DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 379–390

HANTZSCHE–WENDT MANIFOLDS OF DIMENSION 7

R. J. Miatello and J. P. Rossetti

Abstract. We classify Hantzsche-Wendt manifolds in dimension 7, proving that there are exactly 62 diffeomorphism classes. We consider three different notions of isospectrality, showing that all three are equivalent to each other. We also show that a finite part of the spectrum determines the full spectrum. We list the isospectral classes in dimension 7 and give a formula to compute the multiplicities of the eigenvalues.

Introduction

Hantzsche-Wendt manifolds (or HW manifolds) are compact flat Riemannian manifolds which are natural generalizations, to any odd dimension n, of the classical 3-dimensional Hantzsche-Wendt manifold. These manifolds have holonomy group ℤ_{2}^{n−1} and are rational homology spheres. In [MR], by using certain graphs associated to any such manifold, we proved that h(n), the number of diffeomorphism classes of HW manifolds, grows (at least) exponentially with n, showing also that h(5) = 2. Furthermore we proved that for any n ≥ 7, there exist pairs of isospectral manifolds non homeomorphic to each other.

In this paper we shall study in some detail Hantzsche-Wendt manifolds of dimension 7, using the results in [MR]. We shall compare isospectrality and Fisospectrality, a notion having to do with a finite part of the spectrum. We will prove that in dimensions 7 and 9, these two notions are equivalent to each other. We shall also see that if n = 7 or 9, a finite part of the spectrum, namely the multiplicities of those eigenvalues λ such that \frac{λ}{4π^{2}} ≤ \frac{n^{2}−7}{2}, determines the full spectrum. Also, if n = 7, we shall see that these two notions are equivalent to Sunada-isospectrality.

1991 Mathematics Subject Classification. Primary 58G25; Secondary 20H15.

Key words and phrases. Bieberbach groups, isospectral Hantzsche-Wendt manifolds.

Partially supported by Conicet, Conicor and Secyt U.N.C.

This paper is in final form and no version of it will be submitted for publication elsewhere.

At the end we shall give a complete system of representatives of HW manifolds in dimension 7, showing that there are exactly 62 diffeomorphism classes. This classification has been obtained by means of a computer program. We will also include explicit realizations of each manifold, computing the numbers m_{k,l} (see Definition 2.1), which determine the multiplicities of the eigenvalues, and also the Sunada numbers c_{s,t} (see §2).

The authors wish to thank P. Gilkey for useful conversations on the contents of this paper.

1. Preliminaries

In this section we recall some known facts on Hantzsche-Wendt Bieberbach groups. We refer to [MR] for the proofs.

A Bieberbach group is a discrete cocompact subgroup Γ of I(ℝ^{n}) which is torsion-free. Then M_{Γ} = Γ\ℝ^{n} is a compact flat Riemannian manifold with fundamental group Γ. Any element γ ∈ I(ℝ^{n}) decomposes uniquely γ = uL_{v}, with u ∈ O(n) and v ∈ ℝ^{n}. The translations in Γ form a normal abelian subgroup L_{Λ} of Γ, Λ a lattice in ℝ^{n}. The point group F = Λ\Γ coincides with the holonomy group of M_{Γ} and acts on Λ by conjugation. This action defines an integral representation of F, called the holonomy representation.

A Hantzsche-Wendt group (or HW group) Γ is a Bieberbach group with holonomy group F ≃ ℤ_{2}^{n−1} and such that the action of every x ∈ F diagonalizes on a ℤ-basis of Λ with det(x) = 1.

By conjugating by an element of GL_{n}(ℝ), if necessary, we may assume that Λ = ∑_{1}^{n} ℤe_{i}, the canonical lattice in ℝ^{n}; hence F can be identified to the diagonal subgroup {B : Be_{i} = ±e_{i}, 1 ≤ i ≤ n, det B = 1}. Since Γ is torsion-free, then −I ∉ F, hence n must be odd.

If Γ is an HW group, the corresponding manifold M_{Γ} := Γ\ℝ^{n} will be called a Hantzsche-Wendt (or HW) manifold.

We choose B_{i} ∈ F such that B_{i} fixes e_{i} and B_{i}e_{j} = −e_{j} (if j ≠ i), for each 1 ≤ i ≤ n. Clearly, F is generated by B_{1}, B_{2}, …, B_{n−1}. Furthermore B_{n} = B_{1}B_{2} … B_{n−1} (since n is odd).

Thus, if Γ is a Hantzsche-Wendt group, then Γ = ⟨B_{1}L_{b_{1}}, …, B_{n−1}L_{b_{n−1}}, L_{λ} : λ ∈ Λ⟩, for some b_{i} ∈ ℝ^{n}, 1 ≤ i ≤ n − 1. Now, if I = (i_{1}, …, i_{s}) with 1 ≤ i_{1} < i_{2} < ⋯ < i_{s} ≤ n, one can write B_{i_{1}}L_{b_{i_{1}}} … B_{i_{s}}L_{b_{is}} = B_{I}L_{b(I)} with

(1.1) b(I) = b(i_{1}, …, i_{s}) = B_{i_{s}} … B_{i_{2}}b_{i_{1}} + B_{i_{s}} … B_{i_{3}}b_{i_{2}} + ⋯ + B_{i_{s}}b_{i_{s−1}} + b_{i_{s}} .

and B_{I} = B_{i_{1}} … B_{i_{s}}. A complete system of representatives for Λ\Γ is

{B_{i_{1}} … B_{i_{s}}L_{b(i_{1},...,i_{s})} : 1 ≤ i_{1} < i_{2} < ⋯ < i_{s} ≤ n − 1} .

It is shown in [MR] that, by replacing Γ by an isomorphic group, one may assume that the components of b_{i}, b_{ij}, lie in \{0, \frac{1}{2}\}, 1 ≤ i, j ≤ n.

Thus any γ ∈ Γ can be written uniquely

(1.2)  γ = B_{i_{1}} … B_{i_{r}}L_{b_{o}(I)}L_{λ}

where r is odd, 1 ≤ i_{1} < ⋯ < i_{r} ≤ n, λ ∈ Λ and b_{o}(I)_{j} ∈ \{0, \frac{1}{2}\}, for 1 ≤ j ≤ n.

In [DM], Proposition 2.1, it is shown that the condition for Γ to be torsion-free is as follows:

(1.3)  (B_{i_{1}} … B_{i_{s}} + I)b(i_{1}, …, i_{s}) ∈ Λ − (B_{i_{1}} … B_{i_{s}} + I)Λ

for any 1 ≤ i_{1} < i_{2} < ⋯ < i_{s} ≤ n − 1. This condition is equivalent to the condition that the class defined by Γ in H^{2}(ℤ_{2}^{n−1}, Λ) be a special class (see [Ch], Theorem 2.1, p. 79).

We can rephrase (1.3) in terms of the entries of the matrix A := [b_{ij}] associated to Γ as follows (see [MR], Prop. 1.2): Γ is torsion free with translation lattice Λ if and only if for any I ⫋ {1, 2, …, n}, there exists j ∈ I, if |I| is odd (respectively j ∉ I, if |I| is even) such that

(1.4)  card \{i ∈ I : b_{ij} = \frac{1}{2}\} is odd.

In particular, (1.4) implies that for each fixed j, b_{jj} = \frac{1}{2}, and card\{i : 1 ≤ i ≤ n, b_{ij} = \frac{1}{2}\} is even.

In [MR] we saw that to show that Γ is torsion free, it is sufficient to verify that for each j, card \{i : 1 ≤ i ≤ n, b_{ij} = \frac{1}{2}\} is even and that (1.4) holds for all I with |I| odd (or else, for all I with |I| ≤ \frac{n−1}{2}).

Concerning conditions for the isomorphism of two HW groups, we show in [MR] that if Γ, Γ′ are HW groups with associated matrices A = [b_{ij}], A′ = [b′_{ij}] respectively, and b_{ij}, b′_{ij} ∈ \{0, \frac{1}{2}\} for all i, j, then Γ ≃ Γ′ if and only if

(1.5)  A′ = P_{σ}A_{c}P_{σ}^{−1} ,

where σ ∈ S_{n} and A_{c} is a “complementary matrix” of A with respect to a set of columns of A, in the sense that A_{c} is obtained from A by replacing in these columns all \frac{1}{2}’s (except those in the diagonal) by 0’s and all 0’s by \frac{1}{2}’s.

Condition (1.5) implies that when n ≡ 1 mod 4, (resp. n ≡ 3 mod 4), in each isomorphism class of HW groups there is a representative such that the associated matrix has at most \frac{n−1}{2} (resp. \frac{n+1}{2}) non zero entries on each column. If n ≡ 1 mod 4 and we restrict ourselves to such representatives, the above condition for isomorphism reduces to A′ = P_{σ}AP_{σ}^{−1} for some σ ∈ S_{n}.

Let Γ be an HW group with associated matrix A. If A has m_{i} nonzero entries on the i-th column for i = 1, …, n, we shall say that Γ has column pattern (m_{1}, …, m_{n}). The column pattern is an invariant of the isomorphism class of Γ.

When n = 5 there are exactly 2 isomorphism classes, both having column pattern (2, 2, 2, 2, 2). More generally, for any odd n there are exactly two HW manifolds having column pattern (2, …, 2) ([MR], Thm 2.1).

In the case when n = 7 (and also for n = 9), we need only consider n × n matrices A having either 2 or 4 non-zero entries on each column. The possible column patterns are of the type (4, …, 4, 2, …, 2) where 4 and 2 are repeated h and n − h times, respectively. We shall see in Section 3, that if n = 7, necessarily 0 ≤ h ≤ 5, giving a full list of representatives for each value of h.

2. Isospectrality of Hantzsche-Wendt manifolds

The purpose of this section is to study the spectrum of HW manifolds. Given Γ an HW group, we wish to describe the spectrum of the Laplacian on L^{2}(Γ\ℝ^{n}). We first consider the spectrum of Λ\ℝ^{n}.

If v ∈ Λ^{∗}, the dual lattice of Λ, the function f_{v}(x) = e^{2πi v.x} on ℝ^{n} is Λ-invariant and −Δf_{v} = 4π^{2}∥v∥^{2} f_{v}, (Δ the Laplacian on ℝ^{n}). Furthermore, if v, v′ ∈ Λ^{∗}, then ⟨f_{v}, f_{v′}⟩ = δ_{v,v′} and it easily follows from the Stone-Weierstrass theorem that {f_{v} : v ∈ Λ^{∗}} is a complete orthonormal system of L^{2}(Λ\ℝ^{n}). Thus, for each μ ∈ ℕ, the eigenspace of −Δ with eigenvalue 4π^{2}μ, H_{μ} := L^{2}(Λ\ℝ^{n})_{4π^{2}μ} = span{f_{v} : v ∈ Λ_{μ}^{∗}}, where Λ_{μ}^{∗} = {v ∈ Λ^{∗} : ∥v∥^{2} = μ}. The spectrum of −Δ in Λ\ℝ^{n} is thus determined by the cardinality of the sets Λ_{μ}^{∗}, μ ∈ ℕ.

If we now look at M_{Γ} = Γ\ℝ^{n}, we see that for each γ ∈ Γ, μ ∈ ℕ, γ preserves H_{μ}. Indeed, if γ = BL_{b}, B ∈ O(n), b ∈ ℝ^{n}, then γf_{v}(x) = e^{−2πi v.b}f_{Bv}(x). It easily follows that

L^{2}(Γ\ℝ^{n}) ≃ L^{2}(Λ\ℝ^{n})^{F} = ⊕ ∑_{μ} H_{μ}^{F} .

Hence

Spec(Γ\ℝ^{n}) = {(μ, d_{μ,Γ}) : μ = 0, 1, … } ,

where d_{μ,Γ} = dim H_{μ}^{F} , for each μ ∈ ℕ ∪ {0}.

On the other hand, for each μ we have an orthogonal projection p_{μ} : H_{μ} → H_{μ}^{F}, given by p_{μ} = |F |^{−1} ∑_{γ∈Λ\Γ} γ_{|H_{μ}}. Thus we get:

(2.1)  d_{μ,Γ} = |F |^{−1} ( ∑_{γ∈Λ\Γ} tr γ_{|H_{μ}} ) .

If Γ is an HW group, then Λ = Λ^{∗}, the canonical lattice. Furthermore, any γ ∈ Γ can be written (see (1.2)), γ = B_{I}L_{b_{o}(I)}L_{λ}, |I| odd, b_{o}(I)_{j} ∈ \{0, \frac{1}{2}\}, for all 1 ≤ j ≤ n and λ ∈ Λ. Hence γf_{v}(x) = e^{−2πib_{o}(I).v}f_{B_{I}v}(x). For each v ∈ Λ we set

I_{v} = {j : v.e_{j} ≠ 0}, I_{v}^{odd} = {j : v.e_{j} is odd } .

It follows that ⟨γf_{v}, f_{v}⟩ ≠ 0 if and only if B_{I}v = v, that is, iff I ⊃ I_{v}, by the definition of B_{I}, since |I| is odd. If this holds then we get

⟨γf_{v}, f_{v}⟩ = (−1)^{|I_{bo(I)}∩I_{v}^{odd}|} .

Therefore, for each μ ∈ ℕ we have

(2.2)  d_{μ,Γ} = |F |^{−1} ∑_{v∈Λ_{μ}} ∑_{I⊃I_{v}}(−1)^{|I_{bo(I)}∩I_{v}^{odd}|} .

For v, v′ ∈ Λ_{μ}, we say that v ∼ v′ if v′ is obtained from v by a permutation, followed by a sign change of some of the coordinates. We fix for each μ, Λ̃_{μ}, a complete system of representatives of Λ_{μ}. Also, we set for each v ∈ Λ,

(2.3)  m_{v,Γ} = |F |^{−1} ∑_{v′∼v} ∑_{I⊃I_{v′}}(−1)^{|I_{bo(I)}∩I_{v′}^{odd}|} .

We note that m_{v,Γ} is the dimension of the space of F -invariant vectors of the subspace of H_{μ}, generated by f_{v′}, v′ ∼ v, hence m_{v,Γ} ≥ 0.

Thus, for each μ we have d_{μ,Γ} = ∑_{v∈Λ̃_{μ}} m_{v,Γ}, by (2.2) and (2.3).

Now, given v ∈ Λ_{μ}, denote by v̄ ∈ Λ the only vector with the following properties: v̄.e_{j} ∈ {0, 1, 2} for any j, |I_{v}| = |Iv̄|, |I_{v}^{odd}| = |Iodd ; v̄| and the components of v̄ are arranged in non-increasing order.

We have that, for any v, m_{v,Γ} = α_{v}m_{v̄,Γ}, with α_{v} a natural number. Actually, if v has e_{v} (resp. o_{v}) distinct nonzero even (resp. odd) components with e_{1}, …, e_{r} repetitions up to sign, ∑_{j=1}^{r} e_{j} = e_{v} (resp. o_{1}, …, o_{s} repetitions up to sign, ∑_{j=1}^{s} o_{j} = o_{v}), then it is not hard to verify that α_{v} = \binom{e_{v}}{e_{1},...,e_{r}}\binom{o_{v}}{o_{1},...,o_{s}}. For instance, if v = (8, −9, 3, 0, −1, 1, −4, 0, 8, 0, 0), then v̄ = (2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0) and we have m_{v} = \binom{3}{2,1}\binom{4}{1,1,2} mv̄ = 3\frac{4!}{2!} mv̄.

For simplicity, in what follows we shall often omit the group Γ from the notation.

Definition 2.1. For any k, l, with 0 ≤ l ≤ k ≤ n, let m_{k,l} = mv̄ where v is any vector with |I_{v}| = k and |I_{v}^{odd}| = l. We shall say that Γ and Γ′ are F-isospectral if

(F)  m_{k,l,Γ} = m_{k,l,Γ′}, for any 1 ≤ l ≤ k ≤ n .

Summing up we have obtained:

Proposition 2.2. For any μ ∈ ℕ we have:

(2.4)  d_{μ} = ∑_{v∈Λ̃_{μ}} α_{v}m_{k_{v},l_{v}}

where k_{v} = |I_{v}|, l_{v} = |I_{v}^{odd}| and α_{v} ∈ ℕ is as above. Hence, if Γ and Γ′ are HW groups and (F) holds, then M_{Γ} and M_{Γ′} are isospectral.

We shall now consider the converse question, that is, whether isospectrality implies condition (F). By (2.4), the problem reduces to showing that one can invert a system of linear equations. We shall see that this can be done for n = 7, 9 and we believe this should be the case for general n, but at this point we do not know how to handle the linear system in general.

We shall now show that if n = 7, 9, the system in Proposition 2.2 can be inverted, allowing to determine the m_{k,l}’s in terms of the d_{μ}’s.

The number of variables m_{k,l} to be determined is ∑_{j=0}^{n}(j + 1) = \binom{n+2}{2}, that is, there are 36 variables if n = 7 and 55 if n = 9.

We note that if k = n or n − 1, the only element contributing to the sum is the identity element (I = {1, …, n}, I_{b_{o}(I)} = ∅), and one finds that

(2.5)  m_{k,l} = 2^{k−n+1}\binom{n}{k}\binom{k}{l},

so these variables are determined independently of Γ. On the other hand, if l = 0, then I_{v}^{odd} = ∅, hence one computes that

(2.6) m_{k,0} = \frac{2^{k}}{2^{n−1}} ∑_{i=0}^{[ \frac{n−k}{2} ]}\binom{n}{n−2i}\binom{n−2i}{k} = 2^{k−n+1}\binom{n}{k} ∑_{i=0}^{[ \frac{n−k}{2} ]}\binom{n−k}{2i} = \binom{n}{k}

for all Γ. We are thus left with the m_{k,l} corresponding to 1 ≤ l ≤ k ≤ n − 2, a total of \binom{n−1}{2} unknowns. Finally we observe that, for all n, if 1 ≤ μ ≤ 4, m_{μ,μ} is immediately determined by d_{μ}. We are thus left with 11 unknowns to determine, if n = 7 and 24, if n = 9. Assume first n = 7. If we let successively μ = 1, 2, …, 17, 21 and use the partitions of μ as a sum of squares, we get the equations:

d_{5} = m_{5,5} + m_{2,1},  d_{6} = m_{6,6} + m_{3,2},
d_{7} = m_{7,7} + m_{4,3},  d_{8} = m_{5,4} + m_{2,0},
d_{9} = m_{1,1} + m_{6,5} + m_{3,1},  d_{10} = 2m_{2,2} + m_{7,6} + m_{4,2},
d_{11} = 3m_{3,3} + m_{5,3},  d_{12} = 4m_{4,4} + m_{6,4} + m_{3,0},
d_{13} = 5m_{5,5} + m_{2,1} + m_{7,5} + m_{4,1},  d_{14} = 6m_{6,6} + 2m_{3,2} + m_{5,2},
d_{15} = 7m_{7,7} + 3m_{4,3} + m_{6,3},  d_{16} = m_{1,0} + 4m_{5,4} + m_{7,4} + m_{4,0},
d_{17} = m_{2,1} + 5m_{6,5} + m_{3,1} + m_{5,1},
d_{21} = 10m_{5,5} + m_{6,5} + 2m_{3,1} + 5m_{7,5} + m_{4,1} + m_{6,1} .

Now we note that m_{3,2} can be obtained from d_{6}, since m_{6,6} is known. Similarly, m_{4,3} and m_{5,4} are obtained from d_{7}, d_{8} respectively, since m_{7,7} and m_{2,0} are known. By this method we can determine successively all unknowns except for m_{5,5}, m_{2,1}, m_{4,1} and m_{5,1}, which are connected by the equations corresponding to d_{5}, d_{13}, d_{17} and d_{21}. Once the known variables are disregarded, the linear system resulting has the following matrix

[ \begin{matrix}1 & 1 & 0 & 0 \\ 5 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 10 & 0 & 1 & 0 \end{matrix} ]

which is invertible, hence we can solve for the m_{k,l}’s in terms of the d_{μ}’s. This concludes the proof in the case n = 7.

For n = 9 the previous equations remain valid, with equality replaced by congruence modulo terms involving variables of the type m_{k,l} with k = 8 or 9, which are already known. We now need many more equations, namely, those corresponding to d_{μ} with 5 ≤ μ ≤ 33 and μ = 37 (actually not all of them will be used). Ommitting the variables of the type m_{k,l} with k = 8 or 9, m_{k,0} and m_{l,l}, 1 ≤ l ≤ 4, which have already been determined (see (2.5), (2.6)), the resulting equations are:

d_{18} ≡ m_{3,2} + 6m_{7,6} + 2m_{4,2} + m_{6,2},
d_{19} ≡ m_{4,3} + 3m_{5,3} + m_{7,3},
d_{20} ≡ m_{5,4} + 4m_{6,4},
d_{22} ≡ 15m_{6,6} + m_{3,2} + m_{7,6} + 2m_{4,2} + 2m_{5,2} + m_{7,2},
d_{23} ≡ 21m_{7,7} + 3m_{4,3} + 2m_{5,3} + 3m_{6,3},
d_{24} ≡ 6m_{5,4} + 2m_{6,4} + 4m_{7,4},
d_{25} ≡ m_{2,1} + 10m_{6,5} + 2m_{7,5} + 3m_{4,1} + m_{5,1} + m_{7,1},
d_{26} ≡ 2m_{3,2} + 15m_{7,6} + m_{4,2} + 3m_{5,2} + 2m_{6,2},
d_{27} ≡ 3m_{4,3} + 3m_{5,3} + 3m_{6,3} + 3m_{7,3},
d_{28} ≡ 4m_{5,4} + 6m_{6,4} + 3m_{7,4},
d_{29} ≡ 15m_{5,5} + m_{2,1} + 5m_{6,5} + 2m_{3,1} + 10m_{7,5} + 4m_{5,1} + m_{6,1},
d_{30} ≡ 26m_{6,6} + 2m_{3,2} + 6m_{7,6} + 4m_{4,2} + m_{5,2} + 4m_{6,2} + 2m_{7,2},
d_{31} ≡ 42m_{7,7} + 4m_{4,3} + 6m_{5,3} + 3m_{6,3} + 4m_{7,3},
d_{32} ≡ 8m_{5,4} + 8m_{6,4} + 6m_{7,4},
d_{33} ≡ 15m_{6,5} + 2m_{3,1} + 10m_{7,5} + 3m_{4,1} + 5m_{6,1} + m_{7,1},
d_{37} ≡ 25m_{5,5} + m_{2,1} + 10m_{6,5} + 16m_{7,5} + 4m_{4,1} + 4m_{5,1} + 6m_{7,1}.

Now we observe that the system can be split into 4 subsystems according to congruence mod 4. Indeed, for each k, l, the unknown m_{k,l} appears only in the equations corresponding to d_{μ} with μ ≡ l (mod 4).

The subsystem corresponding to μ ≡ 0 (mod 4) can be easily solved. We are thus left with three linear systems: 9×9, if μ ≡ 1, 7×7, if μ ≡ 2, and 5×5, if μ ≡ 3 mod 4. In order to write down the respective matrices of coefficients, it will be convenient to choose the m_{k,l}’s in the lexicographic ordering in (k − l, l), namely m_{5,5}, m_{6,6}, m_{7,7}, m_{2,1}, …, m_{7,6}, m_{3,1}, …, m_{7,1}. The resulting matrices are:

[ \begin{matrix}1 & 1 \\ 0 & 0 & 1 & 1 \\ 5 & 1 & 0 & 0 & 1 & 1 \\ 0 & 2 & 5 & 1 & 0 & 1 & 1 \\ 10 & 0 & 1 & 2 & 5 & 1 & 0 & 1 \\ 0 & 1 & 10 & 0 & 2 & 3 & 1 & 0 & 1 \\ 15 & 1 & 5 & 2 & 10 & 0 & 1 & 1 & 0 \\ 0 & 0 & 15 & 2 & 10 & 3 & 0 & 5 & 1 \\ 25 & 1 & 10 & 0 & 16 & 4 & 4 & 0 & 6 \end{matrix} ];  [ \begin{matrix}1 & 1 \\ 0 & 0 & 1 & 1 \\ 6 & 0 & 0 & 2 & 1 \\ 0 & 1 & 6 & 2 & 0 & 1 \\ 15 & 1 & 1 & 2 & 2 & 0 & 1 \\ 0 & 2 & 15 & 1 & 3 & 2 & 0 \\ 26 & 2 & 6 & 4 & 1 & 4 & 2 \end{matrix} ];  [ \begin{matrix}1 & 1 \\ 0 & 0 & 1 \\ 7 & 3 & 0 & 1 \\ 0 & 1 & 3 & 0 & 1 \\ 21 & 3 & 2 & 3 & 0 \end{matrix} ] .

all of which are invertible. This completes the proof if n = 9.

Remark 2.3. In [MR], we applied a method due to T. Sunada (see [Su]) to produce many examples of isospectral HW manifolds. Given an HW group, we defined for each fixed s < n, s odd, and each t ≤ s, the Sunada number c_{s,t} of Γ, by

c_{s,t}(Γ) = card{I ⊂ {1, …, n} : |I| = s and 2 ∑_{i∈I} b_{o}(I)_{i} = t}.

We have shown in [MR], Proposition 3.3 that if c_{s,t}(Γ) = c_{s,t}(Γ′), for each s, t as above, then M_{Γ} and M_{Γ′} are Sunada-isospectral, hence isospectral. From the explicit values of the numbers m_{k,l} and c_{s,t}, for n = 7 in Table 3.1, we shall see conversely that if n = 7, isospectrality implies Sunada-isospectrality.

3. Hantzsche-Wendt manifolds of dimension 7

In this section we shall give the associated matrices A (see §1) for a full set of representatives of the diffeomorphism classes of Hantzsche-Wendt manifolds in dimension n = 7. For convenience, we shall write 1’s in place of \frac{1}{2}’s, in each matrix. The matrices have been obtained by using the results stated in §1, with the help of computer programs.

We have considered separately the 8 possible column patterns. Each program generates first all the possible matrices with the prescribed column pattern. Then it determines those satisfying the torsion-free condition (1.4) and for each one of them it selects a canonical representative, ordered as follows. The first condition is that the columns having 4 nonzero entries should come first. Secondly, for all the matrices in each given equivalence class (according to (1.5)), the program forms a binary number of n^{2} digits, obtained by putting first the first column of A, then the second column, and so on, up to the n^{th} column, and chooses the one with the greatest such binary number. Finally, each time a new matrix is obtained and reduced to its canonical form, it is compared with those existing so far and added to the list when different from all of them.

As a result, one obtains exactly 2 diffeomorphism classes of type (2, …, 2), 8 classes of type (4, 2, …, 2), 19 classes of type (4, 4, 2, …, 2), 22 classes of type (4, 4, 4, 2, …, 2), 10 classes of type (4, 4, 4, 4, 2, 2, 2), 1 class of type (4, 4, 4, 4, 4, 2, 2) and none of type either (4, 4, 4, 4, 4, 4, 2) or (4, 4, 4, 4, 4, 4, 4). The resulting 62 matrices, listed in each column pattern in decreasing order, according to the ordering above, are as follows:

[ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} ] [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} ]

[ \begin{matrix}1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} ]

[ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} ] [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} ] [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 1 \end{matrix} ]

[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ]

[ \begin{matrix}1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ]

[ \begin{matrix}1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ]

[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} ]

[ \begin{matrix}1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ]

[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ]

[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ]

[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 1 \end{matrix} ]

[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 \end{matrix} ]

[ \begin{matrix}1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 \end{matrix} ]

[ \begin{matrix}1 & 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ]

[ \begin{matrix}1 & 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 1 \end{matrix} ]

By using the information in §2 and Table 3.1, given μ ∈ ℕ, it is a simple matter to compute d_{μ}, once known the different partitions of μ as a sum of squares. As an example, we will find explicitly the multiplicities of some small eigenvalues in the case of the largest isospectral family (42=43=45=51) in Table 3.1. We shall use the equations in §2 relating the d_{μ}’s with the m_{k,l}’s, formulas (2.5) and (2.6) and the information in Table 3.1. We have that:

d_{0} = 1; d_{1} = m_{1,1} = 0; d_{2} = m_{2,2} = 0; d_{3} = m_{3,3} = 2; d_{4} = m_{4,4} + m_{2,0} = 5 + 7 = 12; d_{5} = m_{5,5} + m_{2,1} = 10 + 4 = 14; d_{6} = m_{6,6} + m_{3,2} = 7 + 9 = 16; d_{7} = m_{7,7} + m_{4,3} = 2 + 30 = 32;

d_{17} = m_{2,1} + 5m_{6,5} + m_{3,1} + m_{5,1} = 4 + 10\binom{7}{5} + 20 + 102 = 336.

We next give a table which lists, for each one of the 62 matrices in the given order, the values of the 15 variables m_{k,l}, given in the same ordering as before. We also include the numbers c_{s,t}, for (s, t) = (3, 1); (3, 2); (3, 3); (5, 1); …; (5, 5).

HANTZSCHE–WENDT MANIFOLDS OF DIMENSION 7 389

Table 3.1. Spectral Information.

Matrix m_{k,l} 1 ≤ l ≤ k ≤ 5 c_{s,t} 1 ≤ t ≤ s = 3, 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8
1  0 0 0 7 14 7 7 14 49 35 42 84 70 98 70 7 21 7 7 14 0 0 0
2  0 1 1 4 8 7 12 20 43 35 51 90 70 104 70 8 19 8 10 8 3 0 0
3  0 1 1 4 8 6 12 24 43 30 51 94 62 104 66 12 15 8 9 7 4 1 0
4  0 0 1 7 10 6 7 22 49 30 42 92 62 98 66 11 17 7 7 10 4 0 0
5  0 1 1 3 10 6 13 22 41 30 54 92 62 106 66 11 17 7 9 8 2 2 0
6  0 1 2 3 6 6 13 26 41 30 54 96 62 106 66 11 17 7 10 5 5 1 0
7  0 0 2 6 8 6 8 24 47 30 45 94 62 100 66 10 19 6 8 8 5 0 0
8  0 0 4 4 4 6 10 28 43 30 51 98 62 104 66 8 23 4 10 4 7 0 0
9  0 0 2 6 8 6 8 24 47 30 45 94 62 100 66 10 19 6 8 8 5 0 0
10 0 0 0 8 12 6 6 20 51 30 39 90 62 96 66 12 15 8 6 12 3 0 0
11 0 1 2 2 8 5 14 28 39 25 57 98 54 108 62 14 15 6 9 5 4 3 0
12 0 0 2 5 10 5 9 26 45 25 48 96 54 102 62 13 17 5 7 8 4 2 0
13 0 1 1 5 6 5 15 22 45 25 60 92 54 110 62 13 17 5 9 5 5 1 1
14 0 0 3 4 8 5 10 28 43 25 51 98 54 104 62 12 19 4 8 6 5 2 0
15 0 1 1 5 6 5 11 30 45 25 48 100 54 102 62 17 9 9 8 5 7 1 0
16 0 0 1 7 10 5 7 26 49 25 42 96 54 98 62 15 13 7 6 9 5 1 0
17 0 1 1 3 10 5 13 26 41 25 54 96 54 106 62 15 13 7 8 7 3 3 0
18 0 0 2 6 8 5 8 28 47 25 45 98 54 100 62 14 15 6 7 7 6 1 0
19 0 0 1 6 12 5 8 24 47 25 45 94 54 100 62 14 15 6 6 10 3 2 0
20 0 0 1 7 10 5 7 26 49 25 42 96 54 98 62 15 13 7 6 9 5 1 0
21 0 0 2 6 8 5 8 28 47 25 45 98 54 100 62 14 15 6 7 7 6 1 0
22 0 0 1 7 10 5 7 26 49 25 42 96 54 98 62 15 13 7 6 9 5 1 0
23 0 1 1 3 10 5 13 26 41 25 54 96 54 106 62 15 13 7 8 7 3 3 0
24 0 1 1 5 6 5 15 22 45 25 60 92 54 110 62 13 17 5 9 5 5 1 1
25 0 0 2 4 12 5 10 24 43 25 51 94 54 104 62 12 19 4 7 9 2 3 0
26 0 0 3 5 6 5 9 30 45 25 48 100 54 102 62 13 17 5 8 5 7 1 0
27 0 0 2 5 10 5 9 26 45 25 48 96 54 102 62 13 17 5 7 8 4 2 0
28 0 0 3 3 10 5 11 26 41 25 54 96 54 106 62 11 21 3 8 7 3 3 0
29 0 0 3 4 8 5 10 28 43 25 51 98 54 104 62 12 19 4 8 6 5 2 0
30 0 0 0 8 12 4 10 20 51 20 51 90 46 104 58 16 15 4 5 10 3 2 1
31 0 0 2 6 8 4 12 24 47 20 57 94 46 108 58 14 19 2 7 6 5 2 1
32 0 0 2 7 6 4 11 26 49 20 54 96 46 106 58 15 17 3 7 5 7 1 1
33 0 0 2 7 6 4 11 26 49 20 54 96 46 106 58 15 17 3 7 5 7 1 1
34 0 0 0 9 10 4 5 30 53 20 36 100 46 94 58 21 5 9 4 9 7 1 0
35 0 1 0 5 10 4 11 30 45 20 48 100 46 102 58 21 5 9 6 7 5 3 0
36 0 0 1 6 12 4 8 28 47 20 45 98 46 100 58 18 11 6 5 9 4 3 0
37 0 0 2 7 6 4 7 34 49 20 42 104 46 98 58 19 9 7 6 5 9 1 0
38 0 0 1 6 12 4 8 28 47 20 45 98 46 100 58 18 11 6 5 9 4 3 0

Matrix  m_{k,l} 1 ≤ l ≤ k ≤ 5  c_{s,t} 1 ≤ t ≤ s = 3, 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8
39 0 1 0 5 10 4 15 22 45 20 60 92 46 110 58 17 13 5 7 7 3 3 1
40 0 0 1 7 10 4 11 22 49 20 54 92 46 106 58 15 17 3 6 8 4 2 1
41 0 0 3 4 8 4 10 32 43 20 51 102 46 104 58 16 15 4 7 5 6 3 0
42 0 0 2 5 10 4 9 30 45 20 48 100 46 102 58 17 13 5 6 7 5 3 0
43 0 0 2 5 10 4 9 30 45 20 48 100 46 102 58 17 13 5 6 7 5 3 0
44 0 0 0 9 10 4 9 22 53 20 48 92 46 102 58 17 13 5 5 9 5 1 1
45 0 0 2 5 10 4 9 30 45 20 48 100 46 102 58 17 13 5 6 7 5 3 0
46 0 0 2 7 6 4 11 26 49 20 54 96 46 106 58 15 17 3 7 5 7 1 1
47 0 0 2 7 6 4 7 34 49 20 42 104 46 98 58 19 9 7 6 5 9 1 0
48 0 0 1 7 10 4 11 22 49 20 54 92 46 106 58 15 17 3 6 8 4 2 1
49 0 0 2 6 8 4 12 24 47 20 57 94 46 108 58 14 19 2 7 6 5 2 1
50 0 0 1 6 12 4 8 28 47 20 45 98 46 100 58 18 11 6 5 9 4 3 0
51 0 0 2 5 10 4 9 30 45 20 48 100 46 102 58 17 13 5 6 7 5 3 0
52 0 0 1 9 6 3 9 30 53 15 48 100 38 102 54 21 9 5 5 5 9 1 1
53 0 0 1 7 10 3 11 26 49 15 54 96 38 106 54 19 13 3 5 7 5 3 1
54 0 0 3 3 10 3 11 34 41 15 54 104 38 106 54 19 13 3 6 5 5 5 0
55 0 0 1 7 10 3 11 26 49 15 54 96 38 106 54 19 13 3 5 7 5 3 1
56 0 0 2 6 8 3 12 28 47 15 57 98 38 108 54 18 15 2 6 5 6 3 1
57 0 0 1 7 10 3 7 34 49 15 42 104 38 98 54 23 5 7 4 7 7 3 0
58 0 0 1 7 10 3 7 34 49 15 42 104 38 98 54 23 5 7 4 7 7 3 0
59 0 0 1 7 10 3 7 34 49 15 42 104 38 98 54 23 5 7 4 7 7 3 0
60 0 0 0 8 12 3 10 24 51 15 51 94 38 104 54 20 11 4 4 9 4 3 1
61 0 0 1 7 10 3 11 26 49 15 54 96 38 106 54 19 13 3 5 7 5 3 1
62 0 0 0 9 10 2 9 30 53 10 48 100 30 102 50 25 5 5 3 7 7 3 1

The table shows that manifolds with different column patterns are not isospectral to each other. Furthermore, we have isospectral families, for each column pattern (according to the numbering in the table) as follows: 7=9; 12=27; 13=24; 14=29; 17=23; 18=21; 31=49; 37=47; 40=48; 16=20=22; 32=33=46; 36=38=50; 53=55=61; 57=58=59; 42=43=45=51. Summing up, there are 9 pairs, 5 triples, 1 family of 4 manifolds, and 25 isolated manifolds.

References

[Ch] Charlap, L., Bieberbach groups and flat manifolds, Springer Verlag, 1988.

[DM] Dotti, I., Miatello, R., Isospectral compact flat manifolds, Duke Math. J. 68 (1992), 489–498.

[MR] Miatello, R., Rossetti, J. P., Isospectral Hantzsche-Wendt manifolds, Jour. für die Reine Angewandte Mathematik, to appear.

[Su] Sunada, T., Riemannian coverings and isospectral manifolds, Ann. of Math. 121 (1985), 169–186.

Fa.M.A.F., Universidad Nacional de Córdoba, 5000 Córdoba, ARGENTINA E-mail: miatello@mate.uncor.edu, rossetti@mate.uncor.edu