Polynomial Representation Growth and the Congruence Subgroup Problem

Alexander Lubotzky and Benjamin Martin

September 1, 2003

Dedicated to Andy Magid for his upcoming sixtieth birthday

Abstract

Let Γ be an S-arithmetic group in a semisimple group. We show that if Γ has the congruence subgroup property then the number of isomorphism classes of irreducible complex n-dimensional characters of Γ is polynomially bounded. In characteristic zero, the converse is also true. We conjecture that the converse also holds in positive characteristic, and we prove some partial results in this direction.

1 Introduction

Let Φ be a group. We denote by r_{n}(Φ) (respectively r̂_{n}(Φ)) the number of isomorphism classes of irreducible n-dimensional complex representations (respectively with finite image) of Φ. We call r_{n}(Φ) the representation growth function of Φ. In general r_{n}(Φ) may be infinite. There is no known characterisation of the groups Φ such that r_{n}(Φ) < ∞ for every n; such groups are called rigid in [BLMM02]. For finitely generated Φ, r̂_{n}(Φ) < ∞ for every n if and only if Φ has the property FAb (that is, Φ_{0}^{ab} := Φ_{0}/[Φ_{0}, Φ_{0}] is finite for every finite index subgroup Φ_{0} of Φ) [BLMM02, Proposition 2].

Definition 1.1 We say that Φ has polynomial representation growth (PRG) if the function r_{n}(Φ) is polynomially bounded: that is, if there exist c_{1}, c_{2} ∈ ℝ such that r_{n}(Φ) ≤ c_{1}n^{c_{2}} for every n. (In particular, this implies that every r_{n}(Φ) is finite.)

Let k be a global field and O its ring of integers. Write V for the set of (equivalence classes of) valuations of k, V_{f} for the set of finite (that is, nonarchimedean) valuations, and V_{∞} for the set of infinite (that is, archimedean) valuations. Fix a finite subset S of V containing V_{∞}. Let O_{S} := {x ∈ k | v(x) ≥ 0 ∀v ∈ V −S} be the ring of S-integers. Given v ∈ V, we write k_{v} for the completion of k with respect to v and O_{v} for the valuation ring of k_{v}.

Let G be a semisimple simply connected and connected algebraic group defined over k, with a fixed embedding G↪GL_{N}. Let Γ = G(O_{S}) := G(k) ∩ GL_{N}(O_{S}). We assume that T := {v ∈ V | G(k_{v}) is compact} is disjoint from S−V_{∞}, and that G(O_{S}) is infinite (equivalently, that ∏_{v∈S} G(k_{v}) is noncompact).

We say that Γ has the congruence subgroup property (CSP for short) if C := ker ({ \atop G(O_{S})} →^{π} G(Ô_{S})) is finite; here { \atop G(O_{S})} is the profinite completion

of G(O_{S}) and G(Ô_{S}) = ∏_{v∉S} G(O_{v}) is the congruence completion of G(O_{S}), where G(O_{v}) := G(k_{v}) ∩ GL_{N}(O_{v}). The map π is surjective — that is, Γ is dense in G(Ô_{S}) — by the Strong Approximation Theorem (cf. [PR94, Theorem 7.12] and [Pra77]).

The main goal of this paper is to prove the following theorem.

Theorem 1.2 Let Γ = G(O_{S}) be as above, and assume that if char k = 2 then G contains no factors of type A_{1} or of type C_{m} for any m. If Γ has the congruence subgroup property then Γ has polynomial representation growth.

We conjecture that the converse holds in general, but we can prove this completely (and indeed, in a slightly stronger form) only when char k = 0 (Theorem 1.3). As in [Lub95] and [PR93], we assume for Theorem 1.3 that the Platonov-Margulis Conjecture holds: that is, that G(k) has the standard description of normal subgroups. This is known to hold in almost all cases (see [PR94, Chapter 9], [Seg99]).

Theorem 1.3 Assume that char k = 0. If Γ = G(O_{S}) is as above and r̂_{n}(Γ) is polynomially bounded then Γ has the congruence subgroup property.

A striking feature of the proof is that Theorem 1.2 (or rather, the closely related result Proposition 5.1) is needed to prove Theorem 1.3.

The characterisation of the congruence subgroup property by means of polynomial representation growth joins several previous characterisations of the CSP by purely group-theoretic properties: for example, that the profinite completion Γ̂ is boundedly generated or that log s_{n}(Γ) ≤ o((log n)^{2}) [Lub95], [PR93], where s_{n}(Γ) denotes the number of index n subgroups of Γ. Unfortunately, all of these characterisations so far are true only in characteristic zero and are false in positive characteristic. We believe that the current characterisation is valid in positive characteristic as well, but we were not able to prove this. (The proof of Theorem 1.3 does, however, go a long way in characteristic p.)

The paper is organised as follows. In Section 2 we deal with some preliminary results. In Section 3 we study representations with infinite image. In Section 4 we study representations with finite image; we finish the section with the proof of Theorem 1.2. In Section 5 we prove some consequences of the results in Section 4; these concern estimates for the image size (rather than the number) of n-dimensional representations. Not only do these results have independent interest, but also most of them are needed in Section 6, where Theorem 1.3 is proved.

We are grateful to Nikolay Nikolov and Andrei Jaikin-Zapirain for helpful and informative discussions.

This research was supported by the Israel Science Foundation and the United States Israel Binational Science Foundation. The second author was supported by a Golda Meir Post-Doctoral Fellowship at the Hebrew University of Jerusalem.

2 Preliminaries

All logarithms are to base 2 unless otherwise indicated. We write [x] for the largest integer no greater than x.

Given a group Φ, we write Rep_{n}(Φ) (respectively Irr_{n}(Φ)) for the set of isomorphism classes of n-dimensional complex representations (respectively irreducible representations) of Φ. All representations are complex unless otherwise indicated. We say that a representation is finite if its image is finite, and infinite otherwise.

When we consider subgroups of a profinite group H, we will always mean closed subgroups. We will consider only continuous representations of H; this implies that the kernel of a representation is open and the image is finite, and so r_{n}(H) = r̂_{n}(H). It also implies that if Φ is a discrete group then r̂_{n}(Φ) = r_{n}(Φ̂) = r̂_{n}(Φ̂), where Φ̂ is the profinite completion of Φ.

Definition 2.1 We define R_{n}(Φ) = ∑_{m=1}^{n} r_{m}(Φ).

Note that r_{n}(Φ) is polynomially bounded if and only if R_{n}(Φ) is. On the other hand, even if r_{n}(Φ) = |Irr_{n}(Φ)| is polynomially bounded, in general |Rep_{n}(Φ)| is not.

The following result relates the representation growth of a group to that of a finite index subgroup.

Lemma 2.2 Let Φ′ be a finite index subgroup of Φ. Then for all n ∈ ℕ, we have

R_{n}(Φ′) ≤ [Φ: Φ′]R_{n[Φ:Φ′]}(Φ)

and

R_{n}(Φ) ≤ [Φ: Φ′]R_{n}(Φ′).

Proof For each m ∈ {1, …, n} and each τ ∈ Irr_{m}(Φ′), choose an irreducible component ψ(τ) of the induced representation Ind_{Φ′}^{Φ}τ. This gives a map ψ from ⋃_{m=1}^{n} Irr_{m}(Φ′) to ⋃_{m=1}^{n[ϕ:ϕ′]} Irr_{m}(Φ). Let τ ∈ Irr_{m}(Φ′) and let {τ_{λ} | λ ∈ Λ} be the elements of ψ^{−1}(ψ(τ)). Without loss of generality we assume that dim τ is minimal among the dim τ_{λ}. By Frobenius reciprocity, each τ_{λ} is an irreducible component of ψ(τ)|_{Φ′}; hence, dim ψ(τ) ≥ |Λ|dim τ. But dim ψ(τ) ≤ [Φ: Φ′]dim τ, which implies that |Λ| ≤ [Φ: Φ′]. The first inequality follows.

The proof of the second inequality is similar, and we leave it to the reader.

Corollary 2.3 If Φ′ is a finite index subgroup of Φ then Φ has PRG if and only if Φ′ has PRG.

Definition 2.4 We define A_{n}(Φ) = max{[Φ′: [Φ′, Φ′]] | Φ′ ≤ Φ, [Φ: Φ′] ≤ n}.

We will often use the following important fact.

Lemma 2.5 Every irreducible representation σ of a pronilpotent group H is induced from a one-dimensional representation of some subgroup of index dim σ.

Proof Since σ is continuous by assumption, it factors through some finite nilpotent quotient P of H. Finite nilpotent groups are monomial [CR81, Theorem 11.3], so we have σ = Ind_{Q}^{P}χ for some Q ≤ P with [P: Q] = dim σ and some one-dimensional representation χ of Q. Setting M equal to the preimage of Q in H and regarding χ as a representation of M, it is easily checked that σ = Ind_{M}^{H}χ.

This fact allows us to compare the growth rates of A_{n}(Φ) and R_{n}(Φ).

Lemma 2.6 Let H be a profinite group.

(a) A_{n}(H) ≤ nR_{n}(H). In particular, if A_{n}(H) is not polynomially bounded then H does not have PRG.

(b) If H is pronilpotent then r_{n}(H) ≤ A_{n}(H)s_{n}(H).

(c) If H is compact p-adic analytic then H has PRG if and only if A_{n}(H) is polynomially bounded.

Proof If m ≤ n and M is an index m subgroup of H with abelianisation of size A_{n}(H) then M admits A_{n}(H) one-dimensional representations, so (a) follows from the first inequality of Lemma 2.2. If H is pronilpotent then every irreducible n-dimensional representation is induced from a one-dimensional representation (Lemma 2.5), so r_{n}(H) is bounded by the number of pairs (M, χ), where M ≤ H, [H: M] = n and χ is a one-dimensional representation of H. Part (b) now follows immediately.

Now suppose that H is compact p-adic analytic. If H′ is an open subgroup of H then H has PRG if and only if H′ does (Corollary 2.3), and it is easy to see that A_{n}(H) is polynomially bounded if and only if A_{n}(H′) is. Therefore we can assume that H is a p-adic analytic pro-p group. As s_{n}(H) is polynomially bounded [DdSMS99, Theorem 3.19], (c) follows from (a) and (b).

In fact, for compact p-adic analytic groups we can say something stronger.

Proposition 2.7 Let H be a p-adic analytic group, and let

K_{n}(H) = ⋂{ker ρ | ρ: H → GL_{n}(ℂ) is a representation}.

Suppose that the Lie algebra of H is perfect (we recall the definition of the Lie algebra below). Then [H: K_{n}(H)] is polynomially bounded.

Proof We need to recall some material on uniform pro-p groups and formal group laws; see [DdSMS99, Chapter 8 and Chapter 13] for details. Clearly there is no harm in passing to an open subgroup of H. We assume, therefore, that H is an appropriate open subgroup of a uniform pro-p group, admitting an analytic isomorphism ψ: (pℤ_{p})^{d} → H, where d = dim H, such that the group law with respect to these co-ordinates is given by x.y = F(x, y), where F(x, y) is a d-tuple of power series each with coefficients in ℤ_{p}. We have

F(x, y) = x + y + B(x, y) + O′(3),

where B(x, y) is a bilinear form on (pℤ_{p})^{d} and O′(3) denotes a power series such that each term has total degree at least three and degree at least one in each of the variables. The commutator of a pair of elements in H is given by

[x, y] = B(x, y) − B(y, x) + O′(3).

The map (x, y) ↦ B(x, y) − B(x, y) gives (pℤ_{p})^{d} the structure of a Lie algebra over ℤ_{p}: call this Lie algebra L. The Lie algebra of H is then given by L(ℚ_{p}) := L ⊗_{ℤ_{p}} ℚ_{p} (this is equivalent to the usual definition of the Lie algebra of a compact p-adic analytic group; cf. [DdSMS99, Exercise 9.13]).

We have a filtration H = H_{1} ⊃ H_{2} ⊃ ⋯ of H by normal subgroups, where H_{n} = ψ(p^{n}ℤ_{p}^{d}). Then H_{n+1} = H^{p^{n}} (see [DdSMS99, Theorems 3.6 and 8.3.1]), where H^{p^{n}} denotes the closure of the subgroup generated by the p^{n}-powers in H. It follows that any index p^{n} subgroup of H contains H_{n+1}.

Suppose that L(ℚ_{p}) is perfect. Then there exists r ∈ ℕ such that p^{r}L ⊂ [L, L]. Thus for any n ∈ ℕ, p^{2n+r}L ⊂ [p^{n}L, p^{n}L]. We claim that H_{2n+r} ≤ [H_{n}, H_{n}]H_{3n} for every n ∈ ℕ. To see this, observe that any z ∈ (p^{2n+r}ℤ_{p})^{d} can be written as a sum of Lie commutators [x_{1}, y_{1}] + ⋯ + [x_{s}, y_{s}] for some s, where the x_{i}, y_{i} ∈ (p^{n}ℤ_{p})^{d}. Using the formula for group commutators above, we see that the product of group commutators [x_{1}, y_{1}] ⋯ [x_{s}, y_{s}] is equal to z + (terms in (p^{3n}ℤ_{p})^{d}), and the claim follows. The claim implies that [H_{n}, H_{n}] ⊃ H_{3n} if n ≥ r + 2.

Let ρ be an irreducible representation of H of dimension at most p^{n}, where n ≥ r +2. Then ρ is induced from a one-dimensional representation of a subgroup M of index at most p^{n} (Lemma 2.5). Since M ⊃ H_{n+1}, [M, M] contains H_{3n+3}, so ker ρ ⊃ H_{3n+3}. The index of H_{3n+3} in H is p^{d(3n+2)}, which gives the required bound.

Lemma 2.8 If Φ_{1}, Φ_{2} are groups with PRG then Φ_{1} × Φ_{2} has PRG.

Proof It is well known that any irreducible n-dimensional representation ρ of the product Φ_{1} ×Φ_{2} is isomorphic to a representation of the form ρ = ρ_{1} ⊗ρ_{2}, where ρ_{i} is an irreducible n_{i}-dimensional representation of Φ_{i} for i = 1, 2, and n_{1}n_{2} = n. If Φ_{1}, Φ_{2} have PRG, with r_{n}(Φ_{i}) ≤ c_{i}n^{d_{i}}, say, then

r_{n}(Φ_{1} × Φ_{2}) ≤ ∑_{n_{1}n_{2}=n} r_{n_{1}}(Φ_{1})r_{n_{2}}(Φ_{2}) ≤ ∑_{n_{1}n_{2}=n} c_{1}n_{1}^{d_{1}}c_{2}n_{2}^{d_{2}} ≤ c_{1}c_{2}n^{d_{1}+d_{2}+1},

as required.

Lemma 2.8 implies that in order to prove Theorem 1.2, it suffices to consider simple groups G. Moreover, by changing the field we can further assume that G is absolutely simple. We therefore assume throughout this paper that G is absolutely simple. The same remark applies also to Theorem 1.3, as well as to the other theorems we prove (Theorems 4.2 and 4.3).

3 Infinite Representations

Assume for now that k is a number field. Let Γ = G(O_{S}) be as in Theorem 1.2, and assume that Γ has the congruence subgroup property. Let H = Res_{ℚ}^{k}(G), where Res_{ℚ}^{k} denotes restriction of scalars. Then H is a connected simply connected semisimple algebraic group defined over ℚ, and we have an embedding Γ↪H(ℚ)↪H(ℂ). A representation of a subgroup Δ of H(ℂ) is said to be algebraic if it can be extended to an algebraic representation of H(ℂ).

Lemma 3.1 There is a finite index subgroup Δ of Γ such that every irreducible finite-dimensional representation is isomorphic to a representation of the form ρ = ρ_{1} ⊗ ρ_{2}, where ρ_{1} is finite and ρ_{2} is algebraic.

Proof Let A = A(Γ) be the proalgebraic completion of Γ: that is, A(Γ) is a complex proalgebraic group with an embedding i: Γ↪A(Γ) characterised by the fact that every finite-dimensional complex representation ρ of Γ has a unique extension to an algebraic representation ρ̃ of A(Γ) such that ρ̃∘ i = ρ (cf. [LM85], [BLMM02]). Let A^{0} be the identity component of A, so that we have a short exact sequence

1 → A^{0} → A →^{π} Γ̂ → 1.

Then A has a profinite subgroup P such that A^{0}P = A [BLMM02, Corollary 6]. As Γ satisfies the congruence subgroup property, it has superrigidity: this means that every representation of Γ can be extended on a finite index subgroup to an algebraic representation of H(ℂ) [Rag76, Theorem 7.2]. This implies that A^{0} = H(ℂ). In particular, A^{0} is a finite-dimensional algebraic group, hence its intersection Z with the profinite group P is finite. Let P_{1} be an open subgroup of P with P_{1} ∩ Z = 1, so that P_{1} ⋉ A^{0} has finite index in A. Let C be the centraliser C_{P_{1}}(A^{0}). Since Aut A^{0} = Aut H(ℂ) is a finite-dimensional algebraic group, C is open in P_{1} and so D = C × A^{0} is a finite index open subgroup of A. It is therefore equal to A(Δ) for some finite index subgroup Δ of Γ (in fact, Δ = i^{−1}(D ∩ i(Γ))). As A(Δ) = C × A^{0} = Δ̂ × A^{0}, every irreducible representation of Δ is a tensor product of a finite representation and an algebraic representation, as claimed.

We postpone the treatment of the finite representations to the next section. Here we show that the number of algebraic representations is polynomially bounded.

Proposition 3.2 Let H be a semisimple linear algebraic group over ℂ. Then the number of isomorphism classes of irreducible n-dimensional algebraic representations of H is polynomially bounded.

Proof Below we use standard results on the structure theory and representation theory of H; these may be found in [Hum72]. Fix a maximal torus T of H, let Λ be the group of characters of T and let R ⊂ Λ be the set of roots of H with respect to T. Let (−, −) be the canonical Weyl-group-invariant bilinear form on Λ⊗_{ℤ} ℝ. Choose a decomposition R = R^{+} ∪R^{−} of R into sets of positive and negative roots. We write Λ^{+} = {λ ∈ Λ | (λ, α) ≥ 0 ∀α ∈ R^{+}}.

We recall briefly the classification of finite-dimensional irreducible algebraic representations of H (that is, of irreducible rational H-modules). A nonzero rational H-module M can be written as a direct sum of nonzero T - modules M_{λ} for certain λ ∈ Λ; the λ that appear are called the weights of M. If M is irreducible then there is a unique weight λ_{max}, which belongs to Λ^{+}, such that λ_{max} is the maximum weight with respect to a natural ordering on the set of weights. For any λ ∈ Λ^{+} there is an irreducible rational H-module V (λ) with highest weight λ, and every irreducible M is isomorphic to some V (λ).

By [Hum72, 24.3 Corollary], we have

dim V (λ) = \frac{∏_{α∈R^{+}}(λ + δ, α)}{∏_{α∈R^{+}}(δ, α)},

where δ is half of the sum of the positive roots; moreover, the denominator is independent of λ, and each |(λ + δ, α)| is bounded below by (α, α)/2.

Since H is semisimple, the positive roots span Λ ⊗_{ℤ} ℝ, so we can define a norm || · || on Λ ⊗_{ℤ} ℝ by ||λ|| = max_{α∈R^{+}}|(λ, α)|. For ||λ|| sufficiently large we have max\{|(λ + δ, α)| | α ∈ R^{+}\} ≥ \frac{1}{2}||λ||. The number of weights in the || · ||-ball B(0, n) in Λ ⊗_{ℤ} ℝ is polynomially bounded, so our result follows.

We turn now to the case of positive characteristic. Recall that Γ is always a lattice (that is, a discrete subgroup of finite covolume) in H = ∏_{v∈S} G(k_{v}) [Mar91, Chapter 1, Section 3.2]. If rank(H) := ∑_{v∈S} rank_{k_{v}}G(k_{v}) is one then Γ does not have the CSP [Lub91, Proposition B and Proposition D]. (Indeed, in this case Γ has a finite index subgroup with nonabelian free quotient, so r_{n}(Γ) = ∞ for infinitely many n.) On the other hand, if rank(H) ≥ 2 then by Margulis [Mar91, Chapter VIII, (3.8) Theorem] (see also [Ven88, Theorem 4.10]), Γ has superrigidity and has no infinite complex representations.

Thus in all cases we are reduced to the study of r̂_{n}(Γ), the growth function of the finite representations. This will be the topic of the next section.

4 Finite Congruence Representations

Let k, G, Γ, etc., be as in the Introduction; we assume now that G is absolutely simple. For v ∈ V_{f}, let 𝔪_{v} be the maximal ideal of O_{v}. We write 𝔽_{v} = O_{v}/𝔪_{v} for the residue field, q_{v} = |𝔽_{v}|, p_{v} = char 𝔽_{v} and e_{v} = log_{p_{v}}q_{v}. In general we denote the field with q elements by 𝔽_{q} (so that 𝔽_{v} = 𝔽_{q_{v}}). We write N_{n}^{(v)} for G(O_{v}) ∩ ker (GL_{N}(O_{v}) → GL_{N}(O_{v}/𝔪_{v}^{n})), the nth principal congruence subgroup of G(O_{v}). Define G(𝔽_{v}) to be the image of G(O_{v}) in GL_{N}(𝔽_{v}). Note that |G(𝔽_{v})| ≤ q_{v}^{N^{2}}.

Assumption 4.1 We will always make the following assumption: if char k = 2 then G is not of type A_{1} or of type C_{m} for any m. This ensures that the Lie algebra of G is perfect.

If Γ has the CSP then C := ker ({ \atop G(O_{S})} →^{π} G(Ô_{S})) is finite. Replacing Γ

by a finite index subgroup Γ_{0} such that C ∩ \overline{Γ_{0}} is trivial, we can assume that C is trivial. By Corollary 2.3, the finite representations of Γ_{0} have the same growth as the representations of the congruence completion G(Ô_{S}). The main goal of this section is to show that r_{n}(G(Ô_{S})) grows polynomially. We do not need to assume that Γ has the CSP to prove this: instead we prove that for any arithmetic group, the number of congruence representations is polynomially bounded (where a congruence representation is a representation the kernel of which is a congruence subgroup), since these are precisely the representations that extend to representations of G(Ô_{S}).

Theorem 4.2 Let k, O, S and G be as above. Then r_{n}(G(Ô_{S})) is polynomially bounded.

We recall some properties of the graded Lie algebra associated to the filtration N_{1}^{(v)} ≥ N_{2}^{(v)} ≥ … of N_{1}^{(v)} (compare [LS03, Section 3.5]). Setting L_{n}^{(v)} = N_{n}^{(v)}/N_{n+1}^{(v)}, we obtain a graded Lie algebra L^{(v)} = ∑_{n=1}^{∞} L_{n}^{(v)} over 𝔽_{v}, where addition is given by

xN_{n+1}^{(v)} + yN_{n+1}^{(v)} = xyN_{n+1}^{(v)}

and the Lie bracket by

[xN_{m+1}^{(v)}, yN_{n+1}^{(v)}] = [x, y]N_{n+m+1}^{(v)}.

Each quotient N_{1}^{(v)}/N_{n}^{(v)} is a finite p_{v}-group, and N_{1}^{(v)} is the inverse limit of the N_{1}^{(v)}/N_{n}^{(v)}.

Let H be an open subgroup of N_{1}^{(v)}. Setting L_{n}^{(v)}(H) = (H∩N_{n}^{(v)})N_{n+1}^{(v)}/N_{n+1}^{(v)}, we obtain a graded 𝔽_{p_{v}}-subalgebra L^{(v)}(H) = ∑_{n=1}^{∞} L_{n}^{(v)}(H) of L^{(v)} (this is not an 𝔽_{v}-subalgebra in general). We have

[N_{1}^{(v)}: H] = [L^{(v)}: L^{(v)}(H)]  (1)

(compare [LS94, Lemma 2.13]).

Let P_{1} be the subset of V_{f} consisting of all valuations v such that:

(a) G(𝔽_{v}) is perfect and is a central extension of a finite simple group of Lie type H(q_{v}), where H is of the Lie type of G (either twisted or untwisted);

(b) for every n ∈ ℕ, |N_{n}^{(v)}/N_{n+1}^{(v)}| = q_{v}^{dim G};

(c) for every m, n ∈ ℕ, [N_{m}^{(v)}, N_{n}^{(v)}] = N_{m+n}^{(v)};

(d) G(O_{v}) is perfect.

It follows from the discussion before the proof of Theorem 2.1 of [LL] that conditions (a), (b) and (c) hold for almost all v ∈ V_{f}. Our standing hypothesis Assumption 4.1 implies that the Lie algebra of G is perfect, which implies that [G(O_{v})/N_{2}^{(v)}, N_{1}^{(v)}/N_{2}^{(v)}] = N_{1}^{(v)}/N_{2}^{(v)} for almost all v ∈ V_{f}. This together with conditions (a) and (c) gives condition (d). Therefore P_{2} := V_{f}−P_{1} is finite. Note that (c) implies that [L_{m}^{(v)}, L_{n}^{(v)}] = L_{m+n}^{(v)} for every m, n ∈ ℕ.

To prove Theorem 4.2 we need to prove, in particular, that r_{n}(G(O_{v})) ≤ c_{1}n^{c_{2}} for two constants c_{1}, c_{2} independent of v. In fact, we will prove a stronger result which is of independent interest and will be needed later in Section 6 for the converse theorem.

Theorem 4.3 There exists c = c(G, k) such that for every v ∈ V_{f} and every n ∈ ℕ, [G(O_{v}): K_{n}(G(O_{v}))] ≤ cn^{c}, where for a profinite group H, K_{n}(H) is defined as in Proposition 2.7, namely

K_{n}(H) = ⋂{ker ρ | ρ: H → GL_{n}(ℂ) is a representation}.

Here are two immediate corollaries. Note that the second applies to all representations, not just irreducible ones.

Corollary 4.4 r_{n}(G(O_{v})) ≤ cn^{c} for every v ∈ V_{f}, n ∈ ℕ.

Corollary 4.5 |ρ(G(O_{v}))| ≤ cn^{c} for every v ∈ V_{f}, n ∈ ℕ, ρ ∈ Rep_{n}(G(O_{v})).

To prove Theorem 4.3 we need:

Proposition 4.6 There exists δ > 0 such that for every v ∈ P_{1}, every nontrivial irreducible representation of G(O_{v}) is of dimension at least q_{v}^{δ}.

Proposition 4.6, which we will prove below, is needed in the proof of Theorem 4.3, as well as for proving Theorem 4.2.

Lemma 4.7 (a) For n ∈ ℕ, define f(n) to be the number of tuples (n_{1}, …, n_{t}) of integers such that n = n_{1} ⋯ n_{t} and each n_{i} > 1. Then there exists μ > 0 such that f(n) ≤ n^{μ} for every n ∈ ℕ.

(b) There exists d > 0, depending only on k, such that the number of v ∈ V_{f} with q_{v} ≤ n is bounded by dn, and hence also by n^{b} for some b > 0.

Proof (a) By a result of Kalmar (see [Erd41, Theorem (1)]), we have

∑_{m=1}^{n} f(m) = Dn^{β}(1 + o(1)),

where D is constant and β is the unique positive solution of ζ(β) = 2 (ζ being the Riemann zeta function). The result follows.

(b) This is a consequence of the Prime Number Theorem (see [Ros02, Theorem 5.12] for the function field case).

Let us now show how Corollary 4.4, Proposition 4.6 and Lemma 4.7 imply Theorem 4.2. We write G(Ô_{S}) as A × B, where A = ∏_{v∈P_{2}−S} G(O_{v}) and B = ∏_{v∈P_{1}−S} G(O_{v}). Now A is a finite product of groups, each of which has PRG (Corollary 4.4), so A has PRG by Lemma 2.8. The same lemma implies that G(Ô_{S}) has PRG, once we have shown that B has PRG.

To prove this, let ρ ∈ Irr_{n}(B) be nontrivial. Then for some t and some v_{1}, …, v_{t} ∈ P_{1}−S, we have nontrivial ρ_{i} ∈ Irr_{n_{i}}(G(O_{v_{i}})) such that ρ ≅ ρ_{1} ⊗ ⋯ ⊗ ρ_{t}, where the n_{i} are positive integers such that n = n_{1} ⋯ n_{t}. Each n_{i} is greater than 1 by condition (d) in the definition of P_{1}. The number of possibilities for (n_{1}, …, n_{t}) is bounded by n^{μ}, by Lemma 4.7 (a). Given such a t-tuple (n_{1}, …, n_{t}), then for any fixed i between 1 and t, the number of valuations v_{i} ∈ P_{1} that are possible — that is, such that G(O_{v_{i}}) admits a nontrivial n_{i}-dimensional representation — is bounded by n_{i}^{b/δ}; this follows from Proposition 4.6 and Lemma 4.7 (b). Thus the number of choices for (v_{1}, …, v_{t}) is bounded by ∏_{i=1}^{t} n_{i}^{b/δ} = n^{b/δ}. Now given (n_{1}, …, n_{t}) and (v_{1}, …, v_{t}), for every i = 1, …, t, G(O_{v_{i}}) has at most cn_{i}^{c} irreducible representations of dimension n_{i} (Corollary 4.4). Hence B has at most n^{μ}n^{b/δ}c^{log_{2}n}n^{c} irreducible n-dimensional representations. Thus B has PRG and Theorem 4.2 is proved (modulo Theorem 4.3 and Proposition 4.6).

We turn now to the proofs of Theorem 4.3 and Proposition 4.6. First we need a result that deals with the finitely many bad primes.

Lemma 4.8 For every v ∈ V_{f}, there exists c > 0 such that for every n ∈ ℕ, we have

[G(O_{v}): K_{n}(G(O_{v}))] ≤ cn^{c}.  (2)

Proof As in the proof of Proposition 2.7, we are free to pass to an open subgroup of G(O_{v}). Since G(O_{v}) is an O_{v}-analytic group, it has an open subgroup H which is O_{v}-standard [Ser92, Part II, Chapter IV, Section 8, Theorem]. It can be shown that the Lie algebra L(k_{v}) over k_{v} associated to H (see [DdSMS99, Section 13.3]) is isomorphic to the Lie algebra of the algebraic group G(k_{v}), which is perfect by Assumption 4.1. If char k = 0 then H can be regarded as a ℤ_{p_{v}}-standard group; the Lie algebra L(ℚ_{p_{v}}) is just L(k_{v}) regarded as a Lie algebra over ℚ_{p_{v}}, so it is also perfect. Similarly, if char k = p > 0 then H is an 𝔽_{p}[[t]]-standard group with perfect Lie algebra. The result now follows from Proposition 2.7 in characteristic zero, and from the proof of Theorem 6 of [JZ03] in characteristic p.

We need more for the proof of Theorem 4.3: we need to prove the existence of a c in Eqn. (2) that is independent of v, and for this Proposition 4.6 is also required.

Interestingly enough, the proofs of both Theorem 4.3 and Proposition 4.6 need the following lemma.

Lemma 4.9 Let p be a prime and let q = p^{e}, where e ∈ ℕ. Let V, W, X be finite-dimensional vector spaces over 𝔽_{q}, and let T: V × W → X be an 𝔽_{q}-bilinear map such that T (V × W) spans X over 𝔽_{q}. Suppose that A, B are 𝔽_{p}-subspaces of V, W respectively such that

[V: A][W: B] < q.  (3)

Then T (A × B) spans X over 𝔽_{p}.

Proof First we prove this in the special case that V = W = X = 𝔽_{q} and T is multiplication. Let f: 𝔽_{q} → 𝔽_{p} be any nonzero 𝔽_{p}-linear function. Define an 𝔽_{p}-bilinear form Q_{f}: 𝔽_{q} × 𝔽_{q} → 𝔽_{p} by Q_{f}(x, y) = f(xy). Then Q_{f} is nondegenerate: for there exists z ∈ 𝔽_{q} such that f(z) ≠ 0, and for any 0 ≠ x ∈ 𝔽_{q} we have Q_{f}(x, x^{−1}z) = f(z) ≠ 0. This implies that the orthogonal complement A^{⊥} = {x ∈ 𝔽_{q} | Q_{f}(x, a) = 0 for all a ∈ A} has 𝔽_{p}- dimension equal to the 𝔽_{p}-codimension of A. By Eqn. (3), the sum of the 𝔽_{p}-dimensions of A and B is greater than e, so B ⊄ A^{⊥}, whence f does not vanish on T (A × B). But f is arbitrary, so the 𝔽_{p}-span of T (A × B) must be the whole of 𝔽_{q}.

Now consider the general case. Let Y be the 𝔽_{p}-span of T (A × B). Eqn.

(3) implies that A, B contain 𝔽_{q}-bases for V, W respectively, so Y spans X over 𝔽_{q}. To complete the proof, we therefore need only show that Y is invariant under multiplication by 𝔽_{q}. So let a ∈ A, b ∈ B, w ∈ 𝔽_{q}. We show that wT (a, b) ∈ Y.

Set A′ = {x ∈ 𝔽_{q} | xa ∈ A}, B′ = {y ∈ 𝔽_{q} | yb ∈ B}, regarded as 𝔽_{p}- subspaces of 𝔽_{q}. Then 𝔽_{q}/A′ ≅ 𝔽_{q}.a/(𝔽_{q}.a ∩ A) ≅ (A + 𝔽_{q}.a)/A ⊂ V/A, and similarly 𝔽_{q}/B′ is isomorphic to an 𝔽_{p}-subspace of W/B, so applying Eqn. (3) yields [𝔽_{q}: A′][𝔽_{q}: B′] < q. By the special case above, there exist x_{1}, …, x_{m} ∈ A′ and y_{1}, …, y_{m} ∈ B′ for some m such that w = ∑_{i=1}^{m} x_{i}y_{i}. We have

wT (a, b) = (∑_{i=1}^{m} x_{i}y_{i}) T (a, b) = ∑_{i=1}^{m} T (x_{i}a, y_{i}b) ∈ Y,

as required.

We deduce:

Lemma 4.10 Let v ∈ P_{1} and suppose that H ≤ N_{1}^{(v)} has index smaller than q_{v}^{1/2}. Then [H, H] = N_{2}^{(v)}. In particular, H ⊃ N_{2}^{(v)}, H ⊴ N_{1}^{(v)} and N_{1}^{(v)}/H is elementary abelian.

Proof We have [N_{1}^{(v)}: H] = [L^{(v)}: L^{(v)}(H)], where L^{(v)}, L^{(v)}(H) are the graded Lie algebras defined earlier, so [L^{(v)}: L^{(v)}(H)] < q_{v}^{1/2}; in particular, [L_{i}^{(v)}: L_{i}^{(v)}(H)] < q_{v}^{1/2} for every i ∈ ℕ. Applying Lemma 4.9 to V = L_{1}^{(v)}, W = L_{m}^{(v)}, A = L_{1}^{(v)}(H), B = L_{m}^{(v)}(H) and with T the Lie bracket, we prove by induction that L_{m+1}^{(v)}(H) = L_{m+1}^{(v)} for every m ∈ ℕ. Since [H, H] is closed, we deduce that [H, H] = N_{2}^{(v)}. The other assertions follow immediately.

Lemma 4.11 Let Φ be a perfect finite central extension of a finite simple group of Lie type H(p^{r}), where r ∈ ℕ and p is prime. Then there exists δ_{1} > 0, depending neither on p, r nor on the Lie type, such that every nontrivial complex projective representation of Φ has dimension at least p^{rδ_{1}}. In particular, any proper subgroup Φ′ of Φ has index at least p^{rδ_{1}}.

Proof If Φ′ is a proper subgroup of Φ then the projective representation arising from the permutation representation of Φ on the coset space Φ/Φ′ is nontrivial because Φ is perfect, so the second assertion follows from the first. To prove the first assertion, let ρ: Φ → PGL_{n}(ℂ) be a nontrivial projective representation. Then ρ(Φ) ⊂ PGL_{n}(K) for some number field K, and reducing modulo a suitable prime of K gives a nontrivial projective representation ρ′: Φ → PGL_{n}(F), where F is a field of characteristic l ≠ p (compare [Lan93, Chapter XVIII, Exercise 27]). Since Φ/ker ρ′ is also a perfect central extension of H(p^{r}), we can assume that ρ′ is faithful. The existence of a δ_{1} with the required properties then follows from [KL90, Corollary 5.3.3 and Theorem 5.3.9].

We are now ready for:

Proof of Proposition 4.6 We can assume that δ < 1/2. Let v ∈ P_{1} and let (V, ρ) be a nontrivial n-dimensional irreducible representation of G(O_{v}). Write ρ|_{N_{1}^{(v)}} as a direct sum of irreducible representations W_{1} ⊕ ⋯ ⊕ W_{m}. Without loss of generality, we assume that W_{1} ⊕ ⋯ ⊕ W_{r} is the isotypic component of W_{1}, that is, that W_{1}, …, W_{r} are precisely the summands that are isomorphic to W_{1}. Let H be the stabiliser of W := W_{1}⊕⋯⊕W_{r} in G(O_{v}), and let σ be the representation of H on W. By Clifford’s Theorem, ρ = Ind_{H}^{G(O_{v})}(σ). Now, if W is a proper subspace of V then H is a proper subgroup of G(O_{v}) containing N_{1}^{(v)}; by Lemma 4.11, [G(O_{v}): H] = [G(𝔽_{v}): H/N_{1}^{(v)}] is at least q_{v}^{δ_{1}}, and we have the required bound for dim V.

Assume, therefore, that W = V. Then V is a sum of m isomorphic N_{1}^{(v)}-simple modules, each of dimension s, say. If s ≥ q_{v}^{1/2} then we are done. If not then s < q_{v}^{1/2}, and W_{1}, being an irreducible representation of the pro-p_{v} group N_{1}^{(v)}, is induced from a one-dimensional representation of a subgroup M of N_{1}^{(v)}, of index s < q_{v}^{1/2} (Lemma 2.5). By Lemma 4.10, M contains N_{2}^{(v)} and [M, M] = N_{2}^{(v)}, which implies that the simple N_{1}^{(v)}-module W_{1} is actually a module for the abelian group N_{1}^{(v)}/N_{2}^{(v)}, and hence is onedimensional. It follows that ρ(N_{1}^{(v)}) is central in ρ(G(O_{v})). Hence ρ gives rise to an n-dimensional projective representation of the quasisimple group G(𝔽_{v}), nontrivial because ρ is nontrivial and G(O_{v}) is perfect. Lemma 4.11 implies that n ≥ q_{v}^{δ_{1}}, so the proof of Proposition 4.6 is complete.

Proof of Theorem 4.3 For v ∈ P_{1}, we give a precise estimate for [G(O_{v}): K_{n}(G(O_{v}))]. We claim that for every nontrivial n-dimensional irreducible representation ρ of G(O_{v}), we have ker ρ ⊃ N_{4r+6}^{(v)}, where r = [\frac{log_{pv} n}{e_{v}}]. This will fin-

ish the proof of Theorem 4.3, since [G(O_{v}): N_{4r+6}^{(v)}] = |G(𝔽_{v})|[N_{1}^{(v)}: N_{4r+6}^{(v)}] ≤

q_{v}^{N^{2}}q_{v}^{(4r+5)dim G} ≤ q_{v}^{N^{2}}p_{v}^{(4e_{v} \frac{log_{pv} n}{ev} +5e_{v})dim G} = n^{4dim G}q_{v}^{5dim G+N^{2}}. But since ρ is nontrivial, by Proposition 4.6 we have q_{v} ≤ n^{1/δ}, hence the index of K_{n}(G(O_{v})) is bounded by n^{(4+5/δ)dim G+N^{2}/δ}, as required.

To prove that indeed ker ρ ⊃ N_{4r+6}^{(v)}, we first restrict ρ to N_{1}^{(v)} and decompose it into N_{1}^{(v)}-irreducible representations, each of dimension no greater than n. Fix one of these representations and call it σ. As N_{1}^{(v)} is a pro-p group, σ is induced from a one-dimensional representation of a subgroup H of index at most n (Lemma 2.5). Therefore it suffices to prove:

Proposition 4.12 Let v ∈ P_{1} and let H ≤ N_{1}^{(v)} have index at most n. Then H ⊃ N_{2r+3}^{(v)}, where r = [\frac{log_{pv} n}{e_{v}}], and [H, H] ⊃ N_{4r+6}^{(v)}.

Note that Proposition 4.12 is a stronger version of the standard “index versus level” formula; the usual argument proves only that H ⊃ N_{log_{pv} n+1}^{(v)} [LS03, Proposition 4.3.1]. (Of course, if char k = 0 then the usual argument suffices.)

Proof of Proposition 4.12 We have [N_{1}^{(v)}: H] = [L^{(v)}: L^{(v)}(H)]. Now let s ≥ 2r + 3. Consider the r + 1 pairs of distinct integers (i, s − i) with 1 ≤ i ≤ r + 1. By our hypothesis on the index of H, there must exist at least one such pair (i, s − i) such that [L_{i}^{(v)}: L_{i}^{(v)}(H)][L_{s−i}^{(v)}: L_{s−i}^{(v)}(H)] < q_{v}. By Lemma 4.9 we have [L_{i}^{(v)}(H), L_{s−i}^{(v)}(H)] = [L_{i}^{(v)}, L_{s−i}^{(v)}] = L_{s}^{(v)}. As H is closed, we deduce that H ⊃ N_{2r+3}^{(v)}. It follows from condition (c) in the definition of P_{1} that [H, H] ⊃ [N_{2r+3}^{(v)}, N_{2r+3}^{(v)}] = N_{4r+6}^{(v)}.

This completes the proof of Theorem 4.3 for v ∈ P_{1}. For each of the finitely many valuations v ∈ P_{2}, Lemma 4.8 gives the result we need, so we have proved Theorem 4.3.

Theorem 1.2 is now proved. Indeed, let Γ = G(O_{S}) be as in Theorem 1.2 and suppose that Γ has the congruence subgroup property. By Lemma 2.8 and the arguments in Section 3, passing to a finite index subgroup of Γ if necessary, it suffices to show that r̂_{n}(Γ) is polynomially bounded and that if char k = 0 then the number of algebraic representations of Γ is polynomially bounded. This follows from Theorem 4.2 and Proposition 3.2.

5 Image Growth of Congruence Representations

While proving Theorem 1.2 we showed that the local group G(O_{v}) has a normal subgroup K_{n}(G(O_{v})) of polynomial index which is in the kernel of every finite n-dimensional representation. This is not true for the global group G(Ô_{S}): a weaker property holds, namely that the image size of every finite irreducible representation of dimension n is polynomial in n. For future use (Section 6), we prove a bound for the image size of general finite representations.

Proposition 5.1 There exists γ ∈ (0, 1) such that if n is large enough then for every ρ ∈ Rep_{n}(G(Ô_{S})), we have |ρ(G(Ô_{S}))| ≤ e^{n^{γ}}.

We need two lemmas. In the first lemma, the second inequality is obvious, while the first is proved using induction on d and the Binomial Theorem (we leave the details to the reader).

Lemma 5.2 Let d, a ∈ ℕ. Then \frac{1}{a+1}d^{a+1} ≤ ∑_{i=1}^{d} i^{a} ≤ d^{a+1}.

Let t ∈ ℕ, let H_{1}, …, H_{t} be profinite groups and set H = H_{1} × ⋯ × H_{t}. Given ρ ∈ Rep_{n}(H), let n_{i}(ρ) denote the maximal dimension of an irreducible component of the restriction of ρ to H_{i}.

Lemma 5.3 ∑_{n_{i}(ρ)>1} n_{i}(ρ) ≤ n.

Proof We use induction on n. First we suppose that ρ is irreducible. We can write ρ = ρ_{1} ⊗ ⋯ ⊗ ρ_{t}, where each ρ_{i} ∈ Rep_{n_{i}}(H_{i}) and n_{1} ⋯ n_{t} = n. Clearly ρ|_{H_{i}} is a direct sum of copies of ρ_{i}, so we have n_{i}(ρ) = n_{i}. Then ∑_{n_{i}(ρ)>1} n_{i}(ρ) = ∑_{n_{i}>1} n_{i} ≤ n_{1} ⋯ n_{t} = n.

If ρ is reducible, say ρ = ρ_{1} ⊕ ρ_{2}, then for each i we have n_{i}(ρ) = max{n_{i}(ρ_{1}), n_{i}(ρ_{2})}. Applying the induction hypothesis we have

∑_{n_{i}(ρ)>1} n_{i}(ρ) ≤ ∑_{n_{i}(ρ_{1})>1} n_{i}(ρ_{1}) + ∑_{n_{i}(ρ_{2})>1} n_{i}(ρ_{2}) ≤ dim ρ_{1} + dim ρ_{2} = n,

as required.

Proof of Proposition 5.1 Let ρ ∈ Rep_{n}(G(Ô_{S})). We can regard ρ as a representation of G(O_{v_{1}}) × ⋯ × G(O_{v_{t}}) for some t ∈ ℕ and some v_{1}, …, v_{t} ∈ V_{f}−S, such that each restriction ρ|_{G(O_{v_{i}})} is nontrivial. Since the set of bad primes P_{2} is finite, we can assume (using Theorem 4.3) that each v_{i} belongs to P_{1}; compare the beginning of the proof of Theorem 4.2. This implies that n_{1}(ρ), …, n_{t}(ρ) > 1. We start by finding an upper bound for t in terms of n.

For any m ∈ ℕ with m > 1, Lemma 4.7 (b) and Proposition 4.6 imply that the number of v ∈ P_{1} such that G(O_{v}) admits an irreducible representation of dimension m is at most m^{a}, for some constant a ∈ ℕ. In particular, at most m^{a} of the n_{i}(ρ) can take the value m. We therefore have

n_{1}(ρ) + ⋯ + n_{t}(ρ) ≥ 2^{a}.2 + 3^{a}.3 + ⋯ + d^{a}.d,  (4)

for any d ∈ ℕ such that 2^{a} + 3^{a} + ⋯ + d^{a} ≤ t. Applying Lemma 5.2 and Lemma 5.3 to the RHS and LHS respectively of Eqn. (4) yields \frac{1}{a+2}d^{a+2} ≤ n. By Lemma 5.2, we can take d = [t^{1/(a+1)}]. Setting γ_{0} = \frac{a+1}{a+2} and choosing suitable b > 0, we therefore have

t ≤ bn^{γ_{0}}.  (5)

Consider the function f(x_{1}, …, x_{t}) = x_{1} ⋯ x_{t} defined on the bounded convex domain D = {(x_{1}, … x_{t}) ∈ ℝ^{t} | x_{1}, …, x_{t} ≥ 0, ∑_{i=1}^{t} x_{i} ≤ n}. Elementary calculus shows that the maximum value of f on D is (n/t)^{t}. Now consider the function g(y) = (n/y)^{y} on [1, α]. By elementary calculus, if log_{e}n − log_{e}α ≥ 1 then the maximum value of g(y) is (n/α)^{α}.

It follows that for large enough n and for any γ ∈ (γ_{0}, 1), we have

|ρ(G(Ô_{S}))| ≤ c^{t}(n_{1}(ρ) ⋯ n_{t}(ρ))^{c} by Theorem 4.3
≤ c^{t} (\frac{n}{t})^{tc} by the bound for f(x_{1}, …, x_{t})
≤ c^{bn^{γ_{0}}} (\frac{n}{bn^{γ_{0}}} )^{cbn^{γ_{0}}} by Eqn. (5) and the bound for g(y)
= (b^{−bc}c^{b})^{n^{γ_{0}}}n^{(1−γ_{0})cbn^{γ_{0}}}
≤ e^{n^{γ}},

as required.

To finish the section we prove another “polynomial image growth” result for local groups, this time for representations over a field of positive characteristic. Only case (i) of the following proposition will be needed in the sequel.

Proposition 5.4 Let F be a field of characteristic l ≠ 0, let v ∈ V_{f} and let ρ: G(O_{v}) → GL_{n}(F) be a representation. Set p = p_{v}. Assume that at least one of the following holds:

(i) l ≠ p (ii) char k = 0. Then |ρ(G(O_{v}))| ≤ cn^{c} for some constant c > 0, depending only on G, k and v.

Proof By assumption, ρ is continuous and has finite image. Assume that l ≠ p, and let σ = ρ|_{N_{1}^{(v)}}. Let b = [G(O_{v}): N_{1}^{(v)}]. Since σ(N_{1}^{(v)}) is a finite p-

group, σ can be lifted to a complex representation σ̃ of N_{1}^{(v)} (compare [Lan93, Chapter XVIII, Exercise 27]). Inducing σ̃ gives a complex representation ρ̃ of G(O_{v}), with ker ρ̃ ⊃ K_{bn}(G(O_{v})). We have ker ρ ⊃ ker σ = ker σ̃ ⊃ K_{bn}(G(O_{v})) ∩ N_{1}^{(v)}. By Theorem 4.3, K_{bn}(G(O_{v})) ∩ N_{1}^{(v)} has polynomial index in G(O_{v}), so |ρ(G(O_{v}))| is polynomially bounded. This proves (i).

Assume now that l = p and char k = 0. Then N_{1}^{(v)} is a p-adic analytic pro-p group, so N_{1}^{(v)} is boundedly generated: that is, there exist x_{1}, …, x_{d} ∈ N_{1}^{(v)} such that every g ∈ N_{1}^{(v)} can be written as g = x_{1}^{α_{1}} ⋯ x_{d}^{α_{d}} for some α_{1}, …, α_{d} ∈ ℤ_{p} [DdSMS99, Theorem 3.17]. Now every element of GL_{n}(F) of p-power order must be of order at most pn. (Indeed, if p^{m} ≥ n and y^{p^{m}} = 1 then (y − 1)^{p^{m}} = 0, so y − 1 is a nilpotent element in M_{n×n}(F) and (y − 1)^{n} = 0. This implies that y^{pp^{a}} − 1 = (y − 1)^{pp^{a}} = 0 where a = [log_{p}n], so y has order at most pn.) This implies now that |ρ(N_{1}^{(v)})| ≤ (pn)^{d}, so |ρ(G(O_{v}))| is polynomially bounded.

6 Arithmetic Groups with Polynomial Representation Growth

In this section we analyse groups Γ = G(O_{S}) with polynomial representation growth. The goal is to prove Theorem 1.3.

Recall that the profinite and congruence topologies on G(O_{S}) can be extended to topologies on G(k); these are called the arithmetic and congruence topologies respectively. The congruence subgroup problem is to study the kernel

1 → C → { \atop G(k)} → { \atop G(k)} → 1,

where { \atop G(k)} (respectively { \atop G(k)}) is the completion of G(k) with respect to the arithmetic (respectively congruence) topology (see [Ser70], [Rag76]).

Let C_{0} = [C, { \atop G(k)}]. By the solution to the metaplectic problem (cf. [PR96]), C_{0} is open in C. Note that G(O_{S}) has the congruence subgroup property if and only if C_{0} = 1, so to prove Theorem 1.3, we need to show that C_{0} is trivial.

Proposition 6.1 Suppose that C_{0} has an open normal subgroup M with F := C_{0}/M a nonabelian finite simple group. Then for some ϵ > 0,

R_{n}(G(Ô_{S})) > n^{ϵ log log n}

for infinitely many n. In particular, Γ = G(O_{S}) does not have PRG.

Proof Let N = ⋂_{g∈G(k)} M^{g}. It was shown in [Lub95, (2.4)] that N has infinite index in C_{0}, and since W := C_{0}/N is a subdirect product of copies of F, we have W ≅ ∏_{i=1}^{∞} F. Thus Γ̂, after passing to a subgroup of finite index if necessary, is mapped onto a group E of the following type:

(∗) 1 → W → E →^{π} H → 1,

where H is an open subgroup of G(Ô_{S}). To finish, it is enough to prove the following lemma.

Lemma 6.2 Let E be a profinite group of the form (∗), where F and H are as above. Then for some ϵ > 0, we have

log R_{n}(E) > ϵ log n loglog n

for infinitely many n.

Proof Since F is simple, we can regard F as a subgroup of Aut F. Fix an irreducible representation (V, σ) of Aut F such that σ|_{F} is nontrivial. We can write σ|_{F} as a sum of irreducibles τ_{1} ⊕ ⋯ ⊕ τ_{s} for some s. Note that no τ_{i} is trivial (otherwise, as F ⊴ Aut F, the F -fixed points of V form a nonzero invariant subspace, contradicting the irreducibility of σ). Let c > 1 be the dimension of σ. Clearly there is no harm in taking logarithms to base c in what follows.

Fix constants γ, γ_{1}, γ_{2}, γ_{3} such that 0 < γ < γ_{3} < 1, 0 < γ_{1} < γ_{2} < 1 − γ_{3} and γ is as in Proposition 5.1. Choose an open subgroup U ⊴ E, and define E′ = E/U, W′ = W/(W ∩ U), H′ = H/π(U). Then W′ is the product of a finite number of copies of F, say m copies. We can make m arbitrarily large by an appropriate choice of U. Replacing U by the preimage in E of the centraliser C_{E′}(W′), we can assume that C_{E′}(W′) is trivial (this does not change m, because C_{E′}(W′) ∩ W′ = 1). We can identify Aut F^{m} with 𝔖_{m} ⋉ (Aut F)^{m}, where 𝔖_{m} is the symmetric group. Thus the action of E′ on W′ gives rise to a homomorphism ρ: E′ → 𝔖_{m}. Let K be the kernel of this homomorphism. Since E′ acts faithfully on W′, we can identify K with a subgroup of (Aut F)^{m}, and we have F^{m} ≤ K ≤ (Aut F)^{m}. As 𝔖_{m} embeds into GL_{m}(ℂ), Proposition 5.1 implies that

[E′: K] ≤ e^{m^{γ}}  (6)

for sufficiently large m.

For any S ⊂ {1, …, m}, define an irreducible representation σ′_{S} of (Aut F)^{m} by taking σ′_{S} to be σ on the ith factor of (Aut F)^{m} for every i ∈ S and trivial on the other factors. Choose σ_{S} to be some irreducible component of the restriction of σ′_{S} to K. The dimension of σ_{S} is at most c^{|S|}. We claim that if S_{1} ≠ S_{2} then σ_{S_{1}} ≠ σ_{S_{2}}: in fact, σ_{S_{1}}|_{F^{m}} ≠ σ_{S_{2}}|_{F^{m}}. To see this, observe that any irreducible component of σ_{S}|_{F^{m}} can be written as a tensor product ϕ_{1} ⊗ ⋯ ⊗ ϕ_{m}, where each ϕ_{i} is either trivial or equal to one of τ_{1}, …, τ_{s}, and S is precisely the set of indices i such that ϕ_{i} is nontrivial.

Now let f: ℕ → ℕ be any nondecreasing function such that lim_{m→∞} f(m)/m^{γ} = ∞ and lim_{m→∞} f(m)/m^{γ_{3}} = 0 (for example, if γ < β < γ_{3} then f(m) = [m^{β}] will do). For the rest of the proof, assume that m is sufficiently large. We have

( \begin{matrix}m \\ f(m) \end{matrix} ) = \frac{m!}{(m − f(m))!f(m)!}
≥ \frac{(m − f(m))^{f(m)}}{f(m)^{f(m)}}
≥ (c^{−1} \frac{m}{f(m)})^{f(m)}
= c^{f(m)(log_{c}m−log_{c}f(m)−1)}
≥ c^{f(m)(log_{c}m−log_{c}m^{γ_{3}}−1)}
= c^{f(m)((1−γ_{3})log_{c}m−1)}
≥ c^{γ_{2}f(m)log_{c}m}
= m^{γ_{2}f(m)}.

Thus we have at least m^{γ_{2}f(m)} isomorphism classes of representations of K of the form σ_{S} with |S| = f(m), each of dimension at most c^{f(m)}. This implies that R_{c^{f(m)}}(K) ≥ m^{γ_{2}f(m)}.

Set n = [E′: K]c^{f(m)}. We have

R_{n}(E) ≥ R_{n}(E′)
≥ \frac{1}{[E′: K]}R_{c^{f(m)}}(K) by the first inequality of Lemma 2.2
≥ e^{−m^{γ}}m^{γ_{2}f(m)} by Eqn. (6)
≥ m^{γ_{1}f(m)}.

Taking logarithms gives

log_{c}R_{n}(E) ≥ γ_{1}f(m)log_{c}m ≥ γ_{1}f(m)log_{c}f(m).

Fix κ > 1. Using Eqn. (6), we have n ≤ e^{m^{γ}}c^{f(m)} ≤ c^{κf(m)}, which gives

log_{c}n ≤ κf(m)

and

log_{c}log_{c}n ≤ log_{c}κ + log_{c}f(m) ≤ κlog_{c}f(m).

Combining the previous three equations gives

log_{c}R_{n}(E) ≥ \frac{γ_{1}}{κ^{2}} log_{c}n log_{c}log_{c}n,

and we have our required bound.

For the rest of the section, assume that char k = 0. Assume that Γ = G(O_{S}) does not have the congruence subgroup property. By Rapinchuk’s Lemma (see [Lub95, (2.6)] or [LS03, Section 7.1]), Γ has a finite index subgroup Γ_{0} such that Γ̂_{0} has a quotient E with the following property: there exists a short exact sequence of profinite groups

(∗) 1 → W → E →^{π} H → 1,

where W is the product of countably many copies of a fixed finite simple group F and one of the following three possibilities occurs:

(a) F is nonabelian and H is an open subgroup of G(Ô_{S});

(b) F is abelian, say F = C_{l}, the cyclic group of order l, and H is an open pro-p_{v} subgroup of G(O_{v}) for some v ∈ V_{f}−S, with either:

(b1) l ≠ p_{v}; or

(b2) l = p_{v}.

Thus, to prove Theorem 1.3 it suffices to show by Corollary 2.3 that in each of these three cases, r_{n}(E) does not grow polynomially. Case (a) was exactly the case treated in Lemma 6.2.

Let us now treat case (b1). In this case the sequence (∗) splits by the Schur-Zassenhaus Theorem and E = H ⋉ W. Choose U open and normal in E, and let E′ = E/U, W′ = W/(W ∩ U) and H′ = H/π(U), so that E′ = H′ ⋉ W′. Then W′ ≅ C_{l}^{m} for some m, and m can be made arbitrarily large by a suitable choice of U. Now the centraliser C_{H′}(W′) is a normal subgroup of E′ which has trivial intersection with W′. Replacing U by the inverse image of C_{H′}(W′) in E, we can assume that H′ acts faithfully on C_{l}^{m}, so that we have a faithful representation of H′ into GL_{m}(𝔽_{l}). We deduce from Proposition 5.4 (i) that |H′| ≤ cm^{c} for some constant c. This implies that A_{cm^{c}}(E) ≥ l^{m}. Lemma 2.6 (a) now implies that for any γ ∈ (0, 1/c), R_{n}(E) > e^{n^{γ}} for infinitely many n; in particular, E does not have PRG, and case (b1) is proved.

We haven’t proven Rapinchuk’s Lemma in positive characteristic, but if it is true then the proof of case (b1) works there as well. On the other hand, the following argument, which settles case (b2), works only if char k = 0.

Assume now that we are in case (b2). Set p := p_{v} = l. Then E is a pro-p group. Let E^{p^{n}} be the closed subgroup of E generated by the p^{n}-powers in E and let q_{n}(E) = [E: E^{p^{n}}]. Set E′ = E/E^{p^{n}}, H′ = H/H^{p^{n}} = H/π(E^{p^{n}}). In [DdSMS99, Corollary 11.6] it is shown that if M is a finitely generated pro-p group and q_{n}(M) < p^{p^{n}} for some n ≥ 1 then M is p-adic analytic. Our group E is not p-adic analytic, so q_{n}(E) ≥ p^{p^{n}} for every n. On the other hand, H is p-adic analytic and so is boundedly generated, and q_{n}(H) grows polynomially in n [DdSMS99, Theorem 3.16]: say q_{n}(H) ≤ (p^{n})^{d} for some fixed d. Putting these two facts together we deduce that there is a short exact sequence

1 → W′ → E′ → H′ → 1,

where |E′| ≥ p^{p^{n}} and |H′| ≤ (p^{n})^{d}. We conclude that |W′| ≥ p^{p^{n}−dn}. As W′ is abelian, we deduce that A_{p^{dn}}(E) ≥ p^{p^{n}−dn}, which, again by Lemma 2.6 (a), implies that for any γ ∈ (0, 1/d), R_{n}(E) > e^{n^{γ}}, this time for all sufficiently large n. So E does not have PRG. Thus Theorem 1.3 is proved.

We thank A. Jaikin-Zapirain for the main idea in the proof of case (b2).

References

[BLMM02] Hyman Bass, Alexander Lubotzky, Andy R. Magid, and Shahar Mozes, The proalgebraic completion of rigid groups, Geom. Dedicata 95 (2002), 19–58.

[CR81] Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, John Wiley & Sons Inc, New York, 1981.

[DdSMS99] J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-p groups, second ed., Cambridge Studies in Advanced Mathematics, vol. 61, Cambridge University Press, Cambridge, 1999.

[Erd41] P. Erdös, On some asymptotic formulas in the theory of the “factorisation numerorum”, Ann. of Math. 42 (1941), 989–993.

[Hum72] James E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972, Graduate Texts in Mathematics, Vol. 9.

[JZ03] Andrei Jaikin-Zapirain, The zeta function of representation of pro-p groups, preprint, 2003.

[KL90] Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990.

[Lan93] Serge Lang, Algebra, third ed., Addison-Wesley, Reading, Massachusetts, 1993.

[LL] Michael Larsen and Alexander Lubotzky, Normal subgroup growth of linear groups: the (G_{2}, F_{4}, E_{8})-Theorem, Algebraic Groups and Arithmetic (Mumbai 2001), Tata Inst. Fund. Res. Stud. Math., to appear.

[LM85] Alexander Lubotzky and Andy R. Magid, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 58 (1985), no. 336, xi+117.

[LS94] Alexander Lubotzky and Aner Shalev, On some Λ-analytic pro-p groups, Israel J. Math. 85 (1994), 307–337.

[LS03] Alexander Lubotzky and Dan Segal, Subgroup growth, Progress in Mathematics, vol. 212, Birkhäuser Verlag, Basel, 2003.

[Lub91] Alexander Lubotzky, Lattices in rank one Lie groups over local fields, Geom. and Funct. Anal. (GAFA) 1 (1991), 406–431.

[Lub95] , Subgroup growth and congruence subgroups, Invent. Math. 119 (1995), 267–295.

[Mar91] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991.

[PR93] V. P. Platonov and A. S. Rapinchuk, Abstract properties of Sarithmetic groups and the congruence problem, Russian Acad. Sci. Izv. Math. 40 (1993), 455–476.

[PR94] Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press Inc., Boston, MA, 1994.

[PR96] Gopal Prasad and Andrei S. Rapinchuk, Computation of the metaplectic kernel, Inst. Hautes Études Sci. Publ. Math. (1996), no. 84, 91–187.

[Pra77] Gopal Prasad, Strong approximation for semi-simple groups over function fields, Ann. of Math. 105 (1977), 553–572.

[Rag76] M. S. Raghunathan, On the congruence subgroup problem, Inst. Hautes Études Sci. Publ. Math. (1976), no. 46, 107–161.

[Ros02] Michael Rosen, Number theory in function fields, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, New York, 2002.

[Seg99] Yoav Segev, On finite homomorphic images of the multiplicative group of a division algebra, Ann. of Math. 149 (1999), 219–251.

[Ser70] Jean-Pierre Serre, Le problème des groupes de congruence pour SL2, Ann. of Math. 92 (1970), 489–527.

[Ser92] , Lie algebras and Lie groups, second ed., Lecture Notes in Mathematics, vol. 1500, Springer-Verlag, Berlin, 1992.

[Ven88] T. N. Venkataramana, On superrigidity and arithmeticity of lattices in semisimple groups over local fields of arbitrary characteristic, Invent. Math. 92 (1988), 255–306.

Institute of Mathematics Hebrew University of Jerusalem Jerusalem 91904 Israel

Email address: alexlub@math.huji.ac.il
Email address: benm@math.huji.ac.il