INFINITE SUBGROUPS OF MORAVA STABILIZER GROUPS

V. GORBOUNOV, M. MAHOWALD, AND P. SYMONDS

Abstract. In this note we discuss certain infinite subgroups of the Morava stabilizer groups and outline some applications in homotopy theory.

1. Description of the main result and its applications

First we discuss a theorem about the structure of the group of proper units of the maximal order in a certain class of cyclic division algebras over a local field. This theorem states that such a group contains a free product of finite cyclic subgroups which is dense in the subgroup of elements of norm one. We believe this result should be of interest not only to topologists.

The importance of the above kind of groups for homotopy theory stems from Morava’s work [11] in which he outlined a general algebraic way of studying the homotopy category. The major ingredient in this outline is the chromatic filtration of the homotopy category [13], [14]. The terms of this filtration can be described by the Bousfield localization functors L_{K(n)}: for more details see [6]. An algebraic description of these functors can be given using certain p-adic Lie groups (the so-called Morava stabilizer groups 𝕊_{n}) which are defined for each prime p and each positive integer n. The group 𝕊_{n} can be described as the group of units in the maximal order of the cyclic division algebra over ℚ_{p} of index n with Hasse invariant \frac{1}{n}. Each group 𝕊_{n} comes equipped with a representation θ_{n} in the group of automorphisms of a certain ring E_{n}. The cohomology of 𝕊_{n} with coefficients in this representation is related (via a spectral sequence which often collapses) to the homotopy groups of the Bousfield localization L_{K(n)} of the sphere, which in turn approximate the homotopy groups of spheres.

The group 𝕊_{n} has another algebraic characterization. 𝕊_{n} may be viewed as the group of automorphisms of a certain p-typical formal group F of height n over the algebraic closure of a finite field of p elements. The representation θ_{n} is defined by the canonical action of 𝕊_{n} on the moduli space of deformations of F, which is E_{n}.

Among the applications of the above theorem to homotopy theory are the following. For any prime p the theorem we prove implies an interesting description of the Morava stabilizer group 𝕊_{p−1} and gives a conceptual explanation of some of Ravenel’s computations of the cohomology of this group with trivial coefficients. It also suggests an approach to the computation of the cohomology of the Morava stabilizer group 𝕊_{2} for the prime 3. This has been done in the joint paper by V. Gorbounov, S. Siegel and P. Symonds [8].

This last result is important for the following reasons. The functor L_{K(1)} is completely understood for all primes. The functor L_{K(2)} is currently a topic of intense study. It appears to be closely related to the theory of elliptic curves. The p-typical formal group F, which has the Morava group 𝕊_{2} as its ring of automorphisms, is actually the formal group of a supersingular elliptic curve for the primes 2 and 3. The work of a number of people in algebraic geometry (see for example [9]) suggests a close relation between the cohomology of 𝕊_{2} for the primes 2 and 3 with coefficients in the representation θ_{n} and algebraic properties of the moduli scheme of elliptic curves.

A way to describe this relation in topological terms was suggested recently by M. Hopkins and H. Miller. They constructed the so-called higher real K-theories. Let us describe these briefly. They proved that the Morava group 𝕊_{n} acts in a “nice” way on a certain spectrum E_{n}, so one can talk about the homotopy fixed point spectrum E^{H} of this action for any discrete subgroup H of 𝕊_{n}. A description of the finite subgroups of 𝕊_{n} is known due to S. Amitsur and T. Hewett [1, 5]. Following M. Hopkins and H. Miller, take H to be a maximal finite subgroup of 𝕊_{p−1} and call E^{H} the spectrum of the higher real K-theory for the given n and the prime p. They denote it by EO_{p−1}. For the primes 2 and 3 these maximal finite subgroups happen to be the groups of automorphisms of appropriate supersingular elliptic curves over algebraically closed fields of characteristics 2 and 3, respectively. Thus the construction of EO_{2} at p = 2, 3 suggests that this spectrum is related to the moduli scheme of elliptic curves. The homotopy groups of the Bousfield localization of the sphere, on the other hand, are related to the homotopy groups of EO_{2}. This is how the theory of moduli schemes for elliptic curves should merge into homotopy theory.

Using our results about infinite subgroups of the Morava group we describe an infinite subgroup L of 𝕊_{2} such that H < L < 𝕊_{2}. The spectrum E^{L} is hopefully a better approximation to L_{K(2)}. On the other hand, it is plausible that the group L is the group of automorphisms of an algebraic surface. An appropriate candidate is a K3 surface. If this is the case then one has a chance to use information about the moduli space of K3 surfaces for further study of the Bousfield localization functor L_{K(2)}.

2. Structure of p-adic division algebras

We consider a cyclic algebra 𝔻 over ℚ_{p} of index n with Hasse invariant \frac{1}{n}. We give a brief description here: for more details see [15]. Let 𝕎 be the totally unramified extension of ℚ_{p} of degree n (so 𝕎 ≅ ℚ_{p}[ζ] where ζ is a (p^{n} − 1)-st root of unity). The Galois group of the extension 𝕎/ℚ_{p} is a cyclic group of order n. It is generated by the Frobenius homomorphism σ. We form the crossed product algebra of 𝕎 and Gal(𝕎/ℚ_{p}). This amounts to introducing a variable S which commutes with 𝕎 according to the formula:

wS = Sw^{σ}, w ∈ 𝕎

and S^{n} belongs to ℚ_{p}^{×}. To define 𝔻 we set S^{n} = p. 𝔻 is a division algebra over ℚ_{p} of rank n^{2}. It has a faithful representation

μ: 𝔻 → GL_{n}(𝕎).

If

a = ∑_{i=0}^{n−1} a_{i}S^{i}

then μ(a) is the following matrix:

a_{i+1,j+1} = \begin{cases}a_{j−i}^{σ^{i}} & if i ≤ j \\ pa_{j+n−i}^{σ^{i}} & if i > j \end{cases}

Let det: GL_{n}(𝕎) → ℚ_{p}^{×} be the determinant map.

Let 𝕆 be the maximal order in 𝔻 : it is generated by S and the integers of 𝕎. We are interested in three subgroups of 𝕆. The first is the group of units of 𝕆, which we denote by 𝕊_{n}. The second is the subgroup of strict units in 𝕊_{n}. It consists of the elements a ∈ 𝕊_{n} such that a ≡ 1 mod S. We denote it by 𝕊_{n}^{0}. The third is the kernel of the reduced norm, nrd = det∘μ, restricted to 𝕊_{n}^{0}. We denote this subgroup by 𝕊l. Both 𝕊_{n}^{0} and 𝕊l are pro-p-groups.

Let us recall some facts about the generators and cohomology of pro-p-groups. A set of elements g_{1}, …, g_{n} ∈ G is said to generate G if the subgroup that they generate algebraically is dense in the pro-p topology. Let G^{∗} denote the minimal closed normal subgroup that contains all the commutators and all the p-th powers.

Theorem 2.1. [10], [18, Prop I-25] Let G be a pro-p-group and let g_{1}, …, g_{n} be a subset of G. Then g_{1}, …, g_{n} generate G if and only if they generate G/G^{∗}.

For the pro-p-group 𝕊l the map 𝕊l → 𝕊l/𝕊l^{∗} admits the following elegant description due to [16, Thms 7, 21]. Let H_{i} denote the subgroup of 𝕊l consisting of elements congruent to 1 modulo S^{i}. Note that H_{1} = 𝕊l and H_{2} = 𝕊l^{∗} = [𝕊l, 𝕊l]. For any field k let k^{+} denote its additive group. Then 𝕊l/(S) ≅ 𝔽_{p^{n}}^{+} ≅ 𝕎/(p). Any element a ∈ 𝕊l can be written uniquely as 1 + a_{1}S where a_{1} ∈ 𝕆, and the quotient of 𝕊l by its commutator is isomorphic to 𝔽_{p^{n}}^{+}, via the map ρ: 𝕊l → 𝔽_{p^{n}}^{+} given by the formula

ρ(a) ≡ a_{1} modS ∈ 𝔽_{p^{n}}^{+}

Assume now that n is equal to p−1. It is known [15] that 𝔻 contains any extension of ℚ_{p} of degree p − 1. In particular it contains a p-th root of unity, say z. Since ℚ_{p} does not contain p-th roots of unity different from 1 we conclude that nrd z = 1, so z belongs to 𝕊l. It is also known that 𝔻 contains a (p − 1)-st root of −p, e.g. ω = ζ^{(p−1)/2}S. The following fact from number theory will be useful for us later: ℚ_{p}(ω) ≅ ℚ_{p}(z) [19].

Our main theorem is

Theorem 2.2. There is a p-th root of unity α ∈ 𝕊l such that

(1) {α^{ω^{i}}, 1 ≤ i ≤ p−1} generate an infinite subgroup G of 𝕊l, which is isomorphic to a free product of p − 1 copies of ℤ/p: ℤ/p ∗ ⋯ ∗ ℤ/p

(2) G is dense in 𝕊l.

Remark 2.3. By similar methods we can show that if n = (p−1)p^{r} then 𝕊l contains a free product of n copies of ℤ/p^{(r+1)} which is dense. If n = (p−1)p^{r}a with (a, p) = 1 then all the torsion elements are contained in H_{a}. (If (p − 1) |̸n then 𝕊l is torsion free.)

The proof of the main theorem is in two parts. First we show that “generically” a pair of subgroups of 𝔻 generates a free product, and then we show that 𝕊l is generated by torsion elements.

3. Free products in the group of units of an algebra

We say that two subgroups G and H of a group X generate a projective free product if all the relations in ⟨G, H⟩_{X} come either from G or H or from the fact that G∩Z(X) and H ∩ Z(X) must be central (where Z(X) denotes the centre of X). Equivalently, the images of G and H in X/Z(X) generate a free product.

Theorem 3.1. Let k be a field, A a finitely generated k-algebra and G and H two countable subgroups of A^{×}. Suppose that there is a field extension K of k and an element x ∈ A ⊗_{k} K^{×} such that G and H^{x} generate a projective free product in A ⊗_{k} K^{×}. Then the set of x ∈ A^{×} for which G and H^{x} do not generate a free product all lie within a countable family of subvarieties of A.

Remark 3.2. A is naturally an affine space over k. By subvariety of A we mean the zeros of some non trivial polynomials in the coordinates with coefficients in k.

The hypotheses of the theorem remain in force for the corollaries.

Corollary 3.3. If k is uncountable then the set of x which do yield a projective free product is not empty.

Corollary 3.4. If k is either ℝ or ℂ or a p-adic field then the set of X which do yield a projective free product is dense (in the k-topology). Its complement has Haar measure 0.

Corollary 3.5. Suppose that R is a discrete valuation ring with countable residue class field and quotient field k, and Λ is an R-order in A. Let p denote a generator of the prime ideal of R. Then for any positive integer r there is an x ∈ 1 + p^{r}Λ ⊆ Λ^{×} which yields a projective free product.

Proof. The set of reduced words in the projective free product of G and H is a countable set. Each word is of the form w = g_{1}h_{1}g_{2}h_{2} · · ·. Let r(w) be the number of occurences of an element of H. Define a function ϕ_{w}: A ⊗_{k} K^{×} → A ⊗_{k} K^{×} by ϕ_{w}(x) = g_{1}h_{1}^{x}g_{2}h_{2}^{x} . . .. We are interested in the equation ϕ_{w}(x) = 1, which expresses a relation in the group ⟨G, H^{x}⟩. By hypothesis this equation is not satisfied by all x ∈ A ⊗_{k} K^{×} unless w represents 1 in the projective free product.

It is now convenient to regard A as embedded in an algebra M of matrices over k. Let adj(x) denote the classical adjoint matrix of x (so adj(x) = det(x)x^{−1}) and define hx̂ = adj(x)hx, ϕ̂_{w}(x) = g_{1}hx̂ ; 1g_{2}hx̂ ; 2 · · ·. The equation ϕ_{w}(x) = 1 can be rewritten as ϕ̂_{w}(x) = det(x)^{r(w)}, but the latter can be regarded as a system of polynomial equations (no denominators involving the determinant) over k on the affine space M ⊗_{k} K. By hypothesis this system is non trivial on A ⊗_{k} K so defines a subvariety A_{w} of A for all non-identity w.

If x ∈ A^{×} is not contained in any A_{w} then, by construction, G and H^{x} generate a projective free product.

Proof of corollaries:

3.3: Suppose the contrary. Then A is the union of the subvarieties A_{w} and det = 0, contradicting Lemma 1.6 below.

3.4: This follows from the k-analytic structure [2].

3.5: Let {a_{i}}_{i∈I} be a set of representatives of the residue classes of Λ modulo p^{r}. If the subvarieties A_{w} and det = 0 cover 1 + p^{r}Λ, then the subvarieties a_{i}p^{r}A_{w}, i ∈ I, r ∈ ℤ and det = 0 cover A. This leads to a contradiction as in the proof of 3.4.

Lemma 3.6. If k is an uncountable field, then 𝔸^{n}(k) is not the union of a countable family of subvarieties.

Proof. Use induction on n: the case n = 1 is clear since k is uncountable. It is sufficient to assume that each subvariety is given by just one polynomial. Now the equation of a hyperplane divides any polynomial which is trivial when restricted to it, so a non-trivial polynomial can only be trivial on finitely many hyperplanes. This enables us to choose a hyperplane on which none of the polynomials are trivial, and we apply the induction hypothesis to it.

For our applications we need to construct projective free products in some extension field. Our main tool for this purpose is Nagao’s theorem [12], [17], that for any field K, GL_{2}(K[t]) = GL_{2}(K) ∗_{B(K)} B(K[t]), where B denotes the upper triangular matrices. In fact we can quotient out the centre K^{×} to obtain P GL_{2}(K[t]) = P GL_{2}(K) ∗_{P B(K)} P B(K[t]).Thus if G is a subgroup of GL_{2}(K) and H is a subgroup of B(K[t]) such that G ∩ B(K) ⊆ K^{×} and H ∩ B(K) ⊆ K^{×} then G and H generate a projective free product [[17], ch I, 1.1].

We are particularly interested in the division algebra 𝔻 of section 2, and its maximal order 𝕆. Recall that 𝕆 contains an element ω such that ω^{p−1} = −p, which we shall keep fixed. Let a ∈ 𝕆 be any primitive p-th root of unity.

Proposition 3.7. There is a conjugate α of a in 𝕆^{×} arbitrarily close to a such that ⟨α⟩ and ⟨ω⟩ generate a projective free product in 𝕆^{×}.

Corollary 3.8. 𝕆^{×} contains the free product ⟨α⟩ ∗ ⟨α^{ω}⟩ ∗ ... ∗ ⟨α^{ω^{p−1}}⟩.

Proof of proposition. By Corollary 3.5 applied to ⟨a⟩ and ⟨ω⟩, we only need to find such a projective free product over some extension field K of ℚ_{p}. Let K = ℚ(λ, μ, η), where λ is a primitive p-th root of unity, μ is a primitive (p − 1)-st root of unity, and η is a (p − 1)-st root of −p. K is a splitting field for A, so A ⊗_{ℚ_{p}} K ≅ M_{p−1}(K), the (p−1)×(p−1) matrices over K, and both a and ω are diagonalisable in M_{p−1}(K). It will suffice to find two matrices x and y such that x (resp. y) has the same eigenvalues as a (resp. ω) and also ⟨⟨x⟩, ⟨y⟩⟩ is a projective free product. For then a = x^{u} and ω = y^{v} for some u, v ∈ GL_{p−1}(K) and ⟨⟨a^{u^{−1}v}⟩, ⟨ω⟩⟩ = ⟨⟨x⟩, ⟨y⟩⟩^{v} is a projective free product.

Now a has eigenvalues λ^{i}, 1 ≤ i ≤ p − 1, and S has eigenvalues ημ^{j}, 0 ≤ j ≤ p − 2. Let

x′′ = (\begin{matrix}λ & 0 \\ 0 & λ^{2} \end{matrix}) , c = (\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix}) , x′ = x′′^{c}.

Then

x′^{r} = (\begin{matrix}λ^{r} & 0 \\ λ^{2r} − λ^{r} & λ^{2r} \end{matrix}) , so ⟨x′⟩ ∩ B(K) = 1.

Let

y′′ = η (\begin{matrix}1 & 0 \\ 0 & μ \end{matrix}) , d = (\begin{matrix}1 & t \\ 0 & 1 \end{matrix}) , y′ = y′′^{d}.

Then

y′^{r} = η^{r} (\begin{matrix}1 & t(1 − μ^{r}) \\ 0 & μ^{r} \end{matrix}) , so ⟨y′⟩ ∩ B(K) = ⟨y′^{p−1}⟩ = ⟨p⟩.

So by Nagao’s theorem, as explained above, ⟨x′⟩ and ⟨y′⟩ generate a projective free product in GL_{2}(K[t]). Now extend x′ and y′ to (p − 1) × (p − 1) matrices x and y by adding the other eigenvalues of a or S along the diagonal. Finally, K[t] can be embedded in a p-adic completion of K by sending t to some transcendental over K.

4. Generation by torsion elements

The isomorphism class of a crossed product algebra such as 𝔻 depends only on the p-adic valuation of S^{p−1}. We could instead have used a generator T with T^{p−1} = −p. (In fact this new algebra is isomorphic to 𝔻 by sending T to ω = ζ^{\frac{p−1}{2}} S. It will be convenient in this section to use this description of 𝔻 and the corresponding map ρ : 𝕊l → 𝔽_{p^{p−1}}^{+}. (If a = 1 + a_{1}T, a_{1} ∈ 𝕆, then ρ_{T} (a) = a_{1} mod T.) Note that ρ_{T} (a) = ρ(a)ζ^{\frac{1−p}{2}} and ρ_{T} (a^{ω}) = ρ_{T} (a)^{σ}.

In order to prove the Main Theorem we need to show that we can find a p-th root of unity a ∈ 𝕊l of such that a, a^{ω}, …, a^{ω^{p−1}} generate 𝕊l/𝕊l^{∗}, i.e. such that ρ_{T} (a), .., ρ_{T} (a^{ω^{p−1}}) form a basis for 𝔽_{p^{p−1}} over 𝔽_{p}. In other words, in view of the the formula ρ_{T} (a^{ω}) = ρ_{T} (a)^{σ} above, ρ_{T} (a) and its conjugates should form a normal basis of 𝔽_{p^{p−1}}/𝔽_{p}.

Lemma 4.1. The image under ρ_{T} : 𝕊l → 𝔽_{p^{p−1}} of the p-th roots of unity consists of the elements of the form u^{p−1}. (These are the elements of norm 1.)

Proof. As mentioned in Section 2, ℚ_{p}(T) contains a pth root of unity z. Now ρ_{T} (z) ∈ 𝔽_{p}, and since ρ_{T} (z^{r}) = rρ_{T} (z), we may assume that ρ_{T} (z) = 1. It is a consequence of the Skolem-Noether Theorem that any p-th root of unity in 𝔻 is of the form z^{t}, t ∈ 𝔻^{×}. We may write t = T^{r}u(1 + vT), u ∈ 𝕎 ∩ 𝕆, v ∈ 𝕆. Then ρ_{T} (z^{t}) = u^{p−1} mod p.

Lemma 4.2. The extension 𝔽_{p^{p−1}}/𝔽_{p} has a normal basis of the the form u^{p−1}.

Proof. The characteristic polynomial of σ factorizes over 𝔽_{p}, so 𝔽_{p^{p−1}} decomposes over 𝔽_{p} into eigenspaces for σ. If r is a generator of the multiplicative group 𝔽_{p}^{×}, then the eigenvectors are e_{r}, e_{r}^{2}, .., e_{r}^{p−1} of eigenvalues r, r^{2}, .., r^{p−1} respectively, and they form a basis. Note that σe_{r} = e_{r}^{p−1} by definition of σ, and σe_{r} = re_{r} by definition of e_{r}, so e_{r}^{p−1} = r. The group algebra 𝔽_{p}⟨σ⟩ contains idempotents which project onto each eigenspace, thus x, x^{σ}, .., x^{σ^{p−1}} form a basis if and only if, when we write x = ∑_{i=1}^{p−1} λ_{i}e_{r}^{i}, no λ_{i} is 0.

Now

(1 + e_{r})^{p−1} = 1 + \binom{p − 1}{1}e_{r} + \binom{p − 1}{2}e_{r}^{2} + .. + e_{r}^{p−1}.

Since e_{r}^{p−1} = r, this shows that (1 + e_{r})^{p−1} and its conjugates form a normal basis provided that 1 + r ≠ 0 ie. p ≠ 3. If p = 3 there are four elements of norm 1, so they are not all contained in 𝔽_{3}.

Remark 4.3. If we had used S instead of T, then the pth roots of unity would have had image under ρ of norm −1.

By Corollary 3.8, there is an x ∈ 𝕊l such that α = x^{−1}ax gives rise to a free product G = ⟨α⟩ ∗ ⟨α^{ω}⟩ ∗ … ∗ ⟨yα^{ω^{p−1}}⟩ and ρ(α) = ρ(a). The inclusion of G in 𝕊l induces an isomorphism G/G^{∗} ≅ 𝕊l/𝕊l^{∗}. By proposition 2.1 this implies that G is dense in 𝕊l. This proves 2.2 completely.

Also, since the cohomology of G is generated by H^{1} and its Bocksteins, we have the following easy proposition about the cohomology on continuous cochains of 𝕊l.

Proposition 4.4. There is an epimorphism

H_{c}^{∗}(𝕊l; 𝔽_{p}) → H^{∗}(G; 𝔽_{p}).

5. The proof of the main theorem for p=3

We give now an elementary proof of the main theorem for the prime p = 3. We will be able to give precise formulas for the embedding of ℚ_{3}[\sqrt[3]{1}] into 𝕊l which we believe will be useful for computing the cohomology of 𝕊l. In this case 𝕊_{2} is a subgroup of units of the maximal order of the division algebra (𝕎/ℚ_{3}, σ, 3), where σ, the Frobenius, is the generator of Gal(𝕎/ℚ_{3}).

Since 𝕎 is an unramified extension of degree 2 of ℚ_{3} it can be constructed by adjoining to ℚ_{3} a fourth root of unity, i. Since ℚ_{3} contains a square root of −2, we can set

ζ = \sqrt{−2}\frac{(i + 1)}{2},

and it is easy to check that ζ is an eighth root of unity such that ζ^{2} = −i and

ζ^{S} = ζ^{3} = \sqrt{−2}\frac{(1 − i)}{2}.

Proposition 5.1. In the above notation a cube root of unity z ∈ 𝔻 is given by the formula

z = −\frac{1}{2} + \sqrt{−2}\frac{(i + 1)}{4}S = −\frac{1}{2} + \frac{ζ}{2}S.

Proof. This is a simple computation in number theory. As it was mentioned above 𝕊_{2} contains a square root of −3, e.g. ω = ζS and ℚ_{3}[\sqrt{−3}] = ℚ_{3}[z]. Therefore

z = a + bω (5.2)

where a, b ∈ ℚ_{3}. Using the fact that z^{3} = 1 we immediately obtain that a = −\frac{1}{2}, b = \underline{+}\frac{1}{2}

Remark 5.3. The above method of constructing an embedding of ℚ_{p}[\sqrt[p]{1}] into 𝕊_{p−1} theoretically works for any prime p, but the coefficients in an expansion similar to 5.2 are not rational numbers.

Since ℚ_{3} does not contain a primitive cube root of unity, the norm of z is equal to 1 and we conclude that z ∈ 𝕊l.

Theorem 5.4. The subgroup G of 𝕊l generated by z and z^{S} is isomorphic to a free product of two copies of ℤ/3. Moreover G is dense in 𝕊l.

Proof. We will use the matrix representation μ of this cyclic algebra which was described above. z and z^{S} are two matrices of order three with entries in 𝕎:

z = (\begin{matrix}−\frac{1}{2} & −3\sqrt{−2}\frac{(i−1)}{4} \\ \sqrt{−2}\frac{(i+1)}{4} & −\frac{1}{2} \end{matrix} ) , z^{S} = (\begin{matrix}−\frac{1}{2} & −3\sqrt{−2}\frac{(i+1)}{4} \\ \sqrt{−2}\frac{(i−1)}{4} & −\frac{1}{2} \end{matrix} ) .

Because of the form of the entries, we see that we can also consider these matrices as elements of SL_{2}(ℂ). The subgroup they generate in SL_{2}(ℂ) is isomorphic to the subgroup they generate in SL_{2}(𝕎). Conjugating the two generators by the matrix

(\begin{matrix}1 & 0 \\ 0 & \sqrt{3} \end{matrix})

we can get them to lie in SU(2). Using now the homomorphism SU(2) → SO(3) we reduce ourselves to two matrices in SO(3). Using the formulas in [3] one sees that these are two rotations by 120^{∘} about two perpendicular axes. They generate a subgroup of SO(3) which is isomorphic to ℤ/3 ∗ ℤ/3 due to the following theorem proved by F. Hausdorff [4].

Theorem 5.5. Consider a half turn rotation g (so g^{2} = 1) and a one third rotation h (so h^{3} = 1) of ℝ^{3}, the angle between axes being π/4. Then g and h generate in SO(3) a group isomorphic to ℤ/2 ∗ ℤ/3.

The subgroup we are interested in is the one generated by h and h^{g}. So the statement of 2.2 is an easy corollary of the above theorem.

To prove that G is dense in 𝕊l we will prove that z and z^{S} are topological generators of 𝕊l. According to 2.1 we need to show that ρ(z) and ρ(z^{S}) are generators of

𝕊l/𝕊l^{∗} ≅ 𝔽_{9}^{+}.

Direct computation using the formula from 5.1 shows that

ρ(z) = ζ
ρ(z^{S}) = ζ^{3}

Since ζ and ζ^{3} generate the additive group of 𝔽_{9} the theorem is proven.

6. Applications

We start by describing a certain E_{∞} spectrum E_{n}. The ring of its coefficients is isomorphic to

W_{F_{p^{n}}}[[u_{1}, … u_{n−1}]][u, u^{−1}].

One can obtain such a spectrum by completing the Ravenel - Wilson spectrum E(n) in the appropriate way. M. Hopkins and H. Miller proved that the Morava stabilizer group 𝕊_{n} acts by E_{∞} maps on E_{n}. This implies that for any discrete subgroup H of S_{n} one can form a spectrum E_{n}^{H} of homotopy fixed points of the action of H on E_{n}. This spectrum gives an approximation of the Bousfield localization of the sphere spectrum with respect to Morava K-theory K(n). The latter is denoted by L_{K(n)}S^{0}.

The computation of the homotopy groups of the spectra L_{K(n)}S^{0} can be done using the homotopy fixed point spectral sequence which starts with the continuous Galois cohomology of 𝕊_{n}, H_{c}^{∗}(𝕊_{n}; E_{n,∗})^{Gal}. The spectra E_{n}^{H} for different subgroups H of 𝕊_{n} approximate L_{K(n)}S^{0} and therefore can be useful as a substitution for L_{K(n)}S^{0}. The homotopy groups of E_{n}^{H} can be computed in a similar manner to those of L_{K(n)}S^{0} using the homotopy fixed point spectral sequence which starts with H^{∗}(H; E_{n,∗}).

Now let p be an odd prime and n = p−1. The authors of [7] consider the maximal finite subgroup H of 𝕊_{p−1}, which is unique up to isomorphism, and is isomorphic in this case to ℤ/pℤ/(p − 1)^{2} [5] . They call the spectrum E_{p−1}^{H} the “higher real K-theory” and denote it by EO_{p−1}. The reason for this is that, at the prime 2, 𝕊_{1} is just the group of units in ℤ_{2} and so its maximal finite subgroup is ℤ/2. The KO spectrum is the homotopy fixed point spectrum of this ℤ/2 action on the spectrum of complex K-theory completed at 2, which is E_{1}.

Now we are ready to outline the applications of the main theorem.

7. The spectrum E^{L} for p = 3

We deal here with the case p = 3 and n = 2. The maximal finite subgroup H of 𝕊_{2} in this case is isomorphic to ℤ/3ℤ/4. It is not hard to see that this group is generated by z and ζ^{2}, for example. The infinite subgroup G of 𝕊l defines an infinite subgroup L of 𝕊_{2} which we will describe now. Note that S conjugates z in the same way as the 8-th root of unity ζ. Indeed

z^{S} = (−\frac{1}{2} + \frac{ζ}{2}S)^{S} = −\frac{1}{2} + \frac{ζ^{3}}{2} S

On the other hand

z^{ζ} = ζ^{7}(−\frac{1}{2} + \frac{ζ}{2}S)ζ = −\frac{1}{2} + \frac{ζ^{11}}{2} S = −\frac{1}{2} + \frac{ζ^{3}}{2} S

However z^{S^{2}} = z and z^{ζ^{2}} = z^{2}.

We define the group L to be the semi-direct product

(ℤ/3 ∗ ℤ/3)ℤ/8

where ℤ/8 is the group generated by ζ and the free product is generated by z and z^{ζ}. So we have the following sequence of subgroups of 𝕊_{2}:

𝕊_{2} > (ℤ/3 ∗ ℤ/3)ℤ/8 > ℤ/3ℤ/4,

and therefore the map of spectra E^{L} → E^{H}. As we pointed out in the introduction the spectrum E^{L} should be a better approximation to the spectrum L_{K(2)}S^{0}.

8. Cohomology

Let us describe the computation of H^{∗}(𝕊_{2}^{0}; 𝔽_{3}) from [13]. First we need some background. There is a profinite Hopf algebra S(n) such that

S(n)^{∗} ⊗ 𝔽_{p^{n}} ≅ 𝔽_{p^{n}}[𝕊_{n}^{0}].

It was shown in [13] that

Ext_{S(n)}^{∗}(𝔽_{p}) ⊗ 𝔽_{p^{n}} ≅ H_{c}^{∗}(𝕊_{n}^{0}; 𝔽_{p^{n}}).

Let p = 3, n = 2 and consider

R = E(h_{10}, h_{11}) ⊗ P (b_{10}, b_{11})/I

where

I = (h_{10}h_{11}, b_{10}^{2} + b_{11}^{2}, h_{10}b_{10} − h_{11}b_{11}, h_{11}b_{10} + h_{10}b_{11}). (8.1)

Theorem 8.2. [13] H^{∗}(S(2); 𝔽_{3}) is isomorphic as an algebra to E(ζ_{1}, ζ_{2}) ⊗ R, where R is as defined above, ζ_{1} is an element of H^{1}(S(2); 𝔽_{3}) determined by the determinant map, and ζ_{2} is a certain four-fold Massey product which belongs to H^{2}(S(2); 𝔽_{3}).

Here is the interpretation of a part of this computation in terms of the infinite subgroups described above.

Proposition 8.3. After extending the coefficients to 𝔽_{9}, R becomes isomorphic to H^{∗}(ℤ/3 ∗ ℤ/3; 𝔽_{9}).

Proof. Let i ∈ 𝔽_{9} be an element such that i^{2} = −1. Consider the following basis of R over 𝔽_{9}:

h_{1} = h_{10} + ih_{11}, h_{2} = h_{10} − ih_{11}, b_{1} = b_{10} − ib_{11}, b_{2} = b_{10} + ib_{11}.

Direct computation using 8.1 shows that

R ≅ E(h_{1}) ⊗ P (b_{1}) ⊕ E(h_{2}) ⊗ P (b_{2}) ≅ H^{∗}(ℤ/3 ∗ ℤ/3; 𝔽_{9}),

except in degree 0.

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Mathematics Department, Northwestern University, Evanston IL 60208 E-mail address: vgorb@math.nwu.edu, mark@math.nwu.edu, symonds@math.nwu.edu