Isospectrality III: Arts and Craft

Let \(V\) be any vector space on which \(G\) acts on the right; write \(v.a\) for the action of \(a \in G\) on \(v \in V\). Given \(\Gamma < G\), let \(V^{\Gamma}\) be the set of those vectors in \(V\) fixed by \(\Gamma\). Note that for \(v \in V^{\Gamma}\) and for \(\Gamma a \in \Gamma \backslash G\), one can make sense of \(v. \Gamma a\). If \(\Gamma_1\) and \(\Gamma_2\) are almost conjugate, then the intertwining map \(\tau: F[\Gamma_1\backslash G] \rightarrow F[\Gamma_2\backslash G]\) gives rise to a linear map \(\tau^*_V: V^{\Gamma_2} \rightarrow V^{\Gamma_1}\) given by \(\tau^*(v) = v.\tau(\Gamma_1)\). The mapping \(\tau^*\) is functorial. Thus if \(V\) and \(W\) are two spaces on which \(G\) acts on the right and if \(\psi: V \rightarrow W\) is \(G\)-equivariant, then the following diagram commutes:


In particular, in the setting of Sunada's Theorem, \(G\) acts on the right on \(C^\infty(M)\) by \((f.a)(x) = f(ax)\) for \(a \in G\), \(x \in M\) and we have \(C^\infty(\Gamma_i \backslash M) = C^\infty(M)^{\Gamma_i}\). Letting \(V = W = C^\infty(M)\) and letting \(\psi = \Delta\), Sunada's Theorem follows. The same argument gives strong isospectrality.

In the setting of manifolds constructed from Schreier graphs, which we elaborate on below, a function \(f\) on \(\Gamma_1 \backslash M\) may be viewed as an \(n\)-tuple of functions, indexed by the cosets of \(\Gamma_1 \backslash G\), on the model tile. We write this \(n\)-tuple as a column, denoted \([f]\). Letting \(T\) be the matrix of the intertwining map \(\tau\) with respect to the coset bases of \(\mathbb{R}[\Gamma_1\backslash G]\) and \(\mathbb{R}[\Gamma_2\backslash G]\), then, as first illustrated by Buser, the transplantation is given by the column of functions \(T[f]\), reassembled on the second manifold \(\Gamma_2 \backslash M\).

 Given a vector space \(V\) on which \(G\) acts on the left and a subgroup \(\Gamma\) of \(G\), the space \(V_{\Gamma}\) of \(\Gamma\) co-invariants is given by \(V/S\), where \(S\) is the subspace spanned by all \(\gamma.v - v\) with \(v \in V\) and \(\gamma \in \Gamma\). If \(\Gamma_1\) and \(\Gamma_2\) are almost conjugate subgroups of \(G\), then one obtains a transplantation map \[\tau_{\sharp}:  V_{\Gamma_1} \to V_{\Gamma_2}\]
\[[v] \mapsto [\tau(\Gamma_1) \cdot v]\] Since the boundary map on chains on $M$ is $G$-equivariant, one obtains a transplantation on homology.

Buser showed how to construct Sunada isospectral manifolds using Schreier graphs. Schreier graphs are defined as follows:

Definition: Fix a set of generators \(\{a_1,\ldots,a_k\}\) of \(G\). Given a subgroup \(\Gamma\) of \(G\), the associated Schreier graph has one vertex for each coset of \(\Gamma \setminus G\) and an edge labelled \(a_i\) from vertex \(\Gamma x\) to vertex \(\Gamma x a_i\). 

Isospectral manifolds are obtained by gluing together identical building blocks, the cross pictured below, 

 Today we discuss an alternative proof of Sunada's method that will lead to the resolution of the question 'can one hear the shape of a drum?'.

Recall the following: given a subgroup \(\Gamma\) of \(G\) and the field \(F = \mathbb{Q}\), \(\mathbb{R}\) or \(\mathbb{C}\), consider the vector space \(F[\Gamma\backslash G]\) consisting of all formal linear combinations of cosets. The group \(G\) acts on \(F[\Gamma\backslash G]\) by permuting the basis elements; this is the linear permutation representation of \(G\) defined by the subgroup \(\Gamma\), equivalently, it is the representation \(1_{\Gamma}^G\) induced by the trivial representation of \(\Gamma\) over the field \(F\). The almost conjugacy condition is precisely the condition that the representations of \(G\) on \(F[\Gamma_1\backslash G]\) and \(F[\Gamma_2\backslash G]\) are equivalent.

Buser first illustrated with an example that any choice of intertwining operator \(\tau: F[\Gamma_1\backslash G] \rightarrow F[\Gamma_2\backslash G]\) gives rise to an explicit ``transplantation'' map carrying eigenfunctions of the Laplacian on \(\Gamma_1 \backslash M\) to eigenfunctions of the Laplacian on \(\Gamma_2 \backslash M\), thus giving another proof that the manifolds are isospectral. Bérard systematized the following notion of transplantation.

Transplantation

according to the pattern of the Schreier graphs of \(\Gamma_1 \setminus G\) and \(\Gamma_2 \setminus G\). Their common cover \(M\) is obtained analogously by using the Cayley graph of \(G\). The group \(G\) in Buser's original example is \(SL(3,\mathbb{Z}/2\mathbb{Z})\), which can be generated by two elements denoted \(a\) and \(b\), and the almost conjugate subgroups have index seven.

The result is the following:



To obtain hyperbolic isospectral surfaces, one need only replace this cross by the sphere with 4 discs removed (with the hyperbolic metric) and glue in the same pattern.

We will illustrate this transplantation procedure using this example of isospectral flat surfaces.

The equivalence of the representations \(\rho_{\Gamma_1}\) and \(\rho_{\Gamma_2}\) says that there exists a unitary isomorphism between \(L^2(\Gamma_1 \backslash G)\) and \(L^2(\Gamma_2 \backslash G)\) which intertwines \(\rho_{\Gamma_1}\) and \(\rho_{\Gamma_2}\). The matrix of this isomorphism can be written explicitly:

\[
T = 
\begin{pmatrix}
a & a & a & b & b & b \\
a & b & a & a & a & b \\
a & a & b & b & a & a \\
a & b & b & a & a & b \\
b & a & b & a & a & b \\
b & a & a & b & a & a \\
b & b & a & b & a & a
\end{pmatrix},
\]

where \(a\) and \(b\) are real numbers satisfying the relations \(4a^2 + 3b^2 = 1\) and \(2a^2 + 4ab + b^2 = 0\). (These conditions insure that \(T\) is real orthogonal.)

Bérard's transplantation map is given as follows: A smooth function \(f\) on \(M_1\) may be viewed as a 7-tuple of smooth functions

\[f = \begin{pmatrix} f_1 \\\vdots \\ f_7\end{pmatrix}\]

on the brick \(\mathcal{B}\), subject to compatibility conditions prescribed by the gluing pattern \(\mathcal{G}_1\). Transplant \(f\) to a function \(h = \underline{T}(f)\) on \(M_2\) by multiplying the column vector of functions representing \(f\) by the matrix \(T\) and then identifying the resulting 7-tuple of functions on \(\mathcal{B}\)

with a function \(h\) on \(M_{2}\). From the fact that \(T\) intertwines \(\rho_{\,\Gamma_{1}}\) and \(\rho_{\,\Gamma_{2}}\), one can show that \(h\) is smooth. The transplantation map \(\underline{T}\) intertwines the Laplacians of the two manifolds.

Hearing the shape of a drum

We now observe that each of the isospectral flat surfaces \(M_{i}\) pictured above admits a reflection symmetry \(\tau_{i}\). When we mod out these symmetries, we obtain the plane domains \(\Omega_{1}\) and \(\Omega_{2}\) depicted in the image below.


 (More precisely, the quotient of \(M_{i}\) by \(\tau_{i}\) is an orbifold whose underlying space is the plane domain \(\Omega_{i}\).) It turns out that these plane domains \(\Omega_{1}\) and \(\Omega_{2}\) are both Dirichlet and Neumann isospectral.

With the transplantation map \(\underline{T}\) in hand, one can give a simple proof by picture that the domains are isospectral. The transplantation map

\[\underline{T}:L^{2}(M_{1})\to L^{2}(M_{2})\]

commutes with the symmetries \(\tau_{1}\) and \(\tau_{2}\) and thus carries \(\tau_{1}\)-invariant functions to \(\tau_{2}\)-invariant functions. To get the Neumann transplantation map for the plane domains, lift functions on \(\Omega_{1}\) to \(\tau_{1}\)-invariant functions on \(M_{1}\), use \(\underline{T}\) to transplant them to \(\tau_{2}\)-invariant functions on \(M_{2}\), and then project to \(\Omega_{2}\). For the Dirichlet transplantation map on the domains, ``lift'' functions on \(\Omega_{1}\) to \(\tau_{1}\)-anti-invariant functions on \(M_{1}\) and proceed similarly. (By the ``lift'' of a function \(f\) in this context, we mean to identify one half of \(M_{1}\) with \(\Omega_{1}\), view \(f\) as a function on this half of \(M_{1}\), and then reflect \(f\) to its negative on the other half of \(M_{1}\).)

The Dirichlet transplantation is depicted in the next image.


 Each domain is made up of seven half-crosses (bricks). The letters A--G denote the restrictions of an eigenfunction on \(\Omega_{1}\) to the various bricks. Each brick may be identified with a model half-cross. On each of the bricks making up the second domain, the transplanted function is given as a linear combination of the functions A--G. (For simplicity, we have chosen \(b=1\) and \(a=0\) in the matrix \(T\) given above. The resulting transplantation map is not unitary but still intertwines the Laplacians. Had we used the unitary transplantation map instead, we would have all seven functions appearing in each brick of \(\Omega_{2}\) with appropriate coefficients. The advantage of the unitary transplantation map is that it indicates how to strike the two drums in order to excite each frequency to the same amplitude on both drums, thus resulting in the same sound. In this example, if the first drum is struck at one point, the second drum must be struck at seven points.) The Neumann transplantation is identical except that all the signs of the functions in \(\Omega_{2}\) are positive.

For more pretty pictures the reader is invited to consult the original paper of Gordon, Webb, and Wolpert, from which some of the images here were taken.