MOD-p COHOMOLOGY GROWTH IN p-ADIC ANALYTIC TOWERS OF 3-MANIFOLDS

FRANK CALEGARI AND MATTHEW EMERTON

Dedicated to the memory of Fritz Grunewald

1. Introduction

Let M be a compact, orientable, connected 3-manifold with fundamental group Γ. Let p be a prime, let n ≥ 1 be a positive integer, and let ϕ : Γ → GL_{n}(Z_{p}) be a homomorphism. If we let G denote the closure in GL_{n}(Z_{p}) of the image of Γ, then G is a p-adic analytic group, admitting a normal exhaustive filtration

G_{n} := G ∩ ker(GL_{N}(Z_{p}) → GL_{N}(Z/p^{n}Z)).

This filtration gives rise to a corresponding filtration {Γ_{n}} of Γ via the normal subgroups Γ_{n} := ϕ^{−1}(G_{n}). This filtration may or may not be exhaustive (depending on whether or not ϕ is injective).

Associated to each of the finite index subgroups Γ_{n} of Γ is a finite connected cover M_{n} of M. The main concern of this paper is to estimate the growth of H_{1}(M_{n}, F_{p}) as a function of n, in terms of the geometry of M and the dimension of the group G.

The methods of this paper are very similar to those of [4, 5]. The point of this note, however, is to point out that no assumption on the arithmeticity (or even hyperbolicity) of M is required, and that ϕ need not be a congruence homomorphism. Indeed, our conclusions do not depend at all on the geometry of M, beyond the fact that it has dimension 3.

If H is any pro-p group, let δ(H) := dim_{F_{p}} H/Φ(H) where Φ(H) is the Frattini subgroup of H. If δ(H) is finite, it is equal to the minimal number of topological generators of H. Recall that the dimension d := dim(G) of a p-adic analytic group G is equal to δ(H) for any open uniform subgroup H of G. The sequence of normal subgroups G_{n} ⊂ G constructed above is an exhaustive sequence of subgroups, that is, ⋂ G_{n} = {1}. It follows that for any finite index open subgroup H ⊂ G, there is an inclusion G_{n} ⊂ H for sufficiently large n. In particular, taking H to be uniform, we deduce from Theorem 3.8 of [7] that δ(G_{n}) ≤ d for sufficiently large n.

The following result is the main theorem of this note.

Theorem 1.1. Let d = dim(G). Then either:

(1) dim H_{1}(M_{n}, F_{p}) = λ · p^{dn} + O(p^{(d−1)n}) for some rational constant λ ≠ 0,

(2) dim H_{1}(M_{n}, F_{p}) = λ · p^{(d−1)n} + O(p^{(d−2)n}) for some rational constant λ ≠ 0,

(3) d = 2, and dim H_{1}(M_{n}, F_{p}) = O(1).

(4) d = 3, the boundary of M is a (possibly empty) union of spheres, dim H_{1}(M_{n}, F_{p}) = δ(G_{n}), and dim H_{1}(M_{n}, F_{p}) ≤ 3 for sufficiently large n.

The first author was supported in part by the Sloan foundation and by NSF grants DMS-0701048 and DMS-0846285. The second author was supported in part by NSF grants DMS-0701315 and DMS-1002339.

If d > 3, then Theorem 1.1 implies that the rate of growth of mod-p homology comes in two possible flavours. Since [G : G_{n}] is of order p^{dn}, case (1) of Theorem 1.1 corresponds to “linear growth” of mod-p homology. The question of linear mod-p homology growth is considered in recent work of Lackenby [8, 9], where the existence of such growth in certain circumstances can be used to deduce that some finite index subgroup of Γ is “large” (i.e., admits a surjection onto a free group of rank ≥ 2). (We shall see in Example 5.1 that large subgroups give rise to examples where case (1) applies.) Case (2) of Theorem 1.1, however, corresponds to “sub-linear growth” of mod-p homology. Perhaps the most surprising aspect of our result is that whenever the growth of mod-p homology in a p-adic analytic tower is sub-linear (and dim(G) > 3) the rate of growth is still quite fast, and, moreover, is determined (up to a non-zero scalar) by the single invariant dim(G). A natural question to consider is whether one should expect case (1) or (2) to hold. We conjecture the following:

Conjecture 1.2. Suppose that M is a finite volume hyperbolic manifold and that ϕ is injective, or equivalently, that ⋂ Γ_{n} = {1}. Then case 1 of Theorem 1.1 does not occur.

Although this conjecture is motivated by questions in the theory of automorphic forms (see, for example, [5]), it reflects the general principle that fundamental groups of finite volume hyperbolic 3manifolds are a long way from being free, unlike the fundamental groups of surfaces. (For example, the fundamental group of any closed 3-manifold admits a balanced presentation, i.e., one with an equal number of generators and relations.)

In Section 5, we give examples illustrating our theorem. In particular, every possible case of Theorem 1.1 does occur.

2. Iwasawa Theory

We maintain the notation of the introduction. We define the completed homology groups as follows:

H̃_{i}(F_{p}) = lim_{←} H_{i}(M_{n}, F_{p}).

Example 2.1. Consider the case i = 0. Each H_{i}(M_{n}, F_{p}) is then canonically identified with F_{p}, and the transition maps in the projective limit are just the identity map. Hence H̃_{0}(F_{p}) = F_{p}.

Example 2.2. If M is not closed, then H_{3}(M_{n}, F_{p}) = 0 for each n, and so certainly H̃_{3}(F_{p}) = 0. On the other hand, suppose that M is closed, so that H_{3}(M_{n}, F_{p}) = F_{p} for each value of n, generated by the fundamental class [M_{n}]. If the dimension d of G is at least 1, then for any n ≥ 1, there is n′ > n such that the degree of the map M_{n′} → M_{n} is divisible by p, and hence induces the zero map on H_{3}. Thus, when d ≥ 1, we see that H̃_{3}(F_{p}) = 0 even when M is closed.

Each of the modules H̃_{i}(F_{p}) carries a continuous action of G, and thus is a Λ := F_{p}[[G]]-module. The modules H̃_{i}(F_{p}) are, in fact, finitely generated as Λ-modules. One way to see this, which also sheds light on the nature of the groups H̃_{i}(F_{p}), is as follows: Fix a triangulation of M, and pull this back to obtain a G/G_{n}-equivariant triangulation to M_{n}. We obtain a chain complex of free F_{p}[G/G_{n}]-modules which compute H_{i}(M_{n}, F_{p}) together with the action of G/G_{n}. The rank of these modules is bounded by the number of cells in the triangulation of M, and is hence bounded independently of n. The projective limit of these complexes defines a chain complex of finite rank free Λ = F_{p}[[G]]-modules

S_{•} : S_{3} → S_{2} → S_{1} → S_{0},

whose homology equals H̃_{i}(F_{p}).

In particular, H̃_{1}(F_{p}) can be thought of as an analogue of the Alexander invariant, where the group G plays the role of the covering group Z of a knot complement in the theory of the Alexander invariant, and the ring Λ plays the role of the ring Z[Z] = Z[t, t^{−1}] in that theory.

A basic fact underlying the viewpoint of the paper is that the category of finitely generated modules over Λ := F_{p}[[G]] has good properties whenever G is p-adic analytic; e.g., for any such G, the ring Λ is (left and right) Noetherian [10], and if G is furthermore sufficiently small, then Λ is Auslander regular [14]. For example, If G = Z_{p}^{N}, then Λ ≃ F_{p}[[T_{1}, …, T_{N}]] is a regular local ring of dimension N.

The key technical ingredient that we employ is the notion of codimension for finitely generated Λ-modules. A finitely generated Λ-module A is defined to have codimension c if Ext^{i}(A, Λ) = 0 for all i < c and is non-zero for i = c. A non-zero finitely generated module has codimension ≤ d; we define the codimension of the zero module to be ∞. The utility of this definition is the following result (see [1, § 5]):

Theorem 2.3. Suppose that A is a finitely generated Λ-module of codimension c, and as usual, write dim(G) = d. Then, letting {G_{n}} denote the filtration of G defined above, we have that

dim H_{0}(G_{n}, A) = dim(the space of G_{n}-coinvariants of A) = λ · p^{(d−c)n} + O(p^{(d−c−1)n})

for some non-zero rational number λ.

In order to connect this result to our main theorem, we note the following:

Lemma 2.4. There is an exact sequence:

H_{2}(G_{n}, F_{p}) → H_{0}(G_{n}, H̃_{1}(F_{p})) → H_{1}(M_{n}, F_{p}) → H_{1}(G_{n}, F_{p}) → 0.

Proof. This follows from a version of the Hochschild–Serre spectral sequence for completed homology (where we have taken into account the fact that H̃_{0} = F_{p}, by Example 2.1). □

Note that dim H_{1}(G_{n}, F_{p}) = δ(G_{n}) ≤ d = dim(G) for sufficiently large n, and that H_{2}(G_{n}, F_{p}) is bounded as a function of n (see [6]). The main theorem is thus automatically true for d ≤ 2. Moreover, when d ≥ 3, it suffices to prove either that H̃_{1}(F_{p}) has codimension ≤ 1, or else that the boundary of M is a union of spheres, that d = 3, and that H̃_{1}(F_{p}) = 0. (This equivalence uses the fact that if H̃_{1}(F_{p}) = 0, then H_{1}(M_{n}, F_{p}) = H_{1}(G_{n}, F_{p}) has dimension δ(G_{n}).)

3. The case of closed M

We shall assume in this section that ∂M = ∅. This case contains all the essential ideas of this paper and is unencumbered by the technical modifications required when M has boundary.

Lemma 3.1. There is a spectral sequence

Ext^{i}(H̃_{j}(F_{p}), Λ) ⇒ H̃_{3−i−j}(F_{p}).

Proof. Apply Hom(–, Λ) to the chain complex S_{•} described above. □

Remark. Note that Lemma 3.1 (and its proof) is essentially Poincaré duality “over Λ”, the only difference being that Λ (unlike a field) is not cohomologically trivial, so that one obtains a spectral sequence rather than a simple isomorphism between the Λ-dual of H̃_{i}(F_{p}) and H̃_{3−i}(F_{p}).

Let us now study the spectral sequence. Note that Ext^{i}(F_{p}, Λ) = 0 unless i = d, in which case it equals F_{p}. Thus the E_{2}-page of the spectral sequence has the following form:

Hom(H̃_{2}(F_{p}), Λ)  Ext^{1}(H̃_{2}(F_{p}), Λ)  …

Hom(H̃_{1}(F_{p}), Λ)  Ext^{1}(H̃_{1}(F_{p}), Λ)  …

0  0  …  F_{p}.

The (0, 0) term is stable, and hence, we recover the fact that H̃_{3}(F_{p}) = 0 when d ≥ 1 (which we already observed in Example 2.2 above).

As noted at the end of the preceding section, in order to prove our main theorem, we may assume that d ≥ 3 and that H̃_{1}(F_{p}) has codimension at least 2. We henceforth make these assumptions. From the definition of codimension, we find that Hom(H̃_{1}(F_{p}), Λ) = Ext^{1}(H̃_{1}(F_{p}), Λ) = 0. In particular (since also d > 1) we deduce from the spectral sequence that H̃_{2}(F_{p}) = 0. Then also Hom(H̃_{2}(F_{p}), Λ) = 0, which, together with the fact that Ext^{1}(H̃_{1}(F_{p}), Λ) = 0 and that d > 2, allows us to deduce from the spectral sequence that H̃_{1}(F_{p}) = 0. Thus, the E_{2}-page of the spectral sequence in fact contains only one non-zero term, namely the (d, 0) term F_{p}. We therefore conclude that H̃_{i}(F_{p}) = 0 unless i = 3 − d, and that H̃_{3−d}(F_{p}) = F_{p}. Since we know that H̃_{0}(F_{p}) = F_{p}, we deduce that d = 3. This completes the proof of Theorem 1.1 in the closed case.

4. The case when M has boundary

If M has boundary, then it is still of interest to consider the completed homology groups H̃_{i}(F_{p}). However, we also need to consider homology relative to the boundary, and the corresponding completed homology groups:

H̃_{i}^{BM}(F_{p}) = lim_{←} H_{i}(M_{n}, ∂M_{n}; F_{p}).

(Here the superscript “BM” stands for Borel–Moore homology; the reason for using this notation is that the relative homology H_{i}(M_{n}, ∂M_{n}; F_{p}) coincides with the Borel–Moore homology of the complement of the boundary H_{i}^{BM}(M_{n} \ ∂M_{n}, F_{p}).) Similarly, if we let ∂M denote the boundary of M, then we may define

H̃_{i}(∂, F_{p}) = lim_{←} H_{i}(∂M_{n}, F_{p}).

Lemma 4.1. There are spectral sequences:

(1) Ext^{i}(H̃_{j}^{BM}(F_{p}), Λ) ⇒ H̃_{3−i−j}(F_{p}),

(2) Ext^{i}(H̃_{j}(F_{p}), Λ) ⇒ H̃_{3−i−j}^{BM}(F_{p}),

(3) Ext^{i}(H̃_{j}(∂, F_{p}), Λ) ⇒ H̃_{2−i−j}(∂, F_{p}). Moreover, there is a long exact sequence:

… → H̃_{j}(∂, F_{p}) → H̃_{j}(F_{p}) → H̃_{j}^{BM}(F_{p}) → H̃_{j−1}(∂, F_{p}) → …

Proof. For the various spectral sequences we proceed as in the closed case. The long exact sequence is obtained as the projective limit of the usual long exact sequences for the pairs (M_{n}, ∂M_{n}). □

Suppose that n′ ≥ n. Although ∂M_{n′} is a finite cover of ∂M_{n}, typically neither is connected. We will take a moment to discuss the structure of their component sets in terms of the map ϕ, the group G, and so on.

If Σ is a component of ∂M_{n}, then composing the natural map π_{1}(Σ) → π_{1}(M_{n}) = Γ_{n} with the map ϕ_{| Γ_{n}} : Γ_{n} → G_{n} yields a map π_{1}(Σ) → G_{n}. We let H denote the closure of the image of this map. If n′ ≥ n, and if we let Σ_{n′} denote the preimage of Σ in ∂M_{n′}, then there is a bijection

π_{0}(Σ_{n′}) ≅ G_{n}/G_{n′}H.

We say that the component Σ of ∂M_{n} splits completely up the tower if each component of Σ_{n′} maps homeomorphically to Σ, for each n′ ≥ n, or equivalently, if Σ_{n′} has [G_{n} : G_{n′}] connected components, for each n′ ≥ n, or again equivalently (as one deduces from the preceding description of π_{0}(Σ_{n′})), if H = {1}.

Lemma 4.2. The following are equivalent:

(1) For some value of n, the boundary ∂M_{n} contains a component which splits completely up the tower;

(2) H̃_{0}(∂, F_{p}) has codimension 0 as a Λ-module.

Furthermore, either these equivalent conditions hold, or else H̃_{2}(∂, F_{p}) = 0.

Proof. Let ∂M = Σ^{1} ∐ ⋯ ∐ Σ^{m} be the decomposition of ∂M into a disjoint union of connected components. For each n, let Σ_{n}^{i} denote the preimage of Σ^{i} under the covering map M_{n} → M. If we define A^{i} := lim_{←} H^{0}(Σ_{n}^{i}, F_{p}), then H̃_{0}(∂, F_{p}) = A^{1} ⊕ ⋯ ⊕ A^{m}, and so H̃_{0}(∂, F_{p}) has codimension 0

if and only if at least one A^{i} does.

Now if we write H^{i} to denote the closure of the image of the composite

π_{1}(Σ^{i}) ⟶ π_{1}(M) = Γ ⟶^{ϕ} G,

then we see from the above discussion that π_{0}(Σ_{n}^{i}) = G/G_{n}H^{i}, and thus that

A^{i} = lim_{←} F_{p}[G/G_{n}H^{i}] = F_{p}[[G/H^{i}]]

(with the Λ-module structure being induced by the left action of G on G/H^{i}). From this description of A^{i}, one easily verifies that the codimension of A^{i} is equal to the dimension of H^{i}. Thus A^{i} has codimension 0 if and only if H^{i} has dimension 0, i.e. is finite. This in turn is equivalent to having H^{i} ∩ G_{n} = {1} for sufficiently large n, which is in turn equivalent to one (or equivalently, every) component of Σ_{n}^{i} being completely split up the tower. This establishes the equivalence of (1) and (2).

The final statement follows immediately from a consideration of the (0, 0)-term of the third spectral sequence of Lemma 4.1. We may also prove it more directly: since each ∂M_{n} is a disjoint union of closed surfaces, by arguing just as in the 3-manifold case considered in Example 2.2, we find that H̃_{2}(∂, F_{p}) = 0 unless, for some sufficiently large value of n, there is a component of ∂M_{n} which splits completely up the tower. □

We now turn to proving Theorem 1.1. Since the 2-sphere is simply connected, any 2-sphere in the boundary of M splits completely up the tower of M_{n}. Thus we may cap off all these 2-spheres with 3-balls; note that this doesn’t change H_{1}(M_{n}, F_{p}). Hence we may and do assume that every component of ∂M, and hence every component of ∂M_{n} for each value of n, has positive genus. As noted in the preceding section, the theorem is automatic if d ≤ 2, and so we may assume that d ≥ 3. We have already proved the theorem in the closed case, and thus we may also assume that M has a non-empty boundary (every component of which has positive genus). The theorem will follow if we prove that H̃_{1}(F_{p}) has codimension at most one. Thus we assume that H̃_{1}(F_{p}) has codimension at least 2, and argue by contradiction.

Since M, and so each M_{n}, is not closed, we certainly have that H̃_{3}(F_{p}) = 0, and an argument like that given in the closed case in Example 2.2 shows that also H̃_{3}^{BM}(F_{p}) = 0. (Alternatively, this follows from the second spectral sequence of Lemma 4.1.) Moreover, H̃_{0}^{BM}(F_{p}) = 0 (since each M_{n} is not closed) and H̃_{0}(F_{p}) = F_{p}.

We now collect various consequences of our assumptions:

(1) H̃_{1}^{BM}(F_{p}) = 0, as follows from the second spectral sequence of Lemma 4.1, the fact that d > 1, and the fact that H̃_{1}(F_{p}) has positive codimension.

(2) H̃_{2}(∂, F_{p}) ≅ H̃_{2}(F_{p}), as follows from the long exact sequence of Lemma 4.1 and the vanishing of H̃_{3}^{BM}(F_{p}) (noted above) and H̃_{2}^{BM}(F_{p}) (noted in the preceding point).

(3) H̃^{1}(∂, F_{p}) has positive codimension, as follows from the fact that it embeds into H̃^{1}(F_{p})

(since H̃_{2}^{BM}(F_{p}) vanishes, by (1)), which has positive codimension by assumption. We now deduce that ∂M_{n} has no completely split boundary component, for any value of n. Indeed, suppose that M_{n} did have such a component for some n. By assumption, the genus g of this component would be positive, and so we would find that H̃_{1}(∂, F_{p}) has codimension 0, contradicting point (3) above. Lemma 4.2 then implies that H̃_{2}(∂, F_{p}) = 0, and so point (2) above implies that H̃_{2}(F_{p}) = 0.

A consideration of the second spectral sequence of Lemma 4.1, taking into account our assumptions that d ≥ 3 and that H̃_{1}(F_{p}) has codimension ≥ 2, then shows that H̃_{1}^{BM} = 0. We have already observed that H̃_{i}^{BM} = 0 for i = 0, 2, 3, and so we conclude that the first spectral sequence of Lemma 4.1 is identically zero. Consequently, H̃_{i}(F_{p}) must vanish for every value of i. This contradicts the fact that H̃_{0}(F_{p}) = F_{p}. Thus in fact the codimension of H̃_{1}(F_{p}) is at most one, and the theorem is proved.

5. Examples

Example 5.1. Fix a p-adic Lie group G, let F be a free group of rank ≥ 2, and consider a map F → G with Zariski dense image. If F_{n} denotes the preimage of G_{n}, then F_{n} will be free of rank [G : G_{n}](rank(F) − 1) + 1. In particular, the F_{p}-homology of F_{n} will grow linearly in the degree.

Now suppose that M is a closed 3-manifold such that Γ = π_{1}(M) admits a surjective map Γ → F. If we define ϕ to be the composite ϕ : Γ → F → G, then we are in case (1) of Theorem 1.1. This construction applies more generally: we may take F to be any group admitting a faithful representation into a p-adic Lie group G such that the F_{p}-homology of any finite index subgroup F_{n} has rank at least c · [F : F_{n}] for some constant c independent of n. For example, F could be the fundamental group of a compact surface Σ_{g} of genus g ≥ 2. It seems a difficult problem to characterize such groups F. If M is hyperbolic, one might wonder whether this is the only way in which case (1) can occur.

Example 5.2. The prime 3 splits in Z[\sqrt{−2}], and so there is a surjection Z[\sqrt{−2}] → F_{3} × F_{3}, inducing a surjection SL_{2}(Z[\sqrt{−2}] → SL_{2}(F_{3}) × SL_{2}(F_{3}). If Γ denotes the kernel of this map, then Γ admits a map to SL_{2}(Z_{3}^{2}) ≅ SL_{2}(Z_{3})^{2} with Zariski dense pro-3 image (induced by the embedding Z[\sqrt{−2}] ↪ Z_{3}^{2} given by forming the 3-adic completion of Z[\sqrt{−2}]). The dimension of the target is d = 6 > 3. The computations of [12, §4.1] imply that, with respect to the corresponding pro-3 cover, case (2) of the main theorem occurs.

Example 5.3. Suppose that ∂M is the union of N tori. We obtain a map ϕ : Γ := π_{1}(M) → Z^{N} ↪ Z_{p}^{N} =: G. Let M̃ be the cover of M with covering group Z^{N} corresponding to the kernel of ϕ. The Z[Z^{N}]-module H^{1}(M̃, Z) is the Alexander invariant of M. There is a natural map Z[Z^{N}] → F_{p}[[Z_{p}^{N}]], and the completed homology H̃_{1}(F_{p}) is obtained from the Alexander invariant by base-change along this map:

H̃_{1}(F_{p}) := F_{p}[[Z_{p}^{N}] ⊗_{Z[Z^{N}]} H^{1}(M̃, Z).

Theorem 1.1 shows that H̃_{1}(F_{p}), and hence the Alexander invariant, can be trivial only if N ≤ 2.

If M is a knot complement (in which case N = 1), then in fact H̃_{1}(F_{p}) = 0, as we now show.

Lemma 5.4. If M is a knot complement in S^{3}, then H̃_{1}(F_{p}) = 0, and consequently H_{1}(M_{n}, F_{p}) = F_{p} for all n.

Proof. We give two proofs, one in the spirit of this note, and the other (which was explained to us by Barry Mazur) using the classical theory of the Alexander polynomial.

First proof. We have G = Z_{p}, and G_{n} = p^{n}Z_{p} for each n. Then Λ = F_{p}[[T]], and the exact sequence of Lemma 2.4 simplifies to the following short exact sequence:

0 → H̃_{1}(F_{p})/T^{p^{n}} H̃_{1}(F_{p}) → H_{1}(M_{n}, F_{p}) → F_{p} → 0.

Taking n = 1, and noting that H_{1}(M, F_{p}) = F_{p} (since M is a knot complement), we find that H̃_{1}(F_{p})/T H̃_{1}(F_{p}) = 0. Since H̃_{1}(F_{p}) is finitely generated over F_{p}[[T]], Nakayama’s lemma implies that H̃_{1}(F_{p}) = 0.

Second proof. Write M = S^{3} \ \overline{K}, where \overline{K} is a tubular neighbourhood of the knot K. For any integer e ≥ 1, let X_{e} denote the e-fold cover of S^{3} ramified along K. If Δ(t) denotes the Alexander polynomial of M (note that the variables t and T are related via t = T + 1), then #H_{1}(X_{e}, Z) = ∏_{ζ^{e}=1} Δ(ζ) if the right hand side is non-zero. (If the right hand side vanishes, then H_{1}(X_{e}, Z) has positive rank).

Recall that Δ(1) = 1. Thus, taking e = p^{n}, we find that #H_{1}(X_{p^{n}}, Z) ≡ 1 mod ζ − 1, where ζ is a primitive p^{n}th root of 1. Since ζ − 1 is a non-unit algebraic integer of norm p, while #H_{1}(X_{e}, Z) is an integer, we find that in fact #H_{1}(X_{p^{n}}, Z) ≡ 1 mod p. In particular, H_{1}(X_{p^{n}}, Z) is finite and p-torsion free, and so by the universal coefficient theorem, H_{1}(X_{p^{n}}, F_{p}) = 0. Now M_{n} is equal to the complement in X_{p^{n}} of a tubular neighbourhood of the preimage of \overline{K}, and thus H_{1}(M_{n}, F_{p}) = F_{p}, generated by the class of a meridian in the preimage of \overline{K}.

If n′ > n, then the map of meridians induced by the map M_{n′} → M_{n} has degree p^{n′−n}, and hence the map H_{1}(M_{n′}, F_{p}) → H_{1}(M_{n}, F_{p}) is zero. Thus we also conclude that H̃_{1}(F_{p}) = 0. □

Note that the argument in the second proof of the preceding lemma breaks down if e is divisible by more than one prime, since if ζ is a primitive eth root of unity for such a value of e, then ζ − 1 is a unit in the ring of algebraic integers. And indeed, for such e, the cover X_{e} can have non-zero 1st Betti number, or torsion in H^{1} of order that is not coprime to e. (For example, if K is the trefoil knot and e = 6 then H_{1}(X_{6}, Z) is infinite [13, p. 150].)

If M is the complement of the Hopf link, then N = 2, and it is known that the Alexander invariant, and hence H̃_{1}(F_{p}), vanishes [13, p. 190]. This provides an example in which case (3) of Theorem 1.1 occurs.

Example 5.5. In [3] and [2], a closed arithmetic manifold M is considered which has the property that the 3-adic completion G of its fundamental group Γ is analytic. This implies, for the associated map ϕ : Γ → G, that H̃_{1}(F_{p}) = 0, and hence that we are in case (4) of Theorem 1.1 (with p = 3). Another example is given (with p = 5) by a finite cover of the Weeks manifold (see [3, p. 321]).

Example 5.6. If M is hyperbolic, then Γ can be realized as a torsion free discrete subgroup of SL_{2}(C) and M ≃ Γ\H^{3}. If M has finite volume, then, by Mostow rigidity, we may assume (after possibly conjugating Γ) that Γ ⊂ SL_{2}(E) for some (minimal) number field E. Since Γ is finitely generated, there exists a finite set of primes S in E such that Γ ⊂ SL_{2}(O_{E,S}), where O_{E,S} denotes the ring of S-integers in E. From this description of Γ, it follows that Γ is residually finite, and, for all but finitely many primes 𝔭 ∈ O_{E}, admits an injective map ϕ_{𝔭} : Γ → SL_{2}(Ô_{E,𝔭}) (where Ô_{E,𝔭} denotes the 𝔭-adic completion of the ring of integers of E). This construction provides a natural source of homomorphisms ϕ of the type considered in this paper.

Note that these maps are of quite a different nature to those considered in Example 5.1 (when the latter exist), since the latter cannot be injective. (A finite volume hyperbolic 3-manifold group cannot be free.)

Example 5.7. Let M be any finite volume hyperbolic 3-manifold. Let ϕ be the direct sum of ϕ_{𝔭} for all primes 𝔭 above p. The Zariski closure of the image is a group G of dimension at least 6. It follows that M admits a sequence of covers with large mod-p homology growth. Moreover, we see that any hyperbolic 3-manifold M admits a finite cover M′ such that the fundamental group Γ′ of M′ admits a map to a pro-p analytic group G of dimension > 3. If the pro-p completion of Γ′ were analytic and isomorphic to G, then H̃_{1}(F_{p}) would vanish, contradicting Theorem 1.1. In particular, we see that if Γ is arithmetic, it cannot satisfy the congruence subgroup property. This was first proved for arithmetic lattices in SL_{2}(C) by Lubotzky [11].

Remark 5.8. The assumption that M be orientable has been made primarily for simplicity of exposition, and it should not be difficult to extend the proof of Theorem 1.1 to the non-orientable case. Indeed, any manifold is orientable mod 2, while if p is odd, then we may pass to the orientation double covers M̃_{n} of the M_{n}, each of which is equipped with an orientation reversing involution σ such that M_{n} = M̃_{n}/σ, and work with the σ-fixed part of the F_{p}-homology of M̃_{n} (which is canonically isomorphic to the F_{p}-homology of M_{n}). The reader will easily verify that the arguments of the paper remain valid after restricting to σ-invariants. (The key point is that H^{0}(M̃_{n}, F_{p}) is fixed by σ.)

Acknowledgments. The authors would like to thank Barry Mazur for explaining the second proof of Lemma 5.4 to them.

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