= Sunada unitary equivalence without a finite heat trace
{c}
{title2=$L^2(N)^{U_i}=\bigoplus_\rho V_\rho^{U_i}\otimes\mathcal H_\rho$}
A finite isometry group decomposes $L^2(N)$ into its irreducible representation factors $V_\rho$ and multiplicity Hilbert spaces $\mathcal H_\rho$. The Laplacian commutes with the group, so it acts as the identity on each representation factor. The averaging projection gives $\dim V_\rho^{U_i}=|U_i|^{-1}\sum_{u\in U_i}\chi_\rho(u)$. For <Gassmann equivalent> subgroups these dimensions agree. Choosing unitary identifications of the invariant finite-dimensional factors therefore intertwines the quotient Laplacians, even when a noncompact covering manifold has no finite <heat trace>.
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