p-GROUPS WITH MAXIMAL ELEMENTARY ABELIAN SUBGROUPS OF RANK 2

GEORGE GLAUBERMAN AND NADIA MAZZA

Abstract. Let p be an odd prime number and G a finite p-group. We prove that if the rank of G is greater than p, then G has no maximal elementary abelian subgroup of rank 2. It follows that if G has rank greater than p, then the poset E(G) of elementary abelian subgroups of G of rank at least 2 is connected and the torsion-free rank of the group of endotrivial kG-modules is one, for any field k of characteristic p. We also verify the class-breadth conjecture for the p-groups G whose poset E(G) has more than one component.

1. Introduction

In this article, we prove the following result, which answers a question raised by the second author in [21, § 2]:

Theorem A. Let p be a prime and G be a finite p-group that possesses a maximal elementary abelian subgroup E of order p^{2}. Then G has rank at most p if p is odd.

In other words, if a finite p-group G for an odd prime p has a maximal elementary abelian subgroup of rank 2, then G has no elementary abelian subgroup of rank p + 1. (Recall that the rank of an elementary abelian group of order p^{n} is n, and that the rank of G is the maximum of the ranks of the elementary abelian subgroups of G.)

Surprisingly, groups satisfying the rather narrow hypothesis of Theorem A appear in several different areas of finite group theory. For example, they require a great deal of attention as possible “small” Sylow subgroups in the proof of the Feit-Thompson Odd Order Theorem ([9, pp. 453-454]; [8, pp. 845, 903]) and of many subsequent theorems on classifying simple groups ([10, pp. 67-69]). More recently, they have been important in the study of endotrivial modules in representation theory, as we explain below. Furthermore, in the special case that C_{G}(E) = E, they form part of the family of p-groups of maximal class, by a theorem of M. Suzuki ([15, Satz III.14.23]; [2, Proposition 1.8]).

N. Blackburn studied these groups extensively in [3]. In particular, he noted that for p odd, the centralizer of E in G is a soft subgroup of G, as defined by Héthelyi in [13] (and in Section 2 below). These groups were studied further in [14] and [21].

The condition on E in Theorem A suggests that G must be “small”. Indeed, it is easy to show that G cannot possess a normal elementary abelian subgroup of rank p + 1 (this is a special case of Lemma 2.4(b) below). But what about non-normal subgroups?

Date: October 20, 2009.

By considering subgroups of the wreath product C_{p} ≀ C_{p}, we see that the rank, r, of a group G satisfying the hypothesis of Theorem A can have any of the values 2, 3, …, p. One may show (Remark 3.3) that for p = 2, the values may also be 3 or 4, but not larger. In contrast, Theorem A shows that if p is odd, the only possible values are 2, 3, …, p.

From Theorem A, we obtain an application to representation theory of arbitrary finite groups. The concepts used in the following statement are explained in Section 3.

Corollary B. Let p be an odd prime and G^{∗} a finite group having p-rank greater than p. For any field k of characteristic p, the group T (G^{∗}) of endotrivial kG^{∗}- modules has torsion-free rank one. More precisely, any endotrivial kG^{∗}-module is isomorphic to a direct summand of a module of the form Ω^{n}(k)⊗M, for some integer n and some torsion endotrivial module M.

As an independent result on p-groups with maximal elementary abelian subgroups of rank 2, we end with the proof of the class-breadth conjecture for them ([7]). We define the poset E(G) in Section 2.

Proposition C. Let p be an odd prime and G a finite p-group. Assume that the poset E(G) has more than one component. Then the class-breadth conjecture holds for G.

The paper is organized as follows: In Section 2, we set the notation and definitions. We also review the necessary background, and show that if a finite p-group G has a non-normal maximal elementary abelian subgroup of order p^{2}, then G possesses a unique normal elementary abelian subgroup of order p^{2}, which is hence characteristic in G. In Section 3, we prove Theorem A and Corollary B. We prove Proposition C in Section 4.

2. Generalities: Results old and new

Henceforth in this paper, p denotes a prime number and G a finite p-group, that is, a finite group of order a power of p.

Definition 2.1.

(1) An elementary abelian subgroup of G is an abelian subgroup E of G of exponent at most p. If |E| = p^{n}, the rank of E is the integer n. Hence, the rank of G is the maximum of the ranks of the elementary abelian subgroups of G.

(2) An elementary abelian subgroup E of G is maximal if E is not properly contained in any larger elementary abelian subgroup of G.

(3) The elementary abelian subgroups of G of rank at least 2 form a poset E(G) for the order relation given by inclusion.

Now assume that G has rank at least 2. The groups in Theorem A are important because the study of simple groups and of endotrivial modules for a finite group is usually much easier when E(G) is connected for a Sylow p-subgroup G. In fact, one easily deduces from [10, Lemma 10.21] that E(G) is connected if and only if G has a unique elementary abelian subgroup of rank 2 or G has no maximal elementary abelian subgroups of rank 2.

Assume p is odd. We refer the reader to [10, § 10], and especially [10, Lemmas 10.11 and 10.21], for a detailed description of the structure of the poset E(G). In particular, G possesses a normal elementary abelian subgroup E_{0} of rank 2 and if G has rank at least 3, then all the elementary abelian subgroups of G of rank 3 or more lie in a common connected component of E(G), which contains also a normal elementary abelian subgroup of G of rank 2; the other possible connected components are hence isolated vertices, i.e. maximal elementary abelian subgroups of rank 2. By Lemma 10.21 and Corollary 10.22 of [10], E(G) is connected if the normal rank of G is greater than p, or if the center of G is not cyclic.

We now quote some useful results from [3], [13], [14] and [21]. Let |G| = p^{n}. If G has a non-normal maximal elementary abelian subgroup E of rank 2, then E determines a strictly increasing chain

E ≤ N_{0} < N_{1} < ⋯ < N_{r−1} < N_{r} = G with |N_{i} : N_{i−1}| = |G/N_{r−1}| = p .

Here, N_{0} = C_{G}(E) is a soft subgroup of G (as defined in [13], i.e., C_{G}(N_{0}) = N_{0} and |N_{G}(N_{0})/N_{0}| = p) of the form C_{p^{n−r}} × C_{p}, and N_{i} = N_{G}(N_{i−1}), for all 1 ≤ i ≤ r.

Moreover, |G : N_{0}| = p^{r}, and N_{i} has nilpotence class i + 1, for all 0 ≤ i ≤ r. A striking fact is that the size of N_{0} does not depend on the choice of the non-normal maximal elementary subgroup E. Finally, the centralizer C_{G}(E_{0}) of E_{0} is a maximal subgroup of G, and its intersection H = C_{G}(E_{0}) ∩ N_{r−1} is also independent of E and thus is a characteristic subgroup of index p^{2} in G.

For the remainder of this paper, we refer the reader to one of the books [2], [11], or [15] for the background material and the statements about regular p-groups that we use.

Remark 2.2. For convenience, we single out the mechanics of the Lazard correspondence, as we will repeatedly use them. We refer the reader to [18, Chap. 10], and in particular to the results stated in 10.11, 10.13 and on p. 124.

A celebrated theorem of M. Lazard shows that we may define operations + and [ , ] on any finite p-group H of nilpotence class less than p, in order to make H into a Lie ring H_{L} such that every automorphism of the group H induces an automorphism of the Lie ring H_{L}, and each element of H in H_{L} has the same order under + as its order in the group H. Moreover, each subgroup of the group H is a Lie subring of H_{L}.

The case of interest to us is when H is a subgroup of exponent p of a finite p-group G, say |H| = p^{n}, and x ∈ G has order p and normalizes H. Then conjugation by x induces an automorphism c_{x} of order p of the additive group of H_{L}, which is an elementary abelian group of rank n, and thus a vector space of dimension n over the prime field 𝔽_{p}. By considering the Jordan form of this automorphism, we get the rank of C_{H}(x) as the number of Jordan blocks of c_{x}, which is greater than or equal to n/p.

The following result is a consequence of P. Hall’s Enumeration Principle (see [22, Theorem IV.4.19 (i)] or [12, Theorem 1.4]). Note that Lemma 2.3 generalizes to finite nilpotent groups, because they are the direct product of their Sylow p-subgroups.

Lemma 2.3. Let P be a finite p-group and Q a normal subgroup of order p^{n} of P. For each integer k with 0 ≤ k < n, Q contains a subgroup Q_{k} of order p^{k} that is normal in P.

Proof. We proceed by induction on k. If k = 0, the claim trivially holds. Assume k ≥ 1 and pick a subgroup Q_{1} ≤ Q ∩ Z(P) with |Q_{1}| = p. (Recall that any nontrivial normal subgroup of P intersects Z(P) non-trivially.) Then, Q_{1} ◁ P. Set π : P → P/Q_{1} for the natural projection map and write \overline{K} = π(K) for the image of a subgroup K of P under π. In \overline{P}, we have by induction hypothesis that \overline{Q} contains a normal subgroup \overline{Q_{k}} of \overline{P} of order p^{k−1}. Therefore, Q_{k} = π^{−1}(\overline{Q_{k}}) is a normal subgroup of P contained in Q and |Q_{k}| = p^{k}. □

Lemma 2.4. Suppose that E is a non-normal maximal elementary abelian subgroup of G of rank 2, and H is a subgroup of exponent p in G that is normalized by E. Let |H| = p^{n}. Then:

(a) for each positive integer k less than n, H contains a subgroup H_{k} of order p^{k} that is normalized by E;

(b) n ≤ p; and

(c) if E is not contained in H, then the subgroup H_{k} in part (a) is unique, for each k.

Proof. Let E = ⟨z, x⟩, with z ∈ Z(G). Note that H is normal in HE.

Each part of the lemma is vacuous or obvious if |H| ≤ p or if H = E. So we assume that |H| ≥ p^{2} and that H ≠ E. Then C_{E}(H) = ⟨z⟩ and

(A) C_{H}(x) = C_{H}(E) = H ∩ E .

Part (a) is Lemma 2.3 applied to P = HE and Q = H. Thus, for each 0 ≤ k < n, the group H contains a subgroup H_{k} of order p^{k} that is normalized by E (see also related results in [17, Proposition 0.1]).

For part (b), assume that n ≥ p + 1. We aim for a contradiction. By (a), we may assume that n = p + 1.

Since H has exponent p, it is a regular p-group. Therefore, by a theorem of N. Blackburn ([15, Satz III.14.21]; [2, Theorem 9.5]), H is not a p-group of maximal class.

Suppose first that x ∈ H. Since |H| > p^{2}, we see that

⟨x⟩ < C_{H}(x) = C_{H}(E) = H ∩ E .

Hence

|C_{H}(x)| = |E| = p^{2} , and |H : C_{H}(x)| = |H|/p^{2} .

By Suzuki’s Theorem mentioned in Section 1, H is a p-group of maximal class, a contradiction. Thus, x lies outside H, and

(B) |C_{H}(x)| = |H ∩ E| ≤ p .

Since |H| = p^{p+1} and H does not have maximal class, H has class at most p − 1. We appeal to Remark 2.2. Explicitly, conjugation by x induces an automorphism of order p of the additive group of the Lie ring H_{L}, which is an elementary abelian group of rank p + 1, and thus a vector space of dimension p + 1 over the prime field 𝔽_{p}. By considering the Jordan form of this automorphism, we see that it has at least two Jordan blocks. Therefore, |C_{H}(x)| ≥ p^{2}. But |C_{H}(x)| ≤ p by (B), a contradiction.

For part (c), we assume that E is not contained in H. By (A), we have |C_{H}(x)| = |H ∩E| ≤ p . By part (b), |H| ≤ p^{p}. Hence, H has nilpotence class at most p−1. As in (b), we apply Lazard’s theorem and consider the Jordan form of the automorphism of H_{L} induced by conjugation by x. As |C_{H}(x)| ≤ p, this is a single Jordan block of degree n. Therefore, x preserves a unique k-dimensional subspace of H_{L} over 𝔽_{p}, which proves (c). □

From these technicalities, we draw the following conclusion.

Proposition 2.5. Assume that p is odd and that G has rank greater than 2. If G has some non-normal maximal elementary abelian subgroup of rank 2, then G has a unique normal elementary abelian subgroup of rank 2, which is hence a characteristic subgroup of G.

Proof. Suppose that G contains a normal elementary abelian subgroup F of rank 2 other than the chosen subgroup E_{0} which we introduced after Definition 2.1. Let H = E_{0}F. Then F contains Z, and E_{0}/Z and F/Z are contained in the center of G/Z. Therefore, H has order p^{3} and nilpotence class at most 2, and possesses more than one elementary abelian subgroup of order p^{2}. A review of the groups of order p^{3} for odd p (or an application of [11, Theorem 12.4.3], since H is a regular p-group) shows that H has exponent p.

As E/Z is not normal in G/Z and since H/Z is central in G/Z, we see that E is not contained in H. By Lemma 2.4, E normalizes only one subgroup of order p^{2} in H. But E normalizes E_{0} and F, a contradiction. □

Remark 2.6. We refer the reader to [21, Corollary 2.3] for the case when G has only normal elementary abelian subgroups of rank 2.

3. Proof of the main result

For convenience, we appeal to some additional standard notation. For any finite p-group G of nilpotence class c, we write

1 = Z_{0}(G) ≤ Z_{1}(G) ≤ Z_{2}(G) ≤ ⋯ ≤ Z_{c}(G) = G ,

with Z_{1}(G) = Z(G) , and Z_{i}(G)/Z_{i−1}(G) = Z(G/Z_{i−1}(G)) , ∀ 1 ≤ i ≤ c , for the subgroups in the upper central series of G. Recall that the least positive integer c with Z_{c}(G) = G is the nilpotence class of G. For subgroups H, K of G, the subgroup ⟨H^{K}⟩ of G is generated by the K-conjugates of H. In particular, ⟨H^{G}⟩ is the normal closure of H in G. Also, Ω_{d}(H) is the subgroup ⟨x ∈ H | x^{p^{d}} = 1⟩ of G generated by the elements of order at most p^{d}, for any integer d ≥ 1.

The following lemma is equivalent to [12, Theorem 2.49, (i)].

Lemma 3.1. Suppose that N is a normal subgroup of G and k is an integer, k ≥ 0. (a) If N ∩ Z_{k}(G) = N ∩ Z_{k+1}(G), then N ≤ Z_{k}(G).

(b) If |N| = p^{k}, then N ≤ Z_{k}(G).

Proof. For part (a), let M = N ∩ Z_{k}(G) and \overline{G} = G/M, and let \overline{X} = XM/M for every subgroup X of G. Then, \overline{N} ◁ \overline{G}, and since M ≤ Z_{k}(G), the definition of the upper central series gives

Z(\overline{G}) ≤ Z_{k+1}(G)/M and so \overline{N} ∩ Z(\overline{G}) ≤ (N ∩ Z_{k+1}(G))/M = 1 .

Hence, \overline{N} = 1.

Part (b) follows from (a). Indeed, let N ◁ G. Assume that N ≰ Z_{k}(G). Then

1 = N ∩ Z_{0}(G) < N ∩ Z_{1}(G) < ⋯ < N ∩ Z_{k+1}(G) ,

is a strictly increasing chain of subgroups of G. Thus, we must have |N| > p^{k}. □

In view of [10, Proposition 10.17] (or [17, Theorem]), if p = 3 and G has rank at least 4, then G has normal rank 4. Consequently, Theorem A holds for p = 3, as also observed in [21]. So, we may in addition suppose that p ≥ 5 from now on. Thus, Theorem A follows from our next result.

Theorem 3.2. Let p be a prime greater than 3, and assume that G has order p^{n}. If G has a non-normal maximal elementary abelian subgroup of rank 2, then G has rank at most p.

Proof. We assume that G has rank greater than p and work toward a contradiction.

Let E be a non-normal maximal elementary abelian subgroup of rank 2 in G. By hypothesis, p ≥ 5 and G contains an elementary abelian subgroup A of rank p + 1.

By [1, Theorem D], we may choose A to be normal in its normal closure, ⟨A^{G}⟩, in G. Let N = ⟨A^{G}⟩.

Since A ◁ N, Lemma 2.3 says that A contains a normal subgroup B of N having order p^{p−1}.

Let M = Ω_{1}(Z_{p−1}(N)). Then M ◁ G and B ≤ M by Lemma 3.1. Since Z_{p−1}(N) has class at most p−1, it is a regular p-group. Therefore, M has exponent p because it is a regular p-group generated by elements of order p. Since M ◁ G, Lemma 2.4 yields that |M| ≤ p^{p}. Hence, |M : B| ≤ p.

Let Y = Ω_{1}(Z_{2}(N)) and W = Ω_{1}(Z(N)). Then

W ≤ Y ≤ M and W, Y ◁ G .

Assume first that Y ≤ A. Then C_{G}(Y)◁G and A ≤ C_{G}(Y). Therefore, N = ⟨A^{G}⟩ ≤ C_{G}(Y), and Y ≤ Z(N). More generally, observe similarly that any normal abelian subgroup of G that is contained in any conjugate of A is necessarily contained in Z(N). Then

A ∩ Z_{2}(N) = A ∩ Y = A ∩ Z(N) ,

and A ≤ Z(N), by Lemma 3.1. But then,

A ≤ Ω_{1}(Z_{p−1}(N)) = M and p^{p+1} = |A| ≤ |M| ≤ p^{p} ,

a contradiction. Thus, Y is not contained in A. Therefore, B < BY ≤ M.

Since |M : B| ≤ p, we have M = BY. Moreover, Y/W ≤ Z(N/W). Therefore, M/W is centralized by AW/W. As M ◁ G, it follows that M/W is centralized by ⟨A^{G}⟩W/W, i.e., by N/W. Therefore, M ≤ Z_{2}(N). But now,

A ∩ Z_{3}(N) ≤ A ∩ Z_{p−1}(N) = A ∩ M = A ∩ Z_{2}(N) .

So A ∩ Z_{3}(N) = A ∩ Z_{2}(N). By Lemma 3.1, A ≤ Z_{2}(N). Hence, A ≤ M, and we obtain a contradiction as in the previous paragraph. □

Now we obtain our main result.

Theorem A. Let p be an odd prime and let G be a finite p-group. If G has rank at least p + 1, then G has no maximal elementary abelian subgroup of order p^{2}.

Theorem A contrasts sharply with the situation for p = 2.

Remark 3.3. Suppose G is a 2-group possessing a maximal elementary abelian subgroup of rank 2. By Lemma 2.4, G has no normal elementary abelian subgroup of rank 3. Therefore, by a theorem of Anne MacWilliams Patterson [20], every subgroup of G is generated by 4 or fewer elements. Hence, G has rank at most 4.

Examples in [10, p. 68] show that G may have rank 3. Here, we give an example of rank 4.

Let 𝔽 be the finite field of order 4. For each a, b, c in 𝔽, let M(a, b, c) be the 3 × 3 matrix over 𝔽 given by

M(a, b, c) = (\begin{matrix}1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix})

Let U be the set of all matrices M(a, b, c). Then U is a group under multiplication and is a Sylow 2-subgroup of GL_{3}(4).

For each a in 𝔽, let ā = a^{2}; thus, we obtain the unique non-trivial field automorphism of 𝔽. Let t be the mapping on U given by

M(a, b, c)^{t} = M(b̄, ā, āb̄ + c̄) .

Then t is an automorphism of order two of U (and comes from a unitary automorphism of order two of GL_{3}(4)). Let G be the semi-direct product of U by ⟨t⟩.

Note that C_{U}(t) is the group of all matrices of the form M(a, ā, c) such that c + c̄ = aā. This group is a quaternion group of order 8, and C_{G}(t) = C_{U}(t) × ⟨t⟩. This shows that G possesses a maximal elementary abelian subgroup of rank 2, namely, Z(C_{U}(t)) × ⟨t⟩. However, it is easy to see that U, and hence G, possess an elementary abelian subgroup of rank 4. Therefore, G has rank 4.

We end this section with a consequence of Theorem A concerning some important finitely generated representations of an arbitrary finite group G^{∗} over a field k of characteristic p. (Hence, for the remainder of this section, we let G^{∗} denote an arbitrary finite group.) The relationship between the group of endotrivial kG^{∗}- modules T (G^{∗}) and the result stated in Theorem A is that the torsion-free rank of the group T (G^{∗}) equals the number of conjugacy classes of connected components of the poset E(G^{∗}) ([5, § 3]). By [4] and [21], this number is at most 5 if p = 2 and at most p + 1 if p is odd. In the particular case that T (G^{∗}) has torsion-free rank 1, the description of T (G^{∗}) is much easier, according to the results and notation of [5] (explained below). Indeed, in this case, any endotrivial kG^{∗}-module is isomorphic to a direct summand of a module of the form

Ω^{n}(k) ⊗ M

for some integer n and some torsion endotrivial kG^{∗}-module M. Hence, Theorem A provides a criterion for this to happen which only depends on the p-rank of G^{∗}.

For completeness, we explain the above concepts. We let k denote both a chosen field of characteristic p and the 1-dimensional trivial kG^{∗}-module. The modules Ω^{n}(k) are the syzygies of k. These are defined inductively as follows: Let P_{∗} ↠ k be a minimal projective resolution of k. Then, Ω^{0}(k) = k and for n > 0,

Ω^{n}(k) = ker (P_{n−1} ↠ Ω^{n−1}(k)) .

For n < 0, we set Ω^{n}(k) = Ω^{−n}(k)^{∗}, the k-linear dual of Ω^{−n}(k). Also, M is a torsion endotrivial module if there is a positive integer m and a projective kG^{∗}-module F such that M^{⊗m} ≅ k⊕F. For additional background material on endotrivial modules, we refer the reader to [6] and [5].

Now, to obtain Corollary B, we also recall that for an arbitrary finite group G^{∗} and prime number p, the p-rank of G^{∗} is the rank of a Sylow p-subgroup S_{p} of G^{∗}. Note that the poset E(G^{∗}) has at most as many conjugacy classes of components as the poset E(S_{p}), and E(G^{∗}) is non-empty whenever E(S_{p}) is non-empty. Therefore, if E(S_{p}) is connected, then the components of E(G^{∗}) form a single conjugacy class. This proves:

Corollary B. Let p be an odd prime and G^{∗} a finite group having p-rank greater than p. For any field k of characteristic p, the group T (G^{∗}) of endotrivial kG^{∗}- modules has torsion-free rank one. More precisely, any endotrivial kG^{∗}-module is isomorphic to a direct summand of a module of the form Ω^{n}(k)⊗M, for some integer n and some torsion endotrivial module M.

4. The class-breadth conjecture

We end this note with the class-breadth conjecture for the finite p-groups G whose poset E(G) has more than one component.

Let G be a finite p-group. For x in G, the breadth b(x) of x is given by p^{b(x)} = |G : C_{G}(x)|. In particular, b(x) = 0 if and only if x lies in Z(G). The breadth b(G) of G is the maximum of b(x) as x ranges over G.

Let c(G) denote the nilpotence class of G. The class-breadth conjecture (also known as the Breadth Conjecture) states that the inequality

c(G) ≤ b(G) + 1

always holds. Although counterexamples have been found for p = 2, none is known for p odd. For background and recent results about the class-breadth conjecture, we refer the reader to [19] and [7]. In particular, several cases are known to be true, and moreover, the bound is optimal, in the sense that there are groups for which the equality c(G) = b(G) + 1 holds. The finite abelian p-groups and those of maximal nilpotence class are such instances, and [19] presents further cases.

Proposition C. Let p be an odd prime and G a finite p-group. Assume that the poset E(G) has more than one component. Then the class-breadth conjecture holds for G.

Proof. Write c = c(G) for the nilpotence class of G. Let E = ⟨x, z⟩ be a maximal elementary abelian subgroup of G, with z ∈ Z(G). By [3, Theorem], we obtain the equalities C_{G}(E) = ⟨x⟩ × Z(N_{G}(E)), with Z(N_{G}(E)) cyclic, and

|G : C_{G}(E)| = |G : C_{G}(x)| = p^{c−1}.

Hence c = b(x) + 1. Since b(G) ≥ b(x), the class-breadth conjecture c ≤ b(G) + 1 holds for G. □

Remark 4.1. Observe that a similar proof shows that the class-breadth conjecture holds for any finite p-group G having some soft subgroup A such that, for every proper subgroup H of G containing A, the nilpotence class of N_{G}(H) is one more than the nilpotence class of H.

Acknowledgments. The authors wish to thank the organizers of the conferences at the University of Chicago (March, 2008) and on the Isle of Skye (June, 2009), which allowed them to make extended progress on this project. Therefore, the authors are doubly grateful to Ron Solomon for his involvement in these two events, and also to him and Jon Carlson for suggesting examples for Remark 3.3. The authors also would like to express their sincere gratitude to Y. Berkovich, V. Naik, and the referee for several enlightening suggestions. In addition, the first author thanks the National Security Agency for its support by a grant during the preparation of this paper.

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Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL, 60637, USA

Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, UK