INFINITE PRODUCTS OF FINITE SIMPLE GROUPS II

SIMON THOMAS

Abstract. Let (S_{n} | n ∈ ω) be a sequence of finite simple nonabelian groups. A subgroup H of the complete direct product ∏_{n} S_{n} is said to be of small index if [∏_{n} S_{n} : H] < 2^{ω}. We say that ∏_{n} S_{n} has the small index property if every subgroup H of small index is open in ∏_{n} S_{n}. We classify those sequences (S_{n} | n ∈ ω) of finite simple nonabelian groups such that ∏_{n} S_{n} has the small index property.

In the course of our classification proof, we also show that if K is a field of cardinality 2^{ω} and G is an nontrivial linear group over K, then there exists a subgroup H of G such that 1 < [G : H] ≤ ω.

1. Introduction

Let Ω be a countably infinite set and let G ⩽ Sym(Ω). If Δ ⊆ Ω, then G_{{Δ}} and G_{(Δ)} denote respectively the setwise and pointwise stabilisers of Δ in G. We can regard G as a topological group, by taking the basic open subsets to be the cosets gG_{(Δ)} for Δ ∈ [Ω]^{<ω}. Clearly if H is an open subgroup of G, then [G : H] ≤ ω. G is said to have the small index property (SIP ) if a strong converse of this observation holds; namely, whenever H is a subgroup of G with [G : H] < 2^{ω}, then H is an open subgroup. In 1981, Semmes [18] proved that Sym(Ω) has the small index property. After Neumann [4] rediscovered this result, many people became interested in the following question.

Question 1.1. For which countable structures M = ⟨Ω; … ⟩ does G = Aut(M) have the small index property?

In [6] and [7], Evans proved that if M is a countably infinite vector space over a finite field, possibly equipped with a nondegenerate symplectic, hermitian or orthogonal form, then G = Aut(M) has the small index property. In [22], Truss proved that Aut(ℚ) has the small index property, where ℚ denotes the rational numbers regarded as a linearly ordered set. More recently, Hodges, Hodkinson,

Research partially supported by NSF Grants.

2 SIMON THOMAS

Lascar and Shelah [9] have shown that Aut(M) has the small index property for a large class of structures M, including the cases when M is a countable ω-stable ω-categorical structure and when M is the countable random graph Γ.

In [13], Macpherson and Neumann introduced the apparently unrelated notion of group cofinality. (This notion also appeared implicitly in Serre’s work [19] on groups acting on trees.) Suppose that G is not finitely generated. Then the cofinality of G, written c(G), is defined to be the least cardinal λ such that G can be expressed as the union of a chain of λ proper subgroups. Macpherson and Neumann [13] proved that Sym(Ω) has uncountable cofinality. Then Gourion [8] proved that Aut(ℚ) has uncountable cofinality. And in their paper [9], Hodges, Hodkinson, Lascar and Shelah also proved that Aut(M) has uncountable cofinality, when M is either ωstable or the countable random graph. In each of the above papers, the proof that c(Aut(M)) > ω was closely based on the proof that Aut(M) has the small index property. This raises the question of whether there is any implication between the small index property and uncountable cofinality for G = Aut(M), when M is a countable structure. In this paper, we will consider a class of groups in which the small index property does not coincide with the property of having uncountable cofinality; namely, the class of infinite products of finite simple nonabelian groups.

Let (G_{n} | n ∈ ω) be a sequence of nontrivial finite groups. Then ∏_{n} G_{n} denotes the complete direct product of the groups G_{n}, n ∈ ω. For each subset X ⊆ ω, we define ∏_{n∈X} G_{n} = {g ∈ ∏_{n} G_{n} | g(n) = 1 for all n ∈ ω ∖ X}. It is well-known that there exists a countable structure M such that ∏_{n} G_{n} ≃ Aut(M); essentially, we can take M to be the disjoint union of the Cayley graphs Γ_{n} of the G_{n}. (In order to prevent automorphisms which interchange isomorphic Cayley graphs Γ_{n} and Γ_{m}, we color the vertices of each Cayley graph Γ_{n} with a distinct color c_{n}.) When ∏_{n} G_{n} is regarded as an automorphism group in this manner, a subgroup H is open in ∏_{n} G_{n} iff there exists a cofinite subset X of ω such that ∏_{n∈X} G_{n} ⩽ H. Notice that we obtain the same set of open subgroups if we regard ∏_{n} G_{n} as a profinite group.

Serre conjectured that if S is a finite simple nonabelian group and G_{n} = S for all n ∈ ω, then ∏_{n} G_{n} has uncountable cofinality. This conjecture was confirmed by the following result of Koppelberg and Tits [11].

INFINITE PRODUCTS OF FINITE SIMPLE GROUPS II 3

Theorem 1.2. [11] Let F be a nontrivial finite group and let G_{n} = F for all n ∈ ℕ. Then c(∏_{n} G_{n}) > ω if and only if F is perfect.

□

In [16], Saxl, Shelah and I classified those sequences (S_{n} | n ∈ ω) of finite simple nonabelian groups such that ∏_{n} S_{n} has uncountable cofinality. The following definition covers those sequences (S_{n} | n ∈ ω) for which it is obvious that c(∏_{n} S_{n}) = ω.

Definition 1.3. A sequence (S_{n} | n ∈ ω) of finite simple nonabelian groups is said to satisfy the Malcev condition if there exists an infinite subset I of ω such that the following properties hold.

(1) There exists a fixed (possibly twisted) Lie type L such that for all n ∈ I, S_{n} = L(q_{n}) for some prime power q_{n}.

(2) If n, m ∈ I and n < m, then q_{n} < q_{m}.

We will show that if (S_{n} | n ∈ ω) satisfies the Malcev condition, then c(∏_{n} S_{n}) = ω. To simplify notation, suppose that properties 1.3(1) and 1.3(2) hold for I = ω. Let D be a nonprincipal ultrafilter on ω and let K = ∏_{n} GF (q_{n})/D be the corresponding ultraproduct. Then K is a pseudofinite field of cardinality 2^{ω}. By Point [14],

∏_{n} L(q_{n})/D ≃ L (∏_{n} GF (q_{n})/D) = L(K).

Remember that

∏_{n} L(q_{n})/D = ∏_{n} L(q_{n})/N,

where N = {g ∈ ∏_{n} L(q_{n}) | {n ∈ ω | g(n) = 1} ∈ D}. Since L(K) is a homomorphic image of ∏_{n} S_{n} = ∏_{n} L(q_{n}), it is enough to show that c(L(K)) = ω. Let B be a transcendence basis of K over its prime subfield. Then B is an infinite set. Hence we can express B = ⋃_{n<ω}B_{n} as the union of a chain of proper subsets. For each n < ω, let K_{n} be the algebraic closure of B_{n} in K. Then L(K) = ⋃_{n<ω}L(K_{n}), and each L(K_{n}) is a proper subgroup of L(K).

Theorem 1.4. [16] If (S_{n} | n ∈ ω) is a sequence of finite simple nonabelian groups, then the following are equivalent.

(1) (S_{n} | n ∈ ω) satisfies the Malcev condition.

4 SIMON THOMAS

(2) c(∏_{n} S_{n}) = ω.

□

In this paper, we will consider the problem of classifying those sequences of finite simple nonabelian groups (S_{n} | n ∈ ω) such that ∏_{n} S_{n} has the SIP. The following definition is intended to cover those sequences (S_{n} | n ∈ ω) for which it is obvious that ∏_{n} S_{n} does not have the SIP.

Definition 1.5. A sequence (S_{n} | n ∈ ω) of finite simple nonabelian groups is said to satisfy the Saxl/Wilson condition if there exists an infinite subset I of ω and a fixed group S such that S_{n} = S for all n ∈ I.

If (S_{n} | n ∈ ω) satisfies the Saxl/Wilson condition, then ∏_{n} S_{n} does not have the SIP. For suppose that there exists an infinite subset I of ω and a fixed group S such that S_{n} = S for all n ∈ I. Let D be a nonprincipal ultrafilter on ω such that I ∈ D. Then ∏_{n} S_{n}/D ≃ S. Let N be the normal subgroup of ∏_{n} S_{n} such that ∏_{n} S_{n}/D = ∏_{n} S_{n}/N. Then it is clear that N is a non-open subgroup of finite index. The following result of Saxl and Wilson [17] solves the classification problem for the weak form of the SIP, obtained when we restrict our attention to subgroups of finite index.

Theorem 1.6. [17] If (S_{n} | n ∈ ω) is a sequence of finite simple nonabelian groups, then the following statements are equivalent.

(1) (S_{n} | n ∈ ω) satisfies the Saxl/Wilson condition.

(2) There exists a non-open subgroup of finite index in ∏_{n} S_{n}.

□

At this point, it is clear that uncountable cofinality and the SIP do not coincide for infinite products of finite simple nonabelian groups. For suppose that S is a fixed finite simple nonabelian group and that S_{n} = S for all n ∈ ω. Then ∏_{n} S_{n} is a group of uncountable cofinality which does not have the SIP.

Question 1.7. Does there exist a countable structure M such that Aut(M) is a group of countable cofinality which has the SIP?

In an earlier version of this paper, I conjectured that ∏_{n} S_{n} has the SIP iff the sequence (S_{n} | n ∈ ω) does not satisfy the Saxl/Wilson condition. If this conjecture

INFINITE PRODUCTS OF FINITE SIMPLE GROUPS II 5

had been true, then ∏{P SL(2, p) | p prime } would have been an example of an automorphism group of countable cofinality which has the SIP. However, the following result shows that products of finite simple nonabelian groups cannot be used to answer Question 1.7.

Theorem 1.8. If (S_{n} | n ∈ ω) is a sequence of finite simple nonabelian groups, then ∏_{n} S_{n} has the SIP iff the following two conditions hold.

(1) (S_{n} | n ∈ ω) does not satisfy the Saxl/Wilson condition.

(2) (S_{n} | n ∈ ω) does not satisfy the Malcev condition.

In the remainder of this section, we will sketch the proof of Theorem 1.8. First we will show that if (S_{n} | n ∈ ω) satisfies the Malcev condition, then ∏_{n} S_{n} does not have the SIP. Since every open subgroup of ∏_{n} S_{n} has finite index, it is enough to find a subgroup H such that [∏_{n} S_{n} : H] = ω. As explained above, if (S_{n} | n ∈ ω) satisfies the Malcev condition, then there exists a ( possibly twisted ) Lie type L and a pseudofinite field K of cardinality 2^{ω} such that the simple linear group L(K) is a homomorphic image of ∏_{n} S_{n}. So the following result, which will be proved in Section 2, implies that ∏_{n} S_{n} has a subgroup H such that [∏_{n} S_{n} : H] = ω.

Theorem 1.9. If K is a field of cardinality 2^{ω} and G is an infinite simple linear group over K, then there exists a subgroup H of G such that [G : H] = ω.

The proof of Theorem 1.9 is based on Serre’s analysis of SL(2, ℚ_{p}). Here ℚ_{p} denotes the field of p-adic numbers. Let ℤ_{p} be the ring of p-adic integers. In Chapter II of [19], Serre showed that SL(2, ℚ_{p}) acts on a countable tree via an action in which SL(2, ℤ_{p}) is the stabiliser of a vertex. It follows that

[SL(2, ℚ_{p}) : SL(2, ℤ_{p})] = ω.

In Section 2, we will show that if K is an algebraically closed field of cardinality 2^{ω}, then there exists a valuation ring R of K such that [SL(n, K) : SL(n, R)] = ω for all n ≥ 2. Theorem 1.9 will then follow easily.

Now suppose that (S_{n} | n ∈ ω) is a sequence of finite simple nonabelian groups which satisfies neither the Saxl/Wilson condition nor the Malcev condition. We will eventually show that ∏_{n} S_{n} has the SIP. First note that only finitely many of the S_{n} can be either

6 SIMON THOMAS

(a) one of the 26 sporadic finite simple groups, or

(b) a group of Lie type L, where

L ∈ {^{2}B_{2}, ^{3}D_{4}, E_{6}, ^{2}E_{6}, E_{7}, E_{8}, F_{4}, ^{2}F_{4}, G_{2}, ^{2}G_{2}}.

Now suppose that ω = X ∪ Y is a partition of ω. Then it is easily checked that ∏_{n} S_{n} has the SIP iff both ∏_{n∈X} S_{n} and ∏_{n∈Y} S_{n} have the SIP. Hence, using the classification of the finite simple groups, we can suppose that one of the following cases holds. Case 1. First suppose that each S_{n} is an alternating group. Then the following result, which will be proved in Section 3, shows that ∏_{n} S_{n} has the SIP.

Theorem 1.10. If (Alt(m_{n}) | n ∈ ω) is a sequence of finite alternating groups, then ∏_{n} Alt(m_{n}) has the SIP iff (Alt(m_{n}) | n ∈ ω) does not satisfy the Saxl/Wilson condition.

Case 2. Next suppose that each S_{n} is a projective special linear group P SL(d_{n}, q_{n}). Then d_{n} → ∞ as n → ∞. We will find it more convenient to work with the corresponding sequence of special linear groups (SL(d_{n}, q_{n}) | n ∈ ω). In Section 4, we will prove the following result.

Theorem 1.11. If (SL(d_{n}, q_{n}) | n ∈ ω) is a sequence of finite special linear groups such that d_{n} → ∞ as n → ∞, then ∏_{n} SL(d_{n}, q_{n}) has the SIP.

It follows easily that ∏_{n} S_{n} = ∏_{n} P SL(d_{n}, q_{n}) also has the SIP. Case 3. Now suppose that each S_{n} is a projective symplectic group P Sp(2d_{n}, q_{n}). Then d_{n} → ∞ as n → ∞. Again we will prefer to work with the corresponding sequence (Sp(2d_{n}, q_{n}) | n ∈ ω) of symplectic groups. In Section 5, we will prove the following result.

Theorem 1.12. If (Sp(2d_{n}, q_{n}) | n ∈ ω) is a sequence of finite symplectic groups such that d_{n} → ∞ as n → ∞, then ∏_{n} Sp(2d_{n}, q_{n}) has the SIP.

Again it follows easily that ∏_{n} S_{n} = ∏_{n} P Sp(2d_{n}, q_{n}) has the SIP. Case 4. Finally suppose that either each S_{n} is a projective unitary group or each S_{n} is a projective orthogonal group. Once again, the corresponding analogues of Theorem 1.11 hold; and so we obtain that ∏_{n} S_{n} has the SIP.

INFINITE PRODUCTS OF FINITE SIMPLE GROUPS II 7

Modulo the results which we will prove in later sections, this completes the proof of Theorem 1.8. The proofs of the theorems in Sections 3, 4 and 5 closely follow the proofs of the corresponding theorems in [16]. In [16], the main point of each proof was usually to show that for a certain “large” subgroup H of an infinite product ∏_{n} S_{n}, there exists an element g ∈ ∏_{n} S_{n} such that ∏_{n} S_{n} = ⟨H, g⟩. In this paper, roughly speaking, we will need to prove that there are “many such” elements. More precisely, we will need to find elements g_{α} ∈ ∏_{n} S_{n} for α < 2^{ω} such that whenever α < β < 2^{ω}, then ∏_{n} S_{n} = ⟨H, g_{α}g_{β}^{−1}⟩. It requires only a little more ingenuity to prove these stronger results.

Our notation is standard, but a couple of points should be mentioned. Suppose that G is a subgroup of Sym(Δ). If each nonidentity element g ∈ G is fixedpoint-free, then G is said to act semiregularly on Δ. If G acts transitively and semiregularly, then G is said to act regularly on Δ. In this paper, permutation groups and linear groups always act on the left. If A is a matrix, then A^{T} denotes the transpose of A. If k ∈ ω, then [Ω]^{k} = {Δ ⊆ Ω | |Δ| = k}; and [Ω]^{<ω} denotes the set of finite subsets of Ω. We will sometimes identify the infinite sequence (d_{n} | n ∈ ω) with the corresponding function d : ω → ω such that d(n) = d_{n} for all n ∈ ω.

It is a pleasure to thank Gregory Cherlin, David Evans, Moshe Jarden and Dugald Macpherson for their much-needed assistance with Section 2. In particular, Moshe Jarden explained how to derive Theorem 2.7 from Satz 10.4 of Koch [10]. (I have used some ideas from [15] to express Jarden’s argument in a language that I can fully understand. For this reason, this is no explicit mention of Koch [10].)

2. Subgroups of small index in linear groups

In this section, we will use valuation theory to prove the following result.

Theorem 2.1. If K is a field of cardinality 2^{ω} and G is an nontrivial linear group over K, then there exists a subgroup H of G such that 1 < [G : H] ≤ ω.

Clearly Theorem 1.9 is an immediate consequence of Theorem 2.1.

Definition 2.2. Let K be a field. A valuation υ : K → ℝ ∪ {∞} is said to be a σ_{dc}-valuation iff there exists a countable set of subfields {K_{n} | n ∈ ω} such that the following two conditions are satisfied.

8 SIMON THOMAS

(a) If A is a finite subset of K, then there exists an integer n ∈ ω such that A ⊆ K_{n}.

(b) For each n ∈ ω, υ ↾ K_{n} is a discrete valuation with a countable residue field.

Example 2.3. Let K be a countable field and let t be an indeterminate over K. Then K((t)) is the corresponding field of formal power series; and

P(K) = ⋃_{n≥1} K((t^{1/n}))

is the corresponding field of Puiseux expansions. We can define a valuation υ of P(K) as follows. Let

0 ≠ a = ∑_{k≥M}^{∞} a_{k}t^{k/n} ∈ P(K)

where a_{k} ∈ K, a_{M} ≠ 0, k, M ∈ ℤ and n ≥ 1. Then υ(a) = M/n. For each n ≥ 1, let K_{n} = K((t^{1/n})). Then υ ↾ K((t^{1/n})) is a discrete valuation with residue field K; and for every finite subset A ⊂ P(K), there exists an integer n ≥ 1 such that A ⊂ K_{n}. Hence υ is a σ_{dc}-valuation on P(K).

Proposition 2.4. There exists a σ_{dc}-valuation on ℂ.

Proof. It is well-known that if K is an algebraically closed field, then P(K) is algebraically closed iff K is of characteristic 0. (For example, see [3].) In particular, if \overline{ℚ} be the field of algebraic numbers, then P(\overline{ℚ}) is an algebraically closed field of cardinality 2^{ω}. Hence ℂ ≅ P(\overline{ℚ}); and we have just seen that there exists a σ_{dc}-valuation on P(\overline{ℚ}).

Theorem 2.5. Suppose that K is a field and that υ is a σ_{dc}-valuation on K. Let R = {a ∈ K | υ(a) ≥ 0} be the valuation ring of υ. Then [SL(n, K) : SL(n, R)] = ω for each n ≥ 2.

Proof. Let V = V (n, K) be an n-dimensional vector space over K. Then an Rlattice L of V is a free R-submodule of rank n. Thus L is generated by a K-basis of V. Two R-lattices L_{1}, L_{2} are said to be equivalent iff there exists k ∈ K^{∗} such that L_{1} = kL_{2}. For each R-lattice L, the corresponding equivalence class is denoted by [L]; and we define

X = {[L] | L is an R-lattice of V }.

INFINITE PRODUCTS OF FINITE SIMPLE GROUPS II 9

We can define an action of SL(n, K) on X by φ[L] = [φL] for each φ ∈ SL(n, K). Let {e_{i} | 1 ≤ i ≤ n} be a fixed basis of V and let L_{0} = Re_{1} + ⋯ + Re_{n}. Then SL(n, R) is the stabilizer of [L_{0}] in SL(n, K). Hence it is enough to show that X is countable.

Let {K_{ℓ} | ℓ ∈ ω} be a countable set of subfields of K which witnesses that υ is a σ_{dc}-valuation of K; and for each ℓ ∈ ω, let R_{ℓ} be the discrete valuation ring of υ ↾ K_{ℓ}. For each ℓ ∈ ω, fix an element π_{ℓ} ∈ R_{ℓ} such that υ(π_{ℓ}) generates the value group of υ ↾ K_{ℓ}; and let F_{ℓ} = R_{ℓ}/π_{ℓ}R_{ℓ} be the countable residue field of υ ↾ F_{ℓ}. Let [L] be any R-lattice class; say, L = Rb_{1} + ⋯ + Rb_{n}. For each 1 ≤ i ≤ n, let b_{i} = ∑_{j=1}^{n} r_{ij}e_{j}, where r_{ij} ∈ K. Replacing L by a suitably chosen kL ∈ [L] if necessary, we can suppose that r_{ij} ∈ R for all 1 ≤ i, j ≤ n. Let ℓ be an integer such that r_{ij} ∈ R_{ℓ} for all 1 ≤ i, j ≤ n. Let M_{ℓ} = R_{ℓ}e_{1} + ⋯ + R_{ℓ}e_{n} and N_{ℓ} = R_{ℓ}b_{1} + ⋯ + R_{ℓ}b_{n}. Then it is enough to prove that there are only countably many possibilities for the R_{ℓ}-submodule N_{ℓ} of M_{ℓ}.

By the invariant factor theorem, there exists a basis {v_{i} | 1 ≤ i ≤ n} of M_{ℓ} and a sequence of integers 0 ≤ d_{1} ≤ d_{2} ≤ ⋯ ≤ d_{n} such that

N_{ℓ} = R_{ℓ}π_{ℓ}^{d_{1}}v_{1} + ⋯ + R_{ℓ}π_{ℓ}^{d_{n}}v_{n}.

It follows that there exists a finite sequence C_{1}, …, C_{t} of free R_{ℓ}-submodules of M_{ℓ} such that C_{1} = M_{ℓ}, C_{t} = N_{ℓ} and for each 1 ≤ s ≤ t − 1,

(a) π_{ℓ}C_{s} ⊂ C_{s+1} ⊂ C_{s}, and

(b) C_{s}/C_{s+1} ≅ R_{ℓ}/π_{ℓ}R_{ℓ} = F_{ℓ}. From (a) and (b), we see that C_{s+1} corresponds to one of the countably many hyperplanes of C_{s}/π_{ℓ}C_{s} ≅ V (n, F_{ℓ}). It follows that there are only countably many possiblities for N_{ℓ}. This completes the proof of Theorem 2.5.

We have already seen that there exists a σ_{dc}-valuation on ℂ. Next we need to prove this result for algebraically closed fields of characteristic p > 0 and cardinality 2^{ω}. Let \overline{𝔽}_{p} be the algebraic closure of the prime field 𝔽_{p}. Then there exists a σ_{dc}- valuation υ on the field P(\overline{𝔽}_{p}) of all Puiseux series over \overline{𝔽}_{p}; but unfortunately P(\overline{𝔽}_{p}) is not algebraically closed. However, we will prove that υ extends to a σ_{dc}-valuation of the algebraic closure of P(\overline{𝔽}_{p}).

10 SIMON THOMAS

Lemma 2.6. Suppose that K is a field and that υ is a σ_{dc}-valuation on K. Let R = {a ∈ K | υ(a) ≥ 0} be the valuation ring of υ. Then the additive group ⟨R, +⟩ has countable index in the additive group ⟨K, +⟩.

Proof. Consider the subgroups

R̃ = \{(\begin{matrix}1 & a \\ 0 & 1 \end{matrix}) | a ∈ R\} and K̃ = \{(\begin{matrix}1 & k \\ 0 & 1 \end{matrix}) | k ∈ K\}

of SL(2, K). Since [SL(2, K) : SL(2, R)] = ω, it follows that [K̃ : R̃] ≤ ω.

Theorem 2.7. Let K be an algebraically closed field of characteristic p > 0 and cardinality 2^{ω}. Then there exists a σ_{dc}-valuation on K.

Proof. Let F = P(\overline{𝔽}_{p}) be the field of all Puiseux series over \overline{𝔽}_{p}; and let υ be the natural σ_{dc}-valuation of F which was defined in Example 2.3. Then υ has residue field \overline{𝔽}_{p} and value group ℚ. Furthermore, it is well-known that υ is a Henselian valuation; and so υ extends uniquely to a valuation \overline{υ} of the algebraic closure \overline{F} of F. We will prove that \overline{υ} is a σ_{dc}-valuation of \overline{F}. Since K ≅ \overline{F}, this implies Theorem 2.7.

Let L be an algebraic extension of F and let υ_{L} be the unique valuation of L which extends υ. Then υ_{L} is a Henselian valuation of L with residue field \overline{𝔽}_{p} and value group ℚ. Let R_{L} be the valuation ring of υ_{L}.

Claim 2.8. Any nontrivial finite normal extension N of L has degree p^{n} for some n ≥ 1; and hence contains a normal subextension of degree p which is generated by a root of a polynomial x^{p} − x − a for some a ∈ L.

Proof of Claim 2.8. Let e be the ramification index and f be the residue degree of N over L; and let B be the set of all valuation rings of N lying over R_{L}. By Theorem 20.21 [5], there exists an integer d ∈ ω such that [N : L] = ef|B|p^{d}. Since υ_{L} has value group ℚ and residue field \overline{𝔽}_{p}, it follows that e = f = 1; and |B| = 1, since υ_{L} is Henselian. Thus [N : L] = p^{d} for some d ≥ 1. The lemma now follows from standard results in Galois theory.

Claim 2.9. Suppose that a_{1}, a_{2} ∈ L and that α_{1}, α_{2} are roots of the polynomials x^{p} − x − a_{1}, x^{p} − x − a_{2} respectively. If a_{1} − a_{2} ∈ R_{L}, then L[α_{1}] = L[α_{2}]. Hence if [L : F] is finite, then L has only countably many distinct normal extensions of degree p.

Proof of Claim 2.9. Let b = a_{1} −a_{2} ∈ R_{L}. Applying Hensel’s lemma to the polynomial x^{p} −x−b, we see that there exists an element β ∈ R_{L} such that β^{p} −β −b = 0. Notice that

(β + α_{2})^{p} − (β + α_{2}) = (β^{p} − β) + (α_{2}^{p} − α_{2}) = b + a_{2} = a_{1}.

It follows that L[α_{1}] ⊆ L[α_{2}]. Similarly L[α_{2}] ⊆ L[α_{1}], and so L[α_{1}] = L[α_{2}].

Finally if [L : F] is finite, then υ_{L} is a σ_{dc}-valuation of L. Hence Lemma 2.6 implies that L has only countably many normal extensions of degree p.

It is now easy to complete the proof of Theorem 2.7. Using Claim 2.9, we can construct an increasing sequence of subfields \{F_{n} | n ∈ ω\} of \overline{F} such that

(a) F_{0} = F;

(b) F_{n+1} is an algebraic extension of F_{n} for each n ∈ ω; and

(c) for each n ∈ ω and a ∈ F_{n}, there exist m ≥ n and α ∈ F_{m} such that α^{p} − α − a = 0.

Let F_{ω} = ⋃_{n∈ω} F_{n}. Since each υ_{F_{n}} is a σ_{dc}-valuation of F_{n}, it follows that υ_{F_{ω}} is a σ_{dc}-valuation of F_{ω}. Hence it is enough to show that F_{ω} = \overline{F}. By Claim 2.8, F_{ω} does not have any nontrivial normal extensions. Since F is perfect, it follows that F_{ω} is algebraically closed and so F_{ω} = \overline{F}.

It is now easy to complete the proof of Theorem 2.1. Let K be a field of cardinality 2^{ω} and let G be a nontrivial linear group over K. Clearly we can suppose that K is algebraically closed and that G ⩽ SL(n, K) for some n ≥ 2. We can also suppose that G is uncountable. Hence there exists an element g ∈ G ∖ SL(n, K_{0}), where K_{0} is the algebraic closure of the prime subfield of K. Since Aut(K) acts transitively on K ∖ K_{0}, it follows that there exists a σ_{dc}-valuation υ on K such that g ∉ SL(n, R), where R is the valuation ring of υ. Hence if H = G ∩ SL(n, R), then

1 < [G : H] ≤ [SL(n, K) : SL(n, R)] = ω.

12 SIMON THOMAS

3. Infinite products of alternating groups

In this section, we will characterise those sequences (Alt(m_{n}) | n ∈ ω) of finite alternating groups such that ∏_{n} Alt(m_{n}) has the SIP.

Definition 3.1. D(^{ω}ω) is the set of all functions f : ω → ω such that (i) if n < m, then f(n) ≤ f(m); and (ii) f(n) → ∞ as n → ∞.

Definition 3.2. If f ∈ D(^{ω}ω), then we define Alt(f) = ∏_{n} Alt(f(n)).

Theorem 3.3. If f ∈ D(^{ω}ω), then Alt(f) has the SIP.

Corollary 3.4. If (Alt(m_{n}) | n ∈ ω) is a sequence of finite alternating groups, then ∏_{n} Alt(m_{n}) has the SIP iff (Alt(m_{n}) | n ∈ ω) does not satisfy the Saxl/Wilson condition.

□

The rest of this section will be devoted to the proof of Theorem 3.3. Our argument follows that of Section 2 [16] very closely. First we will reduce to the case when f(n) = 2^{d_{n}} for each n ∈ ω. Then, after defining a suitable notion of a generic sequence of elements of Alt(f), we will adapt the argument of [9].

There will be many points in this paper where we will need to prove results of the form

⟨∏_{n} H_{n}, g_{1}, …, g_{t}⟩ = ∏_{n} G_{n},

where H_{n} ⩽ G_{n} for each n ∈ ω. Clearly such results require “uniform generation” results for the corresponding sequences of groups. We will make repeated use of the following easy observation.

Proposition 3.5. Let (H_{n} | n ∈ ω) and (G_{n} | n ∈ ω) be sequences of groups such that H_{n} ⩽ G_{n} for all n ∈ ω. Suppose that there exists a word w(x_{1}, …, x_{s}, y_{1}, … y_{t}) from the free group on {x_{1}, …, x_{s}, y_{1}, …, y_{t}} such that the following condition is satisfied. (3.5) For all n ∈ ω, there exist θ_{1}(n), …, θ_{t}(n) ∈ G_{n} such that each φ ∈ G_{n} can be

expressed as φ = w(ψ_{1}, …, ψ_{s}, θ_{1}(n), …, θ_{t}(n)) for some ψ_{1}, …, ψ_{s} ∈ H_{n}.

INFINITE PRODUCTS OF FINITE SIMPLE GROUPS II 13

For each 1 ≤ i ≤ t, let g_{i} = (θ_{i}(n))_{n} ∈ ∏_{n} G_{n}. Then ∏_{n} G_{n} = ⟨∏_{n} H_{n}, g_{1}, …, g_{t}⟩.

□

The next lemma will provide a simple example of the way in which we will use Proposition 3.5. Suppose that f, g ∈ D(^{ω}ω) and that g(n) ≤ f(n) for all n ∈ ω. Then we can regard Alt(g(n)) as a subgroup of Alt(f(n)) in the canonical fashion; and hence we can regard Alt(g) as a subgroup of Alt(f).

Lemma 3.6. Suppose that f, g ∈ D(^{ω}ω) and that g(n) = f(n) − 1 for all n ∈ ω. Suppose that the group H satisfies the following conditions.

(a) Alt(g) ⩽ H ⩽ Alt(f); and

(b) [Alt(f) : H] < 2^{ω}. Then H is an open subgroup of Alt(f).

Proof. Since f ∈ D(^{ω}ω), there exists a strictly increasing sequence of integers b_{n}, n ∈ ω, such that

(i) b_{0} = 0; and (ii) f(b_{n}) ≥ 2^{n} for all n ∈ ω. Now we can inductively define elements

π_{η} = (π_{η}(0), …, π_{η}(b_{n})) ∈ ∏_{ℓ=0}^{b_{n}} Alt(f(ℓ)), η ∈ ^{n}2

such that the following conditions are satisfied.

(a) π_{∅}(0) ∈ Alt(f(0)).

(b) π_{η} is an initial segment of both π_{ηˆ 0} and π_{ηˆ 1}.

(c) If η, τ are distinct elements of ^{n}2 and m is the greatest integer such that η ↾ m = τ ↾ m, then π_{η}(ℓ)Alt(g(ℓ)) ≠ π_{τ} (ℓ)Alt(g(ℓ)) for all ℓ such that b_{m+1} ≤ ℓ ≤ b_{n}.

For each η ∈ ^{ω}2, define π_{η} ∈ Alt(f) by

π_{η}(n) = π_{η↾t}(n), where n ≤ b_{t}.

Since [Alt(f) : H] < 2^{ω}, there exist distinct η, τ ∈ ^{ω}2 such that θ = π_{η}π_{τ}^{−1} ∈ H. By construction, there exists n_{0} ∈ ω such that

(a) θ(n) = 1 for all n < n_{0}; and

(b) θ(n) ∉ Alt(g(n)) for all n ≥ n_{0}.

Claim 3.7. Suppose that n ≥ n_{0}. Then for every φ ∈ Alt(f(n)), there exist ψ_{1}, ψ_{2}, ψ_{3} ∈ Alt(g(n)) such that φ = ψ_{1}θ(n)^{−1}ψ_{2}θ(n)ψ_{3}.

Proof of Claim 3.7. Clearly we can suppose that φ ∉ Alt(g(n)). Since θ(n) does not normalise Alt(g(n)), there exists an element ψ_{2} ∈ Alt(g(n)) such that σ = θ(n)^{−1}ψ_{2}θ(n) ∉ Alt(g(n)). As Alt(f(n)) acts 2-transitively on {1, …, f(n)}, we have the double coset decomposition

Alt(f(n)) = Alt(g(n)) ∪ Alt(g(n))σAlt(g(n)).

Thus φ ∈ Alt(g(n))σAlt(g(n)), and the result follows.

It follows that ∏_{n≥n_{0}} Alt(f(n)) ⩽ ⟨Alt(g), θ⟩; and hence H is an open subgroup of Alt(f).

We will also make repeated use of the following easy combinatorial observation.

Lemma 3.8. Suppose that 2n = 2^{d} + r, where 0 ≤ r < 2^{d}. If |Ω| = 2n, then there exist subsets S_{i} ∈ [Ω]^{n}, 1 ≤ i ≤ 2^{d} − 1, such that

\frac{n}{2} < 2^{d−1} = |S_{i} △ S_{j}| ≤ n

for all 1 ≤ i < j ≤ 2^{d} − 1.

Proof. Let V = V (d, 2) be a d-dimensional vector-space over GF (2). Then we can suppose that V ⊆ Ω. Let {U_{i} | 1 ≤ i ≤ 2^{d} − 1} be the set of (d − 1)-dimensional subspaces of V. Let Δ = Ω ∖ V and let Δ_{0} ∈ [Δ]^{\frac{r}{2}} . For each 1 ≤ i ≤ 2^{d} − 1, let S_{i} = U_{i} ∪ Δ_{0}. Then it is easily checked that {S_{i} | 1 ≤ i ≤ 2^{d} − 1} satisfies our requirements.

Let m = 4n for some n ≥ 2. A permutation π ∈ Alt(m) is said to have type 2^{2n} if π is the product of 2n disjoint transpositions. The set of permutations π ∈ Alt(m) of type 2^{2n} forms a single conjugacy class in Alt(m).

Lemma 3.9. [1] Let m = 4n for some n ≥ 2. Let C be the conjugacy class of Alt(m) consisting of all permutations of type 2^{2n}. Then for every ϕ ∈ Alt(m), there exist π_{1}, …, π_{4} ∈ C such that ϕ = π_{1} … π_{4}.

□

INFINITE PRODUCTS OF FINITE SIMPLE GROUPS II 15

Lemma 3.10. Let f ∈ D(^{ω}ω); and suppose that for each n ∈ ω, f(n) = 8m_{n} for some m_{n} ≥ 1. For each n ∈ ω, let Δ_{0}^{n} = {1, …, 4m_{n}}, Δ_{1}^{n} = {4m_{n} + 1, …, 8m_{n}}, and Γ_{n} = Alt(Δ_{0}^{n}) × Alt(Δ_{1}^{n}) ⩽ Alt(f(n)). Suppose that the group H satisfies the following conditions.

(a) ∏_{n} Γ_{n} ⩽ H ⩽ Alt(f); and

(b) [Alt(f) : H] < 2^{ω}. Then H is an open subgroup of Alt(f).

Proof. For each n ∈ ω, let 4m_{n} = 2^{d_{n}} + r_{n}, where 0 ≤ r_{n} < 2^{d_{n}}. By Lemma 3.8, there exist subsets S_{i}^{n} ∈ [Δ_{0}^{n}]^{2m_{n}} for 1 ≤ i ≤ 2^{d_{n}} − 1 such that

m_{n} < |S_{i}^{n} △ S_{j}^{n}| ≤ 2m_{n}

for all 1 ≤ i < j ≤ 2^{d_{n}} − 1. Let

π_{i}(n) = ∏_{ℓ∈S_{i}^{n}}( ℓ 4m_{n} + ℓ) ∈ Alt(f(n)).

Note that d_{n} → ∞ as n → ∞. Hence, arguing as in the proof of Lemma 3.6, we can define elements π_{η} ∈ Alt(f) for η ∈ ^{ω}2 such that the following conditions are satisfied.

(a) For each η ∈ ^{ω}2 and n ∈ ω, there exists 1 ≤ i ≤ 2^{d_{n}} − 1 such that π_{η}(n) = π_{i}(n).

(b) If η, τ ∈ ^{ω}2 are distinct, then there exists n_{0} ∈ ω such that (i) π_{η}(n) = π_{τ} (n) for all n < n_{0}; and (ii) π_{η}(n) ≠ π_{τ} (n) for all n ≥ n_{0}.

Since [Alt(f) : H] < 2^{ω}, there exist distinct η, τ ∈ ^{ω}ω such that σ = π_{η}π_{τ}^{−1} ∈ H. Let n_{0} be the least integer such that π_{η}(n) ≠ π_{τ} (n) for all n ≥ n_{0}. Fix some n ≥ n_{0}. Let π_{η}(n) = π_{i}(n), π_{τ} (n) = π_{j}(n), and T_{n} = S_{i}^{n} △ S_{j}^{n}. Then

σ(n) = ∏_{ℓ∈T_{n}}( ℓ 4m_{n} + ℓ).

Let Δ_{2}^{n} = {2i | 1 ≤ i ≤ 4m_{n}} and Δ_{3}^{n} = {2i − 1 | 1 ≤ i ≤ 4m_{n}}. Since m_{n} < |T_{n}| ≤ 2m_{n}, there exist elements φ_{1}, φ_{2}, φ_{3} ∈ Γ_{n} such that (3.10) (a) φ_{1}[T_{n}] △ φ_{2}[T_{n}] = Δ_{0}^{n} ∩ Δ_{3}^{n}.

(b) φ_{1} [σ(n)[T_{n}]] △ φ_{2} [σ(n)[T_{n}]] = Δ_{1}^{n} ∩ Δ_{2}^{n}.

(c) θ(n) = φ_{3} ∘ φ_{2}σ(n)φ_{2}^{−1} ∘ φ_{1}σ(n)φ_{1}^{−1} is a permutation of type 2^{4m_{n}} such

that θ(n)[Δ_{0}^{n}] = Δ_{2}^{n} and θ(n)[Δ_{1}^{n}] = Δ_{3}^{n}.

Claim 3.11. If φ ∈ Alt(f(n)) is a permutation of type 2^{4m_{n}}, then there exist ψ_{1}, ψ_{2}, ψ_{3} ∈ Γ_{n} such that φ = ψ_{1}θ(n)ψ_{2}θ(n)ψ_{3}.

Proof of Claim 3.11. This was shown in the proof of Lemma 2.10 [16].

Using (3.10), Claim 3.11 and Lemma 3.9, we now see that

∏_{n≥n_{0}} Alt(f(n)) ⩽ ⟨∏_{n} Γ_{n}, σ⟩;

and so H is an open subgroup of Alt(f).

Definition 3.12. D_{exp}(^{ω}ω) is the set of all functions f ∈ D(^{ω}ω) such that for each n ∈ ω, there exists d_{n} ∈ ω such that f(n) = 2^{d_{n}}.

Theorem 3.13. If f ∈ D_{exp}(^{ω}ω), then Alt(f) has the SIP.

Before proving Theorem 3.13, we will show how to complete the proof of Theorem 3.3.

Proof of Theorem 3.3. Let f ∈ D(^{ω}ω); and suppose that H is a subgroup of Alt(f) such that [Alt(f) : H] < 2^{ω}. For each 0 ≤ i ≤ 7, let

P_{i} = ∏{Alt(f(n)) | f(n) ≡ i (mod 8) }

and let H_{i} = H ∩ P_{i}. Then it suffices to show that H_{i} is an open subgroup of P_{i} for each 0 ≤ i ≤ 7. This is trivially true if P_{i} is a finite group. Thus, as [P_{i} : H_{i}] < 2^{ω}, it is enough to prove the following statement for 0 ≤ i ≤ 7. (†)_{i} Suppose that f ∈ D(^{ω}ω), and that f(n) ≡ i (mod 8) for all n ∈ ω. Then

Alt(f) has the SIP. First consider the case when i = 0. Let H be a subgroup of Alt(f) such that

[Alt(f) : H] < 2^{ω}. Fix some integer n ∈ ω. Let f(n) = 8m_{n} and let d_{n} be the integer such that 2^{d_{n}} ≤ 8m_{n} < 2^{d_{n}+1}. Let

Δ_{0}^{n} = {1, …, 4m_{n}} and Δ_{1}^{n} = {4m_{n} + 1, …, 8m_{n}}.

Also let Φ_{0}^{n}, Φ_{1}^{n} be the sets consisting of the first, respectively last 2^{d_{n}} integers in {1, …, 8m_{n}}. Thus Δ_{0}^{n} ⊆ Φ_{0}^{n} and Δ_{1}^{n} ⊆ Φ_{1}^{n}. Let G_{i} = ∏_{n} Alt(Φ_{i}^{n}) and H_{i} = H∩G_{i}

INFINITE PRODUCTS OF FINITE SIMPLE GROUPS II 17

for i = 0, 1. Then [G_{i} : H_{i}] < 2^{ω}. By Theorem 3.13, there exists an integer n_{0} ∈ ω such that ∏_{n≥n_{0}} Alt(Φ_{i}^{n}) ⩽ H_{i} for i = 0, 1. Consequently,

∏_{n≥n_{0}} Alt(Δ_{0}^{n}) × Alt(Δ_{1}^{n}) ⩽ H ∩ ∏_{n≥n_{0}} Alt(f(n)) ⩽ ∏_{n≥n_{0}} Alt(f(n)).

By Lemma 3.10, H ∩ ∏_{n≥n_{0}} Alt(f(n)) is an open subgroup of ∏_{n≥n_{0}} Alt(f(n)); and so H is an open subgroup of Alt(f).

Next consider the case when i = 1. Again let H be a subgroup of Alt(f) such that [Alt(f) : H] < 2^{ω}. Define g ∈ D(^{ω}ω) by g(n) = f(n)−1; so that g(n) ≡ 0 (mod 8) for all n ∈ ω. Then [Alt(g) : H ∩ Alt(g)] < 2^{ω}, and so there exists n_{0} ∈ ω such that

∏_{n≥n_{0}} Alt(g(n)) ⩽ H ∩ ∏_{n≥n_{0}} Alt(f(n)) ⩽ ∏_{n≥n_{0}} Alt(f(n)).

Now Lemma 3.6 implies that H is an open subgroup of Alt(f). Continuing in this fashion, we can successively deal with the remaining cases.

The remainder of this section will be devoted to the proof of Theorem 3.13. Throughout, we fix some function f ∈ D_{exp}(^{ω}ω). Let f(n) = 2^{d_{n}} for each n ∈ ω. First we will introduce a suitable notion of a generic sequence of elements of Alt(f). (This is a slight variant of the notion which was introduced in [16].)

Definition 3.14. The sequence (g_{0}, …, g_{t}) of elements of Alt(f) is a generic sequence if there exists a strictly increasing sequence of integers i_{0} < ⋯ < i_{t} such that the following conditions are satisfied for all 0 ≤ k ≤ t.

(a) g_{k}(n) = 1 iff n < i_{k}.

(b) If n ≥ i_{k}, then ⟨g_{0}(n), …, g_{k}(n)⟩ is an elementary abelian group of order 2^{k+1}, which acts semiregularly on {1, …, f(n)}.

Lemma 3.15. Suppose that H is a subgroup of Alt(f) such that [Alt(f) : H] < 2^{ω}; and let (g_{0}, …, g_{t}) be any generic sequence. Then there exists g_{t+1} ∈ H such that (g_{0}, …, g_{t}, g_{t+1}) is a generic sequence.

Proof. Choose an integer ℓ > i_{t} such that f(ℓ) ≥ 2^{t+2}. For each n ≥ ℓ, let A_{n} = ⟨g_{0}(n), …, g_{t}(n)⟩. Then there exists an elementary abelian group of order 2^{d_{n}} such that

(i) A_{n} < E_{n} < Alt(f(n)); and (ii) E_{n} acts regularly on {1, …, f(n)}.

18 SIMON THOMAS

Choose a subgroup B_{n} of E_{n} such that E_{n} = A_{n}⊕B_{n}. Then we can define elements π_{η} ∈ Alt(f) for η ∈ ^{ω}2 such that the following conditions are satisfied.

(a) π_{η}(n) = 1 for all n < ℓ.

(b) π_{η}(n) ∈ B_{n} for all n ≥ ℓ.

(c) If η, τ ∈ ^{ω}2 are distinct, then there exists n_{0} ≥ ℓ such that (i) π_{η}(n) = π_{τ} (n) for all n < n_{0}; and (ii) π_{η}(n) ≠ π_{τ} (n) for all n ≥ n_{0}.

Since [Alt(f) : H] < 2^{ω}, there exist distinct η, τ ∈ ^{ω}ω such that g_{t+1} = π_{η}π_{τ}^{−1} ∈ H. Clearly (g_{0}, …, g_{t}, g_{t+1}) is a generic sequence.

Lemma 3.16. Suppose that H is a subgroup of Alt(f) such that [Alt(f) : H] < 2^{ω}; and let (g_{0}, …, g_{t}, g_{t+1}) be any generic sequence. If H is not an open subgroup, then there exists an element φ ∈ Alt(f) such that the following conditions are satisfied.

(a) φ(n) = 1 for all n < i_{t}.

(b) φg_{k}φ^{−1} = g_{k} for all 0 ≤ k ≤ t.

(c) φg_{t+1}φ^{−1} ∉ H.

Proof. Suppose that the result fails. Then whenever φ ∈ Alt(f) satisfies conditions (a) and (b), we must have that φg_{t+1}φ^{−1} ∈ H. We will deduce that H is an open subgroup, which is a contradiction.

First we will set up some notation. For each n ≥ i_{t+1}, let A_{n} = ⟨g_{0}(n), …, g_{t}(n)⟩. Then A_{n} is an elementary abelian group of order 2^{t+1} acting semiregularly on {1, …, 2^{d_{n}}}. Let

{1, …, 2^{d_{n}}} = Φ_{1}^{n} ∪ ⋯ ∪ Φ_{2^{dn−t−1}}^{n}

be the decomposition into A_{n}-orbits. Choose representatives α_{i} ∈ Φ_{i}^{n} so that the set {α_{i} | 1 ≤ i ≤ 2^{d_{n}−t−1}} is a union of g_{t+1}(n)-orbits. Let A_{n} = {π_{k} | 1 ≤ k ≤ 2^{t+1}}, where π_{1} = 1, and define

Δ_{k}^{n} = {π_{k}(α_{i}) | 1 ≤ i ≤ 2^{d_{n}−t−1}}

for each 1 ≤ k ≤ 2^{t+1}. Then the diagonal subgroup

D_{n} = Diag(Alt(Δ_{1}^{n}) × ⋯ × Alt(Δ_{2^{t+1}}^{n}))

= { ∏_{i=1}^{2^{t+1}} π_{i}ψπ_{i} | ψ ∈ Alt(Δ_{1}^{n})}

is contained in the centraliser of A_{n} in Alt(2^{d_{n}}). Furthermore, by choosing the set {α_{i} | 1 ≤ i ≤ 2^{d_{n}−t−1}} to be a union of g_{t+1}(n)-orbits, we have ensured that g_{t+1}(n) ∈ D_{n}.

Let n_{0} ≥ i_{t+1} be an integer such that d_{n_{0}} − t − 1 ≥ 3. For each n ≥ n_{0}, let π(n) ∈ D_{n} be any fixed-point-free involution. Then there exists an element φ ∈ Alt(f) such that

(i) φ(n) = 1 for all n < n_{0}; and (ii) φ(n) ∈ D_{n} and φ(n)g_{t+1}(n)φ(n)^{−1} = π(n) for all n ≥ n_{0}. Clearly φ satisfies (a) and (b); and hence φg_{t+1}φ^{−1} ∈ H. Using Lemma 3.9, we now see that ∏_{n≥n_{0}} D_{n} ⩽ H.

Note that [∏_{n≥n_{0}} Alt(Δ_{1}^{n}) : H ∩∏_{n≥n_{0}} Alt(Δ_{1}^{n})] < 2^{ω}. Hence, arguing as in the proof of Lemma 3.15. we see that there exists an element ψ ∈ H ∩ ∏_{n≥n_{0}} Alt(Δ_{1}^{n}) such that for some n_{1} ≥ n_{0}

(1) ψ(n) = 1 for all n < n_{1}; and

(2) ψ(n) acts on Δ_{1}^{n} as a fixed-point-free involution for all n ≥ n_{1}. Notice that {σψσ^{−1} | σ ∈ ∏_{n≥n_{0}} D_{n}} is the conjugacy class of ∏_{n≥n_{0}} Alt(Δ_{1}^{n}) consisting of all elements which satisfy (1) and (2). Applying Lemma 3.9 once more, we obtain that ∏_{n≥n_{1}} Alt(Δ_{1}^{n}) ⩽ H. Continuing in this fashion, we eventually find an integer n_{2} ≥ n_{1} such that ∏_{n≥n_{2}} Alt(Δ_{1}^{n}) × ⋯ × Alt(Δ_{2^{t+1}}^{n}) ⩽ H.

Now note that

∏_{n≥n_{2}} Alt(Δ_{1}^{n}) × Alt(Δ_{2}^{n}) ⩽ H ∩ ∏_{n≥n_{2}} Alt(Δ_{1}^{n} ∪ Δ_{2}^{n}) ⩽ ∏_{n≥n_{2}} Alt(Δ_{1}^{n} ∪ Δ_{2}^{n})

and that

[ ∏_{n≥n_{2}} Alt(Δ_{1}^{n} ∪ Δ_{2}^{n}) : H ∩ ∏_{n≥n_{2}} Alt(Δ_{1}^{n} ∪ Δ_{2}^{n})] < 2^{ω}.

By Lemma 3.10, there exists an integer n_{3} ≥ n_{2} such that ∏_{n≥n_{3}} Alt(Δ_{1}^{n} ∪ Δ_{2}^{n}) is contained in H. Continuing in this manner, we eventually find an integer n_{4} ≥ n_{3} such that ∏_{n≥n_{4}} Alt(f(n)) ⩽ H. But this contradicts the hypothesis that H is not an open subgroup.

The rest of our argument is virtually identical with the proof of Theorem 5.3 [9]. Suppose that H is a non-open subgroup of Alt(f) such that [Alt(f) : H] < 2^{ω}. We will define by induction on s ∈ ^{<ω}2 elements γ_{s}, g_{sˆ 0}, g_{sˆ 1} ∈ Alt(f) such that the following conditions are satisfied. (If t, s ∈ ^{<ω}2, then t ≤ s means that t is an initial segment of s.)

(i) γ_{∅} = 1. (ii) g_{sˆ 0} ∈ H and g_{sˆ 1} ∉ H. (iii) If n > 0 and s ∈ ^{n}2, then (g_{s↾1}, …, g_{s↾n}) is a generic sequence. (iv) γ_{s}g_{t}γ_{s}^{−1} = g_{t} for all t such that ∅ ≠ t ≤ s.

(v) For each ∅ ≠ s ∈ ^{<ω}2, let i_{s} be the integer such that g_{s}(n) = 1 iff n < i_{s}. Then γ_{sˆ 0}(n) = γ_{sˆ 1}(n) = γ_{s}(n) for all n < i_{s}.

Let s ∈ ^{<ω}2 and suppose that γ_{t}, g_{t} have been defined for all ∅ ≠ t ≤ s. By Lemma 3.15, there exists an element g_{sˆ 0} ∈ H such that (g_{s↾1}, …, g_{s}, g_{sˆ 0}) is a generic sequence. By Lemma 3.16, there exists an element φ_{s} ∈ Alt(f) such that the following conditions are satisfied.

(a) φ_{s}(n) = 1 for all n < i_{s}.

(b) φ_{s}g_{t}φ_{s}^{−1} = g_{t} for all ∅ ≠ t ≤ s.

(c) φ_{s}g_{sˆ 0}φ_{s}^{−1} ∉ H. Define g_{sˆ 1} = φ_{s}g_{sˆ 0}φ_{s}^{−1}, γ_{sˆ 0} = γ_{s} and γ_{sˆ 1} = γ_{s}φ_{s}. Then it is easily checked that conditions (i) to (v) continue to hold. Also notice that

γ_{sˆ 1}g_{sˆ 0}γ_{sˆ 1}^{−1} = γ_{s}φ_{s}g_{sˆ 0}φ_{s}^{−1}γ_{s}^{−1}
= γ_{sˆ 0}g_{sˆ 1}γ_{sˆ 0}^{−1}

and hence

γ_{sˆ 0}^{−1}γ_{sˆ 1}g_{sˆ 0}(γ_{sˆ 0}^{−1}γ_{sˆ 1})^{−1} = g_{sˆ 1}.

For each η ∈ ^{ω}2, define γ_{η} ∈ Alt(f) by

γ_{η}(n) = γ_{η↾m}(n), where n < i_{η↾m}.

INFINITE PRODUCTS OF FINITE SIMPLE GROUPS II 21

Suppose that η, τ ∈ ^{ω}2 are distinct; and that n ∈ ω, s ∈ ^{<ω}2 are such that η ↾ n = s 0 and τ ↾ n = s 1. Then it is easily checked that γ_{η}^{−1}γ_{τ} g_{sˆ 0}(γ_{η}^{−1}γ_{τ})^{−1} = g_{sˆ 1}. Consequently, γ_{η}^{−1}γ_{τ} ∉ H. But this contradicts the fact that [Alt(f) : H] < 2^{ω}. This completes the proof of Theorem 3.13.

4. Infinite products of special linear groups

In this section, we will consider infinite products of special linear groups.

Theorem 4.1. If (SL(d_{n}, q_{n}) | n ∈ ω) is a sequence of finite special linear groups such that d_{n} → ∞ as n → ∞, then ∏_{n} SL(d_{n}, q_{n}) has the SIP.

Our strategy in the proof of Theorem 4.1 will be the same as that of Theorem 3.3. We will begin by proving the analogue of Lemma 3.6. Let (SL(d_{n}, q_{n}) | n ∈ ω) be a sequence of special linear groups. Fix some n ∈ ω. Let SL(d_{n}, q_{n}) act on the vector space V (d_{n}, q_{n}) in the natural manner. Extend this action to V (d_{n} + 1, q_{n}) = V (d_{n}, q_{n}) ⊕ ⟨v_{d_{n}+1}⟩ by specifying that π(v_{d_{n}+1}) = v_{d_{n}+1} for all π ∈ SL(d_{n}, q_{n}). Using this action, we can regard SL(d_{n}, q_{n}) as a subgroup of SL(d_{n} + 1, q_{n}).

Lemma 4.2. Suppose that (d_{n} | n ∈ ω) ∈ D(^{ω}ω), and that the group H satisfies the following conditions.

(a) ∏_{n} SL(d_{n}, q_{n}) ⩽ H ⩽ ∏_{n} SL(d_{n} + 1, q_{n}).

(b) [∏_{n} SL(d_{n} + 1, q_{n}) : H] < 2^{ω}. Then H is an open subgroup of ∏_{n} SL(d_{n} + 1, q_{n}).

Proof. We will make use of the Bruhat decomposition

SL(d, q) = ⋃_{w∈W}BwB

of the special linear group, where B is a Borel subgroup and W is the Weyl group. Fix some integer n ∈ ω. Choose a basis {v_{i} | 1 ≤ i ≤ d_{n}} of V (d_{n}, q_{n}). We will regard each element of SL(d_{n} + 1, q_{n}) as a matrix with respect to the basis B_{n} = {v_{i} | 1 ≤ i ≤ d_{n} + 1}. Note that we have identified SL(d_{n}, q_{n}) with the subgroup

S_{n} = \{(\begin{matrix}A & 0 \\ 0 & 1 \end{matrix}) | A ∈ SL(d_{n}, q_{n})\}

of SL(d_{n} + 1, q_{n}).

22 SIMON THOMAS

Let B_{n} be the subgroup of SL(d_{n} + 1, q_{n}) consisting of the upper triangular matrices. Then B_{n} is a Borel subgroup of SL(d_{n} + 1, q_{n}). Let N_{n} be the subgroup of SL(d_{n} + 1, q_{n}) which stabilises the frame {⟨v_{i}⟩ | 1 ≤ i ≤ d_{n} + 1}. Then the Weyl group of SL(d_{n} +1, q_{n}) is W_{n} = N_{n}/B_{n} ∩N_{n}; and W_{n} is isomorphic to Sym(d_{n} +1) acting on the set {v_{i} | 1 ≤ i ≤ d_{n} + 1}. Note that N_{n} ∩ S_{n} corresponds to the subgroup Sym(d_{n}) of W_{n}. Let f : ∏_{n} N_{n} → ∏_{n} Sym(d_{n} + 1) be the canonical homomorphism.

Claim 4.3. There exists an integer ℓ ∈ ω such that the canonical homomorphism f maps ∏_{n≥ℓ} N_{n} ∩ H onto ∏_{n≥ℓ} Sym(d_{n} + 1).

Proof of Claim 4.3. Arguing as in the proof of Lemma 3.6, we can define elements π_{η} ∈ ∏_{n} N_{n} for η ∈ ^{ω}2 such that if η, τ ∈ ^{ω}2 are distinct, then there exists ℓ ∈ ω such that

(i) π_{η}(n) = π_{τ} (n) for all n < ℓ; and (ii) π_{η}(n)π_{τ} (n)^{−1} corresponds to a permutation in Sym(d_{n} + 1) ∖ Sym(d_{n}) for

all n ≥ ℓ. Since [∏_{n} SL(d_{n} + 1, q_{n}) : H] < 2^{ω}, there exist distinct η, τ ∈ ^{ω}ω such that θ = π_{η}π_{τ}^{−1} ∈ H. Let ℓ ∈ ω be such that conditions (i) and (ii) hold. Then, arguing as in the proof of Claim 3.7, we see that ⟨∏_{n≥ℓ} N_{n} ∩ S_{n}, θ⟩ maps onto ∏_{n≥ℓ} Sym(d_{n} + 1).

For each n ∈ ω such that n ≥ ℓ, define

T_{n} = \{(\begin{matrix}1 & 0 \\ 0 & A \end{matrix}) | A ∈ SL(d_{n}, q_{n})\} .

Then there exists π ∈ ∏_{n≥ℓ} N_{n} ∩ H such that π (∏_{n≥ℓ} S_{n}) π^{−1} = ∏_{n≥ℓ} T_{n}.

Claim 4.4. ∏_{n≥ℓ} B_{n} ⩽ ⟨∏_{n≥ℓ} S_{n}, ∏_{n≥ℓ} T_{n}⟩.

Proof of Claim 4.4. This was shown in the proof of Lemma 3.5 [16].

We have now shown that ∏_{n≥ℓ} B_{n} ⩽ H and ∏_{n≥ℓ} N_{n} ⩽ H. Using the Bruhat decomposition of SL(d_{n} + 1, q_{n}), it follows that ∏_{n≥ℓ} SL(d_{n} + 1, q_{n}) ⩽ H.

Next we will prove the analogue of Lemma 3.10. Consider a product of the form ∏_{n} SL(8d_{n}, q_{n}). Fix some n ∈ ω. Let SL(8d_{n}, q_{n}) act on the vector space

INFINITE PRODUCTS OF FINITE SIMPLE GROUPS II 23

V_{n} = V (8d_{n}, q_{n}) in the natural manner, and let B_{n} = {v_{i} | 1 ≤ i ≤ 8d_{n}} be a basis of V_{n}. Let E_{0} = ⟨v_{i} | 1 ≤ i ≤ 4d_{n}⟩ and E_{1} = ⟨v_{i} | 4d_{n} + 1 ≤ i ≤ 8d_{n}⟩. We regard SL(E_{0}) as the subgroup of SL(8d_{n}, q_{n}) consisting of the elements π such that π[E_{0}] = E_{0} and such that π(v_{i}) = v_{i} for all 4d_{n} + 1 ≤ i ≤ 8d_{n}. We also regard SL(E_{1}) as a subgroup of SL(8d_{n}, q_{n}) in the obvious fashion. Let Γ_{n} = SL(E_{0}) × SL(E_{1}) ⩽ SL(8d_{n}, q_{n}).

Lemma 4.5. Suppose that (d_{n} | n ∈ ω) ∈ D(^{ω}ω), and that the group H satisfies the following conditions.

(a) ∏_{n} Γ_{n} ⩽ H ⩽ ∏_{n} SL(8d_{n}, q_{n}).

(b) [∏_{n} SL(8d_{n}, q_{n}) : H] < 2^{ω}. Then H is an open subgroup of ∏_{n} SL(8d_{n}, q_{n}).

Proof. Once again, we will make use of the Bruhat decomposition of the special linear group. Fix some n ∈ ω. We will regard SL(8d_{n}, q_{n}) as a group of matrices with respect to the ordered basis (v_{1}, …, v_{8d_{n}}) of V_{n}. Let U_{n} be the subgroup of strictly upper triangular matrices in SL(8d_{n}, q_{n}), and let H_{n} be the subgroup of diagonal matrices. Then B_{n} = U_{n} ⋊ H_{n} is the Borel subgroup of upper triangular matrices.

First we will show that there exists an integer ℓ_{0} ∈ ω such that ∏_{n≥ℓ_{0}} U_{n} ⩽ H. Fix some n ∈ ω. Let M_{n} be the ring of all 4d_{n} × 4d_{n}-matrices over GF (q_{n}), and let

T_{n} = \{(\begin{matrix}1 & S \\ 0 & 1 \end{matrix}) | S ∈ M_{n}\} .

Then it is enough to find ℓ_{0} ∈ ω such that ∏_{n≥ℓ_{0}} T_{n} ⩽ H. Note that for each A ∈ SL(4d_{n}, q_{n}), we have that

(\begin{matrix}A & 0 \\ 0 & 1 \end{matrix}) (\begin{matrix}1 & S \\ 0 & 1 \end{matrix}) (\begin{matrix}A^{−1} & 0 \\ 0 & 1 \end{matrix}) = (\begin{matrix}1 & AS \\ 0 & 1 \end{matrix}) .

Regard M_{n} as a SL(4d_{n}, q_{n})-module with the natural action, S ⟼^{A} AS; and regard ∏_{n} M_{n} as a ∏_{n} SL(4d_{n}, q_{n})-module. Let M be the subgroup of ∏_{n} M_{n} which corresponds to the subgroup H ∩ ∏_{n} T_{n}. Then M is a ∏_{n} SL(4d_{n}, q_{n})-submodule such that [∏_{n} M_{n} : M] < 2^{ω}. Hence it is enough to prove the following claim.

Claim 4.6. Suppose that M is a ∏_{n} SL(4d_{n}, q_{n})-submodule of ∏_{n} M_{n} such that [∏_{n} M_{n} : M] < 2^{ω}. Then there exists ℓ_{0} ∈ ω such that ∏_{n≥ℓ_{0}} M_{n} ⩽ M.

Proof of Claim 4.6. Throughout the proof of this claim, we will be expressing each

4d_{n} × 4d_{n}-matrix S = (\begin{matrix}A & B \\ C & D \end{matrix}) in terms of 2d_{n} × 2d_{n}-blocks. For each n ∈ ω,

we can represent the field GF (q_{n}^{2d_{n}}) as a collection F_{n} of 2d_{n} × 2d_{n}-matrices over GF (q_{n}). (For example, see Chapter 2.5 [12].) In particular, there exists a collection F_{n} = {A_{i} | 1 ≤ i ≤ q_{n}^{2d_{n}}} of 2d_{n} × 2d_{n}-matrices such that A_{i} − A_{j} is invertible for all 1 ≤ i < j ≤ q_{n}^{2d_{n}}. Thus we can define elements m_{η} ∈ ∏_{n} M_{n} for η ∈ ^{ω}2, where

m_{η}(n) = (\begin{matrix}A_{η(n)} & 0 \\ 0 & 0 \end{matrix}) for some A_{η(n)} ∈ F_{n}, such that if η, τ ∈ ^{ω}2 are distinct,

then there exists ℓ ∈ ω such that (i) if n < ℓ, then A_{η(n)} = A_{τ(n)}; and (ii) if n ≥ ℓ, then A_{η(n)} − A_{τ(n)} is invertible. It follows that there exists an element m_{1} ∈ M and an integer ℓ ∈ ω such that

(a) if n < ℓ, then m_{1}(n) = (\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}); and

(b) if n ≥ ℓ, then m_{1}(n) = (\begin{matrix}A_{n} & 0 \\ 0 & 0 \end{matrix}) for some invertible matrix A_{n}.

If B ∈ GL(2d_{n}, q_{n}), then (\begin{matrix}BA_{n}^{−1} & 0 \\ 0 & A_{n}B^{−1} \end{matrix}) ∈ SL(4d_{n}, q_{n}); and

(\begin{matrix}BA_{n}^{−1} & 0 \\ 0 & A_{n}B^{−1} \end{matrix}) (\begin{matrix}A_{n} & 0 \\ 0 & 0 \end{matrix}) = (\begin{matrix}B & 0 \\ 0 & 0 \end{matrix}) .

By [23], every 2d_{n} × 2d_{n}-matrix is the sum of two invertible ones. It follows that ∏_{n≥ℓ} L_{n}^{1} ≤ M, where L_{n}^{1} is the subgroup of M_{n} consisting of all matrices of the

form (\begin{matrix}C & 0 \\ 0 & 0 \end{matrix}). Continuing in this manner, we can find an integer ℓ_{0} ≥ ℓ such

that ∏_{d_{n}≥ℓ_{0}} L_{n}^{j} ≤ M for 2 ≤ j ≤ 4, where L_{n}^{2}, L_{n}^{3}, L_{n}^{3} are the subgroups of M_{n}

consisting of all matrices of the form (\begin{matrix}0 & C \\ 0 & 0 \end{matrix}), (\begin{matrix}0 & 0 \\ C & 0 \end{matrix}), (\begin{matrix}0 & 0 \\ 0 & C \end{matrix}) respectively.

It follows that ∏_{n≥ℓ_{0}} M_{n} ⩽ M.

INFINITE PRODUCTS OF FINITE SIMPLE GROUPS II 25

For each n ∈ ω, let E_{0}^{n} = {v_{i} | 1 ≤ i ≤ 4d_{n}} and E_{1}^{n} = {v_{i} | 4d_{n} + 1 ≤ i ≤ 8d_{n}}; so that B_{n} = E_{0}^{n} ∪ E_{1}^{n}. Then the groups of permutation matrices Alt(E_{0}^{n}), Alt(E_{1}^{n}) are subgroups of Γ_{n}. By Lemma 3.10, there exists an integer ℓ_{1} ≥ ℓ_{0} such that ∏_{n≥ℓ_{1}} Alt(B_{n}) ⩽ H.

Next we will show that ∏_{n≥ℓ_{1}} H_{n} ⩽ H; and hence ∏_{n≥ℓ_{1}} B_{n} ⩽ H. For each λ ∈ GF (q_{n})^{∗}, let D_{λ} = diag(λ, 1, …, 1) ∈ GL(4d_{n}, q_{n}). Define

F_{n} = \{(\begin{matrix}D_{λ} & 0 \\ 0 & D_{λ}^{−1} \end{matrix}) | λ ∈ GF (q_{n})^{∗}\} .

Since ∏_{n≥ℓ_{1}} Γ_{n} ⩽ H, it is enough to show that ∏_{n≥ℓ_{1}} F_{n} ⩽ H. For each λ ∈ GF (q_{n})^{∗}, let E_{λ} = diag(λ, λ^{−1}, 1, …, 1) ∈ GL(4d_{n}, q_{n}). Define

K_{n} = \{(\begin{matrix}E_{λ} & 0 \\ 0 & 1 \end{matrix}) | λ ∈ GF (q_{n})^{∗}\} .

Then ∏_{n≥ℓ_{1}} K_{n} ⩽ ∏_{n≥ℓ_{1}} Γ_{n} ⩽ H. Also there exists an element π ∈ Alt(B_{n}) such that πK_{n}π^{−1} = F_{n}. It follows that ∏_{n≥ℓ_{1}} H_{n} ⩽ H.

Finally we will prove that ∏_{n≥ℓ_{1}} N_{n} ⩽ H. Let π ∈ ∏_{n≥ℓ_{1}} N_{n}. Let X be the subset of ω consisting of those n such that π(n) corresponds to an odd permutation of B_{n}. Then there exists an element ψ_{X} ∈ ∏_{n≥ℓ_{1}} (N_{n} ∩ Γ_{n}) such that

1. ψ_{X}(n) corresponds to the odd permutation ( v_{1} v_{2} ) if n ∈ X, and

2. ψ_{X}(n) = 1 if n ∉ X. Since ∏_{n≥ℓ_{1}} H_{n} ⩽ H and ∏_{n≥ℓ_{1}} Alt(B_{n}) ⩽ H, it follows that πψ_{X} ∈ H; and hence π ∈ H. Using the Bruhat decomposition of SL(8d_{n}, q_{n}), it follows that ∏_{n≥ℓ_{1}} SL(8d_{n}, q_{n}) ⩽ H.

Now we are ready to begin the proof of Theorem 4.1. Since the proof is very similar to that of Theorem 3.3, we will just sketch the main points. Suppose that (SL(d_{n}, q_{n}) | n ∈ ω) is a sequence of finite special linear groups such that d_{n} → ∞ as n → ∞ . Clearly we can suppose that (d_{n} | n ∈ ω) ∈ D(^{ω}ω). Arguing as in the proof of Theorem 3.3, we can use Lemmas 4.2 and 4.5 to reduce to the case when each d_{n} has the form 2^{m_{n}} for some m_{n} ≥ 1. Now we can introduce a suitable notion of a generic sequence of elements of ∏_{n} SL(2^{m_{n}}, q_{n}).

26 SIMON THOMAS

Definition 4.7. The sequence (g_{0}, …, g_{t}) of elements of ∏_{n} SL(2^{m_{n}}, q_{n}) is generic if there exists a strictly increasing sequence of integers i_{0} < ⋯ < i_{t} such that the following conditions are satisfied for all 0 ≤ k ≤ t.

(a) g_{k}(n) = 1 iff n < i_{k}.

(b) If n ≥ i_{k}, then there exists a basis B_{n} of the vector space V (2^{m_{n}}, q_{n}) such that ⟨g_{0}(n), …, g_{k}(n)⟩ is an elementary abelian subgroup of Alt(B_{n}) of order 2^{k+1} which acts semiregularly on B_{n}.

Arguing as in the proof of Lemma 3.15, we easily obtain the following result.

Lemma 4.8. Suppose that [∏_{n} SL(2^{m_{n}}, q_{n}) : H] < 2^{ω}; and let (g_{0}, …, g_{t}) be any generic sequence. Then there exists g_{t+1} ∈ H such that (g_{0}, …, g_{t}, g_{t+1}) is a generic sequence.

□

We will need the following analogue of Lemma 3.9.

Definition 4.9. Let d = 4ℓ for some ℓ ≥ 1. Then C(d, q) denotes the conjugacy class in SL(d, q) consisting of all elements π such that π is represented by a permutation matrix of type 2^{2ℓ} with respect to some basis B of V (d, q).

Lemma 4.10. [16] Suppose that d = 4ℓ for some ℓ ≥ 1. Then for every φ ∈ SL(d, q), there exist π_{1}, …, π_{10} ∈ C(d, q) such that φ = π_{1} … π_{10}.

□

Now we can prove the analogue of Lemma 3.16.

Lemma 4.11. Suppose that [∏_{n} SL(2^{m_{n}}, q_{n}) : H] < 2^{ω}; and let (g_{0}, …, g_{t}, g_{t+1}) be any generic sequence. If H is not an open subgroup, then there exists an element φ ∈ ∏_{n} SL(2^{m_{n}}, q_{n}) such that the following conditions are satisfied.

(a) If n < i_{t}, then φ(n) = 1.

(b) φg_{k}φ^{−1} = g_{k} for all 0 ≤ k ≤ t.

(c) φg_{t+1}φ^{−1} ∉ H.

Proof. Suppose that the result fails. Then whenever φ ∈ ∏_{n} SL(2^{m_{n}}, q_{n}) satisfies conditions (a) and (b), we must have that φg_{t+1}φ^{−1} ∈ H. We will deduce that H is an open subgroup, which is a contradiction.

INFINITE PRODUCTS OF FINITE SIMPLE GROUPS II 27

For each n ≥ i_{t+1}, let B_{n} be a basis of the vector space V (2^{m_{n}}, q_{n}) chosen so that ⟨g_{0}(n), …, g_{t}(n), g_{t+1}(n)⟩ is an elementary abelian subgroup of Alt(B_{n}) of order 2^{t+2} which acts semiregularly on B_{n}. Let A_{n} = ⟨g_{0}(n), …, g_{t}(n)⟩, and let {Φ_{i}^{n} | 1 ≤ i ≤ 2^{m_{n}−t−1}} be the set of A_{n}-orbits on B_{n}. Choose representatives v_{i}^{1} ∈ Φ_{i}^{n} so that the set {v_{i}^{1} | 1 ≤ i ≤ 2^{d_{n}−t−1}} is a union of g_{t+1}(n)-orbits. Let A_{n} = {π_{k} | 1 ≤ k ≤ 2^{t+1}}, where π_{1} = 1; and define v_{i}^{k} = π_{k}(v_{i}^{1}). For each 1 ≤ k ≤ 2^{t+1}, let V_{k}^{n} = ⟨v_{i}^{k} | 1 ≤ i ≤ 2^{m_{n}−t−1}⟩. Then V (2^{m_{n}}, q_{n}) = V_{1}^{n}⊕⋯⊕V_{2^{t+1}}^{n}, and the diagonal subgroup

D_{n} = Diag (SL(V_{1}^{n}) × ⋯ × SL(V_{2^{t+1}}^{n}))

is contained in the centraliser of A_{n} in SL(2^{m_{n}}, q_{n}). Furthermore, g_{t+1}(n) ∈ D_{n}.

Let ℓ_{0} ≥ max{i_{t+1}, 2^{t+4}}. Then arguing as in the proof of Lemma 3.16, we can use Lemma 4.10 to show that ∏_{n≥ℓ_{0}} D_{n} ⩽ H. Continuing to follow the proof of Lemma 3.16, we next see that there exists an integer ℓ_{1} ≥ ℓ_{0} such that

∏_{n≥ℓ_{1}} SL(V_{1}^{n}) × ⋯ × SL(V_{2^{t+1}}^{n}) ⩽ H.

Finally, by repeatedly applying Lemma 4.5, we see that there exists an integer ℓ_{2} ≥ ℓ_{1} such that ∏_{n≥ℓ_{2}} SL(2^{m_{n}}, q_{n}) ⩽ H. But this contradicts the hypothesis that H is not an open subgroup.

Equipped with Lemmas 4.8 and 4.11, it is now easy to adapt the argument of Theorem 3.13. This completes the proof of Theorem 4.1.

5. Infinite products of classical groups

In this section, we will consider infinite products ∏_{n} S_{n}, where each S_{n} is a classical group of a fixed kind. We will deal successively with products of symplectic groups, unitary groups and orthogonal groups over finite fields. (Clear accounts of the classical groups can be found in [2] and [21].) In each case, the proof is a straightforward variation of the proof of the corresponding result in [16]. For this reason, we will provide details in just two cases; namely, when each S_{n} is a symplectic group or an orthogonal group of the form Ω^{−}(2d_{n} + 2, q_{n}). First we will consider infinite products of symplectic groups. In this case, we will see that the following result is an easy consequence of Theorem 4.1.

Theorem 5.1. If (Sp(2d_{n}, q_{n}) | n ∈ ω) is a sequence of finite symplectic groups such that d_{n} → ∞ as n → ∞, then ∏_{n} Sp(2d_{n}, q_{n}) has the SIP.

Without loss of generality, we can suppose that (d_{n} | n ∈ ω) ∈ D(^{ω}ω), and that d_{n} ≥ 2 for all n ∈ ω. Fix some n ∈ ω. Then there exists a basis e f = (e_{i} | 1 ≤ i ≤ d_{n}) (f_{i} | 1 ≤ i ≤ d_{n}) of the corresponding symplectic space such that (e_{i}, f_{j}) = δ_{ij} and (e_{i}, e_{j}) = (f_{i}, f_{j}) = 0 for all 1 ≤ i, j ≤ d_{n}. (Such a basis is called a normal basis.) We will consider Sp(2d_{n}, q_{n}) as a group of matrices with respect to the ordered basis e f. Let E_{d_{n}} = ⟨e_{1}, …, e_{d_{n}}⟩ and F_{d_{n}} = ⟨f_{1}, …, f_{d_{n}}⟩. Then the setwise stabiliser of the subspaces E_{d_{n}} and F_{d_{n}} in Sp(2d_{n}, q_{n}) contains the subgroup

G_{n} = \{(\begin{matrix}A & 0 \\ 0 & (A^{−1})^{T} \end{matrix} ) | A ∈ SL(d_{n}, q_{n})\} .

Now suppose that [∏_{n} Sp(2d_{n}, q_{n}) : H] < 2^{ω}. Then H ∩ ∏_{n} G_{n} is a subgroup of ∏_{n} G_{n} such that [∏_{n} G_{n} : H ∩ ∏_{n} G_{n}] < 2^{ω}. Since G_{n} ≃ SL(d_{n}, q_{n}), Theorem 4.1 yields an integer ℓ ∈ ω such that ∏_{n≥ℓ} G_{n} ⩽ H ∩ ∏_{n} G_{n}. And so the following lemma implies that H is an open subgroup of ∏_{n} Sp(2d_{n}, q_{n}).

Lemma 5.2. Suppose that the group H satisfies the following conditions.

(a) ∏_{n} G_{n} ⩽ H ⩽ ∏_{n} Sp(2d_{n}, q_{n}).

(b) [∏_{n} Sp(2d_{n}, q_{n}) : H] < 2^{ω}. Then H is an open subgroup of ∏_{n} Sp(2d_{n}, q_{n}).

Proof. We will make use of the Bruhat decomposition

Sp(2d, q) = ⋃_{w∈W}BwB

of the symplectic group, where B is a Borel subgroup and W is the Weyl group. Fix some n ∈ ℕ. Let e f = (e_{i} | 1 ≤ i ≤ d_{n}) (f_{i} | 1 ≤ i ≤ d_{n}) be our distinguished normal basis. For each 1 ≤ i ≤ d_{n}, let E_{i} = ⟨e_{1}, … e_{i}⟩. Then the stabiliser B_{n} of the flag of totally isotropic subspaces

E_{1} ⩽ E_{2} ⩽ ⋯ ⩽ E_{d_{n}}

is a Borel subgroup of Sp(2d_{n}, q_{n}). Let N_{n} be the subgroup of Sp(2d_{n}, q_{n}) which stabilises the symplectic frame {⟨e_{i}⟩, ⟨f_{i}⟩ | 1 ≤ i ≤ d_{n}}. Then the Weyl group of Sp(2d_{n}, q_{n}) is N_{n}/B_{n} ∩ N_{n}. Let H_{n} = B_{n} ∩ N_{n}. Then H_{n} consists of the matrices of the form

(\begin{matrix}D & 0 \\ 0 & D^{−1} \end{matrix})

where D ∈ GL(d_{n}, q_{n}) is a diagonal matrix. Let UT_{n} be the subgroup of strictly upper triangular matrices in SL(d_{n}, q_{n}), and define

U_{n} = \{(\begin{matrix}P & P S \\ 0 & (P^{−1})^{T} \end{matrix} ) | P ∈ UT_{n}, S^{T} = S\} .

Then B_{n} = U_{n} ⋊ H_{n}.

First we will show that there exists an integer ℓ_{0} ∈ ω such that ∏_{n≥ℓ_{0}} U_{n} ⩽ H. Fix some n ∈ ω. Let M_{n} be the additive group of all symmetric d_{n} × d_{n}-matrices over GF (q_{n}), and let

T_{n} = \{(\begin{matrix}1 & S \\ 0 & 1 \end{matrix}) | S ∈ M_{n}\} .

Then it is enough to find ℓ_{0} ∈ ω such that ∏_{n≥ℓ_{0}} T_{n} ⩽ H. Note that for each A ∈ SL(d_{n}, q_{n}), we have that

(\begin{matrix}A & 0 \\ 0 & (A^{−1})^{T} \end{matrix} ) (\begin{matrix}1 & S \\ 0 & 1 \end{matrix}) (\begin{matrix}A^{−1} & 0 \\ 0 & A^{T} \end{matrix} ) = (\begin{matrix}1 & ASA^{T} \\ 0 & 1 \end{matrix} ) .

Regard M_{n} as a SL(d_{n}, q_{n})-module with the natural action, S ⟼^{A} ASA^{T}; and regard ∏_{n} M_{n} as a ∏_{n} SL(d_{n}, q_{n})-module. Let M be the subgroup of ∏_{n} M_{n} which corresponds to the subgroup H ∩ ∏_{n} T_{n}. Then M is a ∏_{n} SL(d_{n}, q_{n})-submodule such that [∏_{n} M_{n} : M] < 2^{ω}. Hence it is enough to prove the following claim.

Claim 5.3. Suppose that M is a ∏_{n} SL(d_{n}, q_{n})-submodule of ∏_{n} M_{n} such that [∏_{n} M_{n} : M] < 2^{ω}. Then there exists ℓ_{0} ∈ ω such that ∏_{n≥ℓ_{0}} M_{n} ⩽ M.

Proof of Claim 5.3. For each n ∈ ω, let p_{n} = char(GF (q_{n})). For the sake of simplicity, we will suppose that p_{n} > 3 for all n ∈ ω. (The corresponding arguments in the proof of Theorem 4.1 [16] can be adapted to deal with the cases when p_{n} = 2, 3.) Since p_{n} is odd, every S ∈ M_{n} is congruent to a diagonal matrix. This easily implies that there exists A ∈ SL(d_{n}, q_{n}) such that ASA^{T} is a diagonal matrix. Thus

30 SIMON THOMAS

we need only consider elements (D_{n})_{n} ∈ ∏_{n} M_{n} such that D_{n} = diag(λ_{1}, …, λ_{d_{n}}) is a diagonal matrix.

For each n ∈ ω, let t_{n} = ⌊d_{n}/2⌋. By Lemma 3.8, there exists a collection of sets S_{n} = {S_{i}^{n} | 1 ≤ i ≤ k_{n}} such that the following conditions are satisfied.

(1) k_{n} → ∞ as n → ∞.

(2) S_{i}^{n} ∈ [{1, …, d_{n}}]^{t_{n}} for all 1 ≤ i ≤ k_{n}.

(3) t_{n}/2 < |S_{i}^{n} △ S_{j}^{n}| ≤ t_{n} for all 1 ≤ i < j ≤ k_{n}. For each n ∈ ω and 1 ≤ i ≤ k_{n}, define

A_{i}^{n} = diag(ε_{i,1}^{n}, …, ε_{i,d_{n}}^{n}) ∈ M_{n}

by

ε_{i,j}^{n} = 1  if j ∈ S_{i}^{n}
= 0 if j ∉ S_{i}^{n}.

Then we can define elements m_{η} ∈ ∏_{n} M_{n} for η ∈ ^{ω}2 such that the following conditions are satisfied.

(a) For each η ∈ ^{ω}2 and n ∈ ω, there exists 1 ≤ i ≤ k_{n} such that m_{η}(n) = A_{i}^{n}.

(b) If η, τ ∈ ^{ω}2 are distinct, then there exists ℓ_{0} ∈ ω such that (i) if n < ℓ_{0}, then m_{η}(n) = m_{τ} (n); and (ii) if n ≥ ℓ_{0}, then m_{η}(n) ≠ m_{τ} (n).

Since [∏_{n} M_{n} : M] < 2^{ω}, there exist distinct η, τ ∈ ^{ω}ω such that m = m_{η} − m_{τ} ∈ M. Fix some n ∈ ω such that n ≥ ℓ_{0}. Let m_{η}(n) = S_{i}^{n}, m_{τ} (n) = S_{j}^{n}, and T_{n} = S_{i}^{n} △ S_{j}^{n}. Let m(n) = diag(τ_{1}^{n}, …, τ_{d_{n}}^{n}). Then τ_{k}^{n} ≠ 0 iff k ∈ T_{n}. Remember that if P ∈ SL(d_{n}, q_{n}) is a permutation matrix, then P^{T} = P^{−1}. Since |T_{n}| ≥ d_{n}/4, we can find permutation matrices P_{1}, …, P_{4} ∈ SL(d_{n}, q_{n}) such that if

m_{i}(n) = P_{i}m(n)P_{i}^{−1} = diag(τ_{i,1}^{n}, …, τ_{i,d_{n}}^{n}) 1 ≤ i ≤ 4

then for each 1 ≤ k ≤ d_{n}, there exists 1 ≤ i ≤ 4 with τ_{i,k}^{n} ≠ 0. Choose a partition {1, …, d_{n}} = X_{1}^{n} ∪ ⋯ ∪ X_{4}^{n} such that for each 1 ≤ i ≤ 4

(i) d_{n}/4 ≤ |X_{i}^{n}| ≤ d_{n}/2; and (ii) X_{i}^{n} ⊆ {k | τ_{i,k}^{n} ≠ 0}.

INFINITE PRODUCTS OF FINITE SIMPLE GROUPS II 31

For each 1 ≤ i ≤ 4, fix some 1 ≤ k_{i} ≤ d_{n} such that τ_{i,k_{i}}^{n} = 0. For each α = (α_{1}, …, α_{d_{n}}) ∈ (GF (q_{n})^{∗})^{d_{n}}, let R_{i}^{n}(α) = diag(α_{i,1}, …, α_{i,d_{n}}), where

α_{i,k} = α_{k}  if k ≠ k_{i}
= ∏_{k≠k_{i}} α_{k}^{−1}  if k = k_{i}.

Then we have that

R_{i}^{n}(α)m_{i}(n)R_{i}^{n}(α)^{T} = diag(α_{1}^{2}τ_{i,1}^{n}, …, α_{d_{n}}^{2}τ_{i,d_{n}}^{n}).

Since p > 3, for each λ ∈ GF (q_{n}), there exist β_{1}, …, β_{4} ∈ GF (q_{n})^{∗} such that λ = ∑_{i=1}^{4} β_{i}^{2}. (For example, see Chapter 4 [20].) Now let D_{n} = diag(λ_{1}, …, λ_{d_{n}}) ∈ M_{n} be any diagonal matrix. For each 1 ≤ i ≤ 4, define D_{i}^{n} = diag(λ_{i,1}, …, λ_{i,d_{n}}) by

λ_{i,k} = λ_{k}  if k ∈ X_{i}^{n}
= 0  if k ∉ X_{i}^{n}.

Thus D_{n} = ∑_{i=1}^{4} D_{i}^{n}; and there exist β_{1}^{i}, …, β_{4}^{i} ∈ (GF (q_{n})^{∗})^{d_{n}} such that

D_{i}^{n} = ∑_{i=1}^{4} R_{i}^{n}(β_{1}^{i})m_{i}(n)R_{i}^{n}(β_{4}^{i})^{T} .

We have now shown that if n ≥ ℓ_{0}, then we can “uniformly generate” each diagonal matrix D_{n} ∈ M_{n} from the generator m(n). It follows that ∏_{n≥ℓ_{0}} M_{n} ⩽ M.

Next we will show that there exists an integer ℓ_{1} ≥ ℓ_{0} such that ∏_{n≥ℓ_{1}} H_{n} ⩽ H. Let LT_{n} be the subgroup of strictly lower triangular matrices in SL(d_{n}, q_{n}), and define

V_{n} = \{(\begin{matrix}Q & 0 \\ SQ & (Q^{−1})^{T} \end{matrix} ) | Q ∈ LT_{n}, S^{T} = S\} .

Note that for each A ∈ SL(d_{n}, q_{n}), we have that

(\begin{matrix}(A^{−1})^{T} & 0 \\ 0 & A \end{matrix}) (\begin{matrix}1 & 0 \\ S & 1 \end{matrix}) (\begin{matrix}A^{T} & 0 \\ 0 & A^{−1} \end{matrix}) = (\begin{matrix}1 & 0 \\ ASA^{T} & 1 \end{matrix}) .

Hence, arguing as above, we see that there exists an integer ℓ_{1} ≥ ℓ_{0} such that ∏_{n≥ℓ_{1}} V_{n} ⩽ H. The proof of Theorem 4.1 [16] shows that

∏_{n≥ℓ_{1}} H_{n} ⩽ ⟨ ∏_{n≥ℓ_{1}} U_{n}, ∏_{n≥ℓ_{1}} V_{n}⟩.

32 SIMON THOMAS

Finally we will show that there exists an integer ℓ_{2} ≥ ℓ_{1} such that ∏_{n≥ℓ_{2}} N_{n} ⩽ H. Using the Bruhat decomposition of Sp(2d_{n}, q_{n}), this implies that ∏_{n≥ℓ_{2}} Sp(2d_{n}, q_{n}) is contained in H, and hence completes the proof of Lemma 5.2. Again fix some n ∈ ω. Then W_{n} is generated by the images of the elements {w_{i} | 1 ≤ i ≤ d_{n}} of N_{n} defined as follows.

(a) If 1 ≤ i < d_{n}, then w_{i} is the permutation matrix corresponding to the permutation ( e_{i} e_{i+1} )( f_{i} f_{i+1} ).

(b) w_{d_{n}}(e_{d_{n}}) = −f_{d_{n}}, w_{d_{n}}(f_{d_{n}}) = e_{d_{n}} and w_{d_{n}} fixes the remaining elements of e f. (Thus w_{d_{n}} corresponds to the odd permutation ( e_{d_{n}} f_{d_{n}} ).)

It follows that W_{n} is isomorphic to ℤ_{2}^{d_{n}} ⋊ Sym(d_{n}), where ℤ_{2}^{d_{n}} is the natural permutation module for Sym(d_{n}). Since ∏_{n} G_{n} ⩽ H, it follows that ∏_{n≥ℓ_{1}} Sym(d_{n}) is included in the image of the canonical homomorphism from H ∩ ∏_{n≥ℓ_{1}} N_{n} into ∏_{n≥ℓ_{1}} W_{n}. Now let W_{n}^{+} be the subgroup of W_{n} corresponding to the even permutations of the set {e_{i}, f_{i} | 1 ≤ i ≤ d_{n}}. Then W_{n}^{+} corresponds to the subgroup E_{n}⋊Sym(d_{n}) of ℤ_{2}^{d_{n}} ⋊Sym(d_{n}), where E_{n} is the submodule of ℤ_{2}^{d_{n}} consisting of the vectors of even weight. Let N_{n}^{+} be the subgroup of N_{n} such that N_{n}^{+}/H_{n} = W_{n}^{+}.

Arguing as in the proof of Claim 5.3, we see that there exists an element v ∈ H ∩ ∏_{n} N_{n}^{+} such that for some integer ℓ_{2} ≥ ℓ_{1}

(a) if n < ℓ_{2}, then v(n) = 1; and

(b) if n ≥ ℓ_{2}, then v(n) corresponds to a vector in ℤ_{2}^{d_{n}} of weight s_{n} for some d_{n}/4 ≤ s_{n} ≤ d_{n}/2.

Note that if z ∈ ℤ_{2}^{d_{n}} is a vector of weight s_{n}, then for all u ∈ E_{n}, there exist π_{1}, …, π_{4} ∈ Sym(d_{n}) such that u = ∑_{i=1}^{4} π_{i}(z). It follows that ∏_{n≥ℓ_{2}} N_{n}^{+} ⩽ H. Since we also have that ∏_{n≥ℓ_{2}} B_{n} ⩽ H, the proof of Theorem 4.1 [16] shows that ∏_{n≥ℓ_{2}} N_{n} ⩽ H.

Next we will consider products ∏_{n} SU(d_{n}, q_{n}) of finite special unitary groups. If d_{n} is even, then the corresponding unitary space has a normal basis e f. Arguing as in the proof of Theorem 5.1, we find that the following result is also an easy consequence of Theorem 4.1.

Theorem 5.4. If (SU(2d_{n}, q_{n}) | n ∈ ω) is a sequence of finite special unitary groups such that d_{n} → ∞ as n → ∞, then ∏_{n} SU(2d_{n}, q_{n}) has the SIP.

INFINITE PRODUCTS OF FINITE SIMPLE GROUPS II 33

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Next consider a product of the form ∏_{n} SU(2d_{n} + 1, q_{n}). Then there exists a basis

e f (w) = (e_{i} | 1 ≤ i ≤ d_{n}) (f_{i} | 1 ≤ i ≤ d_{n}) (w)

of the corresponding unitary space such that

(e_{i}, f_{j}) = δ_{ij} and (e_{i}, e_{j}) = (f_{i}, f_{j}) = 0

for all 1 ≤ i, j ≤ d_{n}; and

(w, w) = 1 and (w, e_{i}) = (w, f_{i}) = 0

for all 1 ≤ i ≤ d_{n}. We can regard SU(2d_{n}, q_{n}) as the subgroup of SU(2d_{n} + 1, q_{n}) consisting of the elements π such that π(w) = w. Arguing as in the proof of Theorem 4.4 [16], we obtain the following result.

Lemma 5.5. Suppose that d_{n} → ∞ as n → ∞ and that the group H satisfies the following conditions.

(a) ∏_{n} SU(2d_{n}, q_{n}) ⩽ H ⩽ ∏_{n} SU(2d_{n} + 1, q_{n}).

(b) [∏_{n} SU(2d_{n} + 1, q_{n}) : H] < 2^{ω}. Then H is an open subgroup of ∏_{n} SU(2d_{n} + 1, q_{n}).

□

Using Theorem 5.4 and Lemma 5.5, we easily obtain the following result.

Theorem 5.6. If (SU(2d_{n} + 1, q_{n}) | n ∈ ω) is a sequence of finite special unitary groups such that d_{n} → ∞ as n → ∞, then ∏_{n} SU(2d_{n} + 1, q_{n}) has the SIP.

□

Finally we will consider products of finite orthogonal groups. First consider the case when each group has the form Ω^{+}(2d, q). Then the corresponding orthogonal space has a normal basis e f. Arguing as in the proof of Theorem 5.1, we obtain the following result.

Theorem 5.7. If (Ω^{+}(2d_{n}, q_{n}) | n ∈ ω) is a sequence of finite orthogonal groups such that d_{n} → ∞ as n → ∞, then ∏_{n} Ω^{+}(2d_{n}, q_{n}) has the SIP.

34 SIMON THOMAS

□

Now we will consider products of the form ∏_{n} Ω(2d_{n} + 1, q_{n}), where d_{n} ≥ 2 for each n ∈ ℕ. Fix some n ∈ ℕ. Let Q be the quadratic form on the corresponding orthogonal space, and let (u, v) = Q(u+v)−Q(u)−Q(v) be the associated bilinear map. We can suppose that there exists a basis

e f (w) = (e_{i} | 1 ≤ i ≤ d_{n}) (f_{i} | 1 ≤ i ≤ d_{n}) (w)

of the orthogonal space such that

(e_{i}, f_{j}) = δ_{ij} and (e_{i}, e_{j}) = (f_{i}, f_{j}) = Q(e_{i}) = Q(f_{i}) = 0

for all 1 ≤ i, j ≤ d_{n}; and

Q(w) = 1 and (w, e_{i}) = (w, f_{i}) = 0

for all 1 ≤ i ≤ d_{n}. Then we can regard Ω^{+}(2d_{n}, q_{n}) as the subgroup of Ω(2d_{n}+1, q_{n}) consisting of the elements π such that π stabilises the subspace ⟨e_{i}, f_{i} | 1 ≤ i ≤ d_{n}⟩ setwise and π(w) = w. Arguing as in the proof of Theorem 4.7 [16], we obtain the following result.

Lemma 5.8. Suppose that such that d_{n} → ∞ as n → ∞ and that the group H satisfies the following conditions.

(a) ∏_{n} Ω^{+}(2d_{n}, q_{n}) ⩽ H ⩽ ∏_{n} Ω(2d_{n} + 1, q_{n}).

(b) [∏_{n} Ω(2d_{n} + 1, q_{n}) : H] < 2^{ω}. Then H is an open subgroup of ∏_{n} Ω(2d_{n} + 1, q_{n}).

□

Using Theorem 5.7 and Lemma 5.8, we easily obtain the following result.

Theorem 5.9. If (Ω(2d_{n} + 1, q_{n}) | n ∈ ω) is a sequence of finite orthogonal groups such that d_{n} → ∞ as n → ∞, then ∏_{n} Ω(2d_{n} + 1, q_{n}) has the SIP.

□

It only remains to consider products of the form ∏_{n} Ω^{−}(2d_{n} + 2, q_{n}). In this case, there exists a basis e f (w, z) of the corresponding orthogonal space such that

(e_{i}, f_{j}) = δ_{ij} and (e_{i}, e_{j}) = (f_{i}, f_{j}) = Q(e_{i}) = Q(f_{i}) = 0

for all 1 ≤ i, j ≤ d_{n}; and

(w, e_{i}) = (w, f_{i}) = (z, e_{i}) = (z, f_{i}) = 0

for all 1 ≤ i ≤ d_{n}; and the subspace ⟨w, z⟩ does not contain any singular vectors. So we can regard Ω^{+}(2d_{n}, q_{n}) as the subgroup of Ω^{−}(2d_{n} + 2, q_{n}) consisting of the elements π such that π stabilises the subspace ⟨e_{i}, f_{i} | 1 ≤ i ≤ d_{n}⟩ setwise, π(w) = w and π(z) = z. While the proof of the following lemma is closely based on that of Theorem 4.8 [16], it contains one significant difference. For this reason, we have provided a complete proof.

Lemma 5.10. Suppose that d_{n} → ∞ as n → ∞ and that the group H satisfies the following conditions.

(a) ∏_{n} Ω^{+}(2d_{n}, q_{n}) ⩽ H ⩽ ∏_{n} Ω^{−}(2d_{n} + 2, q_{n}).

(b) [∏_{n} Ω^{−}(2d_{n} + 2, q_{n}) : H] < 2^{ω}. Then H is an open subgroup of ∏_{n} Ω^{−}(2d_{n} + 2, q_{n}).

Proof. We can suppose that (d_{n} | n ∈ ω) ∈ D(^{ω}ω). Once again, we will make use of the Bruhat decomposition

Ω^{−}(2d + 2, q) = ⋃_{w∈W}BwB,

where B is a Borel subgroup and W is the Weyl group. Fix some n ∈ ω. For each 1 ≤ i ≤ d_{n}, let E_{i} = ⟨e_{1}, …, e_{i}⟩. Then the stabiliser B_{n} of the flag of totally singular subspaces

E_{1} ⩽ E_{2} ⩽ ⋯ ⩽ E_{d_{n}}

is a Borel subgroup of Ω^{−}(2d_{n} + 2, q_{n}). Let N_{n} be the subgroup of Ω^{−}(2d_{n} + 2, q_{n}) which stabilises the polar frame {⟨e_{i}⟩, ⟨f_{i}⟩ | 1 ≤ i ≤ d_{n}}. Then the Weyl group of Ω^{−}(2d_{n}+2, q_{n}) is W_{n} = N_{n}/B_{n}∩N_{n}; and W_{n} corresponds to the permutation group ℤ_{2}^{d_{n}} ⋊ Sym(d_{n}) acting on the set {e_{i}, f_{i} | 1 ≤ i ≤ d_{n}} with blocks of imprimitivity {e_{i}, f_{i}} for 1 ≤ i ≤ d_{n}. The main point is to show that there exists an integer ℓ ∈ ω such that ∏_{d_{n}≥ℓ} B_{n} ⩽ H. Once this is done, the final argument in the proof of Lemma 5.2 shows that there exists an integer ℓ_{1} ≥ ℓ such that ∏_{n≥ℓ_{1}} N_{n} ⩽ H. Hence H is an open subgroup of ∏_{n} Ω^{−}(2d_{n} + 2, q_{n}).

36 SIMON THOMAS

Let U_{n} be the subgroup of unipotent elements of B_{n} and let H_{n} = B_{n} ∩ N_{n}; so that B_{n} = U_{n} ⋊ H_{n}. First we will show that there exists an integer ℓ ∈ ω such that ∏_{n≥ℓ} U_{n} ⩽ H. Note that if π ∈ U_{n}, then there exist vectors x, y ∈ E_{d_{n}} such that π(w) = w + x and π(z) = z + y. Let E_{d_{n}}^{+} = ⟨E_{d_{n}}, w, z⟩ and let Γ_{n} be the setwise stabiliser of E_{d_{n}}^{+} in Ω^{−}(2d_{n} + 2, q_{n}). Let ρ : Γ_{n} → GL(E_{d_{n}}^{+}) be the restriction map. We regard GL(E_{d_{n}}^{+}) as a group of matrices with respect to the ordered basis (e_{1}, …, e_{d_{n}}, w, z). So for each A ∈ SL(d_{n}, q_{n}), we have that

(\begin{matrix}A & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) ∈ ρ [Ω^{+}(2d_{n}, q_{n}) ∩ Γ_{n}] .

Using the fact that [∏_{n} Ω^{−}(2d_{n} + 2, q_{n}) : H] < 2^{ω}, it is easily shown that there exists an element v ∈ H ∩ ∏_{n} U_{n} such that the following conditions are satisfied for some ℓ ∈ ω.

(a) If n < ℓ, then v(n) = 1.

(b) If n ≥ ℓ, then there exist linearly independent vectors x_{n}, y_{n} ∈ E_{d_{n}} such that

ρ(v(n)) = (\begin{matrix}1 & x_{n} & y_{n} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) .

Fix some integer n ≥ ℓ. Note that for each A ∈ SL(d_{n}, q_{n}), we have that

(\begin{matrix}A & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix}1 & x_{n} & y_{n} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix}A^{−1} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) = (\begin{matrix}1 & Ax_{n} & Ay_{n} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) .

Claim 5.11. Suppose that d ≥ 3. Let S = SL(d, q) and V = V (d, q). Let S act on V × V via the action A(x, y) = (Ax, Ay). Suppose that x, y ∈ V are linearly independent. Then for all (a, b) ∈ V × V, there exist A, B ∈ S such that (a, b) = A(x, y) + B(x, y).

Proof of Claim 5.11. Left to the reader.

Now let φ ∈ U_{n} be any element. Then

ρ(φ) = (\begin{matrix}C & a & b \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix})

for some a, b ∈ E_{d_{n}} and some strictly upper triangular matrix C. By Claim 5.11, there exist A, B ∈ SL(d_{n}, q_{n}) such that

(\begin{matrix}1 & −a & −b \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) = (\begin{matrix}1 & Ax_{n} & Ay_{n} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix}1 & Bx_{n} & By_{n} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) .

It follows that there exist elements ψ_{1}, ψ_{2} ∈ Ω^{+}(2d_{n}, q_{n}) ∩ Γ_{n} such that

ψ_{1}v(n)ψ_{1}^{−1}ψ_{2}v(n)ψ_{2}^{−1}φ ∈ Ω^{+}(2d_{n}, q_{n}).

Hence ∏_{n≥ℓ} U_{n} ⩽ H.

Finally we will show that ∏_{n≥ℓ} H_{n} ⩽ H. We will regard Ω^{−}(4, q_{n}) as the subgroup of Ω^{−}(2d_{n} + 2, q_{n}) consisting of the elements π such that π stabilises the subspace ⟨e_{d_{n}}, f_{d_{n}}, w, z⟩ setwise and such that π(e_{i}) = e_{i} and π(f_{i}) = f_{i} for all 1 ≤ i ≤ d_{n} − 1. Since ∏_{n≥ℓ} Ω^{+}(2d_{n}, q_{n}) ⩽ H, it is enough to show that ∏_{n≥ℓ} (H_{n} ∩ Ω^{−}(4, q_{n})) ⩽ H. We will make use of the fact that

Ω^{−}(4, q_{n}) ≃ SL(2, q_{n}^{2})/{±1}.

(For example, see [21, 12.42].) Let p_{n} = char(GF (q_{n})). Then U_{n} ∩ Ω^{−}(4, q_{n}) is a group of order q_{n}^{2}, and hence is a Sylow p_{n}-subgroup of Ω^{−}(4, q_{n}). It is easily checked that SL(2, q_{n}^{2}) is “uniformly generated” by the two Sylow p_{n}-subgroups UT (2, q_{n}^{2}) and LT (2, q_{n}^{2}), consisting of the strictly upper triangular matrices and the strictly lower triangular matrices of SL(2, q_{n}^{2}). (See [2, 6.4.4].) Since SL(2, q_{n}^{2}) acts 2-transitively via conjugation on the set of its Sylow p_{n}-subgroups, it follows that any two of its Sylow p_{n}-subgroups “uniformly generate” SL(2, q_{n}^{2}). So the same is true of Ω^{−}(4, q_{n}). There exists an element w_{n} ∈ Ω^{+}(2d_{n}, q_{n}) ∩ N_{n} such that

(i) w_{n}Ω^{−}(4, q_{n})w_{n}^{−1} = Ω^{−}(4, q_{n}); and (ii) w_{n} (U_{n} ∩ Ω^{−}(4, q_{n})) w_{n}^{−1} ≠ U_{n} ∩ Ω^{−}(4, q_{n}). It follows that ∏_{n≥ℓ} Ω^{−}(4, q_{n}) ⩽ H. Thus ∏_{n≥ℓ} H_{n} ⩽ H and so ∏_{n≥ℓ} B_{n} ⩽ H.

Using Theorem 5.7 and Lemma 5.10, we easily obtain the following result.

Theorem 5.12. If (Ω^{−}(2d_{n} + 2, q_{n}) | n ∈ ω) is a sequence of finite orthogonal groups such that d_{n} → ∞ as n → ∞,, then ∏_{n} Ω^{−}(2d_{n} + 2, q_{n}) has the SIP.

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References

[1] J. L. Brenner, Covering theorems for Finasigs VIII – Almost all conjugacy classes in A_{n} have exponent ≤ 4, J. Austral. Math. Soc. 25 (1978), 210–214.

[2] R. W. Carter, Simple Groups of Lie Type, Wiley, 1972.

[3] C. Chevalley, Introduction to the theory of algebraic functions of one variable, Amer. Math. Soc., New York, 1951.

[4] J. D. Dixon, P. M. Neumann and S. Thomas, Subgroups of small index in infinite symmetric groups, Bull. London Math. Soc. 18 (1986), 580–586.

[5] O. Endler, Valuation Theory, Springer-Verlag, 1972.

[6] D. M. Evans, Subgroups of small index in infinite general linear groups, Bull. London Math. Soc. 18 (1986), 587–590.

[7] D. M. Evans, The small index property for infinite dimensional classical groups, J. Algebra 136 (1991), 248–264.

[8] C. Gourion, A propos du groupe des automorphismes de (ℚ; ≤), C. R. Acad. Sci. Paris 315, Série I (1992), 1329–1331.

[9] W. Hodges, I. Hodkinson, D. Lascar and S. Shelah, The small index property for ω-stable ω-categorical structures and for the random graph, J. London Math. Soc. (2) 48 (1993), 204–218.

[10] H. Koch, Galoissche Theorie der p-Erweiterungen, Mathematische Monographien 10, Berlin 1970.

[11] S. Koppelberg and J. Tits, Une propriété des produits directs infinis groupes finis isomorphes, C. R. Acad. Sci. Paris 279, Série A (1974), 583–585.

[12] R. Lidl and H. Niederreiter, Introduction to finite fields and their applications, Cambridge University Press, 1986.

[13] H. D. Macpherson and P. M. Neumann, Subgroups of infinite symmetric groups, J. London Math. Soc. (2) 42 (1990), 64–84.

[14] F. Point, Ultraproducts of Chevalley groups, preprint.

[15] F. J. Rayner, An algebraically closed field, Glasgow Math. J. 9 (1968), 146–151.

[16] J. Saxl, S. Shelah and S.Thomas, Infinite products of finite simple groups, Trans. Amer. Math. Soc. 348 (1996), 4611–4641.

[17] J. Saxl and J. S. Wilson, A note on powers in simple groups, Math. Proc. Cambridge Phil. Soc. 122 (1997), 91–94.

[18] S. W. Semmes, Endomorphisms of infinite symmetric groups, Abstracts Amer. Math. Soc. 2 (1981), 426.

[19] J. P. Serre, Trees, Springer-Verlag, 1980.

[20] C. Small, Arithmetic of finite fields, Marcel Dekker, 1991.

[21] D. E. Taylor, The Geometry of the Classical Groups, Sigma Series in Pure Mathematics, Vol. 9, Heldermann Verlag, Berlin, 1992.

[22] J. K. Truss, Infinite permutation groups II: subgroups of small index, J. Algebra 120 (1989), 494–515.

[23] D. Zelinsky, Every linear transformation is a sum of non-singular ones, Proc. Amer. Math. Soc 5 (1954), 627–630.

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