Examples Concerning the Main Result

Abstract

We study whether it is possible to apply twisted Laplace spectra (and, if so, how many are necessary) to deduce isometry of some well-known examples of isospectral manifolds in the literature, due to Schüth (simply connected manifolds), Ikeda (lens spaces), Vignéras/Linowitz and Voight (arithmetic surfaces), Milnor (lattice examples), Doyle and Rossetti (Tetra and Didi), Sunada (based on group-theoretical examples from Gerst, Gaßmann and Komatsu), Brooks and Tse (surfaces of small genus, Riemann surfaces of small genus), Barden and Kang (surfaces of genus two), and Miatello and Rossetti (flat manifolds isospectral for all twists by linear characters).

We study whether it is possible to apply twisted Laplace spectra (and, if so, how many are necessary) to deduce isometry of some well-known examples of isospectral manifolds in the literature, due to Schüth (simply connected manifolds), Ikeda (lens spaces), Vignéras/Linowitz and Voight (arithmetic surfaces), Milnor (lattice examples), Doyle and Rossetti (Tetra and Didi), Sunada (based on group-theoretical examples from Gerst, Gaßmann and Komatsu), Brooks and Tse (surfaces of small genus, Riemann surfaces of small genus), Barden and Kang (surfaces of genus two), and Miatello and Rossetti (flat manifolds isospectral for all twists by linear characters).

1 Examples Where Theorem 1.2.1 Does not Apply

We first give some examples where the conditions of Theorem 1.2.1 are not met.

Example 11.1.1

Schüth constructed isospectral, non-isometric simply connected manifolds M ₁ and M ₂ [87]: these are non-isometric manifolds that are indistinguishable by any twisted spectrum on functions, simply because there is nothing to twist by.

From the perspective of our two conditions, this example already violates the first: if a diagram like (1.1) would exist, then by simple connectedness, M ₁ and M ₂ would both be isometric to $\widetilde M_0$, and hence isometric, which they are not.

Sutton [94] and An et al. [3] constructed further examples of isospectral simply connected manifolds by using an extension of Sunada’s method valid for continuous groups, rather than finite groups, as we are considering in this book. $\lozenge$

Example 11.1.2

The isospectral compact Riemann surfaces constructed by Vignéras [97] are described by commensurable arithmetic lattices and by [24, Proposition 3] there does not exist a diagram (1.1), so our first condition is violated. The original examples have Euler characteristic − 201600 (corrected value from [62]) but Linowitz and Voight have constructed similar examples of Euler characteristic − 10 and proven that this is the maximal Euler characteristic that can occur for the class of “unitive” torsion free Fuchsian groups; cf. [62]. $\lozenge$

Example 11.1.3

Ikeda’s isospectral non-isometric lens spaces from Example 4.2.1 have M = S ⁵, and M ₁ = L(11;1, 2, 3) and M ₂ = L(11, 1, 2, 4). In this example, H _i≅Z∕11Z. The action is not Q-homologically wide; actually, the only cyclic cover of M ₁ is M itself.$\lozenge$

Example 11.1.4

We consider Milnor’s example [71] where M _i = Γ_i∖R ¹⁶ with Γ₁ = E ₈ ⊕ E ₈ and Γ₂ = E ₁₆. The group 〈 Γ₁, Γ₂〉 = Γ₁ + Γ₂ is a lattice, and, as Chen [24, §3] observed, there is a diagram of the form (1.2) with M := ( Γ₁ ∩ Γ₂)∖R ¹⁶, G = ( Γ₁ + Γ₂)∕( Γ₁ ∩ Γ₂) and H _i = Γ_i∕( Γ₁ ∩ Γ₂). One easily checks, by writing down explicit lattice bases that G = H ₁ × H ₂ is the Klein four-group with H _i≅Z∕2Z. This can be done, for example, by using the MAGMA [15] code below.

MAGMA Program Code for Computing G

L0:=LatticeWithBasis(8, [2,0,0,0,0,0,0,0, -1,1,0,0,0,0,0,0, 0,-1,1,0,0,0,0,0, 0,0,-1,1,0,0,0,0, 0,0,0,-1,1,0,0,0, 0,0,0,0,-1,1,0,0, 0,0,0,0,0,01,1,0, 1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2]); L1:=DirectSum(L0,L0); L2:=LatticeWithBasis(16, [2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, -1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0, 1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2]); L3:=L1 meet L2; Index(L2,L3); Index(L1,L3); G:=(L1+L2)/L3; G;

Since G is abelian, the subgroups H ₁ and H ₂ cannot be weakly conjugate (since they are not equal as subgroups of G). A cover realising the wreath product cannot exist: the group G acts non-trivially on the coset space G∕H ₁ = H ₂, so the wreath product $C^2 \rtimes G$ is non-commutative, and hence cannot occur as subgroup of the (abelian) fundamental group of M.

Chen also observed that for these M ₁, M ₂, there exists another diagram of the form (1.2) in which the corresponding groups are weakly conjugate, see [24, Prop. 1]. In that example, the group G contains an element involving a non-trivial translation. One easily computes the corresponding lattices bases (e.g., again using the MAGMA [15] code below) to see that in that case, H _i≅(Z∕2Z)¹².

MAGMA Program Code for Computing H1 and H2

Ls:=LatticeWithBasis(4, [2,0,0,0, 0,2,0,0, 0,0,0,2, 1,1,1,1]); Lt:=DirectSum(Ls,Ls); L:=DirectSum(Lt,Lt); H1:=L1/L; H2:=L2/L; H1; H2;

Again, the wreath product cannot be realised for the same reason as above (the corresponding wreath product is non-commutative but the fundamental group of M is commutative).$\lozenge$

Example 11.1.5

Doyle and Rossetti [36] constructed two isospectral, non-isometric closed flat 3-manifolds, called “Tetra” and “Didi”. These are commensurable, given as quotients of R ³∕(Z ² × 2Z) by the action of the two non-isomorphic groups of order 4, but there is no diagram (1.1) (if so, they would be strongly isospectral, but they cannot be isospectral on 1-forms, since they have different first Betti numbers).$\lozenge$

2 Examples from Sunada’s Construction

If M ₁ and M ₂ are isospectral via the Sunada construction, then a diagram of the form (1.2) exists by default (possibly with orbifold M ₀). We discuss some “small” examples of closed surfaces (where a group G is realised by choosing a compact hyperbolic Riemann surface M ₀ whose genus is larger than or equal to the number of generators of G, so that there is a surjection $\pi _1(M_0) \twoheadrightarrow G$). These examples satisfy the requirements of Theorem 1.2.1 and illustrate numerically that, whereas our auxiliary construction involves manifolds and groups of relatively large order and negative Euler characteristic, the dimension of the representations and the number of required twists can be rather small (dictated by the degrees of the corresponding coverings M → M _i).

Example 11.2.1

Sunada lists three examples in [90, §1, Example 1–3], for which we indicate in the first three rows of Table 11.1 the dimensions of the representations by which one needs to twist, as well as how many equalities of multiplicities of zero suffice in Theorem 1.2.1. The examples are

• “Gerst”: $G=({\mathbf {Z}}/{8{\mathbf {Z}}})^* \rtimes {\mathbf {Z}}/{8{\mathbf {Z}}}$, H ₁ = {(1, 0), (3, 0), (5, 0), (7, 0)} and H ₂ = {(1, 0), (3, 4), (5, 4), (7, 0)} (both isomorphic to the Klein four group, but not conjugate in G).

• “Gaßmann”: Example 1.2.2 from the introduction.

• “Komatsu”: $G=S_{p^3}$, H ₁ = (Z∕p Z)³ and H ₂ the Heisenberg group modulo p (i.e., the group $\{ \left ( \begin {smallmatrix} 1 \$ a \$ b \\ \$ 1 \$ c \\ \$ \$ 1 \end {smallmatrix}\right ) \colon a,b,c \in {\mathbf {F}}_p\}$), of order p ³ and exponent p. Both embed in G by the action of left multiplication on themselves, and H ₁ is commutative, whereas H ₂ is not.

Table 11.1 Examples of Sunada triples (with p an odd prime number) from [90, Example 1], [6, 19] and [45, Example 4.1]. We indicate the dimension of the representations involved in Theorem 1.2.1, as well as an upper bound on the number of equalities of spectra that need to be checked. In the last line, we choose ℓ = 2 and use the bound in Remark 5.3.3

The computations of the data in the table in these cases is straightforward, except for the last case, named “Komatsu”; here, we need to find a prime ℓ coprime to |G| = (p ³)!. If we observe that all prime divisors of |G| are < p ³ and use Bertrands postulate/Chebyshev’s theorem that there is a prime between p ³ and 2p ³ − 2 [47, Thm. 418], we can certainly find ℓ ≤ 2p ³ − 3. We also use that $|H_2^{\operatorname {ab}}| \leq p^2$; for this, note the commutator identities

$$\displaystyle \begin{aligned} \$\left[\left( \begin{smallmatrix} 1 \$ 1 \$ \\ \$ 1 \$ \\ \$ \$ 1 \end{smallmatrix}\right), \left( \begin{smallmatrix} 1 \$ \$ \\ \$ 1 \$ 1 \\ \$ \$ 1 \end{smallmatrix}\right)\right] =\left( \begin{smallmatrix} 1 \$ \$ 1 \\ \$ 1 \$ \\ \$ \$ 1 \end{smallmatrix}\right), \\ \$\left[\left( \begin{smallmatrix} 1 \$ 1 \$ \\ \$ 1 \$ \\ \$ \$ 1 \end{smallmatrix}\right), \left( \begin{smallmatrix} 1 \$ \$ 1 \\ \$ 1 \$ \\ \$ \$ 1 \end{smallmatrix}\right)\right]= \left[\left( \begin{smallmatrix} 1 \$ \$ 1 \\ \$ 1 \$ \\ \$ \$ 1 \end{smallmatrix}\right), \left( \begin{smallmatrix} 1 \$ \$ \\ \$ 1 \$ 1 \\ \$ \$ 1 \end{smallmatrix}\right)\right]= \left( \begin{smallmatrix} 1 \$ \$ \\ \$ 1 \$ \\ \$ \$ 1 \end{smallmatrix}\right), \end{aligned} $$

which imply that the abelianisation of the group of upper triangular matrices in GL(3, F _p) with all diagonal entries equal to 1 is isomorphic to (Z∕p Z)². $\lozenge$

We now discuss in some more detail follow-up examples of Brooks–Tse and Barden–Kang that have the smallest known genus (in variable or constant curvature). The results are summarised in the fifth and sixth row of Table 11.1.

Example 11.2.2

Brooks and Tse (see [19] and [17]) constructed closed surfaces of Euler characteristic − 4 (genus 3) that are isospectral but not isometric for well-chosen metrics whose curvature is not constant. Here, G = PSL(2, 7) = (P)SL(3, 2) is the unique simple group of order 168 = 2³ ⋅ 3 ⋅ 7 [51, 6.14(4) & 6.15], the automorphism group of the Fano plane P ²(F ₂), and H ₁, H ₂ are index 7 subgroups given as 3 × 3 matrices in SL(3, 2) in which the first column, respectively the first row, is (1, 0, 0) (stabilisers of a point and a dual hyperplane in P ²(F ₂)).

In this case, M ₀ is an orbifold sphere with 3 singular points of order 7, and χ _M = −2⁵ ⋅ 3. The smallest possible ℓ that can be chosen is ℓ = 5, and then $\widetilde G$ is of order 5⁷ ⋅ 168 ≈ 13 ⋅ 10⁶ and $\chi _{M'} = -2^5 \cdot 3 \cdot 5^7$. The dimension of the required representations in Proposition 5.3.1, on the other hand, is only 7. Recall that S ₄ has a unique normal subgroup isomorphic to the Klein four-group (generated by products of two two-cycles) and quotient isomorphic to S ₃≅GL(2, 2) [51, 5.2]. Since $H_i \cong ({\mathbf {Z}}/2{\mathbf {Z}} \times {\mathbf {Z}}/2{\mathbf {Z}}) \rtimes \mathrm {GL}(2,2) \cong S_4$ has commutator subgroup A ₄, the number of spectral equalities that needs to be checked is just $2\cdot 5 \cdot |{H^{\mathrm {ab}}_2}| =20$ (cf. Proposition 5.3.2).$\lozenge$

Example 11.2.3

In the same reference, Brooks and Tse (see [19] and [17]) used a different representation of the same group G and the same subgroups H _i to construct isospectral non-isometric compact Riemann surfaces (surfaces of constant negative curvature − 1) of Euler characteristic − 6 (genus 4). Here, M ₀ is an orbifold torus with a single singular point of order 7. Then $\chi _{M'} = - 2^4 \cdot 3^2 \cdot 5^7$, but one needs to check only 20 equalities of spectral multiplicities of 7-dimensional representations.$\lozenge$

Example 11.2.4

A similar construction of such surfaces (not of constant curvature) of Euler characteristic − 2 (genus 2) by Barden and Kang [6] has the largest currently known Euler characteristic. In their case, G has order 96, and H ₁ and H ₂ are of index 12 in G, both isomorphic to Z∕2Z ×Z∕4Z. We can choose ℓ = 5 and $\widetilde G$ of order 5¹² ⋅ 96 ≈ 2 ⋅ 10¹⁰. With $\chi _{M_i}=-2$ for i = 1, 2, we have χ _M = −16, $\chi _{M'} = - 2^4 \cdot 5^{12}$, and the dimension of the required representations is 12. In this case, one needs to check 80 equalities of multiplicities of zero in various spectra.$\lozenge$

Example 11.2.5

We consider an example where the order of |G| is odd (the situation can be realised using Riemann surfaces of sufficiently high genus, as above). Note that G is forcedly solvable, by the Feit–Thompson theorem. Let p denote an odd prime number. As in Guralnick [45, Example 4.1], consider the group G of order p ⁵ given as the semidirect product

$$\displaystyle \begin{aligned} G = A \rtimes H, \mbox{ where } A=({\mathbf{Z}}/{p^2{\mathbf{Z}}} \times {\mathbf{Z}}/{p{\mathbf{Z}}}) \mbox{ and } H=({\mathbf{Z}}/{p{\mathbf{Z}}} \times {\mathbf{Z}}/{p{\mathbf{Z}}}) \end{aligned}$$

with the action a↦a ^h of h ∈ H on a ∈ A determined by

$$\displaystyle \begin{aligned} (1,0)^{(1,0)} = (1,1), (0,1)^{(1,0)} = (p,1), (1,0)^{(0,1)} = (p+1,0), (0,1)^{(0,1)} = (0,1). \end{aligned}$$

Now G has two subgroups H _i≅Z∕p Z ×Z∕p Z that are not weakly conjugate; more precisely, in the above description,

$$\displaystyle \begin{aligned} H_1 {=} H {=} \langle ((0,0),(1,0)), ((0,0),(1,0))\rangle \mbox{ and } H_2=\langle ((0,0),(1,0)), ((p,0),(0,1))\rangle. \end{aligned}$$

Using the smallest prime ℓ ≥ 3 coprime to |G| in the main theorem, we need to check the following number of equalities: (a) if p = 3, 90 equalities, using ℓ = 5; (b) if p ≠ 3, 6p ² equalities, using ℓ = 3. In this case, a better result is possible using Pintonello’s method from Remark 5.3.3, where one can choose ℓ = 2 at the cost of adding extra identities, leading to 4p ² + 2 equalities to be checked (this number equals 38 < 90 for p = 3 and is always smaller than 6p ²). Note that an earlier similar example with |G| = p ⁶, |H _i| = p ² can be found in [2, Ch. IV, Ex. 2, p. 63].$\lozenge$

3 Flat Manifolds Isospectral for All Twists by Linear Characters

Example 11.3.1

Miatello and Rossetti constructed non-isometric closed flat manifolds M ₁ and M ₂ that admit non-trivial twists but are twisted isospectral on functions and forms for all twists by linear characters (and hence also any representation that decomposes as a direct sum of such) [70, 4.5]. Twisted Laplacians of such linear characters act on sections of flat line bundles.

The manifolds are constructed as follows (our notations differ from [70]). Consider the group of affine transformations ${\mathbf {R}}^4 \rtimes \mathrm {O}(4)$ of R ⁴. Let τ _v denote the translation by v ∈R ⁴, let

$$\displaystyle \begin{aligned} \Lambda:=\{\tau_v \colon v \in {\mathbf{Z}}^4\} \cong {\mathbf{Z}}^4, \end{aligned}$$

and denote the standard basis vectors as e ₁, …, e ₄. Consider the two orthogonal matrices

$$\displaystyle \begin{aligned} A:=\mathrm{diag}(1,1,-1,-1) \mbox{ and } A^{\prime}:=\mathrm{diag}(1,-1,1,-1) \end{aligned}$$

and the vectors

$$\displaystyle \begin{aligned} b_1=(e_2+e_4)/2, b_1^{\prime}=e_3/2, b_2=e_2/2, b_2^{\prime}=e_1/2. \end{aligned}$$

The manifolds are M _i = Γ_i∖R ⁴ with $\Gamma _i=\langle A \tau _{b_i}, A' \tau _{b^{\prime }_i}, \Lambda \rangle$. The Γ_i are Bieberbach groups fitting into an exact sequence

$$\displaystyle \begin{aligned} 1 \rightarrow \Lambda \cong {\mathbf{Z}}^4 \rightarrow \Gamma_i \rightarrow \langle A, A' \rangle \cong ({\mathbf{Z}}/{{2}{\mathbf{Z}}})^2 \rightarrow 1. \end{aligned}$$

Concerning our two conditions, the situation in this example is as follows. If A ∈O(4) is of order two and b ∈ 1∕2Z ⁴, then (Aτ _b)² = τ _b+Ab ∈ Λ, since b + Ab ∈Z ⁴; hence for any γ ∈ Γ_i, we have γ ² ∈ Λ≤ Γ₁ ∩ Γ₂. Therefore, with Γ₀ := 〈 Γ₁, Γ₂〉 we have Γ₀∕ Γ_i≅(Z∕2Z)², and a diagram such as (1.1) exists with M ₀ := Γ₀∖R ⁴. Then we can set M :=  Γ∖R ⁴ with Γ := Γ₁ ∩ Γ₂ to get a diagram (1.2). From the presentation of Γ_i, one may compute the intersection to be Γ =  Λ≅Z ⁴, so M is a 4-torus. Since G = Γ₀∕ Γ is a group of exponent 2 and order 16, we find G = H ₁ × H ₂≅(Z∕2Z)⁴ with H _i = (Z∕2Z)².

We now look at the second condition. Since G∕H ₁≅H ₂ with left G-action induced by the multiplication in H, it follows that $\operatorname {Ind}_{H_1}^G \mathbf {1} \cong \mathbf {Z}[H_2]$ is the (4-dimensional) regular representation of H ₂, i.e., the direct sum of the four linear characters of the Klein four-group. Note that, similarly, $\operatorname {Ind}_{H_2}^G \mathbf {1} \cong \mathbf {Z}[H_1]$, and hence $\operatorname {Ind}_{H_1}^G \mathbf {1} \cong \operatorname {Ind}_{H_2}^G \mathbf {1},$ meaning that H ₁ and H ₂ are weakly conjugate subgroups of G.

To analyse the homology representation, we note that $\operatorname {H}_1(M) \cong \Gamma ^{\mathrm {ab}} \cong {\mathbf {Z}}^4$. The group G acts on this through conjugation by (outer) automorphisms, i.e., if Aτ _b represents an element of G (with A ∈O(4) of order two, b ∈ 1∕2Z ²), then it acts by τ _−b Aτ _v Aτ _b = τ _Av. We conclude that the representation of G on $\operatorname {H}_1(M)$ factors through the representation of 〈A, A′〉≅(Z∕2Z)² in GL(4, Z). Looking at characters, this is the regular representation of the Klein four-group. We conclude that

$$\displaystyle \begin{aligned} \operatorname{H}_1(M) \cong \operatorname{Ind}_{H_1}^G \mathbf{1} \end{aligned}$$

as G-modules.

We find that the two conditions of our main result are satisfied; although the action of G is not homologically wide (G is of order 16 but the first Betti number of M is only 4), the above calculation shows that condition (∗) does hold. Using ℓ = 2 as in Remark 5.3.3, we find that the manifolds M ₁ and M ₂ can be distinguished by 18 equalities of twisted spectra for 4-dimensional representations. These representation are then forcedly not all direct sums of linear characters.

In a different direction, Gordon, Ouyang and Schüth [41] [86] have shown how to distinguish the manifolds in this example using a non-flat Hermitian line bundle. $\lozenge$

4 An Example that Does not Arise from Sunada’s Construction

Example 11.4.1

We consider a finite group G with two non-weakly conjugate subgroups H ₁ and H ₂. Let M ₀ denote a compact hyperbolic Riemann surface of genus larger or equal to the number of generators of G, and fix a surjective morphism $\pi _1(M_0) \twoheadrightarrow G$. This leads to a diagram of the form (1.2), and homological wideness is satisfied by Proposition 9.1.1. For example, set G = S ₄ with H ₁ = 〈(1234)〉≅Z∕4Z and H ₂ = 〈(12)(34), (13)(24)〉≅Z∕2Z ×Z∕2Z. Since the cycle types in H ₁ and H ₂ are different, they are not weakly conjugate. We choose M ₀ to be a genus two compact Riemann surface and let ℓ = 5; now we can distinguish the genus 7 Riemann surfaces M ₁ and M ₂ using 40 equalities of multiplicities of zero in the spectra of Laplacians twisted by 6-dimensional representations.$\lozenge$

Project

For the “Komatsu” example, find representations of smaller dimension twisting by which implies isometry.

Project

Study the use of spectra of operators on non-flat line bundles to detect isometry.