_{0^{+}}^{K} log λdF (λ) exists, then

0 ≤ −(log ϵ)(F (ϵ)) = −(log ϵ) ∫_{0^{+}}^{ϵ} dF (λ) ≤ − ∫_{0^{+}}^{ϵ} log λdF (λ)

which goes to zero as ϵ → 0. This shows lim_{ϵ→0} −(log ϵ)(F (ϵ)) = 0 and the Lemma follows.

3.4. Homotopy Invariance of L^{2}-Torsion. The Fuglede-Kadison determinant Det_{π} induces a homomorphism from the Whitehead group of π,

Φ_{π} : Wh(π) → ℝ^{+}

which was defined in [16] and [15]. Given a homotopy equivalence f : M → N of compact cell complexes, we choose f to be a cellular homotopy equivalence, and let M_{f} be the cellular mapping cone. Putting π = π_{1}(M), the cochain complex C^{∗}(M_{f}) is an acyclic complex over the group ring ℤ[π], and defines the Whitehead torsion T (f) ∈ Wh(π). Then Φ_{π}(T (f)) ∈ ℝ^{+}.

Now suppose π is residually amenable, so M and N are of determinant class. Let ϕM̃, ϕÑ denote the L^{2}-torsion of M and N. The homotopy equivalence f canonically identifies the determinant lines of L^{2}-cohomology of M̃ and Ñ, so that

ϕM̃ ⊗ ϕ−1 ; Ñ ∈ ℝ^{+}.

Then from [19, Prop. 2.1], one has

ϕM̃ ⊗ ϕ−1 ; Ñ = Φ_{π}(T (f)) ∈ ℝ^{+}.

Therefore, Theorem 0.3 is reduced to the following

Theorem 3.3. Suppose that π is a finitely presented residually amenable group. Then the homomorphism

Φ_{π} : Wh(π) → ℝ^{+}

is trivial.

Proof. We can represent an arbitrary element of Wh(π) as the Whitehead torsion of a homotopy equivalence f : L → K of finite cell complexes, which without loss of generality is an inclusion. Let L̃ and K̃ denote the corresponding regular π covering complexes. The relative cochain complex C^{∗}(K̃, L̃) is acyclic, and so is its L^{2}-completion C_{(2)}^{∗}(K̃, L̃). In particular, the combinatorial Laplacian Δ_{j}^{K̃,L̃} is invertible and we see that

Φ_{π}(T (f)) = ∏_{j=0}^{n} Det_{π}(Δ_{j}^{K̃,L̃})^{\frac{(−1)^{j} j}{2}} > 0.

We claim that Det_{π}(Δ_{j}^{K̃,L̃}) = 1 for each j. The j will be suppressed in the notation.

Form the covering spaces L̃_{n} and K̃_{n} with covering group π/Γ_{n}, and let Δ^{K̃_{n},L̃_{n}} denote the Laplacian on C_{(2)}^{∗}(K̃_{n}, L̃_{n}).

As π/Γ_{n} is amenable, the work of Mathai-Rothenberg applies to the pair (K̃_{n}, L̃_{n}). It follows from their proof [19, Prop. 2.6] that for all n,

Det_{π/Γ_{n}} Δ^{K̃_{n},L̃_{n}} = 1.

Since Δ^{K̃,L̃} is invertible we can choose a common bound K on ∥Δ^{K̃,L̃}∥ and ∥(Δ^{K̃,L̃})^{−1}∥, as in Lemma 2.1. K will also bound ∥Δ^{K̃_{n},L̃_{n}}∥ and ∥(Δ^{K̃_{n},L̃_{n}})^{−1}∥, and therefore and Δ^{K̃_{n},L̃_{n}} has a spectral gap at zero of size at least K^{−1}.

Now by Lemma 2.5,

Det_{π} Δ^{K̃,L̃} = lim_{n→∞} Det_{π/Γ_{n}} Δ^{K̃_{n},L̃_{n}} = 1.

This finishes the proof of Theorem 0.3. As remarked earlier, in both [15] and [19], the approximating spaces are finite. The approximating Laplacians have determinant 1 and spectral gap because they are finite invertible integer matrices, and the previous theorem essentially rests on this fact.

Appendix

Section 3.4 of this paper makes use of a similar argument in [19] which shows that L^{2}-torsion is a homotopy invariant of complexes with amenable fundamental group. The method in [19] contains a gap, and in this Appendix that gap will be corrected. The key idea for this correction is due to M. Rothenberg.

Consider complexes X = Y/π as in this paper, but with π an amenable group. Choose a Følner exhaustion {Λ_{n}}_{n=1}^{∞} of π, and choose a subcomplex F of Y which is a fundamental domain for the action of π. Then Y_{n} is defined to be the finite subcomplex of Y consisting of all translates g.F for g ∈ Λ_{n}, and Δ_{n} is the Laplacian on the complex Y_{n}. Dodziuk and Mathai [8] show that renormalized spectra of Δ_{n} approximate the L^{2}-spectrum of Δ on Y.

Now define maps P_{n} and ι_{n} to be the orthogonal projection and the inclusion map between C_{(2)}(Y) ⊃ C(Y_{n}). Set

Δ̂_{n} = P_{n} ∘ Δ ∘ ι_{n} : C(Y_{n}) → C(Y_{n}).

As the first step towards fixing the problem in [19], notice that the approximation arguments in [8] work equally well with Δ̂_{n}. The two key points to notice are that Δ̂_{n} is still a matrix with integer entries, and still agrees with Δ away from the boundary of Y_{n}.

Next consider the situation of [19], where X, Y, and Y_{n} are replaced by relative complexes (K, L), (K̃, L̃) and (K̃_{n}, L̃_{n}). We use the same notation Δ, Δ_{n}, and Δ̂_{n} for the corresponding operators in this relative situation. Δ is invertible, and there are bounds

∥Δ∥ ≤ K and ∥Δ^{−1}∥ ≤ K.

The projection of Δ_{n} off of its kernel, Δ_{n}^{+}, is invertible. The error in [19] is the claim that there is a bound away from zero of the spectrum of Δ_{n}^{+} which is uniform in n. In general this is not the case.

Here we show that there is a uniform bound away from zero for the spectra of Δ̂_{n}, n = 1, 2, . . .. The rest of the argument in [19] is changed only by the replacement of Δ_{n}^{+} with Δ̂_{n}. Notice that there is no longer any need to project off of the kernel as the operators Δ̂_{n} are already invertible as shown below.

In what follows, ⟨·, ·⟩ and ⟨·, ·⟩_{n} will denote the inner products on C_{(2)}(K̃, L̃) and C(K̃_{n}, L̃_{n}). We have

inf σ(Δ̂_{n}) = inf_{c∈C(K̃_{n},L̃_{n});∥c∥=1} ⟨Δ̂_{n}c, c⟩_{n}

= inf_{c∈C(K̃_{n},L̃_{n});∥c∥=1} ⟨P_{n}Δι_{n}c, c⟩_{n}

= inf_{c∈C(K̃_{n},L̃_{n});∥c∥=1} ⟨Δι_{n}c, ι_{n}c⟩

≥ inf_{c∈C_{(2)}(K̃,L̃);∥c∥=1} ⟨Δc, c⟩

= inf σ(Δ) ≥ K^{−1}.

Thus K^{−1} is a uniform lower bound on the spectra of Δ̂_{n} as required to fix the proof in [19].

References

[1] G. Baumslag and D. Solitar. Some two-generator one-relator non-Hopfian groups. Bull. AMS, 68:199–201, 1962.

[2] K. Brown. Cohomology of Groups. Number 87 in GTM. Springer, 1982.

[3] D. Burghelea, L. Friedlander, and T. Kappeler. Torsion for manifolds with boundary and gluing formulas. IHES preprint, (dg-ga/9510010), Jan. 1996. To appear in Math. Nachrichten, 1998.

[4] D. Burghelea, L. Friedlander, T. Kappeler, and P. McDonald. Analytic and Reidemeister torsion for representations in finite type Hilbert modules. Geom. and Func. Anal., 6:751– 897, 1996.

[5] A. Carey, M. Farber, and V. Mathai. Determinant lines, von Neumann algebras and L^{2} torsion. Crelle Journal, 484:153–181, 1997.

[6] A. Carey and V. Mathai. L^{2} torsion invariants. J. Func. Anal., 110:377–409, 1992.

[7] B. Clair. Residual Amenability and the Approximation of L^{2}-invariants. PhD thesis, U. of Chicago, June 1998.

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

[9] M. Farber. Geometry of growth: Approximation theorems for L^{2} invariants. To appear in Mathematische Annalen, 1997. (dg-ga/9703014).

[10] M. Gromov. Asymptotic Invariants of Infinite Groups. Number 182 in Lecture Notes. LMS, 1993.

[11] M. Gromov and M. A. Shubin. Von Neumann spectra near zero. Geom. and Func. Anal., 1:375–404, 1991.

[12] D. Kazhdan. On arithmetic varieties. In Lie Groups and Their Representations, pages 151– 216. Halsted, 1975.

[13] P. Kropholler. Baumslag-Solitar groups and some other groups of cohomological dimension two. Comment. Math. Helvetici, 65:547–558, 1990.

[14] J. Lott. Heat kernels on covering spaces and topological invariants. J. Diff. Geom., 35:471– 510, 1992.

[15] W. Lück. Approximating L^{2}-invariants by their finite dimensional analogues. Geom. and Func. Anal., 4(4):455–481, 1994.

[16] W. Lück and M. Rothenberg. Reidemeister torsion and the K-theory of von Neumann algebras. K-Theory, 5:213–264, 1991.

[17] W. Magnus. Residually finite groups. Bull. AMS, 75:305–315, 1969.

[18] V. Mathai. L^{2} analytic torsion. J. Func. Anal., 107:369–386, 1992.

[19] V. Mathai and M. Rothenberg. On the homotopy invariance of L^{2} torsion for covering spaces. Proc. AMS, 126(3):887–897, 1998.

[20] A. Paterson. Amenability. Number 29 in Mathematical Surveys and Monographs. AMS, 1988.

[21] E. Raptis and D. Varsos. Residual properties of HNN-extensions with base group an abelian group. J. Pure Appl. Alg., 59:285–290, 1989.

[22] E. A. Scott. A tour around finitely presented infinite simple groups. In Algorithms and Classification in Combinatorial Group Theory, pages 83–119. MSRI Publ., 1992.

Department of Mathematics, City University of New York, New York, NY 10036

   E-mail address: bryan@email.gc.cuny.edu