Isospectrality II: Sunada's method

We continue where we left off in the last post. This time we discuss various constructions of isospectral but non-isometric Riemannian manifolds. 

It has been known for some time that there exist closed Riemannian manifolds which are isospectral but not isometric. Virtually all examples of this can be shown to have this property using an elegant method due to Sunada and its generalisations, which are the focus of this post (Vignéras found the first examples of isospectral, nonisometric hyperbolic manifolds, and are also the first examples where generalisations Sunada's techniques don't apply). Before that, however, a brief discussion of the first examples ever constructed, which relied on explicit computations.

Tori

Define the length spectrum of a lattice \(\Gamma\) to be the collection of lengths of lattice vectors, counted with multiplicities. A computation shows that two flat tori \(\Gamma_1 \backslash \mathbb{R}^n\) and \(\Gamma_2 \backslash \mathbb{R}^n\) are isospectral precisely when \(\Gamma_1^*\) and \(\Gamma_2^*\) have the same length spectrum. Next observe that the geodesic length spectrum of a torus \(\Gamma \backslash \mathbb{R}^n\) coincides precisely with the length spectrum of the lattice \(\Gamma\), if we define the multiplicity of a length in the geodesic length spectrum to be the number of free homotopy classes of loops containing a geodesic of the given length. The Jacobi inversion formula implies that two lattices \(\Gamma_1\) and \(\Gamma_2\) have the same length spectrum if and only if the dual lattices \(\Gamma_1^*\) and \(\Gamma_2^*\) have the same length spectrum. We conclude that two flat tori are isospectral if and only if they have the same geodesic length spectrum.

Witt constructed a pair of noncongruent lattices in \(\mathbb{R}^{16}\) with the same length spectrum. Milnor pointed out that the associated 16-dimensional flat tori are isospectral but not isometric. 

Spherical space forms

The spectra of the round spheres \( S^n \) has eigenvalues \( k(n + k - 1) \) with corresponding eigenspaces the homogeneous harmonic polynomials on \( \mathbb{R}^{n+1} \) of degree \( k \). Ikeda considered spherical space forms, i.e. quotients \( M = \Gamma \setminus S^n \) by finite groups \( \Gamma \) of orthogonal transformations acting without fixed points. The eigenfunctions on \( M \) lift to \( \Gamma \)-invariant eigenfunctions on \( S^n \), so to compute the spectrum of \( M \), one need "only" compute the dimension of the space of \( \Gamma \)-invariant homogeneous harmonic polynomials of each degree. Ikeda gave many examples of isospectral spherical space forms, including lens spaces. (A lens space is a spherical space form with \( \Gamma \) cyclic.) One can also explicitly compute the spectrum of the Laplacian acting on \( p \)-forms for each \( p \). For each positive integer \( k \), Ikeda remarkably constructed examples of pairs of Lens spaces which are \( p \)-isospectral for every \( p \leq k \) but not for \( p = k + 1 \). Here \( p \)-isospectral means that the spectra of the Laplacians acting on \( p \)-forms are the same.

For a more detailed discussion of these examples, see this paper, which is especially convenient for those who cannot read French, which some mathematicians like Vignéras insist on using.

Most known examples of isospectral closed Riemannian manifolds have a common Riemannian covering; i.e. they are of the form \( M_1 = \Gamma_1 \backslash M \) and \( M_2 = \Gamma_2 \backslash M \), where \( M \) is a (not necessarily compact) Riemannian manifold and \( \Gamma_1 \) and \( \Gamma_2 \) are discrete groups of isometries acting freely on \( M \). As in the cases already considered of tori and spherical space forms, the Laplacian of the quotient manifold \( M_i, i = 1, 2 \), may be viewed as the Laplacian of the covering \( M \) restricted to functions which are \( \Gamma_i \)-invariant. 

Before proceeding with the construction of isospectral manifolds of the form \( M_1 = \Gamma_1 \backslash M \) and \( M_2 = \Gamma_2 \backslash M \) as above, we note that \( M_1 \) and \( M_2 \) will be isometric precisely when \( \Gamma_1 \) and \( \Gamma_2 \) are conjugate subgroups of the full isometry group \( \rm{Iso}(M) \). In many cases, one does not know the full isometry group of \( M \), only some subgroup \( G \) containing \( \Gamma_1 \) and \( \Gamma_2 \). In this case, even after making sure that \( \Gamma_1 \) and \( \Gamma_2 \) are not conjugate in \( G \), one must still check, generally by ad hoc methods, whether the manifolds \( M_1 \) and \( M_2 \) are isometric. 

For Sunada's method, we need the following weaker notion of conjugacy.

Definition: Let \( G \) be a finite group and let \(\Gamma_1\) and \(\Gamma_2\) be subgroups of \( G \). We will say that \(\Gamma_1\) is almost conjugate to \(\Gamma_2\) in \( G \) if each \( G \)-conjugacy class \([g]_G\) intersects \(\Gamma_1\) and \(\Gamma_2\) in the same number of elements. 

In the next subsection we explain why this is relevant.

Background on group representations

Let \( G \) be a (possibly finite) Lie group. A subgroup \( \Gamma \) of \( G \) is said to be cocompact or uniform if \( \Gamma \backslash G \) is compact. The existence of a uniform discrete subgroup \( \Gamma \) in \( G \) forces \( G \) to be unimodular; i.e. the Haar measure on \( G \), uniquely defined up to rescaling, is bi-invariant. The Haar measure induces a measure \( \Omega \) on \( \Gamma \backslash G \). The globally defined right translation operators \( R_{\Gamma,a}, a \in G \), given by \( R_{\Gamma,a} (\Gamma x) = (\Gamma xa) \) are volume preserving. Letting \( L^2(\Gamma \backslash G) \) be the space of measurable functions on \( \Gamma \backslash G \) which are square integrable with respect to \( \Omega \), and letting

\[\rho_\Gamma(a) f = f \circ R_{\Gamma,a}\]

for \( a \in G \) and \( f \in L^2(\Gamma \backslash G) \), it follows that \( \rho_\Gamma(a) \) is a unitary isomorphism of \( L^2(\Gamma \backslash G) \). Thus \( \rho_\Gamma \) is a unitary representation of \( G \), called the quasi-regular representation.

In case \( G \) is finite, \( \Omega \) is, up to rescaling, just the counting measure on \( \Gamma \backslash G \) and \( L^2(\Gamma \backslash G) \) is a finite-dimensional vector space. We will be primarily interested in the case of finite \( G \) and, at the opposite extreme, the case of connected \( G \).

Definition: Let \( G \) be a Lie group. We say two uniform discrete subgroups \(\Gamma_1\) and \(\Gamma_2\) of \( G \) are representation equivalent if \(\rho_{\Gamma_1}\) and \(\rho_{\Gamma_2}\) are unitarily equivalent; i.e. if there exists a unitary isomorphism \( T : L^2(\Gamma_1 \backslash G) \rightarrow L^2(\Gamma_2 \backslash G) \) such that \[T (\rho_{\Gamma_1} (x)) T^{-1} = \rho_{\Gamma_2} (x)\] for all \( x \) in \( G \). The isomorphism \( T \) is called an intertwining operator.

In case \( G \) is finite, the condition for \(\Gamma_1\) and \(\Gamma_2\) to be representation equivalent takes a particularly simple form.

Proposition 2: Let \( G \) be a finite group and let \(\Gamma_1\) and \(\Gamma_2\) be subgroups of \( G \). Then \(\Gamma_1\) is almost conjugate to \(\Gamma_2\) in \( G \) if and only if \(\Gamma_1\) is representation equivalent to \(\Gamma_2\).

This standard result is a straightforward consequence of the formula for the characters of the representations.

Sunada's method

Sunada's theorem: Let \(\Gamma_1\) and \(\Gamma_2\) be almost conjugate subgroups of a finite group \(G\). Let \((M, g)\) be a compact Riemannian manifold on which \(G\) acts on the left by isometries. Assume that \(\Gamma_1\) and \(\Gamma_2\) act without fixed points. Then \[\text{spec}(\Gamma_1 \backslash M, g) = \text{spec}(\Gamma_2 \backslash M, g).\]

Proof: Since the Laplacian \(\Delta\) of \(M\) commutes with isometries, the group \(G\) acts on the eigenspaces of \(\Delta\). Let \(\Gamma\) denote either \(\Gamma_1\) or \(\Gamma_2\). The eigenfunctions of the Laplacian of \((\Gamma \setminus M, g)\) are precisely those eigenfunctions of the Laplacian on \(M\) which are \(\Gamma\)-invariant, i.e. which are fixed by the action of \(\Gamma\). To prove the theorem, we need only compare the dimensions of the subspaces of \(\Gamma_1\)-invariant functions and \(\Gamma_2\)-invariant functions in each eigenspace.

Let \(V\) be a finite dimensional vector space (we have the eigenspaces of the Laplacian in mind) and suppose \(G\) acts on \(V\) by linear transformations; i.e. there exists a homomorphism from \(G\) to \(GL(V)\). Denote the action of \(a \in G\) on \(v \in V\) by \(a(v)\). Let \(V^\Gamma\) denote the subspace of vectors fixed by \(\Gamma\). Define a projection map \(P\) from \(V\) to \(V^\Gamma\) by

\[P(v) = \frac{1}{|\Gamma|} \sum_{h \in \Gamma} h(v).\]

Then \[\dim(V^\Gamma) = \text{trace}(P) = \frac{1}{|\Gamma|} \sum_{h \in \Gamma} \text{trace}(h).\]

Since conjugate elements of \(G\) have the same trace as linear operators on \(V\), the almost conjugacy condition implies that \(V^{\Gamma_1}\) and \(V^{\Gamma_2}\) have the same dimension. Thus the manifolds \(M_1\) and \(M_2\) are isospectral. $\blacksquare$

Remarks: 

• Sunada's theorem is also valid for manifolds with boundary as long as the boundary conditions imposed on \(\Gamma_1 \setminus M\) and \(\Gamma_2 \setminus M\) are consistent with those imposed on \(M\). 
•  Isospectral manifolds constructed by this technique are actually strongly isospectral; i.e. any natural strongly elliptic operator on the manifolds -- e.g., the Laplacians on \(p\)-forms -- are isospectral. The proof goes through verbatim. 
• One can actually drop the hypothesis that the action is free, in which case one obtains quotients $\Gamma_i \backslash M$ which might be orbifolds. The spectrum is then the spectrum of the Laplacian of $M$ restricted to the space of $\Gamma_i$-invariant functions.
• The proceedings of Sunada's birthday conference tell the story of a mathematician with an incredibly broad and creative view of mathematics. One of the things he has tried to do was geometrise class field theory, and it is because of this that he was able to come up with this method of creating isospectral manifolds: he was inspired by the analogous construction of number fields with the same Dedekind zeta function. For this, one starts with a Galois field extension$K/ \mathbb{Q}$ and needs to find almost conjugate subgroups of the Galois group. Sunada's method replaces the Dedekind zeta function by the spectral zeta function.
• Ruberman found a curious use of Sunada's technique: one can construct an obstruction to certain finite groups being fundamental groups of closed orientable 3-manifolds. This question was later solved completely by Perelman's proof of geometrisation using Ricci flow, so the idea never gained much attention, but it is rather surprising all the same.   

Isospectral manifolds constructed by Sunada's technique have the same geodesic length spectrum. Indeed, a closed geodesic of length \( l \) in \( \Gamma_1 \setminus M \) lifts to a geodesic \( \sigma \) in \( M \) satisfying \( \sigma (t + l) = \gamma_1 \sigma (t) \) for all \( t \) with \( \gamma_1 \in \Gamma_1 \). The element \( \gamma_1 \) of \( \Gamma_1 \) is conjugate to an element \( \gamma_2 \) in \( \Gamma_2 \), say \( \gamma_2 = a \gamma_1 a^{-1} \), with \( a \in G \). Since \( G \) acts by isometries on \( M \), the curve \( a \sigma \) is a geodesic. It satisfies \( a \sigma (t + l) = \gamma_2 a \sigma (t) \) and thus descends to a closed geodesic of length \( l \) in \( \Gamma_2 \setminus M \). One can strengthen this to take multiplicities into account. Isospectral manifolds constructed by other representation theoretic techniques also have the same length spectra, at least modulo multiplicities.

Generalisations of Sunada

Sunada's technique has been generalised many times in different ways. I'll content myself with describing a couple here.
    
 One can allow \(G\) in Sunada's theorem to be an arbitrary Lie group with \(\Gamma_1\) and \(\Gamma_2\) being uniform discrete representation equivalent subgroups. (The manifold \(M\) will be noncompact if \(G\) is noncompact.) If the subgroups \(\Gamma_1\) and \(\Gamma_2\) act on \(M\) in such a way that the quotients \(\Gamma_1 \setminus M\) and \(\Gamma_2 \setminus M\) are well-defined compact manifolds, then the conclusion of Sunada's theorem remains valid.

Here is one of the special cases of this big generalisation that isn't a priori covered by the Sunada's original theorem:

Theorem 3: Let \(\Gamma_1\) and \(\Gamma_2\) be cocompact discrete subgroups of a Lie group \(G\), and let \(g\) be a left-invariant metric on \(G\). If \(\Gamma_1\) and \(\Gamma_2\) are representation equivalent, then \[\operatorname{spec}(\Gamma_1 \backslash G, g) = \operatorname{spec}(\Gamma_2 \backslash G, g).\]

Proof: The Laplacian of \((G, g)\) may be expressed in terms of left-invariant vector fields. Recall that a left-invariant vector field \(X\) acts on a smooth function \(f\) as

\[Xf(p) = \left.\frac{d}{dt}\right|_{t=0}f\bigl(p \exp(tX)\bigr),\]

where \(\exp\) is the Lie group exponential map. In particular, for \(f \in L^2(\Gamma \backslash G)\),

\[Xf(p) = \left.\frac{d}{dt}\right|_{t=0}\bigl(\rho_{\Gamma}(\exp(tX))f\bigr)(p).\]

Thus the Laplacian on \(\Gamma \backslash G\) acts through the right action \(\rho_{\Gamma}\). When the hypothesis of the theorem holds, the intertwining operator \(T\) between \(\rho_{\Gamma_1}\) and \(\rho_{\Gamma_2}\) intertwines the Laplacians of \((\Gamma_1 \backslash G, g)\) and \((\Gamma_2 \backslash G, g)\). $\blacksquare$

Examples which aren't even locally isometric

Sutton extended the previous theorem to the case where $H_1$ and $H_2$ are nontrivial, connected subgroups of G. This gave rise to the first examples of isospectral, simply-connected, normal homogeneous spaces that are not locally isometric. This is a point where knowing different proofs of Sunada's theorem helped Sutton. One of the proofs, due to Pesce, relies on Frobenius reciprocity. The Lie group $G$ acts on each of the eigenspaces $E_{\lambda}$ of the Laplacian on $C^{\infty}(M)$. Call this action $\rho_{\lambda}: G \to \rm{Hom}(E_{\lambda})$. The dimension of the space of $H_i$-fixed vectors is the multiplicity of the trivial representation in the restriction to $H_i$. If the groups are almost conjugate then one can check this implies that the linear permutation representations $\mathbb{R}[H_i \backslash G]$ are equivalent. Frobenius reciprocity implies that the multiplicities then coincide. Further results on the decomposition of $L^2$ are needed, but this is the main idea.

Sutton requires one more representation-theoretic ingredient, which we now briefly describe.

Results of Larsen-Pink

I think the following is a reasonably natural question: to what extent is a complex Lie group, \( G \), and a finite dimensional representation, \( (\rho, V) \) of \( G \), determined by the dimensions of the various invariant spaces \( W^G \), where the \( W \) are obtained from \( V \) by linear algebra? That is, given \(\dim((\text{Sym}^2(V))^G), \dim((\bigwedge^3 V)^G)\), etc., can one determine \( (G, \rho)? \) 

Definition The dimension data for \( (G, \rho) \) is the data associating \[\dim W^G\]to every Lie group homomorphism \( GL(V) \rightarrow GL(W) \).

Note that this definition makes sense only if \(\dim(V)\) is given. If \(\det(\rho)=1\), we can define dimension data to consist of \[\dim(\text{Hom}_{S_k}(U, V^{\otimes k})^G)\]for every \( k \in \mathbb{N} \) and every irreducible representation of the symmetric group \( S_k \) (which acts on \( V^{\otimes k} \) by permuting the factors). This makes sense even when \(\dim(V)\) is unknown; and it determines \(\dim(V)\) as the largest integer \( k \) such that \( \bigwedge^k V \) has a non-trivial \( G \)-invariant. The classical invariant theory of \( SL(V) \) tells us that the two definitions are equivalent.

Larsen and Pink prove the following results with pretty statements:

Theorem 4: For any faithful finite dimensional representation \( \rho \) of a connected semisimple Lie group \( G \), dimension data uniquely determines \( G \) up to isomorphism. If \( \rho \) is irreducible, dimension data uniquely determines \( \rho \) up to isomorphism.

Using the above, Sutton manages to construct examples in the case where $M=\rm{SU}(n)$ and $H_1$ and $H_2$ are Lie groups acting on $M$. By equipping $\rm{SU}(n)$ with a bi-invariant metric $m$, the simply-connected, normal homogeneous spaces $(\rm{SU}(n) / H_1, m_1)$ and $(\rm{SU}(n) / H_2, m_2)$ are isospectral yet locally non-isometric. It is slightly unfortunate that the examples are of dimension $>10^{10}$, but nonetheless I think this is a very interesting result.

Other ways of distinguishing the manifolds produced

Since the isospectral manifolds constructed by Sunada's theorem have a common cover, they cannot be distinguished by any local geometric invariants. There exist examples using Sunada's original technique that differ by

• fundamental group
• orientability
• diameter
• the number of free homotopy classes of loops containing a geodesic of given length
• spin structure

If one admits the generalisations of Sunada's technique, then even the Betti numbers or being Kaehler need not be preserved.