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  "body": "\n## Statement\n\nLet $M$ be a compact connected [[Riemannian manifold|7CBT4SM83Y]], and let $G$ be a finite group that acts freely on $M$ by isometries.\nLet $H_1$ and $H_2$ be subgroups of $G$ that are *almost conjugate*: every conjugacy class $C$ of $G$ satisfies $\\abs{C \\cap H_1} = \\abs{C \\cap H_2}$.\nGive the quotients $M_1 = M/H_1$ and $M_2 = M/H_2$ the Riemannian metrics for which the projections from $M$ are local isometries.\nThen $M_1$ and $M_2$ are [[isospectral|18MQEGN13R]]: their [[Laplace–Beltrami operators|F4NWZNTRMD]] have the same [[eigenvalues with the same multiplicities|B8TYKV7AA1]].\n\n## Remarks\n\nThe idea of the proof is to compare the spaces of eigenfunctions.\nAn eigenfunction on $M/H$ with eigenvalue $\\lambda$ is the same as an $H$-invariant eigenfunction on $M$.\nSo the multiplicity of $\\lambda$ on $M/H$ is the dimension of the space of $H$-fixed vectors in the representation of $G$ on the $\\lambda$-eigenspace of $M$.\nAlmost conjugate subgroups give permutation representations $\\C[G/H_1]$ and $\\C[G/H_2]$ with equal characters, and by Frobenius reciprocity these dimensions agree for every representation of $G$.\nIf $H_1$ and $H_2$ are conjugate in $G$, then $M_1$ and $M_2$ are isometric, so the theorem is of interest for almost conjugate subgroups that are not conjugate.\nThe projections $M \\to M_i$ are [[covering maps|3XZRRE165N]], so $M_1$ and $M_2$ have a common finite cover.\nThis method gave many pairs of isospectral manifolds that are not isometric, including pairs of Riemann surfaces and, by later work, planar domains that are isospectral for the Dirichlet problem.\n\n## Explanations\n\n### L1 human\n\nTwo shapes can sound the same, that is, have the same frequencies of vibration, without being the same shape.\nSunada's theorem is a recipe that builds such pairs.\nStart with one big symmetric space and fold it up in two different ways by two groups of symmetries.\nIf the two groups are balanced in a precise counting sense, the two folded spaces have the same frequencies.\n\n### L2 human\n\nThe theorem turns a question about the spectrum of the Laplacian into a question in finite group theory.\nThe algebraic input is the same one that produces number fields with the same Dedekind zeta function, which is where almost conjugate subgroups first appeared.\nBefore this method, examples of isospectral non-isometric manifolds were sporadic, such as [[Milnor's flat tori|JNR9YJTT54]] in dimension $16$.\nThe theorem shows that the spectrum of the Laplacian does not determine the metric, so spectral geometry asks which properties the spectrum does determine.\n",
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