The Farrell Cohomology of SP(p − 1, ℤ)

Cornelia Busch

Received: March 24, 2002

Communicated by Günter M. Ziegler

Abstract. Let p be an odd prime with odd relative class number h^{−}. In this article we compute the Farrell cohomology of Sp(p−1, ℤ), the first p-rank one case. This allows us to determine the p-period of the Farrell cohomology of Sp(p−1, ℤ), which is 2y, where p−1 = 2^{r}y, y odd. The p-primary part of the Farrell cohomology of Sp(p−1, ℤ) is given by the Farrell cohomology of the normalizers of the subgroups of order p in Sp(p−1, ℤ). We use the fact that for odd primes p with h^{−} odd a relation exists between representations of ℤ/pℤ in Sp(p − 1, ℤ) and some representations of ℤ/pℤ in U((p − 1)/2).

2000 Mathematics Subject Classification: 20G10 Keywords and Phrases: Cohomology theory

1 Introduction

We define a homomorphism

\begin{aligned}ϕ : & U(n) & ⟶ & Sp(2n, ℝ) \\ & X = A + iB & ⟼ & (\begin{matrix}A & B \\ −B & A \end{matrix}) =: ϕ(X) \end{aligned}

where A and B are real matrices. Then ϕ is injective and maps U(n) on a maximal compact subgroup of Sp(2n, ℝ). This homomorphism allows to consider each representation

ρ̃: ℤ/pℤ ⟶ U((p − 1)/2)

as a representation

ϕ ∘ ρ̃: ℤ/pℤ ⟶ Sp(p − 1, ℝ).

In an article of Busch [6] it is determined which properties ρ̃ has to fulfil for ϕ ∘ ρ̃ to be conjugate in Sp(p − 1, ℝ) to a representation

ρ : ℤ/pℤ ⟶ Sp(p − 1, ℤ).

Theorem 2.2. Let X ∈ U((p − 1)/2) be of odd prime order p. We define ϕ : U((p − 1)/2) → Sp(p − 1, ℝ) as above. Then ϕ(X) ∈ Sp(p − 1, ℝ) is conjugate to Y ∈ Sp(p − 1, ℤ) if and only if the eigenvalues λ_{1}, …, λ_{(p−1)/2} of X are such that

\{λ_{1}, …, λ_{(p−1)/2}, \overline{λ}_{1}, …, \overline{λ}_{(p−1)/2}\}

is a complete set of primitive p-th roots of unity.

The proof of Theorem 2.2 involves the theory of cyclotomic fields. For the p-primary component of the Farrell cohomology of Sp(p − 1, ℤ), the following holds:

Ĥ^{∗}(Sp(p − 1, ℤ), ℤ)_{(p)} ≅ ∏_{P ∈𝔓} Ĥ^{∗}(N(P), ℤ)_{(p)}

where 𝔓 is a set of representatives for the conjugacy classes of subgroups of order p of Sp(p − 1, ℤ) and N(P) denotes the normalizer of P ∈ 𝔓. This property also holds if we consider GL(p−1, ℤ) instead of the symplectic group. This fact was used by Ash in [1] to compute the Farrell cohomology of GL(n, ℤ) with coefficients in 𝔽_{p} for p − 1 ⩽ n < 2p − 2. Moreover, we have

Ĥ^{∗}(N(P), ℤ)_{(p)} ≅ (Ĥ^{∗}(C(P), ℤ)_{(p)})^{N(P)/C(P)}

where C(P) is the centralizer of P. We will determine the structure of C(P) and of N(P)/C(P). After that we will compute the number of conjugacy classes of those subgroups for which N(P)/C(P) has a given structure. Here again arithmetical questions are involved. In the articles of Brown [2] and Sjerve and Yang [9] is shown that the number of conjugacy classes of elements of order p in Sp(p − 1, ℤ) is 2^{(p−1)/2}h^{−} where h^{−} denotes the relative class number of the cyclotomic field ℚ(ξ), ξ a primitive p-th root of unity. If h^{−} is odd, each conjugacy class of matrices of order p in Sp(p − 1, ℝ) that lifts to Sp(p − 1, ℤ) splits into h^{−} conjugacy classes in Sp(p − 1, ℤ). The main results in this article are

Theorem 3.7. Let p be an odd prime for which h^{−} is odd. Then

Ĥ^{∗}(Sp(p − 1, ℤ), ℤ)_{(p)} ≅ ∏_{\begin{matrix}k|p−1 \\ k odd \end{matrix}}( K̃_{k} ; ∏ ; 1 ℤ/pℤ[x^{k}, x^{−k}]),

where K̃_{k} denotes the number of conjugacy classes of subgroups of order p of Sp(p − 1, ℤ) for which |N/C| = k. Moreover K̃_{k} ⩾ K_{k}, where K_{k} is the number of conjugacy classes of subgroups of U((p − 1)/2) with |N/C| = k. As usual N denotes the normalizer and C the centralizer of the corresponding subgroup.

Theorem 3.8. Let p be an odd prime for which h^{−} is odd and let y be such that p − 1 = 2^{r}y and y is odd. Then the period of Ĥ^{∗}(Sp(p − 1, ℤ), ℤ)_{(p)} is 2y.

The Farrell Cohomology of SP(p − 1, ℤ) 241

Corresponding results have been shown for other groups, for example GL(n, ℤ) in the p-rank one case [1], the mapping class group [8] and the outerautomorphism group of the free group in the p-rank one case [7]. This article presents results of my doctoral thesis, which I wrote at the ETH Zürich under the supervision of G. Mislin. I thank G. Mislin for the suggestion of this interesting subject.

2 The symplectic group

2.1 Definition

Let R be a commutative ring with 1. The general linear group GL(n, R) is defined to be the multiplicative group of invertible n×n-matrices over R.

Definition. The symplectic group Sp(2n, R) over the ring R is the subgroup of matrices Y ∈ GL(2n, R) that satisfy

Y^{T}JY = J := (\begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix})

where I_{n} is the n×n-identity matrix.

It is the group of isometries of the skew-symmetric bilinear form

\begin{aligned}⟨ , ⟩ : & R^{2n} × R^{2n} & ⟶ & R \\ & (x, y) & ⟼ & ⟨x, y⟩ := x^{T}Jy. \end{aligned}

It follows from a result of Bürgisser [4] that elements of odd prime order p exist in Sp(2n, ℤ) if and only if 2n ⩾ p − 1.

Proposition 2.1. The eigenvalues of a matrix Y ∈ Sp(p − 1, ℤ) of odd prime order p are the primitive p-th roots of unity, hence the zeros of the polynomial

m(x) = x^{p−1} + ⋯ + x + 1.

Proof. If λ is an eigenvalue of Y, we have λ = 1 or λ = ξ, a primitive p-th root of unity, and the characteristic polynomial of Y divides x^{p} − 1 and has integer coefficients. Since m(x) is irreducible over ℚ, the claim follows.

2.2 A relation between U(\frac{p−1}{2}) and Sp(p − 1, ℤ)

Let X ∈ U(n), i.e., X ∈ GL(n, ℂ) and X^{∗}X = I_{n} where X^{∗} = \overline{X}^{T} and I_{n} is the n×n-identity matrix. We can write X = A + iB with A, B ∈ M(n, ℝ), the ring of real n×n-matrices. We now define the following map

\begin{aligned}ϕ : & U(n) & ⟶ & Sp(2n, ℝ) \\ & X = A + iB & ⟼ & (\begin{matrix}A & B \\ −B & A \end{matrix}) =: ϕ(X). \end{aligned}

The map ϕ is an injective homomorphism. Moreover, it is well-known that ϕ maps U(n) on a maximal compact subgroup of Sp(2n, ℝ).

Theorem 2.2. Let X ∈ U((p − 1)/2) be of odd prime order p. We define ϕ : U((p − 1)/2) → Sp(p − 1, ℝ) as above. Then ϕ(X) ∈ Sp(p − 1, ℝ) is conjugate to Y ∈ Sp(p − 1, ℤ) if and only if the eigenvalues λ_{1}, …, λ_{(p−1)/2} of X are such that

\{λ_{1}, …, λ_{(p−1)/2}, \overline{λ}_{1}, …, \overline{λ}_{(p−1)/2}\}

is a complete set of primitive p-th roots of unity.

Proof. See [5] or [6].

In the proof of Theorem 2.2 we used the following facts. For a primitive p-th root of unity ξ, we consider the cyclotomic field ℚ(ξ). It is well-known that ℚ(ξ + ξ^{−1}) is the maximal real subfield of ℚ(ξ), and that ℤ[ξ] and ℤ[ξ + ξ^{−1}] are the rings of integers of ℚ(ξ) and ℚ(ξ + ξ^{−1}) respectively. Let (𝔞, a) denote a pair where 𝔞 ⊆ ℤ[ξ] and a ∈ ℤ[ξ] are chosen such that 𝔞 ≠ 0 is an ideal in ℤ[ξ] and 𝔞\overline{𝔞} = (a), a principal ideal. Here \overline{𝔞} denotes the complex conjugate of 𝔞. We define an equivalence relation on the set of those pairs by (𝔞, a) ∼ (𝔟, b) if and only if λ, μ ∈ ℤ[ξ] \ \{0\} exist such that λ𝔞 = μ𝔟 and λ\overline{λ}a = μ\overline{μ}b. We denote by [𝔞, a] the equivalence class of the pair (𝔞, a) and by P the set of equivalence classes [𝔞, a]. Let S_{p} denote the set of conjugacy classes of elements of order p in Sp(p−1, ℤ). Sjerve and Yang have shown in [9] that a bijection exists between P and S_{p}. If Y ∈ Sp(p − 1, ℤ) is a matrix of order p, then the equivalence class [𝔞, a] ∈ P corresponding to the conjugacy class of Y in Sp(p − 1, ℤ) can be determined in the following way. Let α = (α_{1}, …, α_{p−1})^{T} be an eigenvector of Y corresponding to the eigenvalue ξ = e^{i2π/p}, that is Y α = ξα. Then α_{1}, …, α_{p−1} is a basis of an ideal 𝔞 ⊆ ℤ[ξ]. Sjerve and Yang [9] proved that this ideal 𝔞 has the property [𝔞, a] ∈ P. Let h and h^{+} be the class numbers of ℚ(ξ) and ℚ(ξ + ξ^{−1}) respectively. Then h^{−} := h/h^{+} denotes the relative class number. Sjerve and Yang [9] showed that the number of conjugacy classes of matrices of order p in Sp(p − 1, ℤ) is h^{−} 2^{(p−1)/2}. The number of conjugacy classes in U((p − 1)/2) of unitary matrices that satisfy the condition in Theorem 2.2 is 2^{(p−1)/2}. Let U_{p} denote the set of conjugacy classes of matrices in U((p−1)/2) that satisfy the condition on the eigenvalues that is given in Theorem 2.2. A consequence of Theorem 2.2 is that it is possible to define a map

Ψ : S_{p} ⟶ U_{p}

and that this map is surjective. Therefore the map

ψ : P ⟶ U_{p}

is surjective either. For a given choice of the ideal 𝔞 (for example 𝔞 = ℤ[ξ]), we denote by P_{𝔞} the set of those classes [𝔞, a] ∈ P, where 𝔞 corresponds to our choice. If the restriction

ψ|_{P_{𝔞}} : P_{𝔞} ⟶ U_{p}

is surjective each conjugacy class in U_{p} of matrices that satisfy Theorem 2.2 yields h^{−} conjugacy classes in Sp(p − 1, ℤ). In general ψ|_{P_{𝔞}} is not surjective. It is a result of Busch, [5], [6], that ψ|_{P_{𝔞}} is surjective if h^{−} is odd. If h^{−} is even and h^{+} is odd, we have no surjectivity of ψ|_{P_{𝔞}}. This happens for example for the primes 29 and 113.

2.3 Subgroups of order p in Sp(p − 1, ℤ)

It follows from Theorem 2.2 that a mapping exists that sends the conjugacy classes of matrices Y ∈ Sp(p − 1, ℤ) of odd prime order p onto the conjugacy classes of matrices X in U((p − 1)/2) that satisfy the condition on the eigenvalues described in Theorem 2.2. This mapping is surjective. It is clear that det X = e^{l2πi/p} for some 1 ⩽ l ⩽ p. If X ∈ U((p − 1)/2) satisfies the condition on the eigenvalues, then so does X^{k}, k = 1, …, p − 1. If det X = e^{l2πi/p} for some 1 ⩽ l ⩽ p − 1, then

{det X, …, det X^{p−1}} = {e^{i2π/p}, …, e^{i(p−1)2π/p}}

and the X^{k} are in different conjugacy classes. If det X = 1, it is possible that some k exists such that X and X^{k} are in the same conjugacy class. In this section we will analyse when and how many times this happens. The number of conjugacy classes of matrices X ∈ U((p − 1)/2) that satisfy the condition required in Theorem 2.2 is 2^{(p−1)/2}. Herewith we will be able to compute the number of conjugacy classes of subgroups of matrices of order p in U((p−1)/2). We remember that the number of conjugacy classes of matrices of order p in Sp(p − 1, ℤ) is 2^{(p−1)/2}h^{−}. If h^{−} = 1, a bijection exists between the conjugacy classes of matrices of order p in Sp(p−1, ℤ) and the conjugacy classes of matrices of order p in U((p − 1)/2) that satisfy the condition required in Theorem 2.2. Let X ∈ U((p − 1)/2) with X^{p} = 1, X ≠ 1. Then X generates a subgroup S of order p in U((p − 1)/2). If det X = 1, it is possible that X is conjugate to X′ ∈ S with X ≠ X′. Two matrices in U((p − 1)/2) are conjugate to each other if and only if they have the same eigenvalues. The set of eigenvalues of X is

{e^{ig_{1}2π/p}, …, e^{ig_{(p−1)/2}2π/p}}

where 1 ⩽ g_{l} ⩽ p − 1 for l = 1, …, \frac{p−1}{2} and for all l ≠ j, l, j = 1, …, (p − 1)/2, g_{l} ≠ p−g_{j} and g_{l} ≠ g_{j}. From now on we consider the g_{j} as elements of (ℤ/pℤ)^{∗}. The matrix X is conjugate to X^{κ} for some κ if the eigenvalues of X and X^{κ} are the same. This is equivalent to

{g_{1}, …, g_{(p−1)/2}} = {κg_{1}, …, κg_{(p−1)/2}} ⊂ (ℤ/pℤ)^{∗}

where g_{j} and κg_{j}, j = 1, …, (p − 1)/2, denote the corresponding congruence classes.

We introduce some notation that will be used in the whole section. Let

G := {g_{1}, …, g_{(p−1)/2}} ⊂ (ℤ/pℤ)^{∗},
κG := {κg_{1}, …, κg_{(p−1)/2}} ⊂ (ℤ/pℤ)^{∗}

for some κ ∈ (ℤ/pℤ)^{∗}. Let x be a generator of the multiplicative cyclic group (ℤ/pℤ)^{∗} and let K be a subgroup of (ℤ/pℤ)^{∗} with |K| = k. Then K is cyclic and k divides p − 1. Let m := (p − 1)/k, then x^{m} generates K. First we prove the following proposition.

Proposition 2.3. Let G ⊂ (ℤ/pℤ)^{∗} be a subset with |G| = (p − 1)/2. The following are equivalent.

i) For all g_{j}, g_{l} ∈ G, g_{j} ≠ −g_{l} and κ ∈ (ℤ/pℤ)^{∗} exists with κG = G, κ ≠ 1.

ii) An integer h ∈ ℕ, 1 ⩽ h ⩽ (p − 1)/2, and n_{j} ∈ (ℤ/pℤ)^{∗}, j = 1, …, h, exist with

G = ⋃_{j=1}^{h} n_{j}K

where

∘ K ⊂ (ℤ/pℤ)^{∗} is the subgroup generated by κ, ∘ the order of K is odd, ∘ for κ′ ∈ K and all j, l = 1, …, h, n_{j} ≠ −n_{l}κ′, ∘ and for all j = 2, …, h, n_{j} ∉ K.

Then we will analyse the uniqueness of this decomposition of G. This will enable us to determine the number of G ⊂ (ℤ/pℤ)^{∗} with |G| = (p − 1)/2 and G = κG for some 1 ≠ κ ∈ (ℤ/pℤ)^{∗}. Herewith we will determine the number of conjugacy classes of subgroups of order p in U((p−1)/2) whose group elements satisfy the condition of Theorem 2.2.

Definition. Let κ ∈ (ℤ/pℤ)^{∗} and let K be the subgroup of (ℤ/pℤ)^{∗} generated by κ. Let G ⊂ (ℤ/pℤ)^{∗} be a subset with |G| = (p − 1)/2. We say that K decomposes G if G, κ and K fulfil the conditions of Proposition 2.3.

So K decomposes G if the order of the group K is odd and G is a disjoint union of cosets n_{1}K, …, n_{h}K of K in (ℤ/pℤ)^{∗} for which for all n_{j}, n_{l}, j, l = 1, …, h, holds n_{j}K ≠ −n_{l}K.

Lemma 2.4. Let G ⊂ (ℤ/pℤ)^{∗} with |G| = (p − 1)/2. Then 1 ≠ κ ∈ (ℤ/pℤ)^{∗} exists with κG = G if and only if 1 ⩽ h ⩽ (p − 1)/2 and n_{j} ∈ (ℤ/pℤ)^{∗}, j = 1, …, h, exist with

G = ⋃_{j=1}^{h} n_{j}K

where n_{j} ∉ K for j = 2, …, h, and K is the subgroup of (ℤ/pℤ)^{∗} that is generated by κ.

Proof. \underline{⇐}: Let κ^{l} ∈ K. Then

κ^{l}G = κ^{l} ⋃_{j=1}^{h} n_{j}K = ⋃_{j=1}^{h} n_{j}κ^{l}K = ⋃_{j=1}^{h} n_{j}K = G.

\underline{⇒}: Without loss of generality we assume that 1 ∈ G. If 1 ∉ G, λ ∈ (ℤ/pℤ)^{∗} exists with 1 ∈ λG because (ℤ/pℤ)^{∗} is a multiplicative group. Of course κλG = λG. Moreover, it is easy to see that if λG is a union of cosets of K, this is also true for G. The equation κG = G implies that KG = G. If 1 ∈ G, then K ⊆ G since KG = G. If K = G, we have finished the proof. If K ≠ G, we consider G′_{1} = G \ K. For all κ^{l} ∈ K we have κ^{l}K = K and

κ^{l}G′_{1} = κ^{l}(G \ K) = G \ K = G′_{1}.

Now λ_{1} ∈ (ℤ/pℤ)^{∗} exists with 1 ∈ λ_{1}G′_{1} =: G_{1}. Then G = K ∪ λ_{1}^{−1}G_{1} and we can repeat the construction on G_{1} instead of G. This procedure finishes after h := (p − 1)/2k steps. Let n_{1} := 1 and for j = 2, …, h let n_{j} := n_{j−1}λ_{j−1}^{−1}. Then G = ⋃_{j=1}^{h} n_{j}K.

Let G = {g_{1}, …, g_{(p−1)/2}} ⊂ (ℤ/pℤ)^{∗} with |G| = (p − 1)/2 and κG = G for some κ ∈ (ℤ/pℤ)^{∗} with κ ≠ 1, κ^{k} = 1. The following lemma will give an answer to the question when G satisfies the conditions g_{l} ≠ g_{j}, g_{l} ≠ −g_{j} for all j ≠ l with j, l = 1, …, \frac{p−1}{2}.

Lemma 2.5. Let G = ⋃_{j=1}^{h} n_{j}K ⊂ (ℤ/pℤ)^{∗} be defined like in Lemma 2.4. Then for all g_{j}, g_{l} ∈ G holds g_{j} ≠ −g_{l} if and only if −1 ∉ K and for all κ ∈ K and all j, l = 1, …, h holds n_{j} ≠ −n_{l}κ.

Proof. \underline{⇒}: Suppose −1 ∈ K. Then −1 = κ^{l} for some l and n_{1} = −n_{1}κ^{l}. But then we have found g_{1} := n_{1} ∈ G and g_{2} := n_{1}κ^{l} ∈ G with g_{1} = −g_{2}. \underline{⇐}: Suppose g_{j}, g_{l} ∈ G exist with g_{j} = −g_{l}. Let g_{j} = n_{j}κ^{j}, g_{l} = n_{l}κ^{l}. Then n_{j}κ^{j} = −n_{l}κ^{l}, and we have found κ^{j−l} ∈ K with n_{l} = −n_{j}κ^{j−l}.

Which subgroups K ⊆ (ℤ/pℤ)^{∗} satisfy the condition −1 ∉ K?

Lemma 2.6. Let K ⊆ (ℤ/pℤ)^{∗} be a subgroup of order k. Then −1 ∉ K if and only if k is odd.

Proof. The group (ℤ/pℤ)^{∗} is cyclic of order p − 1 and K is a cyclic group. Let x be a generator of K, then x^{k} = 1. If k is even, k/2 ∈ ℤ and x^{k/2} ∈ K. But then (x^{k/2})^{2} = x^{k} = 1 and therefore x^{k/2} = −1 ∈ K since −1 is the element of order 2 in (ℤ/pℤ)^{∗}. On the other hand if −1 ∈ K, then K contains an element of order 2. But then k is even, since the order of any element of K divides the order of K.

Proof of Proposition 2.3. A subgroup K decomposes a set G as required in Lemma 2.5 if and only if the order of K is odd. Moreover, the order of K divides p−1. Now Proposition 2.3 follows from Lemma 2.4 and Lemma 2.5.

We did not yet analyse the uniqueness of the decomposition of a set G. It is evident that the n_{j} can be permuted and multiplied with any κ^{l} ∈ K, but we will see that K and h are not uniquely determined. The next lemma states that if K decomposes G then so does any nontrivial subgroup of K.

Lemma 2.7. Let G = ⋃_{j=1}^{h} n_{j}K ⊂ (ℤ/pℤ)^{∗}, |G| = (p − 1)/2, be such that K decomposes G (Proposition 2.3). Let |K| = k be not a prime and let K′ ≠ K be a nontrivial subgroup of K. Then K′ decomposes G.

Proof. Since K′ is a subgroup of K, K can be written as a union of cosets of K′ in K. Moreover, G is a union of cosets of K in (ℤ/pℤ)^{∗}. Therefore

G = ⋃_{j=1}^{h} n_{j}K = ⋃_{i=1}^{h′} n′_{i}K′.

Since K decomposes G, we have n_{l}K ≠ −n_{j}K for all l, j = 1, …, h. This implies that n′_{l}K′ ≠ −n′_{i}K′ for all i, l = 1, …, h′. So K′ decomposes G.

Our next aim is to determine the number of sets G. Therefore we consider for a given G the group K with |K| maximal and K decomposes G.

Lemma 2.8. Let K ⊂ (ℤ/pℤ)^{∗} be a nontrivial subgroup of odd order k. Then 2^{(p−1)/2k} different sets G exist such that K decomposes G and |G| = (p − 1)/2.

Proof. The order of K ⊂ (ℤ/pℤ)^{∗} is odd. Then it follows from Lemma 2.6 that −1 ∉ K. Consider the cosets n_{j}K of K in (ℤ/pℤ)^{∗}. Since −1 ∉ K, we have n_{j}K ≠ −n_{j}K. So n_{j}, j = 1, …, (p − 1)/2k, exist such that

(ℤ/pℤ)^{∗} = ⋃_{j=1}^{(p−1)/2k}(n_{j}K ∪ −n_{j}K).

The group K decomposes G if and only if G is a union of cosets of K and m_{j}K ⊆ G implies that −m_{j}K ⊈ G for m_{j} = ±n_{j}, j = 1, …, (p − 1)/2k. Therefore 2^{(p−1)/2k} sets G exist such that K decomposes G.

Definition. Let K ⊂ (ℤ/pℤ)^{∗} be a group of odd order k. We define N_{k} to be the number of G ⊂ (ℤ/pℤ)^{∗} such that K decomposes G but any K′ with K ⊂ K′ ⊂ (ℤ/pℤ)^{∗}, K ≠ K′, does not decompose G.

To determine N_{k} we have to subtract the number N_{k′} from 2^{(p−1)/2k} for each odd k′ ≠ k with k|k′, k′|p − 1. The integer k′ is the order of the group K′ with K ⊂ K′. Therefore we get a recursive formula

N_{k} = 2^{(p−1)/2k} − ∑_{\begin{matrix}k′ odd, k′>k \\ k|k′, k′|p−1 \end{matrix}} N_{k′}.

Now it remains to determine N_{y}. Let y ∈ ℤ be such that p − 1 = 2^{r}y and y is odd. Then

N_{y} = 2^{(p−1)/2y} = 2^{2^{r−1}}.

Let p − 1 = 2^{r}p_{1}^{r_{1}} … p_{l}^{r_{l}} be a factorisation of p − 1 into primes where p_{1}, …, p_{l} are odd and p_{i} ≠ p_{j} for all i ≠ j with i, j = 1, …, l. Since p − 1 is even, r ⩾ 1. Let K be of order k = p_{1}^{s_{1}} … p_{l}^{s_{l}} where 0 ⩽ s_{j} ⩽ r_{j} for j = 1, …, l. Let x be a generator of (ℤ/pℤ)^{∗}. Then K is generated by x^{m}, m = 2^{r}p_{1}^{r_{1}−s_{1}} … p_{l}^{r_{l}−s_{l}}. If k′ = p_{1}^{t_{1}} … p_{l}^{t_{l}} where s_{j} ⩽ t_{j} ⩽ r_{j} for j = 1, …, l, then K is a proper subgroup of K′ of order k′ if s_{j} < t_{j} for some 1 ⩽ j ⩽ l. Herewith −1+∏_{j=1}^{l}(r_{j} −s_{j} +1) groups K′ exist such that K is a proper subgroup of K′. So the number of sets G that are decomposed by K and for which no K′ ⊋ K exists such that K′ decomposes G is

N_{k} = 2^{(p−1)/2k} − ∑_{y∈T_{k}} N_{y}

where

T_{k} := {y ∈ ℕ | y odd, k|y, y ≠ k and y|p − 1}.

Now we have to determine the number of sets G that satisfy the conditions of Proposition 2.3. Let this be the number N_{G}. One easily sees that

N_{G} = ∑_{\begin{matrix}K⊂(ℤ/pℤ)^{∗} \\ |K|≠1 \\ |K| odd \end{matrix}} N_{|K|} = ∑_{\begin{matrix}k|p−1 \\ k≠1 \\ k odd \end{matrix}} N_{k}.

Now let G ⊂ (ℤ/pℤ)^{∗} with |G| = (p − 1)/2, such that for all g_{i}, g_{j} ∈ G, g_{i} ≠ −g_{j}. Let N_{1} be the number of sets G for which no κ ∈ (ℤ/pℤ)^{∗}, κ ≠ 1, exists such that κG = G. Then

N_{1} = 2^{(p−1)/2} − N_{G} = 2^{(p−1)/2} − ∑_{\begin{matrix}1≠k| p−1 \\ k odd \end{matrix}} N_{k}.

We have seen that each set G corresponds to the set of eigenvalues of a matrix in U((p − 1)/2) that satisfies Theorem 2.2.

Definition. We define a matrix X_{G} ∈ U(\frac{p−1}{2}) with the eigenvalues

{e^{ig_{1}2π/p}, …, e^{ig_{(p−1)/2}2π/p}}

where G = {g_{1}, …, g_{(p−1)/2}} ⊂ (ℤ/pℤ)^{∗}. We used the same notation for the elements of (ℤ/pℤ)^{∗} and their representatives in ℤ.

Let the maximal order of K that decomposes G be k. Then G yields k elements of the group generated by X_{G}. As a result we have:

Proposition 2.9. The number of conjugacy classes of subgroups of order p in U((p−1)/2) whose group elements satisfy the necessary and sufficient condition is

K(p) = \frac{1}{p − 1} ∑_{\begin{matrix}k odd \\ k|p−1 \end{matrix}} kN_{k}.

3 The Farrell cohomology

3.1 An introduction to Farrell cohomology

An introduction to the Farrell cohomology can be found in the book of Brown [3]. The Farrell cohomology is a complete cohomology for groups with finite virtual cohomological dimension (vcd). It is a generalisation of the Tate cohomology for finite groups. If G is finite, the Farrell cohomology and the Tate cohomology of G coincide. It is well-known that the groups Sp(2n, ℤ) have finite vcd.

Definition. An elementary abelian p-group of rank r ⩾ 0 is a group that is isomorphic to (ℤ/pℤ)^{r}.

It is well-known that Ĥ^{i}(G, ℤ) is a torsion group for every i ∈ ℤ. We write Ĥ^{i}(G, ℤ)_{(p)} for the p-primary part of this torsion group, i.e., the subgroup of elements of order some power of p. We will use the following theorem.

Theorem 3.1. Let G be a group such that vcd G < ∞ and let p be a prime. Suppose that every elementary abelian p-subgroup of G has rank ⩽ 1. Then

Ĥ^{∗}(G, ℤ)_{(p)} ≅ ∏_{P ∈𝔓} Ĥ^{∗}(N(P), ℤ)_{(p)}

where 𝔓 is a set of representatives for the conjugacy classes of subgroups of G of order p and N(P) denotes the normalizer of P.

Proof. See Brown’s book [3].

We also have

Ĥ^{∗}(G, ℤ) ≅ ∏_{p} Ĥ^{∗}(G, ℤ)_{(p)}

where p ranges over the primes such that G has p-torsion. A group G of finite virtual cohomological dimension is said to have periodic cohomology if for some d ≠ 0 there is an element u ∈ Ĥ^{d}(G, ℤ) that is invertible in the ring Ĥ^{∗}(G, ℤ). Cup product with u then gives a periodicity isomorphism Ĥ^{i}(G, M) ≅ Ĥ^{i+d}(G, M) for any G-module M and any i ∈ ℤ. Similarly we say that G has p-periodic cohomology if the p-primary component Ĥ^{∗}(G, ℤ)_{(p)}, which is itself a ring, contains an invertible element of non-zero degree d. Then we have

Ĥ^{i}(G, M)_{(p)} ≅ Ĥ^{i+d}(G, M)_{(p)},

and the smallest positive d that satisfies this condition is called the p-period of G.

Proposition 3.2. The following are equivalent:

i) G has p-periodic cohomology.

ii) Every elementary abelian p-subgroup of G has rank ⩽ 1.

Proof. See Brown’s book [3].

3.2 Normalizers of subgroups of order p in Sp(p − 1, ℤ)

In order to use Theorem 3.1, we have to analyse the structure of the normalizers of subgroups of order p in Sp(p − 1, ℤ). We already analysed the conjugacy classes of subgroups of order p in Sp(p − 1, ℤ). Let N be the normalizer and let C be the centralizer of such a subgroup. Then we have a short exact sequence

1 ⟶ C ⟶ N ⟶ N/C ⟶ 1.

Moreover, it follows from the discussion in the paper of Brown [2] that for p an odd prime

C ≅ ℤ/pℤ × ℤ/2ℤ ≅ ℤ/2pℤ,

and therefore N is a finite group. We will use the following proposition.

Proposition 3.3. Let

1 ⟶ U ⟶ G ⟶ Q ⟶ 1

be a short exact sequence with Q a finite group of order prime to p. Then

Ĥ^{∗}(G, ℤ)_{(p)} ≅ (Ĥ^{∗}(U, ℤ)_{(p)})^{Q} .

Proof. See Brown [3], the Hochschild-Serre spectral sequence.

Applying this to our case, we get

Ĥ^{∗}(N, ℤ)_{(p)} ≅ (Ĥ^{∗}(C, ℤ)_{(p)})^{N/C} .

Therefore we have to determine N/C and its action on C ≅ ℤ/2pℤ. From now on, if we consider subgroups or elements of order p in U((p − 1)/2), we mean those that satisfy the condition of Theorem 2.2. In what follows we assume that p is an odd prime for which h^{−} = 1, because in this case we have a bijection between the conjugacy classes of subgroups of order p in U((p − 1)/2) and those in Sp(p − 1, ℤ). Therefore, in order to determine the structure of the conjugacy classes of subgroups of order p in Sp(p − 1, ℤ), we can consider the corresponding conjugacy classes in U((p − 1)/2). We have already seen that in a subgroup of U((p − 1)/2) of order p different elements can be in the same conjugacy class. Let N_{k} be the number of conjugacy classes of elements of order p in U((p − 1)/2) where k powers of one element are in the same conjugacy class. Let K_{k} be the number of conjugacy classes of subgroups of U((p − 1)/2) with |N/C| = k, where N denotes the normalizer and C the centralizer of this subgroup. Then the number K(p) of conjugacy classes of subgroups of order p in U((p − 1)/2) is

K(p) = ∑_{\begin{matrix}k|p−1, \\ k odd \end{matrix}} K_{k}.

If |N/C| = k, then

N/C ≅ ℤ/kℤ ⊆ ℤ/(p − 1)ℤ ≅ Aut(ℤ/2pℤ)

where k|p − 1 and k is odd. This means that N/C is isomorphic to a subgroup of Aut(ℤ/pℤ). So we get the short exact sequence

1 ⟶ ℤ/2pℤ ⟶ N ⟶ ℤ/kℤ ⟶ 1.

Moreover, we have an injection ℤ/pℤ ↪ ℤ/2pℤ ↪ N. Applying the proposition to this case yields

Ĥ^{∗}(N, ℤ)_{(p)} ≅ (Ĥ^{∗}(ℤ/2pℤ, ℤ)_{(p)})^{ℤ/kℤ} .

The action of ℤ/kℤ on ℤ/2pℤ is given by the action of ℤ/kℤ as a subgroup of the group of automorphisms of ℤ/pℤ ⊂ ℤ/2pℤ.

Lemma 3.4. The Farrell cohomology of ℤ/lℤ is

Ĥ^{∗}(ℤ/lℤ, ℤ) = ℤ/lℤ [x, x^{−1}]

where deg x = 2, x ∈ Ĥ^{2}(ℤ/lℤ, ℤ), and ⟨x⟩ ≅ ℤ/lℤ.

Proof. See Brown’s book [3]. For finite groups the Farrell cohomology and the Tate cohomology coincide.

Proposition 3.5. Let p be an odd prime and let k ∈ ℤ divide p − 1. Then

(Ĥ^{∗}(ℤ/2pℤ, ℤ)_{(p)})^{ℤ/kℤ} ≅ ℤ/pℤ[x^{k}, x^{−k}]

where x ∈ Ĥ^{2}(ℤ/2pℤ, ℤ).

Proof. For an odd prime p

Ĥ^{∗}(ℤ/2pℤ, ℤ)_{(p)} = (ℤ/2pℤ[x, x^{−1}])_{(p)} = ℤ/pℤ[x, x^{−1}].

We have to consider the action of ℤ/kℤ on ℤ/pℤ[x, x^{−1}]. We have px = 0 and x ∈ Ĥ^{2}(ℤ/2pℤ, ℤ). The action is given by x ↦ qx with q such that (q, p) = 1, q^{k} ≡ 1 (mod p) and k is the smallest number such that this is fulfilled. The action of ℤ/kℤ on

Ĥ^{2m}(ℤ/2pℤ, ℤ)_{(p)} ≅ (⟨x^{m}⟩) ≅ ℤ/pℤ

is given by

x^{m} ↦ q^{m}x^{m}.

The ℤ/kℤ-invariants of Ĥ^{∗}(ℤ/2pℤ, ℤ)_{(p)} are the x^{m} ∈ Ĥ^{2m}(ℤ/2pℤ, ℤ)_{(p)} with x^{m} ↦ x^{m}, or equivalently q^{m} ≡ 1 (mod p). Herewith we get

Ĥ^{∗}(N, ℤ)_{(p)} ≅ (Ĥ^{∗}(ℤ/2pℤ, ℤ)_{(p)})^{ℤ/kℤ} ≅ (ℤ/pℤ[x, x^{−1}])^{ℤ/kℤ}
≅ ℤ/pℤ[x^{k}, x^{−k}].

Proposition 3.6. Let p be an odd prime for which h^{−} = 1. Then

Ĥ^{∗}(Sp(p − 1, ℤ), ℤ)_{(p)} ≅ ∏_{\begin{matrix}k|p−1 \\ k odd \end{matrix}}(∏_{1}^{K_{k}} ℤ/pℤ[x^{k}, x^{−k}]),

where K_{k} is the number of conjugacy classes of subgroups of U((p − 1)/2) with |N/C| = k. As usual N denotes the normalizer and C the centralizer of this subgroup.

Proof. Let p be a prime with h^{−} = 1. Then a bijection exists between the conjugacy classes of matrices of order p in U((p − 1)/2) that satisfy the conditions of Theorem 2.2 and the conjugacy classes of matrices of order p in Sp(p − 1, ℤ). Now this proposition follows from Theorem 3.1.

Now it remains to determine K_{k}, the number of conjugacy classes of subgroups of U((p − 1)/2) of order p with N/C ≅ ℤ/kℤ. Therefore we need N_{k}, the number of conjugacy classes of elements X ∈ U((p − 1)/2) of order p for which 1 = j_{1} < ⋯ < j_{k} < p exist such that the X^{j_{l}}, l = 1, …, k, are in the same conjugacy class than X and k is maximal. One such class yields k elements in a group for which |N/C| = k and therefore

K_{k} = kN_{k} \frac{1}{p − 1}.

We recall the formula for N_{k}:

N_{k} = 2^{\frac{p−1}{2k}} − ∑_{\begin{matrix}k′ odd, k′>k \\ k|k′, k′|p−1 \end{matrix}} N_{k′}.

Now we have everything we need to compute the p-primary part of the Farrell cohomology of Sp(p − 1, ℤ) for some examples of primes with h^{−} = 1.

252 C. Busch

3.3 Examples with 3 ⩽ p ⩽ 19 p = 3 : It is Sp(2, ℤ) = SL(2, ℤ). One conjugacy class exists with N = C.

Therefore

Ĥ^{∗}(Sp(2, ℤ), ℤ)_{(3)} ≅ ℤ/3ℤ[x, x^{−1}],

and Sp(2, ℤ) has 3-period 2. p = 5 : One conjugacy class exists with N = C. Therefore

Ĥ^{∗}(Sp(4, ℤ), ℤ)_{(5)} ≅ ℤ/5ℤ[x, x^{−1}],

and Sp(4, ℤ) has 5-period 2. p = 7 : One conjugacy class exists with N/C ≅ ℤ/3ℤ, and one class exists with

N = C. Therefore

Ĥ^{∗}(Sp(6, ℤ), ℤ)_{(7)} ≅ ℤ/7ℤ[x^{3}, x^{−3}] × ℤ/7ℤ[x, x^{−1}],

and Sp(6, ℤ) has 7-period 6. p = 11 : One conjugacy class exists with N/C ≅ ℤ/5ℤ and 3 classes exist with

N = C. Therefore

Ĥ^{∗}(Sp(10, ℤ), ℤ)_{(11)} ≅ ℤ/11ℤ[x^{5}, x^{−5}] × ∏_{1}^{3} ℤ/11ℤ[x, x^{−1}],

and Sp(10, ℤ) has 11-period 10. p = 13 : One conjugacy class exists with N/C ≅ ℤ/3ℤ and 5 classes exist with

N = C. Therefore

Ĥ^{∗}(Sp(12, ℤ), ℤ)_{(13)} ≅ ℤ/13ℤ[x^{3}, x^{−3}] × ∏_{1}^{5} ℤ/13ℤ[x, x^{−1}],

and Sp(12, ℤ) has 13-period 6. p = 17 : 16 conjugacy classes exist with N = C. Therefore

Ĥ^{∗}(Sp(16, ℤ), ℤ)_{(17)} ≅ ∏_{1}^{16} ℤ/17ℤ[x, x^{−1}],

and Sp(16, ℤ) has 17-period 2. p = 19 : One conjugacy class exists with N/C ≅ ℤ/9ℤ, one class exists with

N/C ≅ ℤ/3ℤ, and 28 classes exist with N = C.

Ĥ^{∗}(Sp(18, ℤ), ℤ)_{(19)} ≅ ℤ/19ℤ[x^{9}, x^{−9}] × ℤ/19ℤ[x^{3}, x^{−3}]

× ∏_{1}^{28} ℤ/19ℤ[x, x^{−1}],

and Sp(18, ℤ) has 19-period 18.

Documenta Mathematica 7 (2002) 239–254

3.4 The p-primary part of the Farrell cohomology of Sp(p − 1, ℤ)

Let p be an odd prime and let ξ be a primitive p-th root of unity. Let h^{−} be the relative class number of the cyclotomic field ℚ(ξ). In this section we compute Ĥ^{∗}(Sp(p − 1, ℤ), ℤ)_{(p)} and its period for any odd prime p for which h^{−} is odd.

Theorem 3.7. Let p be an odd prime for which h^{−} is odd. Then

Ĥ^{∗}(Sp(p − 1, ℤ), ℤ)_{(p)} ≅ ∏_{\begin{matrix}k|p−1 \\ k odd \end{matrix}}( K̃_{k} ; ∏ ; 1 ℤ/pℤ[x^{k}, x^{−k}]),

where K̃_{k} denotes the number of conjugacy classes of subgroups of order p of Sp(p − 1, ℤ) for which |N/C| = k. Moreover K̃_{k} ⩾ K_{k}, where K_{k} is the number of conjugacy classes of subgroups of U((p − 1)/2) with |N/C| = k. As usual N denotes the normalizer and C the centralizer of the corresponding subgroup.

Proof. We have seen in Section 2.2 that if h^{−} is odd, a bijection exists between the conjugacy classes of matrices of order p in U((p − 1)/2) that satisfy the conditions of Theorem 2.2 and the conjugacy classes of matrices of order p in Sp(p − 1, ℤ) that correspond to the equivalence classes [ℤ[ξ], u] ∈ P. Each conjugacy class of subgroups of order p in U((p − 1)/2) whose group elements satisfy the condition required in Theorem 2.2 yields at least one conjugacy class in Sp(p − 1, ℤ). This implies that the p-primary part of the Farrell cohomology of Sp(p − 1, ℤ) is a product

∏_{\begin{matrix}k|p−1, \\ k odd \end{matrix}}( K̃_{k} ; ∏ ; 1 ℤ/pℤ[x^{k}, x^{−k}])

where K̃_{k} denotes the number of conjugacy classes of subgroups of order p of Sp(p − 1, ℤ) that satisfy |N/C| = k. Let K_{k} be the number of such subgroups in U((p − 1)/2). Because h^{−} is odd, each such subgroup gives at least one such subgroup of Sp(p − 1, ℤ). Therefore, if h^{−} is odd, K̃_{k} ⩾ K_{k}. If h^{−} is even, it may be possible that no subgroup of Sp(p − 1, ℤ) of order p exists for which |N/C| = k.

Theorem 3.8. Let p be an odd prime for which h^{−} is odd and let y be such that p − 1 = 2^{r}y and y is odd. Then the period of Ĥ^{∗}(Sp(p − 1, ℤ), ℤ)_{(p)} is 2y.

Proof. By Theorem 3.7 we know that the p-primary part of the Farrell cohomology of Sp(p − 1, ℤ) is

Ĥ^{∗}(Sp(p − 1, ℤ), ℤ)_{(p)} ≅ ∏_{\begin{matrix}k|p−1 \\ k odd \end{matrix}}( K̃_{k} ; ∏ ; 1 ℤ/pℤ[x^{k}, x^{−k}]).

Moreover, K̃_{k} ⩾ 1 and the period of ℤ/pℤ[x^{k}, x^{−k}] is 2k. Herewith the period of the p-primary part of the Farrell cohomology is 2y.

If p is a prime for which h^{−} is even, the p-period of Ĥ^{∗}(Sp(p − 1, ℤ), ℤ) is 2z where z is odd and divides p−1. The period is not necessarily 2y because there may be no subgroup of order p in which y elements are conjugate in Sp(p−1, ℤ) even if we know that they are conjugate in Sp(p − 1, ℝ).

References

[1] A. Ash, Farrell cohomology of GL(n, ℤ), Israel J. of Math. 67 (1989), 327-336.

[2] K. S. Brown, Euler characteristics of discrete groups and G-spaces, Invent. Math. 27 (1974), 229-264.

[3] K. S. Brown, Cohomology of Groups, GTM, vol. 87, Springer, 1982.

[4] B. Bürgisser, Elements of finite order in symplectic groups, Arch. Math. 39 (1982), 501-509.

[5] C. Busch, Symplectic characteristic classes and the Farrell cohomology of Sp(p − 1, ℤ), Diss. ETH No. 13506, ETH Zürich (2000).

[6] C. Busch, Symplectic characteristic classes, L’Ens. Math. t. 47 (2001), 115-130.

[7] H. H. Glover, G. Mislin, On the p-primary cohomology of Out(F_{n}) in the p-rank one case, J. Pure Appl. Algebra 153 (2000), 45-63.

[8] H. H. Glover, G. Mislin, Y. Xia, On the Farrell cohomology of mapping class groups, Invent. Math. 109 (1992), 535-545.

[9] D. Sjerve and Q. Yang, Conjugacy classes of p-torsion in Sp_{p−1}(ℤ), J. Algebra 195, No. 2 (1997), 580-603.

Cornelia Busch

Katholische Universität Eichstätt

- Ingolstadt Mathematisch-Geographische Fakultät Ostenstr. 26–28 D-85072 Eichstätt Cornelia.Busch@ku-eichstaett.de