Growth of Betti Numbers

Bryan Clair

Department of Mathematics, Saint Louis University, St. Louis, MO 63103

Kevin Whyte

Department of Mathematics, University of Illinois at Chicago, Chicago, IL 60607

Introduction

Let X = X̃/Γ be a finite simplicial complex. We study the growth rate of the Betti numbers of finite regular covers of X. Let X_{i} = X̃/Γ_{i}, i = 1, 2, …, be a sequence of finite regular coverings of X. It is easy to see that the sequence of Betti numbers {b_{q}(X_{i})} can grow at most linearly in [Γ : Γ_{i}]. A theorem of Lück describes exactly when the linear growth rate is achieved, settling a conjecture of Kahzdan. Lück’s Theorem [8] states that

b_{q}^{(2)}(X̃; Γ) = lim_{i→∞} \frac{b_{q}(X_{i})}{[Γ : Γ_{i}]}

when the Γ_{i} are a tower of finite index normal subgroups of Γ with ∩Γ_{i} = {e}.

This shows that linear growth of Betti numbers occurs if and only if the L^{2}- Betti number, b_{q}^{(2)}(X̃; Γ), is non zero. Nonzero L^{2}-Betti number means that the Laplacian in dimension q has kernel, and is therefore equivalent to the existence of non-trivial L^{2} harmonic q-cochains on X̃.

In this paper, we assume b_{q}^{(2)}(X̃; Γ) vanishes and control the growth of the {b_{q}(X_{i})}. In particular, the sequence cannot grow arbitrarily close to linear in [Γ : Γ_{i}] — there is a universal upper bound, and even better bounds given assumptions on X.

The upper bounds use information about the spectrum of the Laplacian near zero: finite covers with large Betti numbers force the L^{2} Laplacian to have lots of small spectrum. For example, the best bounds are achieved when X̃ has a spectral gap. Our results suggest that one might use finite covers to understand the spectrum of X̃ and its associated data, such as Novikov-Shubin invariants.

Preprint submitted to Elsevier Science 19 June 2002

Our main proof follows Lück’s in outline, with more involved estimates. There are two key factors in estimating the Betti numbers of covers of X. The first measures how X is “unrolled” in the cover:

Definition 0.1 Let Γ be a finitely generated group, with the word metric from some fixed generating set. For any subgroup Γ′ of Γ, define short(Γ′) to be the length of the shortest non-identity element of Γ′.

It is clear from the proof of Lück’s Theorem that instead of a tower of Γ_{i}, one could allow any sequence of Γ_{i} ◁ Γ with short(Γ_{i}) → ∞.

The second ingredient in our estimates is the behavior of the L^{2} spectrum of X̃. Just as Lück’s theorem relates linear growth of Betti numbers to the kernel of the L^{2} Laplacian, our results relate slower growth rates to the behavior of the spectrum of the L^{2} Laplacian near zero.

To say “growth rates” suggests a tower, or at least a sequence of coverings. But the main theorems in this paper are stated as bounds on given finite covers X′ of X:

Theorem 0.2 Let X be a finite simplicial complex, and X̃ an infinite regular covering with covering group Γ. Suppose that b_{q}^{(2)}(X̃; Γ) = 0, or equivalently that there are no L^{2} harmonic q-cochains on X̃.

(1) (Spectral Gap) Suppose there is a gap near 0 in the L^{2} spectrum of X̃ in dimension q. Then there are C > 0 and M > 0 so that for any finite regular cover X′ = X̃/Γ′ of X:

b_{q}(X′) ≤ C \frac{[Γ : Γ′]}{e^{Mshort(Γ′)}} .

(2) (Positive Novikov-Shubin Invariant) If X̃ has Novikov-Shubin invariant α_{q} > 0, then for any ε > 0 there is a C_{ε} > 0 so that for any finite regular cover X′ = X̃/Γ′ of X:

b_{q}(X′) ≤ C_{ε} \frac{[Γ : Γ′]}{(short(Γ′))^{α_{q}−ε}} .

(3) (General Case) For any X there is a C so that for any finite regular cover X′ = X̃/Γ′ of X:

b_{q}(X′) ≤ C \frac{[Γ : Γ′]}{log(short(Γ′))}.

The three cases of this theorem are proved later as Theorem 4.1, Theorem 5.2, and Theorem 6.1, respectively. They are all special cases of a more general statement (Proposition 3.4) which involves the spectrum of Δ near 0 in a more technical way.

For a family of covers X_{i} = X̃/Γ_{i} with short(Γ_{i}) → ∞, Theorem 0.2 can be interpreted as giving sublinear upper bounds on the rate of growth of b_{q}(X_{i}) in terms of [Γ : Γ_{i}].

We can relate short(Γ_{i}) directly to [Γ : Γ_{i}] if we assume the covers of X “unroll” evenly in all directions, which we make precise in Definition 7.1. We call such a family a uniform family. As an example, congruence subgroups of arithmetic groups are uniform.

The uniform assumption is most interesting in the case of spectral gap, where it leads to particularly clean bounds of a form previously studied by Sarnak and Xue ([13],[15]).

Theorem 0.3 Let X be a finite simplicial complex, and X̃ an infinite regular covering with covering group Γ. Suppose that Γ has exponential growth, {Γ_{i}} is a uniform family of finite index normal subgroups, and that Δ_{q} on X̃ has no spectrum below λ_{0}.

• There is some C > 0 and β < 1 so that for all X_{i} = X̃/Γ_{i},

b_{q}(X_{i}) < C[Γ : Γ_{i}]^{β}.

• Fix λ < λ_{0}. There is some C > 0 and β < 1 so that for all i,

#{μ ≤ λ | μ is an eigenvalue of Δ_{q} on X_{i}} < C[Γ : Γ_{i}]^{β}.

The second part of Theorem 0.3 is proved by a slight alteration of the main argument for Betti numbers. Although one could make this sort of eigenvalue bound in a more general setting, we only carry out the calculations in the most interesting case.

To produce geometric applications, one needs a good understanding of the spectrum of the Laplacian near zero. The locally symmetric spaces provide a wide class of interesting examples, as the L^{2}-Betti numbers and Novikov-Shubin invariants are known. For these spaces short(Γ) is essentially the injectivity radius of the cover, and [Γ : Γ′] is the volume of the cover. Specifically, for hyperbolic manifolds:

Theorem 0.4 Let X be a compact hyperbolic n manifold. There are constants C > 0 and β_{q} > 0 so that for any finite regular cover X′ of X one has

• For n odd:

· If q ≠ \frac{n±1}{2}, then

b_{q}(X′) ≤ C \frac{Vol(X′)}{e^{β_{q} Inj(X′)}} .

· For q = \frac{n±1}{2},

b_{q}(X′) ≤ C \frac{Vol(X′) · log Inj(X′)}{Inj(X′)}.

• For n even:

· If q ≠ \frac{n}{2}, then

b_{q}(X′) ≤ C \frac{Vol(X′)}{e^{β_{q} Inj(X′)}} .

· For q = \frac{n}{2}, the L^{2}-Betti number b_{q}^{(2)}(ℍ^{n}; Γ) is non-zero.

We investigate the case Γ = ℤ^{n} in depth, using different techniques:

Theorem 0.5 Suppose X̃ is a regular ℤ^{n} covering of a finite simplicial complex X, and assume the L^{2}-Betti number b_{q}^{(2)}(X̃; ℤ^{n}) = 0. Then there is a constant C > 0 so that for any finite cover X′ = X̃/Γ′ of X:

b_{q}(X′) ≤ C \frac{[ℤ^{n} : Γ′]}{short(Γ′)}.

This is similar to the bound Theorem 0.2 would give if α_{q}(X), the Novikov-Shubin invariants of X, satisfied α_{q} ≥ 1. However, Lott [7, Ex. 42] gives examples with Γ = ℤ and α_{q}(X) arbitrarily small.

1 Stripes

To understand the growth of Betti numbers of covers it helps to have some examples. In particular, one would like a method for constructing complexes with a given fundamental group where it is easy to describe finite covers and their Betti numbers. The construction described in this section starts with an arbitrary finite complex and attaches pieces which do not change the fundamental group and which contribute in a straightforward way to the Betti numbers of finite covers.

Given complexes X and Y, and a complex Z which includes as a subcomplex of both X and Y, build a complex X ∪_{Z} Y which is the quotient of the disjoint union of X and Y by the relation which identifies the copies of Z. More generally, given a Z with maps f : Z → X and g : Z → Y, build a new complex, which we describe as “gluing X and Y along Z”, by carrying out the previous construction where we replace X and Y by the mapping cylinders of f and g, which are homotopy equivalent to X and Y and contain Z as a subcomplex.

Assume that π_{1}(X) = Γ. By Van Kampen’s theorem, π_{1}(X ∪_{Z} Y) is also equal to Γ provided that π_{1}(Z) ↪ π_{1}(X) and π_{1}(Z) = π_{1}(Y). With these assumptions on fundamental groups, put

W = X ∪_{Z} Y.

Let Γ′ be a finite index normal subgroup of Γ, and let W′ be the cover of W with π_{1}(W′) = Γ′. We can describe W′ as the result of a gluing X′ ∪_{Z′} Y′ where X′, Y′, and Z′ are the induced finite covers of X, Y, and Z.

To understand the homology of W′, we need to understand the homology of X′, Y′, and Z′, as well as the maps on homology induced by the inclusions. The special case we want to consider here is where all of the homology comes from the parts we have glued on. So assume that X has dimension n, and that we want to compute H_{q}(W′) for q > n. This homology is simply the homology of Y′.

If π_{1}(Z) = Γ then we have gained nothing by this construction, as it reduces to the same problem for one of the pieces. However, if π_{1}(Z) is a proper subgroup of Γ then the examples coming from this construction give useful insight into the general problem.

Example 1.1 As a simple example, suppose π_{1}(Z) = {e}. Then Y′ is the disjoint union of [Γ : Γ′] copies of Y. If H_{q}(Y) = 0, then H_{q}(W′) = 0 as well, for any Γ′. If H_{q}(Y) ≠ 0, then the Betti numbers b_{q}(W′) grow linearly in [Γ : Γ′]. Neither case is interesting from the point of view of our problem.

In general, the number of components of Y′ is

b_{0}(Y′) = [π_{1}(Y) : π_{1}(Y) ∩ Γ′],

each of which is the cover of Y with fundamental group π_{1}(Y) ∩ Γ′. Thus, to understand the Betti numbers of these covers, we need to understand the the Betti numbers of the covers of Y and the number of components. The key source of examples is the following:

Definition 1.2 Put Z = S^{1} ↪ X, and Y = S^{1} × S^{q}. The construction of W = X ∪_{Z} Y is called gluing a stripe to X along Z. We will often refer to Z simply by the corresponding element of Γ = π_{1}(X).

For the stripe construction, the components of the cover Y′ of Y are S^{1} × S^{q}, so their q-dimensional homology has rank one. Thus

dim(H_{q}(W′)) = [π_{1}(Z) : π_{1}(Z) ∩ Γ′].

Let γ be a generator of π_{1}(Z) = ℤ, and let o(γ) denote the order of γ in the quotient Γ/Γ′. Then

dim(H_{q}(W′)) = \frac{[Γ : Γ′]}{o(γ)}.

To get examples of fast sublinear growth of Betti numbers, we want o(γ) to grow, but as slowly as possible. The following Lemma demonstrates the key piece of geometry of Γ′ that enters our bounds:

Lemma 1.3 Suppose γ ∈ Γ has infinite order. There is a C > 0 so that for any complex W built by gluing a stripe along γ, and for any finite index normal Γ′:

dim(H_{q}(W′)) ≤ C \frac{[Γ : Γ′]}{short(Γ′)}.

PROOF. By the earlier discussion, this amounts to the claim that the order of γ in Γ/Γ′ is at least short(Γ′)/C. If γ^{k} = e in Γ/Γ′ then γ^{k} ∈ Γ′. Since γ has infinite order γ^{k} is a non-trivial element of Γ′, and so has length at least short(Γ′). On the other hand, the length of γ^{k} is at most k times the length of γ. Thus the Lemma follows with C equal to the length of γ. □

Example 1.4 Let Γ = ℤ×ℤ = ⟨a⟩×⟨b⟩, and take a family of subgroups Γ_{i} = m_{i}ℤ × n_{i}ℤ. Put X = 𝕋^{2}, and glue a q-stripe to X along a. Then b_{q}(W_{i}) = n_{i}, while [Γ : Γ_{i}] = m_{i}n_{i}.

If n_{i} > m_{i}, then short(Γ_{i}) = m_{i} and the bound in Lemma 1.3 is achieved. If m_{i} > n_{i}, however, it is clear that the stripe along a fails to capture the geometry. A stripe along b would give a larger Betti number.

In the previous example, we could have glued two stripes so that for any fixed family of groups Γ_{i} = m_{i}ℤ×n_{i}ℤ, growth of order [Γ : Γ_{i}]/short(Γ_{i}) is achieved. However, for some families Γ_{i}, there is no stripe construction that achieves this rate of growth:

Example 1.5 Let Γ = ℤ × ℤ, let {p_{i}} be the sequence of primes which are equivalent to 1 modulo 4. It is well known that such primes are the sum of two squares. Put p_{i} = a_{i}^{2} + b_{i}^{2}, and let Γ_{i} be the lattice spanned by (a, b) and (−b, a) in ℤ × ℤ, so that [Γ : Γ_{i}] = p_{i}. For any γ ∈ ℤ × ℤ, γ generates Γ/Γ_{i} unless γ ∈ Γ_{i}. Since short(Γ_{i}) → ∞, any given γ is in only finitely many of the Γ_{i}. Therefore, the Betti numbers of the Γ_{i} covers of the complex obtained by gluing a stripe (or any finite number of stripes) are bounded over all i.

We will prove that this example is not a defect of the stripe construction. Indeed, for Γ = ℤ^{n}, and any family of coverings, the stripes examples achieve the fastest possible growth of Betti numbers (see Remark 2.4). We do not know if this is true for general Γ.

2 The Abelian Case

In this section, we investigate in depth the special case where Γ = ℤ^{n}. The techniques are different from the general situation considered in Section 3. Here, the key step is an algebraic estimate of rational solutions to a trigonometric polynomial. The results are of a similar form to the general theorems, and illustrate the strength of the stripe construction.

Theorem 2.1 Suppose X̃ is a regular ℤ^{n} covering of a finite simplicial complex X, and assume the L^{2}-Betti number b_{q}^{(2)}(X̃; ℤ^{n}) = 0. Then there is a constant C(X) so that for any Γ′ < ℤ^{n}, we have

b_{q}(X′) ≤ C(X)\frac{[ℤ^{n} : Γ′]}{short(Γ′)}.  (1)

Here X′ = X̃/Γ′.

Suppose that X has a cells in dimension q. All work is done in the fixed dimension q, but the q is suppressed in most of the notation. Put Γ = ℤ^{n}.

Choosing lifts of cells from X to X̃, we identify C_{q}^{(2)}(X̃) with ⨁_{1}^{a} l^{2}(Γ). In this basis, Δ is represented by an a × a matrix B with entries in ℤ[Γ].

Since Γ = ℤ^{n} is abelian, we can form det(B) ∈ ℤ[Γ]. The determinant det(B) will not depend on the choice of lifts of cells, so we will simply write det(Δ).

Identify the n-torus 𝕋^{n} with the irreducible unitary representations of ℤ^{n}. Let

K = {ρ ∈ 𝕋^{n} | ρ(det(Δ)) = 0}.

Call K the pattern of X on 𝕋^{n}.

We want to calculate the Betti numbers for the finite cover X′ of X. The Laplacian on q-chains on X′ is also given by the matrix B, now acting on ⨁_{1}^{a} l^{2}(Γ/Γ′). The space l^{2}(Γ/Γ′) splits as a direct sum of irreducible representations ρ, where ρ lies on a lattice Λ of rational points in 𝕋^{n}. Λ is (noncanonically) isomorphic to Γ/Γ′, and

Λ = {ρ ∈ 𝕋^{n} | ρ is trivial on Γ′} .

Lemma 2.2

|Λ ∩ K| ≤ b(X′) ≤ a · |Λ ∩ K|

PROOF. We have

⨁_{1}^{a} l^{2}(Γ/Γ′) = ⨁_{ρ∈Λ} (⨁_{1}^{a} V_{ρ}) ,

where V_{ρ} = ℂ. The Laplacian Δ′ preserves the a-dimensional space ⨁_{1}^{a} V_{ρ}, and has kernel there precisely when ρ(det(Δ)) = 0. The dimension of the kernel is obviously bounded by a. □

The goal, then, is to count certain rational points in 𝕋^{n} which lie on the pattern of X. The pattern of X is the set of root-of-unity solutions to a polynomial equation, and these are fully described by Conway and Jones [3].

Choose generators g_{1}, …, g_{n} for ℤ^{n}. Then det(Δ) is a polynomial in the g_{k} (with integer coefficients). For ρ = (x_{1}, …, x_{n}) ∈ 𝕋^{n}, ρ(g_{k}) = e^{2πix_{k}}. Therefore,

ρ(det(Δ)) = ∑_{I} A_{I}e^{2πil_{I}(x)},

where l_{I}(x) is some integer linear combination of the x_{i}. It follows from [3] that the rational solutions to

∑_{I} A_{I}e^{2πil_{I}(x)} = 0

lie on one of a finite family of rational linear subspaces of 𝕋_{n}.

We have shown that the rational points in the pattern of X are contained in a finite collection of rational linear subspaces. Fix one such space L ⊂ 𝕋^{n}. If L is not itself a subgroup, then it is a coset of some subgroup L′ < 𝕋^{n}. The intersection Λ ∩ L is either empty or itself a coset of Λ ∩ L′. Therefore,

|Λ ∩ L| ≤ |Λ ∩ L′|.

So, to estimate |Λ ∩ L| we may as well assume that L is a subgroup of 𝕋^{n}.

There is a subgroup L̂ ⊂ Γ so that

L = {ρ ∈ 𝕋^{n} | ρ is trivial on L̂} .

Note that dim(L) + rank_{ℤ} L̂ = n.

Choose any g ∈ L̂. For optimal constants, one should make ∥g∥ as small as possible. Let o(g) denote the order of g in Γ/Γ′. We have

o(g) ≥ \frac{short(Γ′)}{∥g∥}.

Construct a representation ρ : Γ → S^{1} which is trivial on Γ′ and sends g to a primitive o(g)-th root of unity (this is easy, since Γ/Γ′ is a finite abelian group). For k = 1, 2, …, o(g), each representation ρ^{k} is in Λ, but all assume different values on g. In particular, ρ^{1}, ρ^{2}, …, ρ^{o(g)} all lie in different cosets of Λ ∩ L < Λ. Therefore

|Λ ∩ L| ≤ \frac{|Λ|}{o(g)} ≤ \frac{∥g∥ · [Γ : Γ′]}{short(Γ′)}.

Summing over the various L which appear in K (the pattern of X), we conclude that

|Λ ∩ K| ≤ \frac{C_{1}(X) · [Γ : Γ′]}{short(Γ′)}.  (2)

Here C_{1}(X) is a constant depending on X, or more precisely, depending on the rational linear subspaces appearing in the pattern of X.

Combining (2) with Lemma 2.2 completes the proof of Theorem 2.1. □

As a corollary of the proof, we can make an even better statement for Γ = ℤ:

Corollary 2.3 Suppose X̃ is a regular ℤ covering of a finite simplicial complex X. Put X_{i} = X̃/iℤ, the i-fold covering of X. Then exactly one of the following possibilities occurs:

(1) b_{q}^{(2)}(X̃; ℤ) ≠ 0, so the sequence of Betti numbers b_{q}(X_{i}) is asymptotically linear in i.

(2) There is a constant C(X) so that b_{q}(X_{i}) ≤ C(X) for all i.

PROOF. The torus 𝕋^{1} is just S^{1}, so a linear subspace is either a single point or the entire circle. If the pattern of X contains the entire circle, we are in the first case. Otherwise, the pattern of X consists of a finite collection of k points (counted with multiplicities), and so the betti numbers of X_{i} are bounded by k times the number of cells in X. □

Remark 2.4 For Γ = ℤ^{n} and for a given sequence of subgroups Γ_{i} ⊂ Γ, we need only consider the stripe construction when looking for large Betti numbers. This is because given X = X̃/Γ, the Betti numbers b_{q}(X_{i}) depend only on the rational linear subspaces contained in the pattern of X. We can reproduce these rational linear subspaces by repeatedly gluing stripes to an n-torus.

Example 1.4 then suggests that Theorem 2.1 has the right geometric ingredients for an upper bound, but that these do not completely determine the growth rate. See Example 1.5.

Remark 2.5 The q^{th} Novikov-Shubin invariant for a space with a q-stripe is 1, the same as for ℝ. Compare the bound (1) for Γ abelian with the general bound (8) for Novikov-Shubin invariant 1.

3 General Bounds

In this section, we derive a general bound on Betti numbers of coverings, in terms of the L^{2} spectral density function. The argument follows the proof of Lück’s Theorem [8], but we carefully control the estimates throughout.

3.1 Preliminaries

Let X be a finite simplicial complex, and X̃ an infinite regular cover, with covering transformation group Γ. Suppose we have a normal subgroup Γ′ of finite index in Γ. Form X′ = X̃/Γ′, a covering of X of order [Γ : Γ′].

Let Δ be the combinatorial Laplacian on C_{q}^{(2)}(X̃), the space of q-dimensional l^{2}-cochains. Let Δ′ be the Laplacian on q-cochains of X′.

Suppose X has a cells in dimension q. Lifting these cells to X̃ gives a basis over l^{2}(Γ) for C_{q}^{(2)}(X̃). These lifts descend to give a basis over l^{2}(Γ/Γ′) for C_{q}(X′).

In this basis, Δ is represented by an a×a matrix B with entries in ℤ[Γ], acting by right multiplication on ⨁_{j=1}^{a} l^{2}(Γ). The same matrix B describes Δ′, now acting by right multiplication on ⨁_{j=1}^{a} l^{2}(Γ/Γ′).

Lemma 3.1 There exists a number K > 1 so that ∥Δ∥ ≤ K and so that for any group Γ′ as above, ∥Δ′∥ ≤ K.

PROOF. This is identical to [8, Lemma 2.7]. The number K can be defined from the coefficients of group elements appearing in B. □

Lemma 3.2 Let

R = max{∥g∥ | g ∈ Γ appears with nonzero coefficient in some B_{ij}}.

Then for any polynomial p with deg(p) < \frac{short(Γ′)}{R},

Tr_{Γ} p(Δ) = Tr_{Γ/Γ′} p(Δ′)

PROOF. Write

∑_{j=1}^{a}(p(B))_{jj} = ∑_{g∈Γ} λ_{g}g (∈ ℤ[Γ]).

Then

Tr_{Γ} p(Δ) = λ_{e}

and

Tr_{Γ/Γ′} p(Δ′) = ∑_{g∈Γ′} λ_{g}.

However, B contains group elements of length at most R, so B^{n} contains group elements of length at most nR. Therefore p(B) contains group elements of length less than short(Γ′), and so λ_{g} = 0 for all g ∈ Γ′, g ≠ e. □

3.2 Density functions and Betti numbers

Definition 3.3 Let

P_{n} = \{ Polynomials p in one variable | \begin{matrix}deg p ≤ n; \\ p non-negative on [0, 1]; \\ p(0) = 1 \end{matrix} \}

For a probability measure dμ on [0, 1], put

J(n, μ) = inf_{p∈P_{n}} ∫_{0}^{1} p(x)dμ(x).

The measure of interest to us is the spectral density function of the Laplacian, suitably rescaled. To define it, let {E(λ) : λ ∈ [0, ∞)} denote the family of spectral projections of Δ. Then the spectral density function of Δ is

F (λ) = Tr_{Γ} E(λ).

Proposition 3.4 Let

n < \frac{short(Γ′)}{R}

and

μ(x) = \frac{F (Kx)}{a}.

Then

b_{q}(X′) ≤ a[Γ : Γ′]J(n, μ).

PROOF. Let p ∈ P_{n}. For any r > 1, there is some λ > 0 so that every x ∈ [0, λ] satisfies

rp(\frac{x}{K}) ≥ 1.

Let χ be the characteristic function of the interval [0, λ]. We have rp(\frac{x}{K}) ≥ χ(x) on [0, K]. Then

\frac{b_{q}(X′)}{[Γ : Γ′]} ≤ Tr_{Γ/Γ′} χ(Δ)

≤ Tr_{Γ/Γ′} rp(\frac{Δ}{K})

= r Tr_{Γ} p(\frac{Δ}{K})

using Lemma 3.2. Sending r → 1,

b_{q}(X′) ≤ [Γ : Γ′] Tr_{Γ} p(\frac{Δ}{K})

= [Γ : Γ′] ∫_{0}^{K} p(\frac{λ}{K})dF (λ)

= a[Γ : Γ′] ∫_{0}^{1} p(x)dμ(x).

Since p ∈ P_{n} was arbitrary, the proof is done. □

3.3 Choosing a polynomial

We now proceed to estimate J(n, μ), with an eye towards using the decay of μ(x) as x → 0. Fix z ∈ (0, 1). Let p ∈ P_{n} be bounded by 1 on [0, z] and bounded by M on [z, 1]. If p(z) < M, the estimates will be strictly improved by moving z to the left, so we may as well restrict attention to the situation where p(z) is the maximal value of p on [z, 1]. We have

J(n, μ) ≤ ∫_{0}^{1} p(x)dμ(x)
= ∫_{0}^{z} p(x)dμ(x) + ∫_{z}^{1} p(x)dμ(x)
≤ μ(z) + p(z).

To minimize μ(z) + p(z), we want a polynomial which drops as quickly as possible from x = 0 to x = z and then stays low until x = 1. Via a linear transformation, this is equivalent to finding a polynomial which is bounded by ±1 on [−1, 1] and grows as quickly as possible for x > 1. Among polynomials of degree n, the Chebyshev polynomial T_{n} is the optimal solution [12].

The Chebyshev polynomials are defined by T_{n}(cos(θ)) = cos(nθ). The first few are 1, x, 2x^{2} − 1, 4x^{3} − 3x. We need two facts: first, T_{n}(1) = 1 for all n; second, the Chebyshev polynomials satisfy a recurrence relation

T_{n+1}(x) = 2xT_{n}(x) − T_{n−1}(x)

which yields

T_{n}(x) = \frac{1}{2} [(x + \sqrt{x^{2} − 1})^{n} + (x − \sqrt{x^{2} − 1})^{n}] .

Now, put

l(x) = (\frac{−2}{1 − z} ) x + (\frac{1 + z}{1 − z} ) ,

so l(1) = −1, l(z) = 1, and l(0) = \frac{1+z}{1−z}. Set

p_{n}(x) = \frac{T_{n}(l(x)) + 1}{T_{n}(l(0)) + 1}.

It is easy to see that p_{n} ∈ P_{n}. Therefore

J(n, μ) ≤ μ(z) + p_{n}(z)

≤ μ(z) + \frac{2}{T_{n} (\frac{1+z}{1−z} )}

= μ(z) + \frac{4}{(\frac{1+\sqrt{z}}{1−\sqrt{z}} )^{n} + (\frac{1+\sqrt{z}}{1−\sqrt{z}} )^{−n}}

≤ μ(z) + 4(\frac{1 + \sqrt{z}}{1 − \sqrt{z}} )^{−n}

≤ μ(z) + 4e^{−2n\sqrt{z}}.

This proves:

Proposition 3.5 For any n > 0, probability measure dμ, and z ∈ [0, 1],

J(n, μ) ≤ μ(z) + 4e^{−2n\sqrt{z}}  (3)

To choose z appropriately requires information about μ. We handle three specific cases in the next sections.

4 Spectral Gap

Suppose that the L^{2} spectrum for X̃ has a gap at zero. This means that there exists λ_{0} > 0 with F (λ) = 0 for all λ < λ_{0}.

Continuing notation from Section 3, μ(x) = F (Kx)/a has a gap at zero of size λ_{0}/K. Letting z → λ_{0}/K in (3) yields

J(n, μ) ≤ 4e^{−2n\sqrt{λ_{0}/K}}.

Combining with Proposition 3.4 proves the following:

Theorem 4.1 If X̃ has spectral gap of size λ_{0} in dimension q, then

b_{q}(X′) ≤ 4a\frac{[Γ : Γ′]}{e^{Mshort(Γ′)}} ,  (4)

where M = \frac{2}{R} \sqrt{\frac{λ_{0}}{K}}. Here R is the maximum length of group elements appearing in the matrix for Δ, and K is a bound on the norms of Δ and Δ_{i} as in Lemma 3.1.

If Γ has exponential growth, [Γ : Γ′] will be at least exponential in short(Γ′). For a family of subgroups Γ_{i}◁Γ, a natural assumption is that short(Γ_{i}) actually does grow like log[Γ : Γ_{i}]. This is the assumption of the following Corollary.

Corollary 4.2 Let {Γ_{i}} be a family of normal, finite index subgroups of Γ. Suppose there exists D > 0 so that

short(Γ_{i}) > D log[Γ : Γ_{i}] − constant.  (5)

Put X_{i} = X̃/Γ_{i}. Then there is some C > 0 so that

b_{q}(X_{i}) < C · [Γ : Γ_{i}]^{1−MD}

for all i.

This also proves the first part of Theorem 0.3. The estimate (5) comes directly from the assumption that Γ has exponential growth and that the family {Γ_{i}} is uniform. Applying (5) to the next Proposition proves the second part of Theorem 0.3.

Proposition 4.3 Fix λ < λ_{0}, and let

E(λ) = {μ ≤ λ | μ is an eigenvalue of Δ′_{q} on X′}.

Then there is a C > 0 and M > 0 so that

#E(λ) < C \frac{[Γ : Γ′]}{e^{Mshort(Γ′)}} .

PROOF. We continue the notation of Section 3.3, where p_{n} is a linear transformation of the n^{th} Chebyshev polynomial. Note that p_{n} is decreasing on [0, λ_{0}/K]. Then

#E(λ) · p_{n}(λ/K) ≤ ∑_{μ∈E(λ)} p_{n}(μ/K) ≤ [Γ : Γ′] Tr_{Γ/Γ′} p_{n}(Δ′/K).

If n < short(Γ′)/R, Lemma 3.2 applies, and we get

#E(λ) ≤ [Γ : Γ′]\frac{Tr_{Γ} p_{n}(Δ/K)}{p_{n}(λ/K)}.

The spectrum of Δ is empty below λ_{0}, so

#E(λ) ≤ [Γ : Γ′]\frac{p_{n}(λ_{0}/K)}{p_{n}(λ/K)}.

Finally, a simple calculation shows that the ratio \frac{p_{n}(λ_{0}/K)}{p_{n}(λ/K)} decays exponentially in n as n → ∞ (although the base approaches 1 when λ nears λ_{0}). Replacing n with short(Γ′)/R completes the proof. □

5 Positive Novikov-Shubin Invariant

The q^{th} Novikov-Shubin invariant of X̃ describes the decay of the spectral density function F (λ) near λ = 0. Suppose there is some C > 0 so that

C^{−1}λ^{α_{q}/2} < F (λ) < Cλ^{α_{q}/2}  (6)

for small λ. Then α_{q} is the q^{th} Novikov-Shubin invariant of X̃. In general, one defines

α_{q} = 2 lim inf_{λ→0^{+}} \frac{log(F (λ) − F (0))}{log(λ)} ∈ [0, ∞]

when F has no spectral gap. The 2 is a normalization which is discarded by some authors. With our definition, all Novikov-Shubin invariants of ℝ^{n} are n.

For the moment, suppose that μ(x) < Cx^{β} for some C > 0 and β > 0.

To make J(n, μ) small, we choose z so that the two terms in the bound (3) are roughly the same size for large n. We ask that

z^{β} = e^{−2n\sqrt{z}}

or

β log(z) = −2n\sqrt{z}.

Substituting x = z^{−1/2} yields

x log(x) = n/β,

which has the solution log(x) = W (n/β), where W is the Lambert W-function. For large n, W (y) is asymptotic to log(y/ log(y)). Based on this, we choose

z = (\frac{log(n/β)}{n/β})^{2}.

Now plug in to the bound (3) for J(n, μ):

J(n, μ) ≤ C(\frac{log(n/β)}{n/β})^{2β} + 4 exp (−2n\frac{log(n/β)}{n/β})

= C(\frac{log(n/β)}{n/β})^{2β} + 4 (\frac{1}{n/β} )^{2β} .

Therefore, there is some constant C′ depending on C and β so that for all n,

J(n, μ) ≤ C′ · (\frac{log(n)}{n})^{2β}.

Combining this result with Proposition 3.4 gives the following:

Theorem 5.1 Suppose the q^{th} spectral density function of X̃ satisfies

F_{q}(λ) < Cλ^{β}

for some β > 0, C > 0. Then there is a constant C_{1} > 0 so that

b_{q}(X′) ≤ C_{1}[Γ : Γ′](\frac{log(short(Γ′))}{short(Γ′)})^{2β}.  (7)

Now, we interpret this for Novikov-Shubin invariants:

Theorem 5.2 Suppose b_{q}^{(2)}(X̃; Γ) = 0 and the Novikov-Shubin invariant α_{q}(X̃) is positive. Then for every ε > 0, there is some constant C_{ε} depending on ε and X so that for any finite covering X′ = X̃/Γ′ of X,

b_{q}(X′) ≤ C_{ε} \frac{[Γ : Γ′]}{(short(Γ′))^{α_{q}(_{X})−ε}} .  (8)

PROOF. If α_{q}(X̃) > 0, we have a bound of the form

F_{q}(λ) < Cλ^{β}

for any β < α_{q}(X̃)/2. Since we can always make β a little larger than needed for a given ε, the log(short(Γ′)) term in (7) is absorbed into the constant. □

6 Sublogarithmic Decay

The most general estimate known for spectral density functions is

F (λ) < \frac{a log(K)}{− log(λ)}.

This estimate holds whenever Γ belongs a large class of groups G. The class G includes all residually finite groups. For details, see [4], [2], [14].

Assume that μ(x) < \frac{C}{− log(x)}. As in Section 5, we want to choose z so that the two terms in the bound (3) are roughly the same size for large n. We ask that

\frac{1}{− log(z)} = e^{−2n\sqrt{z}}.

Substitute x = 2n\sqrt{z} to get

e^{x} + 2 log(x) = 2 log(2n),

which has the approximate solution x = log(2 log(2n)) for large n. As the precise constants won’t matter in the end, choose

z = (\frac{log(log n)}{n})^{2}.

Now plug in to the bound (3) for J(n, μ):

J(n, μ) ≤ \frac{C}{−2(log(log(log n)) − log(n))} + 4e^{− log(log n)}

Therefore, there is some constant C′ so that for all n,

J(n, μ) ≤ \frac{C′}{log(n)}.  (9)

Theorem 6.1 Given Γ and X = X̃/Γ, with b_{q}^{(2)}(X̃; Γ) = 0. There is a constant C so that for any finite covering X′ = X̃/Γ′ of X,

b_{q}(X′) ≤ C \frac{[Γ : Γ′]}{log(short(Γ′))}.

PROOF. If Γ is not residually finite, short(Γ′) is uniformly bounded over all Γ′, so the theorem is vacuously true. Otherwise, from [8], one has decay of the L^{2} spectrum of X̃ of the form

F (λ) < \frac{a log(K)}{− log(λ)},

for λ < ε, with some ε > 0. Now μ(x) < \frac{C}{− log(x)} for some C > 0, and we apply (9) to Proposition 3.4. □

Remark 6.2 If Γ is not residually finite, form Γ_{f} = ⋂ Γ′, where the intersection is over all finite index normal subgroups of G. Then all finite covers of X are covered by X_{f} = X̃/Γ_{f}. The group Γ/Γ_{f} acts on X_{f} and is residually finite, so we can still bound Betti numbers of X′ in a non-trivial way. The subtle point is that short(Γ′) is replaced by the shortest element of Γ′/Γ_{f} as a subgroup of Γ/Γ_{f}.

7 Applications And Questions

In applying Theorem 0.2 to specific spaces there are two main obstacles: controlling the L^{2} spectral data and calculating the growth of short(Γ′) for various covers.

If Γ acts on X properly discontinuously and cocompactly by isometries, then X and Γ are quasi-isometric, hence, up to a constant factor we can measure distances in X rather than in Γ. For any γ in Γ this gives the estimate that the minimal length in Γ of a conjugate of γ is comparable to the minimal distance a point is moved in X, which is the same as the minimal length of a loop in X in the free homotopy class of γ. In this way we can interpret short(Γ′) as the minimal length of a nontrivial loop in X/Γ′. If X is non-postively curved then this loop is the shortest closed geodesic, and short(Γ′) is comparable to the injectivity radius of X′/Γ′.

Balls of diameter short(Γ′) are embedded, so their volume cannot exceed the volume of X/Γ′, which is proportional to the index of Γ′ in Γ. When Γ has exponential growth, one gets short(Γ′) ≤ C log[Γ : Γ′], and for Γ = ℤ^{n} one has short(Γ′) ≤ C[Γ : Γ′]^{\frac{1}{n}}. In the Euclidean case this bound is sharp for covers which have a fundamental domain a cube (for example Γ′ = kℤ^{n} for any k), and way off if the fundamental domain is long and thin. Similarly, in general, this estimate is sharp when the covers X/Γ′ unroll all directions equally. Precisely, we define:

Definition 7.1 A family {Γ_{i}} of finite index normal subgroups of Γ is uniform if there is a C > 0 so that

[Γ : Γ_{i}] ≤ Vol(B_{Γ}(C · short(Γ_{i})))

for all i. Here B_{Γ}(r) is the ball in Γ of radius r, in the word metric.

Say that a family of regular covers {X_{i}} is uniform if the corresponding family of groups is uniform.

As examples of uniform coverings, generalizing the kℤ^{n} ⊂ ℤ^{n}, we have:

Lemma 7.2 If Γ is arithmetic, the collection of congruence subgroups {Γ_{n}} is uniform.

PROOF. We show this when Γ has exponential growth. The proof for polynomial growth is no more difficult. View Γ as a group of integral matrices. Let m be the largest matrix entry appearing in all generators of Γ. A word in generators of Γ must be of length at least log_{m}(n) to have an entry of size n. Therefore, short(Γ_{n}) ≥ log_{m}(n). As the volume growth of balls in Γ is exponential, we can choose a C so that Vol(B_{Γ}(C · short(Γ_{n}))) is larger than any given polynomial in n, and in particular is larger than the index [Γ : Γ_{n}]. □

All of this discussion allows for geometric estimates on Betti numbers in some cases:

Example 7.3 Suppose X is an n-dimensional manifold of pinched negative curvature, −1 ≤ κ < −(\frac{n−2d}{n−1})^{2}. Then the universal cover X̃ of X has spectral gap for |q − n/2| ≥ d [5]. For such a q, one has C > 0 and M > 0 so that for all finite regular covers X′ of X:

b_{q}(X′) < C \frac{Vol(X′)}{e^{M Inj(X′)}}

For a uniform family of covers {X_{i}}, Theorem 0.3 applies and gives C > 0, β < 1 so that

b_{q}(X_{i}) < C Vol(X_{i})^{β}  (10)

for all i.

In particular, (10) holds for congruence covers of arithmetic hyperbolic manifolds, outside the middle dimension. Here, Sarnak and Xue [13] have conjectured an upper bound of the form (10), with β = 2q/(n − 1) + ε. For these manifolds, lower bounds of the same type are known. Xue [15] has shown:

b_{q}(X_{p}) > C_{ε} · Vol(X_{p})^{δ−ε}

for any ε > 0, and with an explicit δ.

For more general symmetric spaces, the spectrum near zero is well understood. The L^{2} Betti numbers vanish except possibly in the middle dimension. Near the middle dimension, the spectral density obeys a power law decay, and away from the middle range there is spectral gap. See [11] for the exact results. This allows us to give upper bounds for the growth rates of Betti numbers of locally symmetric manifolds, for example Theorem 0.4.

Adams and Sarnak [1] have precise results for similar questions, calculating multiplicities of representations instead of Betti numbers.

Question 7.4 Is there a C > 0 so that for any compact hyperbolic 3-manifold M,

b_{1}(M) ≤ C \frac{Vol(M)}{Inj(M)}?

More generally, for a compact locally symmetric space modeled on G/K, how big can the Betti numbers be in terms of volume and injectivity radius?

For more general spaces, computing the L^{2} spectral data is very hard. Even the question of whether all spaces have positive Novikov-Shubin invariants is open. If they do, the bounds for arbitrary spaces in Theorem 6.1 would get significantly better, changing from

\frac{[Γ : Γ′]}{log short(Γ′)}

to

\frac{[Γ : Γ′]}{short(Γ′)^{β}}  (11)

(for some β depending on X). These results imply that any space violating the bounds (11) must have vanishing Novikov-Shubin invariant, and so our results could be used to detect such a space.

Question 7.5 Is there a space X with b_{q}^{(2)}(X) = 0 and a sequence of finite normal covers X_{i} = X/Γ_{i} so that the q^{th} Betti numbers of X_{i} grow faster than:

\frac{[Γ : Γ_{i}]}{(short(Γ_{i}))^{β}} .

for all β > 0?

We know of no example with Betti numbers growing anywhere near that quickly:

Question 7.6 Is there a space X with b_{q}^{(2)}(X) = 0 and a sequence of finite normal covers X_{i} = X/Γ_{i} so that the Betti numbers b_{q}(X_{i}) grow faster than any multiple of

\frac{[Γ : Γ_{i}]}{short(Γ_{i})}?

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