Approximating L^2 invariants of amenable covering spaces: A combinatorial approach

14 Pages·0.17 MB·English

by  Jozef Dodziuk

APPROXIMATING L2 INVARIANTS OF AMENABLE COVERING
SPACES: A COMBINATORIAL APPROACH.
JOZEF DODZIUK AND VARGHESE MATHAI
7 Abstract. Inthispaper,weprovethat theL2 Bettinumbersof anamenable covering
9 spacecanbeapproximatedbytheaverageBettinumbersofaregularexhaustion,proving
9
aconjecturein[DM]. Wealsoprovethatanarbitraryamenablecoveringspaceofafinite
1
simplicial complex is of determinant class.

n
a
J

9
Introduction

3
Let Y be a connected simplicial complex. Suppose that π acts freely and simplicially
v
3 on Y so that X = Y/π is a finite simplicial complex. Let a finite subcomplex of Y,
F
0 which is a fundamental domain for the action of π on Y.
0
We assume that π is amenable. The Følner criterion for amenability of π enables one
9
∞
0 to get, cf. [Ad], a regular exhaustion Ym m=1, that is a sequence of finite subcomplexes
6 of Y such that
9 (cid:8) (cid:9)
(1) Y consists of N translates g. of for g π.
a/ m ∞ m F F ∈
g (2) Y = Y .
m
-
g m=1
[
d (3) If N˙ denotes the number of translates of which have distance (with respect to
m,δ
F
: the word metric in π) less than or equal to δ from a translate of having a non-empty
v
F
i intersection with the topological boundary ∂Y of Y (we identify here g. with g) then,
X m m F
for every δ > 0,
r
a N˙
m,δ
lim = 0.
m→∞ Nm
One of our main results is
Theorem 0.1 (Amenable Approximation Theorem). Let Y be a connected simplicial
complex. Suppose that π is amenable and that π acts freely and simplicially on Y so
∞
that X = Y/π is a finite simplicial complex. Let Y be a regular exhaustion of Y.
m m=1
Then
bj(Y ) (cid:8) (cid:9)
m j
lim = b (Y :π) for all j 0.
m→∞ Nm (2) ≥
bj(Y ,∂Y )
m m j
lim = b (Y : π) for all j 0.
m→∞ Nm (2) ≥
Date: SEPTEMBER 1996.
1991 Mathematics Subject Classification. Primary: 58G11, 58G18 and 58G25.
Key words and phrases. L2 Betti numbers,approximation theorems, amenable groups.
1
2 JOZEF DODZIUK AND VARGHESE MATHAI
Here bj(Y ) denotes the jth Betti number of Y , bj(Y ,∂Y ) denotes the jth Betti
m m m m
number of Y relative its boundary ∂Y and bj (Y :π) denotes the jth L2 Betti number
m m (2)
of Y. (See the next section for the definition of the L2 Betti numbers of a manifold)
Remarks. This theorem proves the main conjecture in the introduction of an earlier
paper [DM]. The combinatorial techniques of this paper contrasts with the heat kernel
approach used in [DM]. Under the assumption dim Hk(Y) < , a special case of the
∞
Amenable Approximation Theorem above is obtained by combining proofs of Eckmann
[Ec] and Cheeger-Gromov [CG]. The assumption dim Hk(Y) < is very restrictive
∞
and essentially says that Y is a fiber bundle over a Bπ with fiber a space with finite
fundamental group. Cheeger-Gromov use this to show that the Euler characteristic of
a finite Bπ, where π contains an infinite amenable normal subgroup, is zero. Eckmann
proved the same result in the special case when π itself is an infinite amenable group.
There is a standing conjecture that any normal covering space of a finite simplicial
complex is of determinant class (cf. section 4 for the definition of determinant class and
for a more detailed discussion of what follows). Let M be a smooth compact manifold,
and X triangulation of M. Let M be any normal covering space of M, and Y be the
triangulation of M which projects down to X. Then on M, there are two notions of
determinant class, one analytic afnd the other combinatorial. Using results of Efremov
[E], Gromov andfShubin [GS], one observes as in [BFKM]fthat the combinatorial and
analytic notions of determinant class coincide. It was proved in [BFK]) using estimates
of Lu¨ck [Lu] that any residually finite normal covering space of a finite simplicial complex
is of determinant class, which gave evidence supporting the conjecture. Our interest in
this conjecture stems from work on L2 torsion [CFM], [BFKM]. The L2 torsion is a
well defined element in the determinant line of the reduced L2 cohomology, whenever the
covering space is of determinant class. Our next main theorem says that any amenable
normal covering space of a finite simplicial complex is of determinant class, which gives
further evidence supporting the conjecture.
Theorem 0.2 (Determinant Class Theorem). Let Y be a connected simplicial complex.
Suppose that π is amenable and that π acts freely and simplicially on Y so that X = Y/π
is a finite simplicial complex. Then Y is of determinant class.
The paper is organized as follows. In the first section, some preliminaries on L2 coho-
mology andamenablegroupsarepresented. Insection 2, themainabstractapproximation
theorem is proved. We essentially use the combinatorial analogue of the principle of not
feeling the boundary (cf. [DM]) in Lemma 2.3 and a finite dimensional result in [Lu], to
prove this theorem. Section 3 contains the proof of the Amenable Approximation Theo-
rem and some related approximation theorems. In section 4, we prove that an arbitrary
amenable normal covering space of a finite simplicial complex is of determinant class.
Thesecond author warmly thanks ShmuelWeinberger for someusefuldiscussions. This
paper has been inspired by Lu¨ck’s work [Lu] on residually finite groups.
1. Preliminaries
Let π be a finitely generated discrete group and (π) be the von Neumann algebra
U
generated by the action of π on ℓ2(π) via the left regular representation. It is the weak (or
APPROXIMATING L2 BETTI NUMBERS OF AMENABLE COVERING SPACES 3
strong) closureof thecomplex group algebra ofπ, C(π) acting on ℓ2(π) by left translation.
The left regular representation is then a unitary representation ρ : π (π). Let Tr
U(π)
→ U
be the faithful normal trace on (π) defined by the inner product Tr (A) (Aδ ,δ )
U(π) e e
U ≡
for A (π) and whereδ ℓ2(π) is given by δ (e) = 1 and δ (g) = 0 for g π and g = e.
e e e
∈ U ∈ ∈ 6
LetY beasimplicialcomplex, and Y denotethesetofallp-simplicesinY. Regarding
j
| |
the orientation of simplices, we use the following convention. For each p-simplex σ Y ,
j
∈ | |
we identify σ with any other p-simplex which is obtained by an even permutation of the
vertices in σ, whereas we identify σ with any other p-simplex which is obtained by an
−
odd permutation of the vertices in σ. Suppose that π acts freely and simplicially on Y
so that X = Y/π is a finite simplicial complex. Let a finite subcomplex of Y, which
F
is a fundamental domain for the action of π on Y. Consider the Hilbert space of square
summable cochains on Y,
Cj (Y) = f Cj(Y) : f(σ)2 <
(2) ∈ | | ∞
n σaj−Xsimplex o
Since π acts freely on Y, we see that there is an isomorphism of Hilbert ℓ2(π) modules,
Cj (Y)= Cj(X) ℓ2(π)
(2) ∼ ⊗
Here π acts trivially on Cj(X) and via the left regular representation on ℓ2(π). Let
j j+1
d : C (Y) C (Y)
j (2) → (2)
denote the coboundary operator. It is clearly a bounded linear operator. Then the (re-
duced) L2 cohomology groups of Y are defined to be
ker(d )
j j
H (Y) = .
(2)
im(d )
j−1
Let d ∗ denote the Hilbert space adjoint of d . One defines the combinatorial Laplacian
j j
∆ :Cj (Y) Cj (Y) as ∆ = d (d )∗+(d )∗d .
j (2) → (2) j j−1 j−1 j j
BytheHodgedecompositiontheoreminthiscontext, thereisanisomorphismofHilbert
ℓ2(π) modules,
j
H (Y) = ker(∆ ).
(2) ∼ j
j
Let P :C (Y) ker∆ denote the orthogonal projection to the kernel of the Laplacian.
j (2) → j
Then the L2 Betti numbers bj (Y : π) are defined as
(2)
j
b (Y :π) = Tr (P ).
(2) U(π) j
(m)
In addition, let ∆ denote the Laplacian on the finite dimensional cochain space
j
Cj(Y ) or Cj(Y ,∂Y ). We do use the same notation for the two Laplacians since all
m m m
proofs work equally well for either case. Let D (σ,τ) = ∆ δ ,δ denote the matrix coef-
j j σ τ
h i
(m) (m)
ficients of the Laplacian ∆ and D (σ,τ) = ∆ δ ,δ denote the matrix coefficients
j j j σ τ
of the Laplacian ∆(m). Let d(σ,τ) denote thDe distance bEetween σ and τ in the natural
j
simplicial metric on Y, and d (σ,τ) denote the distance between σ and τ in the natural
m
simplicial metric on Y . This distance (cf. [Elek]) is defined as follows. Simplexes σ and
m
4 JOZEF DODZIUK AND VARGHESE MATHAI
τ are one step apart, d(σ,τ) = 1, if they have equal dimensions, dimσ = dimτ = j, and
there exists either a simplex of dimension j 1 contained in both σ and τ or a simplex of
−
dimension j +1 containing both σ and τ. The distance between σ and τ is equal to k if
there exists a finite sequence σ = σ ,σ ,... ,σ = τ, d(σ ,σ ) = 1 for i = 0,... ,k 1,
0 1 k i i+1
−
and k is the minimal length of such a sequence.
Then one has the following, which is an easy generalization of Lemma 2.5 in [Elek] and
follows immediately from the definition of combinatorial Laplacians and finiteness of the
complex X = Y/π.
(m)
Lemma 1.1. D (σ,τ) = 0 whenever d(σ,τ) > 1 and D (σ,τ) = 0 whenever d (σ,τ) >
j j m
1. There is also a positive constant C independent of σ,τ such that D (σ,τ) C and
j
≤
(m)
D (σ,τ) C.
j ≤
Let Dk(σ,τ) = ∆kδ ,δ denote the matrix coefficient of the k-th power of the Lapla-
j j σ τ
D E k
cian, ∆k, and D(m)k(σ,τ) = ∆(m) δ ,δ denote the matrix coefficient of the k-th
j j j σ τ
(cid:28) (cid:29)
(m)k (cid:16) (cid:17)
power of the Laplacian, ∆ . Then
j
Dk(σ,τ) = D (σ,σ )...D (σ ,τ)
j j 1 j k−1
σ1,...σXk−1∈|Y|j
and
(m)k (m) (m)
D (σ,τ) = D (σ,σ )...D (σ ,τ).
j j 1 j k−1
σ1,...σkX−1∈|Ym|j
Then the following lemma follows easily from Lemma 1.1.
Lemma 1.2. Let k be a positive integer. Then Dk(σ,τ) = 0 whenever d(σ,τ) > k and
j
(m)k
D (σ,τ) = 0 whenever d (σ,τ) > k. There is also a positive constant C independent
j m
of σ,τ such that Dk(σ,τ) Ck and D(m)k(σ,τ) Ck.
j ≤ j ≤
Since π commutes with the Laplacian ∆k, it follows that
j
(1.1) Dk(γσ,γτ) = Dk(σ,τ)
j j
for all γ π and for all σ,τ Y . The von Neumann trace of ∆k is by definition
∈ ∈ | |j j
(1.2) Tr (∆k) = Dk(σ,σ),
U(π) j j
σ∈X|X|j
where σ˜ denotes an arbitrarily chosen lift of σ to Y. The trace is well-defined in view of
(1.1).
1.1. Amenable groups. Let d be the word metric on π. Recall the following charac-
1
terization of amenability due to Følner, see also [Ad].
Definition 1.3. A discrete group π is said to be amenable if there is a sequence of finite
∞
subsets Λ such that for any fixed δ > 0
k k=1
(cid:8) (cid:9) # ∂δΛk
lim { } = 0
k→∞ # Λk
{ }
APPROXIMATING L2 BETTI NUMBERS OF AMENABLE COVERING SPACES 5
where ∂ Λ = γ π : d (γ,Λ ) < δ and d (γ,π Λ ) < δ is a δ-neighborhood of the
δ k 1 k 1 k
{ ∈ ∞ − }
boundary of Λ . Such a sequence Λ is called a regular sequence in π. If in addition
k k k=1
∞
Λ Λ for all k 1 and Λ(cid:8) =(cid:9) π, then the sequence Λ ∞ is called a regular
k ⊂ k+1 ≥ k k k=1
k=1
exhaustion in π. [ (cid:8) (cid:9)
Examples of amenable groups are:
(1) Finite groups;
(2) Abelian groups;
(3) nilpotent groups and solvable groups;
(4) groups of subexponential growth;
(5) subgroups, quotient groups and extensions of amenable groups;
(6) the union of an increasing family of amenable groups.
Free groups and fundamental groups of closed negatively curved manifolds are not
amenable.
∞
Let π be a finitely generated amenable discrete group, and Λ a regular exhaus-
m m=1
∞
tion in π. Then it defines a regular exhaustion Y of Y.
m m=1 (cid:8) (cid:9)
Let P (λ) : λ [0, ) denote the right continuous family of spectral projections of
j
{ ∈ ∞ } (cid:8) (cid:9)
the Laplacian ∆ . Since ∆ is π-equivariant, so are P (λ) = χ (∆ ), for λ [0, ). Let
j j j [0,λ] j
∈ ∞
F : [0, ) [0, ) denote the spectral density function,
∞ → ∞
F(λ) = Tr (P (λ)).
U(π) j
Observe that the j-th L2 Betti number of Y is also given by
j
b (Y : π)= F(0).
(2)
We have the spectral density function for every dimension j but we do not indicate ex-
plicitly this dependence. All our arguments are performed with a fixed value of j.
(m)
Let E (λ) denote the number of eigenvalues µ of ∆ satisfying µ λ and which are
m j ≤
(m)
counted with multiplicity. We may sometimes omit the subscript j on ∆ and ∆ to
j j
simplify the notation.
We next make the following definitions,
E (λ)
m
F (λ) =
m
N
m
F(λ) = limsupF (λ)
m
m→∞
F(λ) = liminfF (λ)
m
m→∞
+
F (λ) = lim F(λ+δ)
δ→+0
F+(λ) = lim F(λ+δ).
δ→+0
2. Main Technical Theorem
Our main technical result is
6 JOZEF DODZIUK AND VARGHESE MATHAI
Theorem 2.1. Let π be countable, amenable group. In the notation of section 1, one has
(1) F(λ) = F+(λ) = F+(λ).
(2) F and F are right continuous at zero and we have the equalities
F(0) = F+(0) = F(0) = F(0) = F+(0)
# E (0)
m
= lim F (0) = lim { } .
m
m→∞ m→∞ Nm
(3) Suppose that 0 < λ< 1. Then there is a constant K > 1 such that
logK2
F(λ) F(0) a .
− ≤ − logλ
To prove this Theorem, we will first prove a number of preliminary lemmas.
Lemma 2.2. There exists a positive number K such that the operator norms of ∆ and
j
of ∆(m) for all m = 1,2... are smaller than K2.
j
Proof. The proof is similar to that in [Lu], Lemma 2.5 and uses Lemma 1.1 together with
uniformlocalfinitenessofY. Morepreciselyweusethefactthatthenumberofj-simplexes
in Y at distance at most one from a j-simplex σ can be bounded independently of σ, say
# τ Y : d(τ,σ) 1 b. A fortiori the same is true (with the same constant a)
j
{ ∈ | | ≤ } ≤
for Y for all m. We now estimate the ℓ2 norm of ∆κ for a cochain κ = a σ (having
m σ σ
identified a simplex σ with the dual cochain). Now
P
∆κ= D(σ,τ)a σ
τ
!
σ τ
X X
so that
2
D(σ,τ)a D(σ,τ)2 a2 C2b a2,
τ ≤   τ ≤ τ
!
σ τ σ d(σ,τ)≤1 d(σ,τ)≤1 σ d(σ,τ)≤1
X X X X X X X
  
where we have used Lemma 1.1 and Cauchy-Schwartz inequality. In the last sum above,
for every simplex σ, a2 appears at most b times. This proves that ∆κ 2 C2b2 κ 2.
σ k k ≤ k k
Identical estimate holds (with the same proof) for ∆(m) which yields the lemma if we set
K = √Cb.
Observe that ∆ can be regarded as a matrix with entries in Z[π], since by definition,
j
the coboundary operator d is a matrix with entries in Z[π], and so is its adjoint d∗ as it
j j
is equal to the simplicial boundary operator. There is a natural trace for matrices with
entries in Z[π], viz.
TrZ[π](A) = TrU[π](Ai,i).
i
X
Lemma 2.3. Let π be an amenable group and let p(λ) = d a λr be a polynomial.
r=0 r
Then,
1 P
(m)
TrZ[π](p(∆j)) = ml→im∞ Nm TrC p ∆j .
(cid:16) (cid:16) (cid:17)(cid:17)
APPROXIMATING L2 BETTI NUMBERS OF AMENABLE COVERING SPACES 7
Proof. First observe that if σ Y is such that d(σ,∂Y ) > k, then Lemma 1.2 implies
m j m
∈ | |
that
Dk(σ,σ) = ∆kδ ,δ = ∆(m)kδ ,δ = D(m)k(σ,σ).
j j σ σ j σ σ j
By (1.1) and (1.2) D E D E
1
TrZ[π](p(∆j)) = p(∆j)σ,σ .
N h i
m
σ∈X|Ym|j
Therefore we see that
1
(m)
TrZ[π](p(∆j))− N TrC p ∆j ≤
(cid:12) m (cid:12)
(cid:12) (cid:16) (cid:16) (cid:17)(cid:17)(cid:12)
1 d (cid:12) (cid:12)
(cid:12)a Dr(σ,σ)+D(cid:12)(m)r(σ,σ) .
r
N | |
m
Xr=0 σ XYm j (cid:16) (cid:17)
∈ | |
d(σ,∂Y ) d
m
≤
Using Lemma 1.2, we see that there is a positive constant C such that
1 N˙ d
TrZ[π](p(∆j))− N TrC p ∆j(m) ≤2 Nm,d |ar|Cr.
(cid:12)(cid:12) m (cid:16) (cid:16) (cid:17)(cid:17)(cid:12)(cid:12) m Xr=0
The proof of th(cid:12)e lemma is completed by taking the(cid:12)limit as m .
(cid:12) (cid:12) → ∞
We next recall the following abstract lemmata of Lu¨ck [Lu].
Lemma 2.4. Let p (µ) be a sequence of polynomials such that for the characteristic
n
function of the interval [0,λ], χ (µ), and an appropriate real number L,
[0,λ]
lim p (µ)= χ (µ) and p (µ) L
n [0,λ] n
n→∞ | | ≤
holds for each µ [0, ∆ 2]. Then
j
∈ || ||
lim TrZ[π](pn(∆j)) = F(λ).
n→∞
Lemma 2.5. Let G : V W be a linear map of finite dimensional Hilbert spaces V and
→
W. Let p(t) = det(t G∗G) be the characteristic polynomial of G∗G. Then p(t) can be
−
written as p(t)= tkq(t) where q(t) is a polynomial with q(0) = 0. Let K be a real number,
6
K max 1, G and C > 0 be a positive constant with q(0) C > 0. Let E(λ) be the
≥ { || ||} | | ≥
number of eigenvalues µ of G∗G, counted with multiplicity, satisfying µ λ. Then for
≤
0 < λ< 1, the following estimate is satisfied.
E(λ) E(0) logC logK2
{ }−{ } − + .
dimCV ≤ dimCV( logλ) logλ
− −
Proof of theorem 2.1. Fix λ 0 and define for n 1 a continuous function f : R R
n
≥ ≥ →
by
1+ 1 if µ λ
n ≤
f (µ) = 1+ 1 n(µ λ) if λ µ λ+ 1
n  n − − ≤ ≤ n

 1 if λ+ 1 µ
n n ≤



8 JOZEF DODZIUK AND VARGHESE MATHAI
Then clearly χ (µ) < f (µ) < f (µ) and f (µ) χ (µ) as n for all µ
[0,λ] n+1 n n [0,λ]
→ → ∞ ∈
[0, ). For each n, choose a polynomial p such that χ (µ) < p (µ) < f (µ) holds for
n [0,λ] n n
∞
allµ [0,K2]. We canalways findsuchapolynomialby asufficiently close approximation
∈
of f . Hence
n+1
χ (µ) < p (µ) <2
[0,λ] n
and
lim p (µ)= χ (µ)
n [0,λ]
n→∞
for all µ [0,K2]. Recall that E (λ) denotes the number of eigenvalues µ of ∆(m)
∈ m j
satisfying µ λ and counted with multiplicity. Note that ∆(m) K2 by Lemma 2.2.
≤ || j ||≤
1 1
(m)
N TrC pn(∆j ) = N pn(µ)
m m
(cid:0) (cid:1) µ∈X[0,K2]
E (λ) 1
m
= + (p (µ) 1)+ p (µ)
n n
N N  −
m m
µ∈X[0,λ] µ∈(λX,λ+1/n]
+  p (µ)
n

µ∈(λ+X1/n,K2] 
Hence, we see that 
E (λ) 1
m (m)
(2.1) Fm(λ) = N ≤ N TrC pn(∆j ) .
m m
(cid:0) (cid:1)
In addition,
1 E (λ) 1
(m) m
N TrC pn(∆j ) ≤ N + N sup{pn(µ)−1 :µ ∈ [0,λ]} Em(λ)
m m m
(cid:0) (cid:1)
1
+ sup p (µ) :µ [λ,λ+1/n] (E (λ+1/n) E (λ))
n m m
N { ∈ } −
m
1
+ sup p (µ) :µ [λ+1/n, K2] (E (K2) E (λ+1/n))
n m m
N { ∈ } −
m
E (λ) E (λ) (1+1/n)(E (λ+1/n) E (λ))
m m m m
+ + −
≤ N nN N
m m m
(E (K2) E (λ+1/n))
m m
+ −
nN
m
E (λ+1/n) 1 E (K2)
m m
+
≤ N n N
m m
a
F (λ+1/n)+
m
≤ n
APPROXIMATING L2 BETTI NUMBERS OF AMENABLE COVERING SPACES 9
sinceE (K2)= dimCj(Y ) aN forapositiveconstant aindependentof m. Itfollows
m m m
≤
that
1 a
(m)
(2.2) N TrC pn(∆j ) ≤ Fm(λ+1/n)+ n.
m
(cid:0) (cid:1)
Taking the limit inferior in (2.2) and the limit superior in (2.1), as m , we get that
→ ∞
a
(2.3) F(λ) TrZ[π] pn(∆j) F(λ+1/n)+ .
≤ ≤ n
Taking the limit as n in (2.3) a(cid:0)nd using(cid:1)Theorem 2.4, we see that
→ ∞
F(λ) F(λ) F+(λ).
≤ ≤
For all ε > 0 we have
F(λ) F+(λ) F(λ+ε) F(λ+ε) F(λ+ε).
≤ ≤ ≤ ≤
Since F is right continuous, we see that
F(λ) = F+(λ) = F+(λ)
proving the first part of theorem 2.1.
(m)
Next we apply theorem 2.5 to ∆ . Let p (t) denote the characteristic polynomial
j m
of ∆(m) and p (t) = trmq (t) where q (0) = 0. The matrix describing ∆(m) has integer
j m m m 6 j
entries. Hence p is a polynomial with integer coefficients and q (0) 1. By Lemma
m m
| | ≥
2.2 and Theorem 2.5 there are constants K and C = 1 independent of m, such that
F (λ) F (0) logK2
m m
−
a ≤ logλ
−
That is,
alogK2
(2.4) F (λ) F (0) .
m m
≤ − logλ
Taking limit inferior in (2.4) as m yields
→ ∞
alogK2
F(λ) F(0) .
≤ − logλ
Passing to the limit as λ +0, we get that
→
F(0) = F+(0) and F(0) = F+(0).
+
We have seen already that F (0) = F(0) = F(0), which proves part ii) of Theorem 2.1.
alogK2
Since is right continuous in λ,
− logλ
alogK2
+
F (λ) F(0) .
≤ − logλ
Hence part iii) of Theorem 2.1 is also proved.
We will need the following lemma in the proof of Theorem 0.2 in the last section. We
follow the proof of Lemma 3.3.1 in [Lu].
10 JOZEF DODZIUK AND VARGHESE MATHAI
Lemma 2.6.
K2 F(λ) F(0) K2 F (λ) F (0)
m m
− dλ liminf − dλ
λ ≤ m→∞ λ
Z0+ (cid:26) (cid:27) Z0+ (cid:26) (cid:27)
Proof. By Theorem 2.1, and the monotone convergence theorem, one has
K2 F(λ) F(0) K2 F(λ) F(0)
− dλ = − dλ
λ λ
Z0+ (cid:26) (cid:27) Z0+ (cid:26) (cid:27)
K2 F (λ) F (0)
m m
= liminf − dλ
m→∞ λ
Z0+ (cid:26) (cid:27)
K2 F (λ) F (0)
n n
= lim inf − n m dλ
m→∞ λ | ≥
Z0+ (cid:18) (cid:26) (cid:27)(cid:19)
K2 F (λ) F (0)
n n
= lim inf − n m dλ
m→∞ λ | ≥
Z0+ (cid:26) (cid:27)
K2 F (λ) F (0)
m m
liminf − dλ.
≤ m→∞ λ
Z0+ (cid:26) (cid:27)
3. Proofs of the main theorems
In this section, we will prove the Amenable Approximation Theorem (Theorem 0.1) of
the introduction. We will also prove some related spectral results.
Proof of Theorem 0.1 (Amenable Approximation Theorem). Observe that
(m)
bj(Ym) dimC ker(∆j )
=
N (cid:16)N (cid:17)
m m
= F (0).
m
Also observe that
j
b (Y : π) = dim ker(∆ )
(2) π j
= F(0).(cid:16) (cid:17)
Therefore Theorem 0.1 follows from Theorem 2.1 after taking the limit as m .
→ ∞
j
Suppose that M is a compact Riemannian manifold and Ω (M) denote the Hilbert
(2)
space of square integrable j-forms on a normal covering space M, with transformation
j j f
group π. The Laplacian ∆ : Ω (M) Ω (M) is essentially self-adjoint and has a
j (2) → (2)
f
spectral decomposition P (λ) : λ [0, ) where each P (λ) has finite von Neumann
j j
{ e ∈f ∞ } f
trace. The associated von Neumann spectral density function, F(λ) is defined as
e e
F : [0, ) [0, ), F(λ) = Tr (P (λ)).
U(π) j
∞ → ∞ e
e e e